Time Series Good
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ﺟﺎﻣﻌﺔ اﻟﻤﻠﻚ ﺳﻌﻮد آﻠﻴﺔ اﻟﻌﻠﻮم ﻗﺴﻢ اﻹﺣﺼﺎء وﺑﺤﻮث اﻟﻌﻤﻠﻴﺎت
ﻃﺮق اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ) اﻟﺠﺰء اﻷول(
ﺗﺄﻟﻴﻒ د .ﻋﺪﻧﺎن ﻣﺎﺟﺪ ﻋﺒﺪاﻟﺮﺣﻤﻦ ﺑﺮي أﺳﺘﺎذ اﻹﺣﺼﺎء وﺑﺤﻮث اﻟﻌﻤﻠﻴﺎت اﻟﻤﺸﺎرك
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ﺑﺴﻢ اﷲ اﻟﺮﺣﻤﻦ اﻟﺮﺣﻴﻢ اﻟﺤﻤﺪ ﷲ رب اﻟﻌﺎﻟﻤﻴﻦ واﻟﺼﻼة واﻟﺴﻼم ﻋﻠﻰ اﺷﺮف ﺧﻠﻖ اﷲ ﺳﻴﺪﻧﺎ وﻧﺒﻴﻨﺎ ﻣﺤﻤﺪ وﻋﻠﻰ ﺁﻟﻪ وﺻﺤﺒﻪ وﺳﻠﻢ. أﻣﺎ ﺑﻌﺪ. هﺬﻩ هﻲ اﻟﻤﺴﻮدة اﻷوﻟﻰ ﻟﻜﺘﺎب ﻃﺮق اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﻟﻄﻼب ﻣﺮﺣﻠﺔ اﻟﺒﻜﺎﻟﻮرﻳﻮس. هﺬا اﻟﻜﺘﺎب ﺳﻴﻈﻞ ﻣﺴﻮدة إﻟﻰ ﻣﺎﺷﺎء اﷲ ﻷﻧﻲ وﺑﺈذن اﷲ ﺗﻌﺎﻟﻰ ﺳﻮف أﻗﻮم ﺑﺘﻄﻮﻳﺮﻩ وﺗﺠﺪﻳﺪﻩ وﺗﺤﺴﻴﻨﻪ ﺑﺸﻜﻞ ﻣﺴﺘﻤﺮ وﺳﻴﻈﻞ ﺑﺸﻜﻠﻪ اﻹﻟﻜﺘﺮوﻧﻲ هﺬا ﻷﻧﻲ أﻋﺘﻘﺪ ان اﻟﻌﻠﻮم واﻟﺘﻘﻨﻴﺔ ﺗﺘﻄﻮر ﻳﻮﻣﻴﺎ وﺑﺸﻜﻞ ﻣﺘﺴﺎرع ﺑﺤﻴﺚ ان وﺿﻌﻬﺎ ﻓﻲ آﺘﺎب ﺟﺎﻣﺪ ﺳﺘﺎﺗﻴﻜﻲ ﻻﻳﺘﻨﺎﺳﺐ ﻣﻊ دﻳﻨﺎﻣﻴﻜﻴﺔ اﻟﻤﻮﺿﻮع وﺧﺎﺻﺔ ﻓﻲ ﻋﺼﺮ ﺛﻮرة اﻟﻤﻌﻠﻮﻣﺎت واﻹﻧﺘﺮﻧﺖ. ﻳﻐﻄﻲ اﻟﺠﺰء اﻷول ﻣﻦ اﻟﻜﺘﺎب اﻷﺳﺎﺳﻴﺎت اﻷوﻟﻴﺔ ﻟﻠﻤﻮﺿﻮع وﻳﺘﻄﺮق إﻟﻰ ﻣﻮﺿﻮع اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﺑﺈﺳﺘﺨﺪام ﻧﻤﺎذج ARIMAواﻟﺘﻲ آﺎﻧﺖ اول ﻣﻌﺎﻟﺠﺔ رﻳﺎﺿﻴﺔ ﺟﺎدة وﻣﺤﻜﻤﺔ ﻟﻠﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ Time Seriesآﻤﺎ ﺗﻄﺮﻗﺖ ﻓﻲ ﺁﺧﺮ اﻟﻜﺘﺎب إﻟﻰ ﺑﻌﺾ اﻟﻄﺮق اﻟﺘﻘﻠﻴﺪﻳﺔ اﻟﻬﻮرﺳﺘﻴﻜﻴﺔ ﻟﻠﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ وﻓﻲ ﺟﻤﻴﻊ أﺟﺰاء اﻟﻜﺘﺎب ﻗﻤﺖ ﺑﺘﻮﺿﻴﺢ اﻷﻣﺜﻠﺔ واﻟﺤﺎﻻت اﻟﺪراﺳﻴﺔ ﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ Minitabوهﻲ ﺑﺮاﻣﺞ ﺣﺎﺳﺐ ﻃﻮرت ﺧﺎﺻﺔ ﻟﺘﻌﻠﻴﻢ ﻋﻠﻢ اﻹﺣﺼﺎء ﺑﺠﻤﻴﻊ ﻓﺮوﻋﻪ وهﺬﻩ اﻟﺤﺰﻣﺔ ﻣﺘﻮﻓﺮة ﻟﻠﻄﻼب ﺑﺎﻟﻤﺠﺎن. اﻟﺠﺰء اﻟﺜﺎﻧﻲ ﻣﻦ اﻟﻜﺘﺎب وﻣﻮﺟﻪ ﻟﻄﻼب اﻟﺪراﺳﺎت اﻟﻌﻠﻴﺎ ﺳﻮف ﻳﺘﻄﺮق ﺑﺈذن اﷲ ﻟﻤﻮاﺿﻴﻊ ﻣﺜﻞ ﺗﺤﻠﻴﻞ اﻟﺘﺪﺧﻞ Intervention Analysisوﻧﻤﺎذج داﻟﺔ اﻟﺘﺤﻮﻳﻞ Transfer Function Modelsوﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﺘﻌﺪدة Multivariate Time Series Modelsوﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺟﻬﻪ Vector Time Series Modelsوﻧﻤﺎذج ﻓﻀﺎء اﻟﺤﺎﻟﺔ وﻣﺮﺷﺢ آﺎﻟﻤﻦ State Space Models and Kalman Filteringوﻧﻤﺎذج اﻟﺤﺪ Threshold Time Series Modelsوﻧﻤﺎذج ARCHوﻧﻤﺎذج GARCHوﺗﻄﺒﻴﻘﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ اﻟﻤﺎﻟﻲ Finantial Time Series Forecastingآﻤﺎ ﺳﻨﺘﻄﺮق إﻟﻰ اﻟﺸﺒﻜﺎت اﻟﻌﺼﺒﻴﺔ Neural Networksوإﺳﺘﺨﺪاﻣﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ. هﺬا وارﺟﻮا ﻣﻦ اﷲ ان ﻳﻮﻓﻘﻨﻲ ﻓﻲ إﻧﺠﺎز هﺬا اﻟﻌﻤﻞ ﻟﻮﺟﻬﻪ اﻟﻜﺮﻳﻢ وﻹﺛﺮاء اﻟﻤﻜﺘﺒﺔ اﻟﻌﺮﺑﻴﺔ اﻟﻔﻘﻴﺮة إﻟﻰ ﻣﺜﻞ هﺬا اﻟﻜﺘﺎب. ﺳﻴﻜﻮن هﺬا اﻟﻜﺘﺎب ﻣﺠﺎﻧﻲ ﻷي ﻃﺎﻟﺐ ﻋﻠﻢ وهﻮ ﺳﻴﻜﻮن ﻣﺘﻮاﺟﺪ ﻋﻠﻰ ﺷﺒﻜﺔ اﻹﻧﺘﺮﻧﺖ ﻓﻲ اﻟﻤﻮﻗﻊ http://www.abarry.net/or/or٢٢١book١.pdf واﷲ اﻟﻤﻮﻓﻖ. اﻟﻤﺆﻟﻒ د .ﻋﺪﻧﺎن ﻣﺎﺟﺪ ﻋﺒﺪ اﻟﺮﺣﻤﻦ ﺑﺮي ﺟﺎﻣﻌﺔ اﻟﻤﻠﻚ ﺳﻌﻮد ذو اﻟﻘﻌﺪة ١٤٢٢هـ ﻳﻨﺎﻳﺮ ٢٠٠٢م ٣
اﻟﻤﺤﺘﻮﻳﺎت ﻣﻘﺪﻣﺔ -١اﻟﻔﺼﻞ اﻷول :ﻣﻘﺪﻣﺔ وﺗﻌﺎرﻳﻒ١٠.................................................................. ١-١أﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ١٠....................................................... ٢-١اﻟﻐﺮض ﻣﻦ دراﺳﺔ وﺗﺤﻠﻴﻞ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ١٠ ................................. ٣-١اﻟﺨﻄﻮات اﻟﻤﺘﺨﺬة ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ١٠ . ................................................ ١-٣-١ﺗﻌﻴﻴﻦ اﻟﻨﻤﻮذج ١٠ .. ................................................................... ٢-٣-١ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج ١١ ................................................................... ٣-٣-١ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج ١١...................................................... ٤-٣-١ﺗﻮﻟﻴﺪ اﻟﺘﻨﺒﺆات ١١..................................................................... ٥-٣-١إﺳﺘﺨﺪام اﻟﺘﻨﺒﺆات ووﺿﻊ اﻟﻘﺮارات ١١ ........................................... ٤-١ﺗﻌﺎرﻳﻒ وﻣﺒﺎدئ أوﻟﻴﺔ ١١..................................................................... ١-٤-١ﺗﻌﺮﻳﻒ ﻣﺎﺿﻲ أو ﺗﺎرﻳﺦ اﻟﻈﺎهﺮة ١١ ............................................. ٢-٤-١ﺗﻌﺮﻳﻒ اﻟﺤﺎﺿﺮ أو اﻵن ١١ ......................................................... ٣-٤-١ﺗﻌﺮﻳﻒ أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ١١ ......................................................... ٤-٤-١ﺗﻌﺮﻳﻒ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ١٢ ............................................................. ٥-٤-١ﺗﻌﺮﻳﻒ اﻹﺳﺘﻘﺮار ١٢ ................................................................. ٦-٤-١ﺗﻌﺮﻳﻒ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٢ ......................................................... ٧-٤-١ﻣﺜﺎل : ١اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ١٢ ................................................... ٨-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ١٣ .................................................... ٩-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ١٣ ................................................... ١٠-٤-١ﻣﺜﺎل : ٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٣ ......................... ١١-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ١٤ ...................................... ١٢-٤-١ﻣﺜﺎل :٣داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٥.................. ١٣-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ ١٦......................................... ١٤-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ١٧ .............................. ١٥-٤-١ﻣﺜﺎل :٤داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ١٨ ......... -٢اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ :ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ARMAوإﺳﺘﺨﺪاﻣﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ ٢٢................................................................................................ ١-٢ﺗﻌﺮﻳﻒ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )٢٢ ........ (p,q ٢-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ ٢٢......................................................... ٣-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻷﻣﺎﻣﻲ ٢٢ ..................................................... ٤-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ ٢٢.................................................................. ٥-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻟﺘﺠﻤﻴﻊ ٢٢ ................................................................. ٦-٢أﻣﺜﻠﺔ ٢٣ ..................................................................................... ٧-٢ﺧﺼﺎﺋﺺ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ٢٤ ............................. ١-٧-٢ﻧﻤﻮذج )٢٤ ........................................................... ARMA(٠،٠ ٢-٧-٢ﻧﻤﻮذج )٢٧ .................................................................................. AR(١ ٣-٧-٢ﻧﻤﻮذج )٣١ .................................................................... AR(٢ ٤
٤-٧-٢ﻧﻤﻮذج )٣٦ ..................................................................... MA(١ ٥-٧-٢ﻧﻤﻮذج )٣٩ .................................................................... MA(٢ ٦-٧-٢ﻧﻤﻮذج )٤٠ ............................................................. ARMA(١،١ ٧-٧-٢ﺧﻮاص ﻧﻤﺎذج )٤٧ ................................................ ARMA(p,q -٣اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ :ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ٤٩ ...................................... ١-٣ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ ٤٩ .............................................................. ٢-٣ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﺘﺒﺎﻳﻦ ٥٠ ................................................................ ٣-٣ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )٥٢ ...... (p,d,q ١-٣-٣ﻧﻤﻮذج )٥٢ ....................................................... ARIMA(١،١،٠ ٢-٣-٣ﻧﻤﻮذج )٥٢ ........................................................ ARIMA(٠،١،١ ٣-٣-٣ﻧﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ﺑﺈﻧﺠﺮاف ٥٣ ................................................. ٤-٣داﻟﺔ اﻷوزان ) y (Bوﺗﻤﺜﻴﻞ ﻧﻤﺎذج )٥٣ ............................... ARMA(p,q ٥-٣اﻣﺜﻠﺔ ﻟﺪاﻟﺔ اﻷوزان ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٥٤ ..................................................... ١-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٤ .................................................... AR(١ ٢-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٥ .................................................... MA(١ ٣-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٥ .................................................... AR(٢ ٤-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٦ ................................................... MA(٢ ٥-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٦ ........................................... ARMA(١،١ ٦-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٧ ................................................... ARI(١ ٧-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ )٥٧ .................. ARIMA(١،٠،١ ٦-٣ﺑﻌﺾ ﺧﻮاص داﻟﺔ اﻷوزان ) ٥٨ .................................................... y (B -٤اﻟﻔﺼﻞ اﻟﺮاﺑﻊ :اﻟﺘﻨﺒﺆات ذات ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ﻟﻨﻤﺎذج )٥٩ ....... ARMA(p,q ١-٤ﻧﻈﺮﻳﺔ :٢أﺧﻄﺎء اﻟﺘﻨﺒﺆ ٦١ ......................................................................... ٢-٤ﻣﺠﻤﻮﻋﺔ اﻟﻤﻌﻠﻮﻣﺎت ٦٢ ................................................. Information Sets ٣-٤ﻧﻈﺮﻳﺔ :٣اﻟﻤﺘﻨﺒﺊ ذا ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ٦٢ ............................................ ٤-٤ﻗﺎﻋﺪة ٦٢ ............................................................................................. ٢ ٥-٤ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﻨﺒﺆ ٦٢ ................................................................................. ٦-٤دوال اﻟﺘﻨﺒﺆ ﻟﻨﻤﺎذج )٦٢ ....................................................... ARIMA(p,d,q ١-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٣ ............................................................... AR(١ ٢-٦-٤ﺷﺮط اﻹﺳﺘﻤﺮار ٦٣ ............................................................................ ٣-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٤ .............................................................. AR(٢ ٤-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٥ ................................................. ARIMA(٠،١،١ ٥-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٦ .............................................................. MA(١ ٦-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٦ .............................................................. MA(٢ ٧-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٧ ..................................................... ARMA(١،١ ٧-٤ﺣﺪود اﻟﺘﻨﺒﺆ ٦٨ ......................................................................................... ١-٧-٤ﺗﻌﺮﻳﻒ ﻓﺘﺮة ﺗﻨﺒﺆ ﻟﻠﻘﻴﻤﺔ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ٦٨ ..................................................... ٢-٧-٤ﻣﺜﺎل ٦٩ ........................................................................................ -٥اﻟﻔﺼﻞ اﻟﺨﺎﻣﺲ :ﺗﺼﻤﻴﻢ وﺑﻨﺎء ﻧﻈﺎم ﺗﻨﺒﺆ إﺣﺼﺎﺋﻲ ٧١ ............................................ ١-٥ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ٧١ ....................................................................... ١-١-٥ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ ٧١ .............................................................................. ٢-١-٥إﺧﺘﻴﺎر درﺟﺔ اﻟﺘﻔﺮﻳﻖ ٧١ ................................................................ d
٥
٣-١-٥ﺗﺤﺪﻳﺪ ٧١ ............................................................................... p,q ٤-١-٥إﺿﺎﻓﺔ ﻣﻌﻠﻢ إﻧﺠﺮاف ٧١ ................................................................... ٢-٥ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج ٧٢ .............................................................................. ١-٢-٥ﻃﺮﻳﻘﺔ اﻟﻌﺰوم ٧٢ ........................................................................ ٢-٢-٥ﺗﻘﺪﻳﺮ اﻟﻌﺰوم ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٧٣ ....................................................... ١-٢-٢-٥ﻟﻨﻤﻮذج )٧٣ .............................................................. AR(١ ٢-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ ......................................................... MA(١ ٣-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ .......................................................... AR(٢ ٤-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ ......................................................... MA(٢ ٥-٢-٢-٥ﻟﻨﻤﻮذج )٧٥ ................................................ ARMA(١،١ ٣-٢-٥ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ٧٥ .............................................. ٤-٢-٥ﺗﻘﺪﻳﺮات اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٧٦ ......................... ١-٤-٢-٥ﻟﻨﻤﺎذج )٧٦ .........................................................AR(١ ٢-٤-٢-٥ﻟﻨﻤﺎذج )٧٧ ....................................................... MA(١ ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج٧٨ ......................................................... ٣-٥ ١-٣-٥ﻓﺤﺺ اﻟﺒﻮاﻗﻲ ٧٨ .................................................................... ١-١-٣-٥إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ ﻟﻠﺒﻮاﻗﻲ ٧٩ .................................................. ٢-١-٣-٥إﺧﺘﺒﺎر اﻟﻌﺸﻮاﺋﻴﺔ ﻟﻠﺒﻮاﻗﻲ ٧٩ ................................................. ٣-١-٣-٥إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ أو اﻹﺳﺘﻘﻼل ﻟﻠﺒﻮاﻗﻲ ٧٩ ................................... ٤-١-٣-٥إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ ٧٩ .................................................... ٢-٣-٥ﺑﻌﺾ اﻟﻤﻌﺎﻳﻴﺮ اﻻﺧﺮى ﻹﺧﺘﻴﺎر ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ٧٩ ............................. ١-٢-٣-٥إﺣﺼﺎﺋﻴﺔ آﻴﻮ ﻟﻠﺠﻨﻖ وﺑﻜﺲ ٧٩ ........................................... ٢-٢-٣-٥ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ ٨٠ ......................................... AIC ٣-٣-٥أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ٨٠ .......................................................... -٦اﻟﻔﺼﻞ اﻟﺴﺎدس :ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻴﺔ ١١٣ ...... ١-٦دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ ١١٤ ..... ١-١-٦ﻟﻨﻤﻮذج ١١٤ ......................................... SARMA(٠،١)(١،١)١٢ ٢-٦دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ ١١٥ .................................. ١١٥............................................. SARIMA(٠,d,٠)(٠,D,١)s ١-٢-٦ ١١٥ ........................................... SARIMA(٠,d,٠)(١,D,١)s ٢-٢-٦ ١١٥ ............................................ SARIMA(٠,d,١)(٠,D,١)s ٣-٢-٦ ١١٦ ........................................... SARIMA(٠,d,٠)(١,D,١)s ٤-٢-٦ ١١٦ ........................................... SARIMA(٠,d,١)(١,D,٠)s ٥-٢-٦ ١١٦ ............................................ SARIMA(٠,d,٢)(٠,D,١)s ٦-٢-٦ ٣-٦داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ ١١٨ .................... ٤-٦أﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ١١٩ ....................................... ٥-٦إﺷﺘﻘﺎق دوال ﺗﻨﺒﺆ ﻟﺒﻌﺾ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ ١٢٣ .......... ١-٥-٦داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج ١٢٣ .................... SARIMA(٠،٠،٠)(٠،١،١)١٢ ٢-٥-٦داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج ١٢٤ .................... SARIMA(٠،١،١)(٠،١،١)١٢ ٦-٦أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ١٢٥ .......................... اﻟﺠﺰء اﻟﻌﻤﻠﻲ: -٧اﻟﻔﺼﻞ اﻟﺴﺎﺑﻊ :ورﻗﺔ ﺗﺪرﻳﺐ ﻋﻤﻠﻲ ﻋﻠﻰ اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ١٤٢ ............................................................................................ ٦
-٨اﻟﻔﺼﻞ اﻟﺜﺎﻣﻦ :ﻣﺜﺎل ﺗﺤﻠﻴﻞ اﻟﺒﻮاﻗﻲ وﻣﻌﻴﻴﺮ إﺧﺘﻴﺎر ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ١٥٥ .................. -٩اﻟﻔﺼﻞ اﻟﺘﺎﺳﻊ :ﺗﺤﻠﻴﻞ أو ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ إﻟﻰ ﻣﺮآﺒﺎت ١٦٢ ..................... -١٠اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ١٧٨ ........................................... ١-١٠اﻟﻮﺳﻴﻂ اﻟﺠﺎري ١٨١ ............................................................. -١١اﻟﻔﺼﻞ اﻟﺤﺎدي ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﺒﺴﻴﻂ ١٨٤ ............. -١٢اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﻤﺰدوج ١٩٠ ........... ١-١٢ﻃﺮﻳﻘﺔ ﺑﺮاون ١٩٠ ........................................................................ ٢-١٢ﻃﺮﻳﻘﺔ هﻮﻟﺖ ١٩٠ ........................................................................ ٣-١٢أﻣﺜﻠﺔ ١٩١ .................................................................................. -١٣اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﺜﻼﺛﻲ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ ١٩٨ ............................................................................. ١-١٣اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ ١٩٩ ............................................................... ٢-١٣اﻟﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ ٢٠١ .............................................................. ٣-١٣ﻣﺜﺎل ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ٢٠٤ ........................................................... ٤-١٣ﻣﺜﺎل ﺁﺧﺮ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ٢١٠ ..................................................... اﻟﻤﺮاﺟﻊ ٢١٩ ...........................................................................................
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اﻟﻔﺼﻞ اﻷول ﻣﻘﺪﻣﺔ وﺗﻌﺎرﻳﻒ ﺗﻌﺮﻳﻒ :١اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ Time Series
ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺮﺗﺒﺔ ﻣﻊ اﻟﺰﻣﻦ ) او ﻣﺮﺗﺒﺔ ﻣﻊ اﻟﻤﻜﺎن (
اﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ: -١ﺳﻌﺮ اﻗﻔﺎل ﺳﻬﻢ ﺑﻨﻚ اﻟﺮﻳﺎض ﻳﻮﻣﻴﺎ. -٢ﻋﺪد اﻟﻮﺣﺪات اﻟﻤﻄﻠﻮﺑﺔ اﺳﺒﻮﻋﻴﺎ ﻣﻦ اﻧﺘﺎج ﺳﻠﻌﺔ ﻣﻌﻴﻨﺔ. -٣ﺣﺠﻢ اﻟﻤﺒﻴﻌﺎت ﺷﻬﺮﻳﺎ ﻣﻦ ﺳﻠﻌﺔ ﻣﺎ. -٤ﺣﺠﻢ اﻹﻧﺘﺎج اﻟﻴﻮﻣﻲ ﻟﻠﻨﻔﻂ اﻟﺨﺎم ﺑﺎﻟﻤﻤﻠﻜﺔ. واﻟﻐﺮض ﻣﻦ دراﺳﺔ وﺗﺤﻠﻴﻞ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ هﻮ: -١ﻓﻬﻢ وﻧﻤﺬﺟﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﻈﺎهﺮة اﻟﻤﺸﺎهﺪة. -٢اﻟﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻟﻠﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ. -٣اﻟﺘﺤﻜﻢ ﺑﺎﻟﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ إذا اﻣﻜﻦ ذﻟﻚ. اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة وهﻲ ﻋﺒﺎرة ﻋﻦ اﻹﻧﺘﺎج اﻟﻴﻮﻣﻲ ﻟﻠﺤﻠﻴﺐ ﺑﺎﻟﺮﻃﻞ ﻟﺒﻘﺮة ﻣﺎ
850
750
C1 650
550
70
60
50
30
40
20
10
Index
اﻟﺨﻄﻮات اﻟﻤﺘﺨﺬة ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: إن إﻳﺠﺎد ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ﺗﻨﻄﺒﻖ ﻋﻠﻴﺔ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة ﻳﻌﺘﺒﺮ ﻣﻦ اﻟﻤﻬﺎم اﻟﺼﻌﺒﺔ واﻟﺘﻲ ﺗﺤﺘﺎج اﻟﻰ اﻟﻜﺜﻴﺮ ﻣﻦ اﻟﺒﺤﺚ واﻟﺨﺒﺮة .ﺳﻮف ﻧﺴﺘﻌﺮض ﺑﻌﺾ اﻟﺨﻄﻮات اﻟﻌﺮﻳﻀﺔ ﻟﺒﻨﺎء ﻧﻤﻮذج رﻳﺎﺿﻲ ﻟﻠﺘﻨﺒﺆ ﻋﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺎ: -١ ﺗﻌﻴﻴﻦ اﻟﻨﻤﻮذج أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج :Model Identificationوهﺬا ﻳﺘﻢ ﺑﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﻓﻴﻤﺎ ﻳﺴﻤﻰ Time Plotﺣﻴﺚ ﻳﻜﻮن اﻹﺣﺪاﺛﻲ اﻻﻓﻘﻲ هﻮ اﻟﺰﻣﻦ واﻟﺮأﺳﻲ ﺣﺠﻢ اﻟﻈﺎهﺮة اﻟﻤﺸﺎهﺪة وﻣﻦ ﺛﻢ إﺧﺘﻴﺎر ﻧﻤﻮذج رﻳﺎﺿﻲ ﻣﻌﺘﻤﺪﻳﻦ ﻋﻠﻲ ﺑﻌﺾ اﻟﻤﻘﺎﻳﻴﺲ اﻹﺣﺼﺎﺋﻴﺔ اﻟﺘﻰ ﺗﻤﻴﺰ ﻧﻤﻮذج ﻋﻦ ﺁﺧﺮ وﻋﻠﻰ اﻟﺨﺒﺮة اﻟﻤﺴﺘﻤﺪة ﻣﻦ اﻟﺪراﺳﺎت واﻻﺑﺤﺎث.
٨
-٢
-٣
-٤
-٥
ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج :Model Fittingﺑﻌﺪ ﺗﺮﺷﻴﺢ ﻧﻤﻮذج او اآﺜﺮ آﻨﻤﻮذج ﻣﻨﺎﺳﺐ ﻟﻮﺻﻒ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ﻧﻘﻮم ﺑﺘﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ هﺬا اﻟﻨﻤﻮذج ﻣﻦ اﻟﺒﻴﺎﻧﺎت اﻟﻤﺸﺎهﺪة ﺑﺈﺳﺘﺨﺪام ﻃﺮق اﻟﺘﻘﺪﻳﺮ اﻹﺣﺼﺎﺋﻲ اﻟﺨﺎﺻﺔ ﺑﺎﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ وهﺬا اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ ﻳﺆﺧﺬ آﻨﻤﻮذج اوﻟﻲ ﻗﺎﺑﻞ ﻟﻠﺘﻌﺪﻳﻞ ﻻﺣﻘﺎ. ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج :Model Diagnosticsإﺟﺮاء إﺧﺘﺒﺎرات ﺗﻔﺤﺼﻴﺔ ﻋﻠﻰ أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ Fitting Errorsﻟﻤﻌﺮﻓﺔ ﻣﺪى ﺗﻄﺎﺑﻖ اﻟﻤﺸﺎهﺪات ﻣﻊ اﻟﻘﻴﻢ اﻟﻤﺤﺴﻮﺑﺔ ﻣﻦ اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ وﻣﺪى ﺻﺤﺔ ﻓﺮﺿﻴﺎت اﻟﻨﻤﻮذج .ﻓﻲ ﺣﺎﻟﺔ إﺟﺘﻴﺎز اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ ﻟﻬﺬﻩ اﻹﺧﺘﺒﺎرات ﻧﻘﻮم ﺑﺈﻋﺘﻤﺎدة ﻋﻠﻰ اﻧﻪ اﻟﻨﻤﻮذج اﻟﻨﻬﺎﺋﻲ وﻳﺴﺘﺨﺪم ﻟﺘﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ وإﻻ ﻧﻌﻮد ﻟﻠﺨﻄﻮة اﻻوﻟﻰ ﻟﺘﻌﻴﻴﻦ ﻧﻤﻮذج ﺟﺪﻳﺪ. ﺗﻮﻟﻴﺪ اﻟﺘﻨﺒﺆات :Forecast Generationﻳﺴﺘﺨﺪم اﻟﻨﻤﻮذج اﻟﻨﻬﺎﺋﻲ ﻟﺘﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ وﻣﻦ ﺛﻢ ﺣﺴﺎب أﺧﻄﺎء اﻟﺘﻨﺒﺆ Forecast Errorsآﻠﻤﺎ اﺳﺘﺠﺪت ﻗﻴﻢ ﺟﺪﻳﺪة ﻣﺸﺎهﺪة ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ وﻣﺮاﻗﺒﺔ هﺬﻩ اﻷﺧﻄﺎء ﻓﻰ ﻣﺎ ﻳﺴﻤﻰ ﺑﻤﺨﻄﻄﺎت اﻟﻤﺮاﻗﺒﺔ Control Chartsواﻟﺘﻲ ﺗﻮﺿﻊ ﻟﻠﻘﺒﻮل ﺑﻨﺴﺒﺔ ﺧﻄﺄ ﻣﻌﻴﻦ إذا ﺗﺠﺎوزﺗﺔ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﻳﻌﺎد اﻟﻨﻈﺮ ﻓﻲ اﻟﻨﻤﻮذج وﺗﻌﺎد اﻟﺪورة ﻣﻦ ﺟﺪﻳﺪ ﺑﺘﺤﺪﻳﺪ ﻧﻤﻮذج ﻣﺮﺷﺢ ﺁﺧﺮ. إﺳﺘﺨﺪام اﻟﺘﻨﺒﺆات ووﺿﻊ اﻟﻘﺮارات :Implementation and Decision making ﺗﻘﺪم اﻟﺘﻨﺒﺆات ﻓﻰ ﺗﻘﺮﻳﺮ ﻟﺼﺎﻧﻌﻲ اﻟﻘﺮار ﻟﻠﻨﻈﺮ ﻓﻲ إﺳﺘﺨﺪاﻣﻬﺎ ﺑﺎﻟﺸﻜﻞ اﻟﻤﻨﺎﺳﺐ.
ﺗﻌﺎرﻳﻒ وﻣﺒﺎدئ اوﻟﻴﺔ: ﺳﻮف ﻧﺮﻣﺰ ﻟﻠﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ أو اﻟﻌﻤﻠﻴﺔ اﻟﻌﺸﻮاﺋﻴﺔ اﻟﺘﻲ ﺗﻮﻟﺪ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﺑﺎﻟﺮﻣﺰ } {L, Z −1 , Z 0 , Z1 , Z 2 ,Lاو اﺧﺘﺼﺎرا }} {Z t , t ∈{L, −1, 0,1, 2,Lاو ﺑﺒﺴﺎﻃﺔ
وﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة ﺑﺎﻟﺮﻣﺰ } {z1 , z2 ,L, zn−1 , zn
} {Z t
ﺗﻌﺮﻳﻒ :٢اﻟﻘﻴﻢ z1 , z2 ,L , zn −1ﺗﺴﻤﻰ ﺑﺎﻟﻤﺎﺿﻰ او ﺗﺎرﻳﺦ اﻟﻈﺎهﺮة History
واﻟﺘﺎرﻳﺦ ﻣﻬﻢ ﺟﺪا ﻓﻲ ﻋﻤﻠﻴﺔ اﻟﻨﻤﺬﺟﺔ
ﺗﻌﺮﻳﻒ : ٣اﻟﻘﻴﻤﺔ znﺗﺴﻤﻰ اﻟﺤﺎﺿﺮ او اﻵن وهﻲ اﻟﻤﺸﺎهﺪة اﻷﺧﻴﺮة .
ﺗﻌﺮﻳﻒ :٤أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ et = zt − zˆt , t = 1, 2,..., nﺣﻴﺚ zˆtهﻲ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ) اﻟﻘﻴﻢ اﻟﺘﻲ ﻧﺘﺤﺼﻞ ﻋﻠﻴﻬﺎ ﻣﻦ اﻟﻨﻤﻮذج( وﺗﺴﻤﻲ أﻳﻀﺎ اﻟﺮواﺳﺐ Residuals وﻳﻼﺣﻆ ان اﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ﻧﺤﺼﻞ ﻋﻠﻴﻬﺎ دﻓﻌﺔ واﺣﺪة ﺑﻌﺪ ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج. ﻣﻼﺣﻈﺔ :ﺳﻮف ﻧﺮﻣﺰ ﻟﻠﻤﺸﺎهﺪات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺑﺎﻟﺮﻣﻮز zn +1 , zn + 2 , zn +3 ,...او ﺑﺸﻜﻞ ﻋﺎم zn +l , l ≥ 0وﻧﺮﻣﺰ ﻟﺘﻨﺒﺆاﺗﻬﺎ ﺑﺎﻟﺮﻣﺰ zn (1) , zn ( 2 ) , zn ( 3) ,...او ﺑﺸﻜﻞ ﻋﺎم zn ( l ) , l ≥ 0
٩
en ( l ) = zn +l − zn ( l ) , l ≥ 0
ﺗﻌﺮﻳﻒ :٥أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ
وأﺧﻄﺎء اﻟﺘﻨﺒﺆ ﺗﻨﺘﺞ اﻟﻮاﺣﺪة ﺗﻠﻮ اﻻﺧﺮى آﻠﻤﺎ ﺗﻘﺪم اﻟﺰﻣﻦ وﺷﻮهﺪت اﻟﻘﻴﻢ اﻟﺤﻘﻴﻘﻴﺔ ﺗﻌﺮﻳﻒ :٦ﻳﻘﺎل ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة } {z1 , z2 ,L, zn−1 , znﻣﺴﺘﻘﺮة Stationary
إذا ﺣﻘﻘﺖ اﻟﺸﺮوط اﻟﺘﺎﻟﻴﺔ:
1) E ( zt ) = constant = µ , ∀t ⎧constant = γ 0 , ∀t , ∀s, t = s ⎨ = ) 2) cov ( zt , zs ⎩ f ( s − t ) , ∀t , ∀ s , t ≠ s
اﻵن ﺳﻮف ﻧﻌﺮف ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﻬﻤﺔ ﺟﺪا ﻟﻜﻮﻧﻬﺎ ﺣﺠﺮة او ﻃﻮب اﻟﺒﻨﺎء Building Blocks ﻟﺠﻤﻴﻊ اﻟﻨﻤﺎذج اﻟﺘﻲ ﺳﻮف ﻧﺪرﺳﻬﺎ
ﺗﻌﺮﻳﻒ :٧ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء White Noise Seriesاوﻋﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء {at } White Noise Processهﻲ ﻋﺒﺎرة ﻋﻦ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻟﻌﺸﻮاﺋﻴﺔ ﻏﻴﺮ اﻟﻤﺘﺮاﺑﻄﺔ ) واﺣﻴﺎﻧﺎ ﻧﻔﺘﺮض اﻧﻬﺎ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻤﺘﻐﻴﺮات اﻟﻌﺸﻮاﺋﻴﺔ اﻟﺘﻲ ﺗﻜﻮن ﻣﺴﺘﻘﻠﺔ وﻟﻬﺎ ﺗﻮزﻳﻌﺎت ﻣﺘﻄﺎﺑﻘﺔ ) ( Independent, Identically Distributed (IIDﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ ﺛﺎﺑﺖ σ 2أي:
1) E ( at ) = 0, ∀t
وﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ ) WN ( 0, σ 2
⎧σ 2 , ∀t , ∀s, t = s ⎨ = ) 2) cov ( at , as ⎩ 0 , ∀t , ∀ s , t ≠ s at
ﻣﺜﺎل :١ﻣﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ : Random Walk ﺳﻮف ﻧﺒﻨﻲ ﻋﻤﻠﻴﺔ ﻋﺸﻮاﺋﻴﺔ } {Z tآﺎﻟﺘﺎﻟﻲ:
Z1 = a1 Z 2 = a1 + a2 M
Z t = a1 + a2 +L + at أو
Z t = Z t −1 + at أي ﻟﻮ اﻋﺘﺒﺮﻧﺎ ان a jهﻮ ﺣﺠﻢ اﻟﺨﻄﻮة اﻟﺘﻲ ﺗﺆﺧﺬ اﻟﻲ اﻻﻣﺎم او اﻟﺨﻠﻒ ﻋﻨﺪ اﻟﺰﻣﻦ jﻓﺎن Z t
هﻲ ﻣﻮﻗﻊ ﻣﺎﺷﻲ ﻋﺸﻮاﺋﻲ ﻋﻨﺪ اﻟﺰﻣﻦ t ﻣﻼﺣﻈﺔ :هﺬﻩ اﻟﻌﻤﻠﻴﺔ او اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﻦ اﻟﻨﻤﺎذج اﻟﻬﺎﻣﺔ ﺟﺪا اﻟﺘﻲ ﺗﺼﻒ اﺳﻮاق اﻟﻤﺎل اﻟﻌﺎﻟﻤﻴﺔ ﺗﻤﺮﻳﻦ :اوﺟﺪ ) E ( Z tو ) cov ( Z t , Z sﻟﺠﻤﻴﻊ ﻗﻴﻢ t , sوهﻞ اﻟﻌﻤﻠﻴﺔ ﻣﺴﺘﻘﺮة؟ ١٠
ﺗﻌﺮﻳﻒ :٨داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ Autocovariance Functionوﺗﻌﺮف آﺎﻟﺘﺎﻟﻲ: = cov ( Z t , Z s ) , ∀t , ∀s
γ t ,s
= E ⎡⎣( Z t − µ )( Z s − µ ) ⎤⎦ , ∀t , ∀s
وإذا ﻋﺮﻓﻨﺎ اﻟﺘﺨﻠﻒ kﻋﻠﻲ اﻧﻪ اﻟﻔﺘﺮة اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﺗﻔﺼﻞ ﺑﻴﻦ Z tوﺑﻴﻦ Z t −kأو Z t +kﻓﺈن داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: γ k = cov ( Z t , Z t −k ) , k = 0, ±1, ±2,L = E ⎡⎣( Z t − µ )( Z t −k − µ ) ⎤⎦ , k = 0, ±1, ±2,L ﻣﻼﺣﻈﺔ :ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﺘﻌﺮﻳﻒ اﻟﺜﺎﻧﻲ داﺋﻤﺎ
ﺗﻌﺮﻳﻒ :٩داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ) Autocorrelation Function (ACFوﺗﻌﺮف آﺎﻟﺘﺎﻟﻲ: γ ρ k = k , k = 0, ±1, ±2,L γ0 وﻟﻬﺎ اﻟﺨﻮاص اﻟﺘﺎﻟﻴﺔ:
1. ρ 0 = 1 2. ρ − k = ρ k
ρk ≤ 1
3.
ﻣﺜﺎل :٢ﺳﻮف ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء هﻲ: 2 ⎧σ , k = 0 ⎨ = ) γ k = cov ( at , at −k ⎩ 0, k ≠0 وذﻟﻚ ﻣﻦ اﻟﺘﻌﺮﻳﻒ ٧وﻣﻨﻬﺎ ﻧﺠﺪ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ: ⎧1 , k = 0 γ ⎨ = ρk = k γ 0 ⎩0 , k ≠ 0 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: Autocorrelation function of White Noise 1.0
0.0 9
8
7
6
5
4
Lag
١١
3
2
1
0
Autocorr
0.5
ﺗﻌﺮﻳﻒ :١٠داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation Function )(PACF وﺗﻌﻄﻲ ﻣﻘﺪار اﻟﺘﺮاﺑﻂ ﺑﻴﻦ Z tو Z t −kﺑﻌﺪ إزاﻟﺖ ﺗﺄﺛﻴﺮ اﻟﺘﺮاﺑﻂ اﻟﻨﺎﺗﺞ ﻣﻦ اﻟﻤﺘﻐﻴﺮات Z t −1 , Z t −2 ,..., Z t −k +1اﻟﻮاﻗﻌﺔ ﺑﻴﻨﻬﻤﺎ وﻳﺮﻣﺰ ﻟﻬﺎ ﻋﻨﺪ اﻟﺘﺨﻠﻒ kﺑﺎﻟﺮﻣﺰ φkkوأﺣﺪ ﻃﺮق ﺣﺴﺎﺑﻬﺎ ﺗﻘﻮم ﻋﻠﻲ ﺣﺴﺎب ﻣﻌﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺠﺰﺋﻲ φkkﻓﻲ اﻟﺘﻤﺜﻴﻞ: Z t = φk 1Z t −1 + φk 2 Z t −2 + L + φkk Z t −k + at
ﺣﺴﺎب : φ11
Z t = φ11Z t −1 + at
ﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﺑـ Z t −1وأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
) E ( Z t −1Z t ) = φ11 E ( Z t −1Z t −1 ) + E ( Z t −1at
أي
γ 1 = φ11γ 0 ﺣﻴﺚ ) E ( Z t −1at ) = 0ﺑﺸﻜﻞ ﻋﺎم E ( Z t −k at ) = 0, k = 1, 2,...آﻤﺎ ﺳﻨﺒﻴﻦ ﻻﺣﻘﺎ ( وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ γ 0ﻧﺠﺪ φ11 = ρ1
ﺗﻌﺮﻳﻒ :١١ﺑﺸﻜﻞ ﻋﺎم ﺗﻌﺮف داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ آﺎﻟﺘﺎﻟﻲ: k =0 k =1
k = 2,3,...
1,
ρ1 , ρ1 ρ2
ρ k −2 ρ k −3
M
M
ρk , ρ k −1 ρ k −2 M 1
ﺣﻴﺚ
ρ1 ρ k −2 ρ k −3 M
ρ1
L L L L L L L L
ρ1
1
1 M
ρ1 M
ρ k −1 ρ k −2 1 ρ1 1 ρ1 M
M
ρ k −1 ρ k −2
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = φkk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪
ﺗﺮﻣﺰ اﻟﻲ ﻣﺤﺪدة ﻣﺼﻔﻮﻓﺔ
اﻟﺘﻌﺮﻳﻒ اﻟﺴﺎﺑﻖ ﺻﻌﺐ اﻹﺳﺘﺨﺪام ﻟﻘﻴﻢ kاﻟﻜﺒﻴﺮة وﻟﻬﺬا ﺳﻮف ﻧﻌﻄﻲ ﺗﻌﺮﻳﻒ ﺁﺧﺮ ﻟﺤﺴﺎب داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﻜﺮارﻳﺎ: ١٢
ﺗﻌﺮﻳﻒ ١١ب :ﺗﺤﺴﺐ φkkﺗﻜﺮارﻳﺎ ﻣﻦ اﻟﻌﻼﻗﺎت
φ00 = 1, by definition φ11 = ρ1 k −1
, k = 2,3,...
ρ k − ∑φk −1, j ρ k − j j =1 k −1
1 − ∑φk −1, j ρ j
= φkk
j =1
ﺣﻴﺚ φkj = φk −1, j − φkkφk −1,k −1 ,
j = 1, 2,..., k − 1 ﺣﺴﺎب : φ22 ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب:
ρ 2 − φ11 ρ1 ρ 2 − ρ12 = φ22 = 1 − φ11 ρ1 1 − ρ12
وذﻟﻚ ﻷن . φ11 = ρ1
ﻣﺜﺎل :٣ﺳﻮف ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء: ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب φ00 = 1, by definition φ11 = ρ1 = 0 وذﻟﻚ ﻣﻦ ﻣﺜﺎل ١اﻟﺴﺎﺑﻖ وﺑﺎﻟﺘﻌﻮﻳﺾ ﻓﻲ ﺗﻌﺮﻳﻒ ١١ب ﻋﻦ φkkﻧﺠﺪ φ22 = φ33 = L = 0 وهﻜﺬا: ⎧1, k = 0 ⎨ = φkk ⎩0, k ≠ 0 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: Partial Autocorrelation function of White Noise 1.0
0.0
PACF
0.5
ﻣﻼﺣﻈﺔ :ﻻﺣﻆ أن آﻞ ﻣﻦ داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء Lag ﺗﺴﺎوي اﻟﺼﻔﺮ ﻣﻦ اﻟﺘﺨﻠﻒ اﻷول .وهﺬﻩ ﺧﺎﺻﻴﺔ ﺟﻤﻴﻊ اﻟﻤﺘﻐﻴﺮات اﻟﻌﺸﻮاﺋﻴﺔ ﻏﻴﺮ اﻟﻤﺘﺮاﺑﻄﺔ او اﻟﻤﺴﺘﻘﻠﺔ .ﻹﺧﺘﺒﺎر ﻋﺪم اﻟﺘﺮاﺑﻂ ﺑﻴﻦ ﻗﻴﻢ ﻣﺸﺎهﺪة ﻟﻤﺘﻐﻴﺮ ﻋﺸﻮاﺋﻲ ﺗﺴﺘﺨﺪم داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺬﻟﻚ. 9
8
7
6
5
4
١٣
3
2
1
0
ﺗﻌﺮﻳﻒ : ١٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ Sample Autocorrelation Function SACF ﻟﻤﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة z1 , z2 ,L , zn −1 , znوﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ rk , k = 0,1, 2,...وﺗﻌﻄﻰ
ﺑﺎﻟﻌﻼﻗﺔ:
, k = 0,1, 2, ...
) − z )( zt +k − z 2
)−z
n −k
t
∑( z t =1
n
t
∑( z
= rk
t =1
1 n ﺣﻴﺚ z = ∑ zt n t =1 وهﻲ ُﻣﻘ ﱢﺪر Estimatorﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ أي ρˆ k = rk , k = 0,1, 2,...وﺑﻤﺎ اﻧﻬﺎ ُﻣﻘ ﱢﺪر ﻓﻬﻲ إذا ﺗﺘﻐﻴﺮ ﻋﺸﻮاﺋﻴﺎ ﻣﻦ ﻋﻴﻨﺔ ﻻﺧﺮى وﻟﻬﺬا ﻓﺈن ﻟﻬﺎ اﻟﺨﻮاص اﻟﻌﻴﻨﻴﺔ اﻟﺘﺎﻟﻴﺔ: -١إذا آﺎﻧﺖ ρ k = 0, k > qﻓﺈن q ⎛1 ⎞ V ( rk ) ≅ ⎜ 1 + 2∑ ρ k2 ⎟ , k > q ⎝n k =1 ⎠ 1 وﻓﻲ اﻟﺤﺎﻟﺔ اﻟﺨﺎﺻﺔ ﻋﻨﺪﻣﺎ ρ k = 0, k > 0ﻓﺈن V ( rk ) ≅ , k > 0 n -٢ﻟﻘﻴﻢ nاﻟﻜﺒﻴﺮة و ρ k = 0ﻓﺈن rkﻳﻜﻮن ﻟﻬﺎ ﺗﻘﺮﻳﺒﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ وﺑﺎﻟﺘﺎﻟﻲ ﻧﺴﺘﻄﻴﻊ اﻟﻘﻴﺎم ﺑﺎﻻﺧﺘﺒﺎر اﻟﺘﺎﻟﻲ:
H 0 : ρk = 0 H1 : ρ k ≠ 0 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻹﺣﺼﺎﺋﺔ: = n rk
rk − 12
n
وذﻟﻚ ﻋﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05وﺗﺮﻓﺾ H 0إذا آﺎﻧﺖ n rk > 1.96 -٣ﺗﺤﺖ اﻟﻔﺮﺿﻴﺔ H 0 : ρ k = 0, ∀kﻓﺈن corr ( rk , rk − s ) ≅ 0, s ≠ 0 ُ -٤ﺗﻘ ﱠﺪر اﻟﺘﺒﺎﻳﻨﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ آﺎﻟﺘﺎﻟﻲ:
⎛1 ⎞ Vˆ ( rk ) ≅ ⎜ 1 + 2∑ rk2 ⎟ , k > q ⎝n k =1 ⎠ ﺗﻌﺮﻳﻒ :١٣داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ Sample Partial Autocorrelation Function SPACFﻟﻤﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة z1 , z2 ,L , zn −1 , znوﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ rkk , k = 0,1, 2,...ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: q
١٤
k =0 k =1 rk −2 r1 rk −3 r2 M M r1 rk , rk −2 rk −1 rk −3 rk −2 M M r1 1
k = 2,3,...
L L L L L L L L
⎧ 1, ⎪ r, ⎪ 1 ⎪ 1 r1 ⎪ 1 ⎪ r1 ⎪ M M ⎪ rkk = ⎨ rk −1 rk −2 ⎪ 1 r1 ⎪ 1 ⎪ r1 ⎪ M M ⎪ ⎪ rk −1 rk −2 ⎪ ⎩⎪
و ﻟﺤﺴﺎب داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ﺗﻜﺮارﻳﺎ:
ﺗﻌﺮﻳﻒ ١٣ب :ﺗﺤﺴﺐ rkkﺗﻜﺮارﻳﺎ ﻣﻦ اﻟﻌﻼﻗﺎت
r00 = 1, by definition r11 = r1 k −1
, k = 2,3, ...
rk − ∑ rk −1, j rk − j j =1 k −1
1 − ∑ rk −1, j rj
= rkk
j =1
j = 1, 2,..., k − 1
ﺣﻴﺚ
rkj = rk −1, j − rkk rk −1,k −1 ,
وهﻲ اﻳﻀﺎ ﻣﻘﺪﱠر Estimatorﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ أي φˆkk = rkk , k = 0,1, 2,...وﺑﻤﺎ اﻧﻬﺎ ُﻣﻘ ﱢﺪر ﻓﻬﻲ إذا ﺗﺘﻐﻴﺮ ﻋﺸﻮاﺋﻴﺎ ﻣﻦ ﻋﻴﻨﺔ ﻻﺧﺮى وﻟﻬﺬا ﻓﺈن ﻟﻬﺎ اﻟﺨﻮاص اﻟﻌﻴﻨﻴﺔ اﻟﺘﺎﻟﻴﺔ: 1 V ( rkk ) ≅ , k > 0 -١ n -٢ﻟﻘﻴﻢ nاﻟﻜﺒﻴﺮة ﻓﺈن rkkﻳﻜﻮن ﻟﻬﺎ ﺗﻘﺮﻳﺒﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ وﺑﺎﻟﺘﺎﻟﻲ ﻧﺴﺘﻄﻴﻊ اﻟﻘﻴﺎم ﺑﺎﻻﺧﺘﺒﺎر اﻟﺘﺎﻟﻲ:
H 0 : φkk = 0 H1 : φkk ≠ 0 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻹﺣﺼﺎﺋﺔ: ١٥
= n rkk
rkk − 12
n
وذﻟﻚ ﻋﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05وﺗﺮﻓﺾ H 0إذا آﺎﻧﺖ n rkk > 1.96
-٣ﺗﺤﺖ اﻟﻔﺮﺿﻴﺔ H 0 : φkk = 0, ∀kﻓﺈن corr (φkk ,φk − s ,k −s ) ≅ 0, s ≠ 0
ُ -٤ﺗﻘ ّﺪر اﻟﺘﺒﺎﻳﻨﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ آﺎﻟﺘﺎﻟﻲ: 1 Vˆ ( rkk ) ≅ , k > 0 n ﻣﺜﺎل :٤اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﺗﻤﺜﻞ اﻟﻄﻠﺐ ﻋﻠﻲ ﻣﻨﺘﺞ ﻣﻌﻴﻦ ﻳﻮﻣﻴﺎ: 158 222 248 216 226 239 206 178 169 أﺣﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ وارﺳﻤﻬﻤﺎ: 1 1 اوﻻ :ﻧﺤﺴﺐ اﻟﻤﺘﻮﺳﻂ zt = (158 + 222 + L + 169 ) = 206.89 ∑ 9 n t =1 ﺛﺎﻧﻴﺎ :ﻧﺤﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻣﻦ اﻟﻌﻼﻗﺔ n
, k = 0,1,2,...
=z n −k
) − z )( zt +k − z 2
)−z
t
∑( z
n
t
∑( z
t =1
= rk
t =1
) (158 × 222 + 222 × 248 + L + 178 × 169 = 0.265116 ) (158 × 158 + 222 × 222 + L + 169 × 169 ) (158 × 248 + 222 × 216 + L + 206 × 169 = r2 = -0.212 ) (158 × 158 + 222 × 222 + L + 169 × 169 ) (158 × 216 + 222 × 226 + L + 239 × 169 = r3 = −0.076 ) (158 × 158 + 222 × 222 + L + 169 × 169 = r1
وهﻜﺬا r8 = 0.230, r7 = 0.104, r6 = −0.242, r5 = −0.387, r4 = −0.183 ﺛﺎﻟﺜﺎ :ﻧﺤﺴﺐ اﻟﺘﺒﺎﻳﻨﺎت ﻣﻦ q ⎛1 ⎞ Vˆ ( rk ) ≅ ⎜ 1 + 2∑ rk2 ⎟ , k > q ⎝n k =1 ⎠ 1 ≅ ) Vˆ ( r1 9 1 1 2 Vˆ ( r2 ) ≅ (1 + 2 r12 ) = 1 + 2 ( 0.265) = 0.1267 n 9 1 1 2 2 Vˆ ( r3 ) ≅ 1 + 2 ( r12 + r22 ) = 1 + 2 ( 0.265) + ( −0.212 ) = 0.1367 n 9
)
))
(
(
١٦
( )
(
Vˆ ( r4 ) ≅ 0.138 Vˆ ( r5 ) ≅ 0.1454 Vˆ ( r6 ) ≅ 0.1787 اﻟﺦ… راﺑﻌﺎ :ﻧﺤﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ:
r00 = 1, by definition r11 = r1 = 0.265
ﺛﻢ ﻧﺤﺴﺐ ﺑﺎﻗﻲ اﻟﺘﺮاﺑﻄﺎت ﻣﻦ اﻟﻌﻼﻗﺎت اﻟﺘﻜﺮارﻳﺔ k −1
, k = 2,3, ...
rk − ∑ rk −1, j rk − j j =1 k −1
1 − ∑ rk −1, j rj
= rkk
j =1
ﺣﻴﺚ
j = 1, 2,..., k − 1
rkj = rk −1, j − rkk rk −1,k −1 , 1
r2 − r11r1 ( −0.212 ) − ( 0.265)( 0.265) −0.282225 = = 1 − r11r1 )1 − ( 0.265)( 0.265 0.929775
=
r2 − ∑ r1, j r2− j j =1 1
1 − ∑ r1, j rj
= r22
j =1
= −0.30354 ﻟﺤﺴﺎب r33ﻧﺤﺘﺎج اﻟﻰ r21وﺗﺤﺴﺐ ﻣﻦ r21 = r11 − r22 r11 = 0.265 − ( −0.303)( 0.265) = 0.345295
2
) r3 − ( r21r2 + r22 r1 ) 1 − ( r21r1 + r22 r2
=
r3 − ∑ r2, j r3− j j =1 k −1
1 − ∑ r2, j rj
= r33
j=1
) )( −0.076 ) − ( ( 0.345)( −0.212 ) + ( −0.303)( 0.265 ) ) 1 − ( ( 0.345)( 0.265) + ( −0.303)( −0.212
=
= 0.092
وهﻜﺬا ﻧﺤﺴﺐ ﺑﺎﻗﻲ اﻟﺘﺮاﺑﻄﺎت اﻟﺠﺰﺋﻴﺔ ﻟﻠﻌﻴﻨﺔ r88 = 0.042, r77 = 0.013, r66 = −0.207, r55 = −0.294, r44 = −0.298 وﻟﻬﺎ ﺟﻤﻴﻌﺎ اﻟﺘﺒﺎﻳﻨﺎت ﺗﺴﺎوي ﺗﻘﺮﻳﺒﺎ 1 = 0.1111 9 ﺧﺎﻣﺴﺎ :رﺳﻢ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ Autocorrelation Function for Demand
6
١٧
3
4
2
Corr
1
Lag
LBQ
T
Corr
Lag
LBQ
T
13.66
0.52
0.23
8
0.87 1.50
1 0.27 0.80 2 -0.21 -0.59
Autocorrelation
8
7
5
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
Partial Autocorrelation
Partial Autocorrelation Function for Demand 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
5
6
Lag PAC
T
Lag PAC
T
0.27 -0.30 0.09 -0.30 -0.29 -0.21 0.01
0.80 -0.91 0.27 -0.89 -0.88 -0.62 0.04
8 0.04
0.13
1 2 3 4 5 6 7
١٨
7
8
اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Autoregressive-Moving Average Modelsوإﺳﺘﺨﺪاﻣﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ: هﻨﺎك ﻋﺎﺋﻠﺔ آﺒﻴﺮة ﻣﻦ اﻟﻨﻤﺎذج اﻟﺘﻲ ﻳﻄﻠﻖ ﻋﻠﻴﻬﺎ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Autoregressive-Moving Average Modelsواﻟﺘﻲ اﺛﺒﺘﺖ اﻷﺑﺤﺎث اﻟﻜﺜﻴﺮة ﻓﻲ ﻣﺨﺘﻠﻒ اﻟﻤﻴﺎدﻳﻦ اﻟﺘﻄﺒﻴﻘﻴﺔ ﻋﻠﻲ ﺗﻔﻮﻗﻬﺎ اﻟﻬﺎﺋﻞ ﻋﻠﻲ اﻟﻄﺮق اﻟﺘﻘﻠﻴﺪﻳﺔ ﻓﻲ اﻟﺘﻨﺒﺆ.
ﺗﻌﺮﻳﻒ :١٤ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qوﻳﺮﻣﺰ ﻟﻪ ) ARMA ( p, qﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة } {z1 , z2 ,K, zn−1 , znﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: zt = δ + φ1 zt −1 + φ2 zt −2 + L + φ p zt − p + at − θ1at −1 − θ 2 at −2 − L − θ q at −q
ﺣﻴﺚ ) at WN ( 0,σ 2ﻣﺘﺴﻠﺴﻠﺔ ﺿﺠﺔ ﺑﻴﻀﺎء و ∞ < −∞ < δﻣﻌﻠﻢ ﺛﺎﺑﺖ ﻳﻤﺜﻞ اﻟﻤﺴﺘﻮي و φ1 ,φ2 ,K,φ pهﻲ ﻣﻌﺎﻟﻢ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ Autoregressive Parametersو θ1 ,θ 2 ,K,θ qهﻲ ﻣﻌﺎﻟﻢ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Moving Average Operators
ﺳﻮف ﻧﺴﺘﻌﻴﻦ ﺑﺠﺒﺮ اﻟﻌﻤﺎل Operators Algebraﻟﺘﺒﺴﻴﻂ هﺬﻩ اﻟﻨﻤﺎذج ﻟﻜﻲ ﻳﺴﻬﻞ اﻟﺘﻌﺎﻣﻞ ﻣﻌﻬﺎ
ﺗﻌﺮﻳﻒ :١٥ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ Backshift Operatorوﻳﺮﻣﺰ ﻟﻪ Bوﻟﻪ اﻟﺨﻮاص اﻟﺘﺎﻟﻴﺔ: 1 − Bzt = zt −1 2 − B m zt = B m−1 ( Bzt ) = B m−2 ( B ( Bzt ) ) = L = zt −m
3 − Bc = c, c is a constant ﺑﺎﻹﺿﺎﻓﺔ اﻟﻲ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ ﺗﻮﺟﺪ ﻋﻤﺎل اﺧﺮي ﻧﺤﺘﺎج اﻟﻴﻬﺎ ﻻﺣﻘﺎ هﻲ:
ﺗﻌﺮﻳﻒ ١٥ب: -١ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻷﻣﺎﻣﻲ Forewardshift Operatorوﻳﺮﻣﺰ ﻟﻪ Fوﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: F = B −1 -٢ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ Difference Operatorوﻳﺮﻣﺰ ﻟﻪ ∇ وﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: ) ∇ = (1 − B -٣ﻋﺎﻣﻞ اﻟﺘﺠﻤﻴﻊ Sum Operatorوﻳﺮﻣﺰ ﻟﻪ Sوﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: −1 ) S = ∇ −1 = (1 − B اﻵن ﻧﻌﻮد اﻟﻲ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qوﻧﻜﺘﺒﻪ ﻋﻠﻲ اﻟﺸﻜﻞ:
١٩
zt − φ1 zt −1 − φ2 zt −2 − L − φ p zt − p = δ + at − θ1at −1 − θ 2 at −2 − L − θ q at −q zt − φ1 Bzt − φ2 B 2 zt − L − φ p B p zt = δ + at − θ1 Bat − θ 2 B 2 at − L − θ q B q at
(1 − φ B − φ B 1
2
2
− L − φ p B p ) zt = δ + (1 − θ1 B − θ 2 B 2 − L − θ q B q ) at
أو
φ p ( B ) zt = δ + θ q ( B ) at
Autoregressive هﻮ ﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲφ p ( B ) = 1 − φ1B − φ2 B 2 − L − φ p B p ﺣﻴﺚ هﻮ ﻋﺎﻣﻞ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮكθ q ( B ) = 1 − θ1 B − θ 2 B 2 − L − θ q B q وOperator Moving Average Operator
:أﻣﺜﻠﺔ وﻳﻜﺘﺐARMA ( 0,0 ) وﻳﺮﻣﺰ ﻟﻪConstant Mean Model ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ-١ :ﻋﻠﻲ اﻟﺸﻜﻞ φ0 ( B ) zt = δ + θ 0 ( B ) at او (1) zt = δ + (1) at WN ( 0,σ 2 )
z t = δ + a t , at
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA (1,0 ) ≡ AR (1) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ-٢ φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
zt = δ + φ1 zt −1 + at , at
WN ( 0,σ 2 )
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA ( 0,1) ≡ MA (1) ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ-٣ φ0 ( B ) zt = δ + θ1 ( B ) at zt = δ + (1 − θ1 B ) at
zt = δ + at − θ1at −1 , at
WN ( 0,σ 2 )
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA ( 2,0 ) ≡ AR ( 2 ) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ-٤ φ 2 ( B ) z t = δ + θ 0 ( B ) at
(1 − φ B − φ B ) z 2
1
2
t
= δ + at
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at
WN ( 0,σ 2 )
: وﻧﻜﺘﺒﻪ ﻋﻠﻲ اﻟﺸﻜﻞARMA (1,1) (١و١) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ-٥
φ1 ( B ) zt = δ + θ1 ( B ) at (1 − φ1B ) zt = δ + (1 − θ1B ) at
zt = δ + φ1 zt −1 + at − θ1at −1 , at
WN ( 0,σ 2 )
٢٠
ﺧﺼﺎﺋﺺ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك: ﺳﻮف ﻧﺪرس اﻟﺨﺼﺎﺋﺺ اﻹﺣﺼﺎﺋﻴﺔ اﻟﺘﻲ ﺗﻤﻴﺰ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك وﻣﻌﺮﻓﺔ آﻴﻔﻴﺔ اﻟﺘﻌﺮف ﻋﻠﻲ اﺣﺪ هﺬﻩ اﻟﻨﻤﺎذج ﻣﻦ ﻋﻴﻨﺔ ﻣﺸﺎهﺪة وذﻟﻚ ﻟﺘﻌﻴﻴﻦ او ﺗﺤﺪﻳﺪ ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ﻳﺼﻒ اﻟﻤﺸﺎهﺪات.
أوﻻ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ):ARMA(٠،٠ وﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ
φ0 ( B ) zt = δ + θ 0 ( B ) at
او
) WN ( 0, σ 2
z t = δ + a t , at
ﺳﻮف ﻧﺸﺘﻖ اﻟﺨﻮاص اﻹﺣﺼﺎﺋﻴﺔ ﻟﻬﺬا اﻟﻨﻤﻮذج وذﻟﻚ ﺑﺈﻳﺠﺎد اﻟﺘﻮﻗﻊ )اﻟﻤﺘﻮﺳﻂ( وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ آﺎﻟﺘﺎﻟﻲ: ) E ( z t ) = δ + E ( at وذﻟﻚ ﻷن ) WN ( 0, σ 2
=δ at
ﺳﻮف ﻧﺮﻣﺰ ﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ ) E ( ztﺑﺎﻟﺮﻣﺰ µأي ) µ = E ( ztوﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن δ = µ
وﻳﻜﺘﺐ اﻟﻨﻤﻮذج:
zt − µ = at ﻹﺷﺘﻘﺎق داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻧﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي
⎦⎤ E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at
وﻟﻜﻦ E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ = γ kﻣﻦ ﺗﻌﺮﻳﻒ ٨إذا γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L وﻧﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ: ⎦⎤ k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ﻹﻳﺠﺎد اﻟﻄﺮف اﻷﻳﻤﻦ ﻧﻀﺮب ﻃﺮﻓﻲ zt − µ = atﻓﻲ atوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي وذﻟﻚ ﻷن ) WN ( 0, σ 2
E ⎡⎣( zt − µ ) at ⎤⎦ = E ( at at ) = σ 2
atإذا
k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ⎤⎦ = σ 2 k = 1: γ 1 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ = 0
وذﻟﻚ ﻷن
zt −1 − µ = at −1 E ⎡⎣( zt −1 − µ ) at ⎤⎦ = E ( at −1at ) = 0
ﻓﻲ اﻟﺤﻘﻴﻘﺔ ﻓﺈن
٢١
zt −k − µ = at −k , k = 1, 2,K
E ⎡⎣( zt −k − µ ) at ⎦⎤ = E ( at −k at ) = 0, k = 1, 2,K أي
:١ ﻗﺎﻋﺪة ⎧σ , k = 0 E ⎡⎣ ( zt −k − µ ) at ⎤⎦ = E ( at −k at ) = ⎨ ⎩ 0, k = 1, 2,.. 2
أي
γ0 =σ γ k = 0, k = ±1, ±2,K 2
:وﺗﻮﺿﻊ ﻋﻠﻲ ﺷﻜﻞ داﻟﻲ
⎧σ , k = 0 γk = ⎨ 2
⎩ 0, k ≠ 0
ﻧﺠﺪγ 0 = σ 2 وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ
γ k ⎧1, k = 0 =⎨ γ 0 ⎩0, k ≠ 0 :وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ Autocorrelation function of Constant Mean Model 1.0
Autocorr
ρk =
0.5
0.0 0
1
2
3
4
5
6
7
8
9
Lag
ﻧﺠﺪ١١ ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﻦ اﻟﺘﻌﺮﻳﻒ
٢٢
φ00 = 1, by definition φ11 = ρ1 , by definition φ11 = 0
1
ρ1 1 0 ρ2 0 0 = =0 ρ1 1 0
ρ1
1
1
ρ1
ρ1 ρ2
1
1
φ22 =
φ33 =
ρ1
0 1
1
ρ1 1 0 ρ2 0 1 ρ3 0 0 = ρ2 1 0 ρ1 0 1
ρ1
1
ρ1 ρ1
1
ρ1 ρ2
0 0 0 =0 0 0 0 0 1
M
1
ρ1
ρ1
1 M
M
ρ k −1 ρ k −2 ρ1 1 ρ1 1
φkk =
M
M
ρ k −1 ρ k −2
L L M L L L M L
ρ1 ρ2
1 0 M 0 = 1 0 M 0
M
ρk ρ k −1 ρ k −2 M 1
0 1 M 0 0 1 M 0
L L M L L L M L
0 0 M 0 0 = = 0, k = 2,3,L 0 1 0 M 1 :وﺗﻮﺿﻊ ﻋﻠﻲ ﺷﻜﻞ داﻟﻲ
⎧1, k = 0 ⎩0, k ≠ 0
φkk = ⎨
Partial Autocorrelation function of Constant Mean Model
:وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
PACF
1.0
0.5
0.0 0
1
2
3
4
5
6
7
8
9
Lag اﻟﺒﻴﻀﺎء اﻻ ﻓﻲ ان ﻟﻪ ﻣﺘﻮﺳﻂ ﻏﻴﺮ ﺻﻔﺮي ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ ﻻﻳﻔﺘﺮق ﻋﻦ ﻧﻤﻮذج اﻟﻀﺠﺔ:ﻣﻼﺣﻈﺔ
٢٣
ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ )ARMA(١،٠) = AR(١ وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞ: φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
) WN ( 0,σ 2
zt = δ + φ1 zt −1 + at , at
آﺎﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﺳﻮف ﻧﻮﺟﺪ اﻟﺘﻮﻗﻊ )اﻟﻤﺘﻮﺳﻂ( وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ:
(1 − φ1B ) zt = δ + at
+ (1 − φ1 B ) at
δ
−1
) (1 − φ1
δ
−1 ⎤ + E ⎡(1 − φ1 B ) at ⎦ ⎣ ) (1 − φ1
= zt
= ) E ( zt
اﻟﺤﺪ اﻟﺜﺎﻧﻲ ﻓﻲ اﻟﻄﺮف اﻷﻳﻤﻦ هﻮ
⎤ ⎞ ⎡⎛ ∞ j j −1 ⎡ ⎤ ⎥ E (1 − φ1 B ) at = E ⎢⎜ ∑φ1 B ⎟ at ⎣ ⎦ ⎢⎣⎝ j =0 ⎦⎥ ⎠ ﻹدﺧﺎل اﻟﺘﻮﻗﻊ ﻋﻠﻲ اﻟﻤﺠﻤﻮع اﻟﻶﻧﻬﺎﺋﻲ ﻳﺠﺐ ان ﺗﻜﻮن اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻶﻧﻬﺎﺋﻴﺔ ∞ < B j
j
∞
∑φ
1
ﻣﺘﻘﺎرﺑﺔ وذﻟﻚ
j =0
ﻳﺘﺤﻘﻖ إذا آﺎﻧﺖ φ1 < 1وذﻟﻚ إذا اﻋﺘﺒﺮﻧﺎ اﻟﻌﺎﻣﻞ Bاﻵن ﻳﻠﻌﺐ دور ﻣﺘﻐﻴﺮ ﻣﺮآﺐ Complex Variableﻟﻪ اﻟﺸﻜﻞ B = a + ibوﻟﻪ اﻟﻘﻴﺎس B = 1ﻓﻲ اﻟﺤﻘﻴﻘﺔ ﻻﺑﺪ ان ﻧﺘﻄﻠﺐ ان ﺗﻜﻮن ﺟﺰور او اﺻﻔﺎر (1 − φ1 B ) = 0ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة أي B > 1أي 1 − φ1 B = 0 1
φ1 > 1 ⇒ φ1 < 1
1
φ1
=B
⇒B >1
وهﺬا هﻮ ﺷﺮط اﻹﺳﺘﻘﺮار .ﻧﻌﻮد اﻟﻲ اﻟﻌﻼﻗﺔ
⎤ ⎞ ⎡⎛ ∞ j −1 ⎥ E ⎡(1 − φ1 B ) at ⎤ = E ⎢⎜ ∑φ1 B j ⎟ at ⎣ ⎦ ⎥⎦ ⎠ ⎣⎢⎝ j =0 ⎞ ⎡⎛ ∞ j j ⎤ ⎥ ) = ⎢⎜ ∑φ1 B ⎟ E ( at ⎢⎣⎝ j =0 ⎦⎥ ⎠ =0, ∀t وﻳﻜﻮن
٢٤
δ
) (1 − φ1 او
= ) E ( zt
δ
=µ
) (1 − φ1 ) ∴δ = µ (1 − φ1
وﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻓﻲ ﺻﻴﻐﺔ اﻟﻨﻤﻮذج ﻧﺠﺪ
zt = δ + φ1 zt −1 + at
= µ (1 − φ1 ) + φ1 zt −1 + at = µ + φ1 ( zt −1 − µ ) + at ﻧﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي
( zt − µ ) − φ1 ( zt −1 − µ ) = at
E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ − φ1E ⎡⎣( zt −k − µ )( zt −1 − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L أي
γ k − φ1γ k −1 = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L
وذﻟﻚ ﻣﻦ ﺗﻌﺮﻳﻒ ٨و ﺗﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ آﻤﺎ ﻳﻠﻲ: ⎦⎤ k = 0 : γ 0 − φ1γ 1 = E ⎡⎣( zt − µ ) at ﻹﻳﺠﺎد اﻟﻄﺮف اﻷﻳﻤﻦ ﻧﻘﻮم ﺑﺎﻟﺘﺎﻟﻲ: ) E ⎡⎣ at ( zt − µ ) ⎤⎦ − φ1 E ⎡⎣ at ( zt −1 − µ ) ⎤⎦ = E ( at at E ⎡⎣ at ( zt − µ ) ⎤⎦ − φ1 × ( 0 ) = σ 2 ∴ E ⎡⎣ at ( zt − µ ) ⎤⎦ = σ 2 إذا
γ 0 − φ1γ 1 = σ k = 1: γ 1 − φ1γ 0 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ = 0 2
ﻓﻲ اﻟﺤﻘﻴﻘﺔ
ﺑﻘﺴﻤﺔ اﻟﻤﻌﺎدﻟﺔ اﻷﺧﻴﺮة ﻋﻠﻲ γ 0
γ k − φ1γ k −1 = 0, k = 1, 2,L ﻧﺠﺪ
ρ k − φ1 ρ k −1 = 0, k = 1, 2,L
أو
وﺑﻤﺎ ان ρ 0 = 1
ρ k = φ1 ρ k −1 , k = 1, 2,L ﻓﺈن:
ρ1 = φ1 ρ 0 = φ1 ρ 2 = φ1 ρ1 = φ12
٢٥
M
ρ k = φ1k أو ﺑﺸﻜﻞ داﻟﺔ
ρ k = φ , k = 0, ±1, ±2,L وذﻟﻚ ﻷن ρ − k = ρ k , ∀kﺳﻮف ﻧﻨﻈﺮ ﻣﻦ اﻵن وﺻﺎﻋﺪا ﻟﻠﺸﻖ اﻟﻤﻮﺟﺐ ﻣﻦ ρ kأي k ρ k = φ1 , k = 0,1, 2,L هﺬﻩ اﻟﺪاﻟﺔ ﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 > 0 k 1
Autocorrelation function of AR(1) Model 0.5 0.4
ACF
0.3 0.2 0.1 0.0 10
9
8
7
6
4
5
3
2
1
0
Lag
-٢ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 < 0 Autocorrelation function of AR(1) Model 0.3 0.2 0.1 0.0
-0.2 -0.3 -0.4 -0.5 10
9
8
7
6
4
5
Lag
ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﻦ ﺗﻌﺮﻳﻒ ١١ﻧﺠﺪ
٢٦
3
2
1
0
ACF
-0.1
φ00 = 1, by definition φ11 = ρ1 = φ1 , by definition 1
φ22 =
ρ1 1
ρ1
1 φ1 ρ1 ρ 2 φ1 φ12 0 = = =0 1 φ1 1 − φ12 ρ1 1 φ1 1
M 1
ρ1
ρ1
1 M
M
ρ ρ k −2 φkk = k −1 1 ρ1 1 ρ1 M
M
ρ k −1 ρ k −2
L L L L L L L L
ρ1 ρ2 M
1
φ1
φ1
1 M
M
ρk φ = ρ k −1 1 ρ k −2 φ1
φ1k −2 φ1 L
k −1 1
M 1
φ
L φ1 L φ12 L M L φ1k
M
φ
k −1 1
φ1k L φ1k −1
1 M
L L
k −2 1
=
0 >0
M 1
وﻧﻜﺘﺐφ1 ﻣﺤﺪدة اﻟﺒﺴﻂ ﺗﺴﺎوي ﺻﻔﺮا ﻷن اﻟﻌﺎﻣﻮد اﻷﺧﻴﺮ ﻳﺴﺎوي اﻟﻌﺎﻣﻮد اﻷول ﻣﻀﺮوﺑﺎ ﻓﻲ :داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻋﻠﻲ اﻟﺸﻜﻞ اﻟﺪاﻟﻲ ⎧ 1, k = 0 ⎪ φkk = ⎨φ1 , k = 1 ⎪ 0, k ≥ 2 ⎩ :وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ φ1 > 0 ﻋﻨﺪﻣﺎ ﺗﻜﻮن-١ Partial Autocorrelation function of AR(1) Model 0.5
PACF
0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
Lag
φ1 < 0 ﻋﻨﺪﻣﺎ ﺗﻜﻮن-٢
٢٧
Partial Autocorrelation function of AR(1) Model 0.0 -0.1
-0.3
PACF
-0.2
-0.4 -0.5
10
9
8
7
6
4
5
3
2
1
0
Lag
ﻣﻼﺣﻈﺔ :داﺋﻤﺎ ﻻﺗﺮﺳﻢ أي ﻣﻦ ρ 0 = 1او φ00 = 1ﻓﻲ اﻷﺷﻜﺎل اﻟﺒﻴﺎﻧﻴﺔ.
ﻣﻨﺎﻗﺸﺔ اﻟﻨﻤﻮذج: -١ -٢ -٣ -٤
ﻋﻨﺪﻣﺎ ﺗﻜﻮن ) φ1 < 1ﺷﺮط اﻹﺳﺘﻘﺮار( ﻓﺈن ) E ( zt ) = δ (1 − φ1وهﻮﺛﺎﺑﺖ ﻟﺠﻤﻴﻊ ﻗﻴﻢ t داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ داﻟﺔ ﻟﻠﺘﺨﻠﻒ kﻓﻘﻂ وﻻﺗﻌﺘﻤﺪ ﻋﻠﻲ اﻟﺰﻣﻦ t داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ إﺑﺘﺪاءا ﻣﻦ ρ1ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 > 0وﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻣﺘﺮددة ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 < 0 داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ ﻗﻴﻤﺔ واﺣﺪة ﻏﻴﺮ ﺻﻔﺮﻳﺔ ) ﻣﻊ ﻋﺪم اﻟﻨﻈﺮ اﻟﻲ ( φ00وﻳﻜﻮن إﺗﺠﺎهﻬﺎ ﺣﺴﺐ إﺷﺎرة φ1وﻣﻘﺪارهﺎ ﻳﺴﺎوي φ1
ﺛﺎﻟﺜﺎ :ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ): ARMA(٢،٠) = AR(٢ وﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ 2 ( B ) zt = δ + θ 0 ( B ) a t = δ + at ) WN ( 0,σ 2
(1 − φ B + φ B ) z 2
t
2
1
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at
آﺎﻟﺴﺎﺑﻖ ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: (1 − φ1B − φ2 B ) zt = δ + at 2
−1 δ + (1 − φ1 B − φ2 B 2 ) at ) (1 − φ1 − φ2 −1 δ ⎤ + E ⎡(1 − φ1 B − φ2 B 2 ) at ⎦⎥ ⎣⎢ ) (1 − φ1 − φ2
٢٨
= zt
= ) E ( zt
∞ ⎛ ⎞ اﻟﺤﺪ اﻟﺜﺎﻧﻲ ﻓﻲ اﻟﻄﺮف اﻷﻳﻤﻦ ﻣﺠﻤﻮع ﻻﻧﻬﺎﺋﻲ ﻋﻠﻰ اﻟﺸﻜﻞ ⎟ E ⎜ ∑ψ j at − jوﻟﻜﻲ ﻧﺪﺧﻞ اﻟﺘﻮﻗﻊ ⎝ j =0 ⎠ ∞
داﺧﻞ اﻟﺘﺠﻤﻴﻊ اﻟﻼﻧﻬﺎﺋﻲ ﻻﺑﺪ ان ﺗﻜﻮن
∑ψ a
j t− j
ﻣﺘﻘﺎرﺑﺔ ﻓﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺮﺑﻊ وهﺬا ﻳﺘﺤﻘﻖ إذا
j =0
وﻓﻘﻂ إذا آﺎن ∞ <
2 j
∞
∑ψ
وهﺬا ﻳﺘﺤﻘﻖ إذا ﺣﻘﻘﺖ ﻣﻌﺎﻟﻢ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺸﺮوط اﻟﺘﺎﻟﻴﺔ:
j =0
φ2 − φ1 < 1 φ2 + φ1 < 1 −1 < φ2 < 1 واﻟﺘﻲ ﺗﺴﻤﻲ ﺑﺸﺮوط اﻹﺳﺘﻘﺮار ) هﺬﻩ اﻟﺸﺮوط ﺗﻨﺘﺞ اﻳﻀﺎ ﻣﻦ آﻮن ﺟﺰور او أﺻﻔﺎر (1 − φ1 B − φ2 B 2 ) = 0ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ( .إذا ﺗﺤﻘﻘﺖ ﺷﺮوط اﻹﺳﺘﻘﺮار ﻓﺈن −1 −1 E ⎡(1 − φ1 B − φ2 B 2 ) at ⎤ = ⎡(1 − φ1 B − φ2 B 2 ) E ( at ) ⎤ = 0, ∀t ⎢⎣ ⎦⎥ ⎦⎥ ⎢⎣
وﻳﻜﻮن
δ ) (1 − φ1 − φ2
و ﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻓﻲ ﺻﻴﻐﺔ اﻟﻨﻤﻮذج ﻧﺠﺪ
= ) µ = E ( zt
δ = (1 − φ1 − φ2 ) µ zt = (1 − φ1 − φ2 ) µ + φ1 zt −1 + φ2 zt −2 + at = µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at
( zt − µ ) − φ1 ( zt −1 − µ ) − φ2 ( zt −2 − µ ) = at ﻧﻀﺮب اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ:
⎦⎤ ) E ⎡⎣( zt − µ )( zt −k − µ ) − φ1 ( zt −1 − µ )( zt −k − µ ) − φ2 ( zt −2 − µ )( zt −k − µ = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
أي ⎦⎤ ) E ⎡⎣( zt − µ )( zt −k − µ ) ⎦⎤ − φ1 E ⎡⎣ ( zt −1 − µ )( zt −k − µ ) ⎤⎦ − φ2 E ⎡⎣( zt −2 − µ )( zt −k − µ = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
أو
γ k − φ1γ k −1 − φ2γ k −2 = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
وذﻟﻚ ﻣﻦ ﺗﻌﺮﻳﻒ ٨اﻵن ﻧﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ آﻤﺎ ﻳﻠﻲ:
٢٩
k = 0 : γ 0 − φ1γ −1 − φ2γ −2 = E ⎡⎣ at ( zt − µ ) ⎤⎦ = σ 2 ⇒ γ 0 = φ1γ 1 − φ2γ 2 + σ 2 وذﻟﻚ ﻣﻦ ﻗﺎﻋﺪة ١ k = 1: γ 1 − φ1γ 0 − φ2γ 1 = 0 ⇒ γ 1 = φ1γ 0 − φ2γ 1 k = 2 : γ 2 − φ1γ 1 − φ2γ 0 = 0 ⇒ γ 2 = φ1γ 1 − φ2γ 0 وﺑﺸﻜﻞ ﻋﺎم k ≥ 1: γ k = φ1γ k −1 + φ2γ k −2 ﺑﻘﺴﻤﺔ اﻟﻄﺮﻓﻴﻦ ﻋﻠﻲ γ 0ﻧﺠﺪ ρ k = φ1 ρ k −1 + φ2 ρ k −2 , k = 1, 2, ... ) ﻣﻼﺣﻈﺔ :ﺑﻮﺿﻊ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻋﻠﻲ اﻟﺸﻜﻞ ρ k − φ1 ρ k −1 − φ2 ρ k −2 = 0, k = 1, 2,... ﻧﺠﺪ اﻧﻬﺎ ﻣﻌﺎدﻟﺔ ﻓﺮوﻗﻴﺔ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ واﻟﺘﻲ ﻳﻤﻜﻦ ﺣﻠﻬﺎ ﺑﺸﻜﻞ ﻣﻐﻠﻖ ﺑﺈﺳﺘﺨﺪام ﻃﺮق ﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻔﺮوﻗﻴﺔ وﻟﻜﻦ هﺬا ﺧﺎرج ﻧﻄﺎق اﻟﻤﻘﺮر اﻟﺤﺎﻟﻲ( ﺳﻮف ﻧﺤﻞ اﻟﻌﻼﻗﺔ اﻟﺴﺎﺑﻘﺔ ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺘﻜﺮارﻳﺔ واﻟﺘﻲ ﺗﺤﺘﺎج اﻟﻲ ﻗﻴﻤﺘﻴﻦ اوﻟﻴﺘﻴﻦ: 1 − ρ0 = 1 φ 2 − ρ1 = φ1 ρ 0 + φ2 ρ −1 ⇒ ρ1 = 1 1 − φ2 وﻣﻨﻬﺎ ﻧﺠﺪ 2 φ ρ 2 = φ1 ρ1 + φ2 ρ0 ⇒ ρ 2 = 1 + φ2 1 − φ2 وهﻜﺬا اﻟﺦ… اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ ) AR ( 2 -١اﻟﺸﻜﻞ )φ1 = 0.4, φ2 = 0.4 (١ -٢اﻟﺸﻜﻞ )φ1 = 1.5, φ2 = −0.8 (٢ -٣اﻟﺸﻜﻞ )φ1 = 0.5, φ2 = −0.6 (٣ ﺷﻜﻞ )(١ ACF 0.7 0.6 0.5
0.3 0.2 0.1 0.0 20
10
Lag
٣٠
0
ACF
0.4
(٢) ﺷﻜﻞ ACF 1.0
ACF
0.5
0.0
-0.5
0
10
20
Lag
(٣) ﺷﻜﻞ ACF
ACF
0.5
0.0
-0.5
0
10
20
Lag
: آﺎﻟﺘﺎﻟﻲAR ( 2 ) اﻵن ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ
φ00 = 1, by definition φ11 = ρ1 , by definition 1 ρ1 ρ1 ρ 2 ρ 2 − ρ12 = ≠0 φ22 = 1 ρ1 1 − ρ12 ρ1 1
٣١
=0
ρ1 = φ1 + φ2 ρ1 ρ 2 = φ1 ρ1 + φ2 ρ3 = φ1 ρ 2 + φ2 ρ1
ρ1 1
ρ1
>0
ρ1 1 ρ2 ρ1 ρ3 ρ 2 = ρ2 ρ1
ρ1
1
1
ρ1 ρ2
ρ1 ρ1 1
ρ1
1
= φ33
ρ1 ρ2
1 وذﻟﻚ ﻷن اﻟﻌﻤﻮد اﻷﺧﻴﺮ ﻓﻲ ﻣﺤﺪدة اﻟﺒﺴﻂ هﻮ ﺗﺮآﻴﺐ ﺧﻄﻲ ﻣﻦ اﻟﻌﻤﻮدﻳﻦ اﻷول واﻟﺜﺎﻧﻲ ،آﺬﻟﻚ 1 ρ1 L ρ1 1 ρ1 L ρ1 = φ1 ρ 0 + φ2 ρ1 ρ1 1 L ρ2 ρ1 1 L ρ 2 = φ1 ρ1 + φ2 ρ 0 M M L M M M L M ρ ρ k −2 L ρ k ρ ρ k −2 L ρ k = φ1 ρ k −1 + φ2 ρ k −2 φkk = k −1 = k −1 = 0, k = 3, 4,... 1 ρ1 L ρ k −1 >0 1 L ρ k −2 ρ1 M M L M ρ k −1 ρ k −2 L 1 وذﻟﻚ اﻳﻀﺎ ﻟﻨﻔﺲ اﻟﺴﺒﺐ اﻟﺴﺎﺑﻖ .إذا
k =0 ⎧ 1, ⎪ ρ, k =1 1 ⎪⎪ φkk = ⎨ ρ 2 − ρ12 ⎪ 1− ρ2 , k = 2 1 ⎪ k ≥3 ⎪⎩ 0,
اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ ) AR ( 2 -٤اﻟﺸﻜﻞ )φ1 = 0.4, φ2 = 0.4 (٤ -٥اﻟﺸﻜﻞ )φ1 = 1.5, φ2 = −0.8 (٥ -٦اﻟﺸﻜﻞ )φ1 = 0.5, φ2 = −0.6 (٦ ﺷﻜﻞ )(٤
PACF 0.7 0.6 0.5
0.3 0.2 0.1 0.0 20
10
Lag
٣٢
0
PACF
0.4
ﺷﻜﻞ )(٥ PACF 1
PACF
0
-1
20
0
10
Lag
ﺷﻜﻞ )(٦ PACF 0.3 0.2 0.1
-0.2
PACF
0.0 -0.1
-0.3 -0.4 -0.5 -0.6 20
0
10
Lag
راﺑﻌﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ ): ARMA(٠،١) = MA(١ وﺗﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ0 ( B ) zt = δ + θ1 ( B ) at
) WN ( 0,σ 2
zt = δ + (1 − θ1 B ) at zt = δ + at − θ1at −1 , at
اﻵن ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: E ( zt ) = E (δ + at − θ1at −1 ) = δ ∴µ = δ
وﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﺑﻀﺮب هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻓﻲ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
zt − µ = at − θ1at −1
E ⎡⎣( zt − µ )( zt −k − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
٣٣
او
γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ
⎦⎤ k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt − µ ) at −1
ﻧﻮﺟﺪ آﻞ ﻣﻦ ⎦⎤ E ⎡⎣( zt − µ ) atو ⎦⎤ E ⎡⎣( zt − µ ) at −1آﺎﻵﺗﻲ: E ⎡⎣( zt − µ ) at ⎤⎦ = E ( at at ) − θ1 E ( at −1at ) = σ 2 E ⎡⎣( zt − µ ) at −1 ⎤⎦ = E ( at at −1 ) − θ1E ( at −1at −1 ) = −θ1σ 2
) ∴γ 0 = σ 2 − θ1 ( −θ1σ 2 ) = σ 2 (1 + θ12
⎦⎤ k = 1: γ 1 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −1 − µ ) at −1 −θ1 γ1 = γ 0 1 + θ12 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻟﻘﺎﻋﺪة ١
= ∴ γ 1 = −θ1σ 2 ⇒ ρ1
⎦⎤ k = 2 : γ 2 = E ⎡⎣( zt −2 − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −2 − µ ) at −1 ∴ γ 2 = 0 ⇒ ρ2 = 0
أﻳﻀﺎ ﻣﻦ ﻗﺎﻋﺪة ١وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن
k ≥ 2 : γ k = 0 ⇒ ρk = 0 وهﻜﺬا ﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ ) MA (1هﻲ ﻋﻠﻲ اﻟﺸﻜﻞ: ⎧ 1, k =0 ⎪ ⎪ −θ ρk = ⎨ 1 2 , k = 1 ⎪ 1 + θ1 ⎪⎩ 0 k ≥2 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ θ1 = 0.8 ACF 0.0 -0.1
-0.3 -0.4 -0.5 20
0
10
Lag
٣٤
ACF
-0.2
-٢ﻋﻨﺪﻣﺎ θ1 = −0.8 ACF 0.5 0.4
0.2
ACF
0.3
0.1 0.0
20
10
0
Lag
اﻵن ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ )MA (1 φ00 = 1, by definition
φ11 = ρ1 , by definition 1 ρ1 1 ρ1 ) −θ12 (1 − θ12 ρ1 ρ 2 ρ1 0 − ρ12 −θ12 = φ22 = = = = 1 ρ1 1 − ρ12 1 − ρ12 1 + θ12 + θ14 1 − θ16 ρ1 1 ρ1 1 ρ1 ρ1 ρ2 ρ1 1 0 ) −θ13 (1 − θ12 ρ3 0 ρ1 0 ρ13 = = = ρ2 1 ρ1 0 1 − 2 ρ12 1 − θ18 ρ1 ρ1 1 ρ1 1 0 ρ1 1 وﺑﺸﻜﻞ ﻋﺎم k >0
وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ ﻋﻨﺪﻣﺎ θ1 = −0.8
٣٥
ρ1
1
1
ρ1 ρ2
ρ1 ρ1 1
ρ1
(1 − θ ) , 2 1
)2 k +1
1
= φ33
ρ1 ρ2 k 1
−θ
( 1 − θ1
= φkk
PACF 0.5 0.4 0.3
0.1 0.0
PACF
0.2
-0.1 -0.2 -0.3 20
0
10
Lag
-٢ﻋﻨﺪﻣﺎ θ1 = 0.8 PACF 0.0 -0.1
-0.3
PACF
-0.2
-0.4 -0.5 20
0
10
Lag
ﺧﺎﻣﺴﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ): ARMA(٠،٢) = MA(٢ وﺗﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ 0 ( B ) zt = δ + θ 2 ( B ) a t zt = δ + (1 − θ1 B − θ 2 B 2 ) at
) WN ( 0,σ 2
zt = δ + at − θ1at −1 − θ 2 at −2 , at
اﻵن ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: E ( zt ) = E (δ + at − θ1at −1 − θ 2 at −2 ) = δ ∴µ = δ
٣٦
وﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﺑﻀﺮب هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻓﻲ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
zt − µ = at − θ1at −1 − θ 2 at −2
⎦⎤ E ⎡⎣( zt − µ )( zt −k − µ ) ⎤⎦ = E ⎡⎣ ( zt −k − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −k − µ ) at −1 − θ 2 E ⎡⎣( zt −k − µ ) at −2 ⎤⎦ , k = 0, ±1, ±2,...
او − µ ) at −2 ⎤⎦ , k = 0, ±1, ±2,...
γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ − θ 2 E ⎡⎣( zt −k
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ ﻧﺠﺪ 2
γ 0 = (1 + θ + θ ) σ 2 2
2 1
γ 1 = ( −θ1 + θ1θ 2 ) σ 2 γ 2 = −θ 2σ 2 γ k = 0, k > 2
وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ γ 0ﻧﺠﺪ
−θ1 + θ1θ 2 1 + θ12 + θ 22
= ρ1
−θ 2 1 + θ12 + θ 22
= ρ2
ρ k = 0, k > 2 وﺗﻜﺘﺐ ﻋﻠﻲ ﺷﻜﻞ داﻟﺔ 1, k =0 ⎧ ⎪ −θ + θ θ ⎪ 1 2 1 22 , k = 1 ⎪ 1 + θ1 + θ 2 ⎨ = ρk ⎪ −θ 2 , k =2 ⎪1 + θ12 + θ 22 ⎪ 0, k>2 ⎩
اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ ) MA ( 2 -٧اﻟﺸﻜﻞ )θ1 = 0.4, θ 2 = 0.4 (٧ -٨اﻟﺸﻜﻞ )θ1 = 1.5, θ 2 = −0.8 (٨ -٩اﻟﺸﻜﻞ )θ1 = 0.5, θ 2 = −0.6 (٩
٣٧
(٧) ﺷﻜﻞ
ACF 0.0
ACF
-0.1
-0.2
-0.3
0
10
20
Lag
(٨) ﺷﻜﻞ
ACF 0.2 0.1 0.0
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0
10
20
Lag
(٩) ﺷﻜﻞ ACF 0.4 0.3
0.2 0.1
ACF
ACF
-0.1
0.0 -0.1 -0.2 -0.3 -0.4 -0.5 0
10
Lag
٣٨
20
ﻣﻦ اﻟﺼﻌﺐ ﺟﺪا إﻳﺠﺎد ﺷﻜﻞ ﻣﻐﻠﻖ ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ) MA ( 2وﻟﻬﺬا ﺳﻮف ﻧﺴﺘﺨﺪم ﺗﻌﺮﻳﻒ ١١ب ﻟﺤﺴﺎﺑﻬﺎ ورﺳﻤﻬﺎ ﺗﻜﺮارﻳﺎ ﻟﻘﻴﻢ اﻟﻤﻌﺎﻟﻢ اﻟﺘﺎﻟﻴﺔ: -١٠اﻟﺸﻜﻞ )θ1 = 0.4, θ 2 = 0.4 (١٠ -١١اﻟﺸﻜﻞ )θ1 = 1.5, θ 2 = −0.8 (١١ -١٢اﻟﺸﻜﻞ )θ1 = 0.5, θ 2 = −0.6 (١٢
ﺷﻜﻞ )(١٠
PACF 0.0
-0.1
PACF
-0.2
-0.3
20
0
10
Lag
ﺷﻜﻞ )(١١ PACF 0.2 0.1 0.0 -0.1
-0.3 -0.4 -0.5 -0.6 -0.7
20
10
Lag
٣٩
0
PACF
-0.2
ﺷﻜﻞ )(١٢ PACF 0.4 0.3 0.2 0.1
PACF
0.0 -0.1 -0.2 -0.3 -0.4 -0.5 20
0
10
Lag
ﺳﺎدﺳﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك-اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ ): ARMA(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ:
φ1 ( B ) zt = δ + θ1 ( B ) at
(1 − φ1B ) zt = δ + (1 − θ1B ) at WN ( 0, σ 2 ) , φ1 ≠ θ1
zt − φ1 zt −1 = δ + at − θ1at −1 zt = δ + φ1 zt −1 + at − θ1at −1 , at
ﺷﺮط اﻹﺳﺘﻘﺮار φ1 < 1وﺷﺮط اﻹﻧﻘﻼب θ1 < 1وهﻨﺎك ﺷﺮط ﺁﺧﺮ ﻳﺴﻤﻰ ﺷﺮط اﻹﻣﺘﺴﺎخ Degeneracy Conditionوهﻮ φ1 ≠ θ1وهﺬا اﻟﺸﺮط ﻳﻀﻤﻦ ﻋﺪم إﻣﺘﺴﺎخ اﻟﻨﻤﻮذج إﻟﻰ ﻧﻤﻮذج أﻗﻞ درﺟﺔ ﻓﻔﻲ ﺣﺎﻟﺔ آﻮن φ1 = θ1ﻓﻤﻦ اﻟﻌﻼﻗﺔ (1 − φ1B ) zt = δ + (1 − θ1B ) atوﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ) (1 − φ1Bﻧﺠﺪ أن اﻟﻨﻤﻮذج ﻳﺼﺒﺢ zt = δ ′ + atﺣﻴﺚ ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ آﺎﻟﺘﺎﻟﻲ:
δ 1 − φ1
= δ ′وهﻮ ) ARMA ( 0,0
(1 − φ1B ) zt = δ + (1 − θ1B ) at (1 − θ1B ) a δ = zt + t ) 1 − φ1 (1 − φ1B (1 − θ1B ) E a δ = ) E ( zt + )( t ) 1 − φ1 (1 − φ1B
وذﻟﻚ ﻷن φ1 < 1وهﻜﺬا δ
أي
δ 1 − φ1
1 − φ1
= ) E ( zt
= E ( zt ) = µأو ) δ = µ (1 − φ1وﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻧﺠﺪ zt = µ (1 − φ1 ) + φ1 zt −1 + at − θ1at −1
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1 وﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ ﺑﺎﻟﺤﺪ ( zt −k − µ ) , k = 0, ±1, ±2,...وأﺧﺬ اﻟﺘﻮﻗﻊ ﻟﻠﻄﺮﻓﻴﻦ ﻧﺠﺪ E ⎡⎣( zt −k − µ )( zt − µ )⎤⎦ − φ1E ⎡⎣ ( zt −k − µ )( zt −1 − µ )⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﻣﻨﻬﺎ ٤٠
γ k − φ1γ k −1 = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣ ( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ ﻧﺠﺪ
k = 0 γ 0 − φ1γ 1 = E ⎡⎣ ( zt − µ ) at ⎤⎦ − θ1E ⎡⎣ ( zt − µ ) at−1 ⎤⎦
ﺑﻀﺮب اﻟﻌﻼﻗﺔE ⎡⎣( zt − µ ) at −1 ⎤⎦ وE ⎡⎣( zt − µ ) at ⎤⎦ ﻧﻮﺟﺪ اﻵن آﻞ ﻣﻦ
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1
وأﺧﺬ اﻟﺘﻮﻗﻊat −1 وat ﻓﻲ آﻞ ﻣﻦ E ⎡⎣ ( zt − µ ) at ⎤⎦ − φ1E ⎡⎣ ( zt −1 − µ ) at ⎤⎦ = E [at at ] − θ1 E [at −1at ]
ﻧﺠﺪ١ وﻣﻦ اﻟﻘﺎﻋﺪة
E ⎡⎣( zt − µ ) at ⎤⎦ − φ1 ( 0 ) = σ − θ1 ( 0 ) 2
E ⎡⎣( zt − µ ) at ⎤⎦ = σ 2 E ⎡⎣( zt − µ ) at −1 ⎤⎦ − φ1E ⎡⎣( zt −1 − µ ) at −1 ⎤⎦ = E [at at −1 ] − θ1E [at −1at −1 ]
و
E ⎡⎣( zt − µ ) at −1 ⎤⎦ − φ1σ 2 = 0 − θ1σ 2 ∴ E ⎡⎣( zt − µ ) at −1 ⎤⎦ = σ 2 (φ1 − θ1 )
وﺑﺎﻟﺘﻌﻮﻳﺾ ﻓﻲ اﻟﺼﻴﻐﺔ اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ k = 0 γ 0 − φ1γ 1 = σ 2 − θ1σ 2 (φ1 − θ1 ) ∴γ 0 − φ1γ 1 = σ 2 ⎡⎣1 − θ1 (φ1 − θ1 )⎤⎦ k = 1 γ 1 − φ1γ 0 = E ⎡⎣ ( zt −1 − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −1 − µ ) at −1 ⎤⎦
و
∴ γ 1 − φ1γ 0 = −θ1σ 2 k = 2 γ 2 − φ1γ 1 = E ⎡⎣ ( zt −2 − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −2 − µ ) at −1 ⎤⎦ = 0
و
∴ k ≥ 2 γ k − φ1γ k −1 = 0
وﻣﻦ اﻟﻤﻌﺎدﻻت
γ 0 − φ1γ 1 = σ ⎡⎣1 − θ1 (φ1 − θ1 )⎤⎦ 2
و γ 1 − φ1γ 0 = −θ1σ
2
ﻧﺠﺪ γ0 = γ1 =
ρ1 =
1 + θ − 2φ1θ1 2 σ 1 − φ12 2 1
(1 − φ1θ1 )(φ1 − θ1 ) σ 2 1 − φ12
وﻣﻦ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ
γ 1 (1 − φ1θ1 )(φ1 − θ1 ) = 1 + θ12 − 2φ1θ1 γ0
٤١
وﻣﻦ اﻟﻌﻼﻗﺔ
γ k − φ1γ k −1 = 0, k ≥ 2
وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ γ 0ﻧﺠﺪ ρ k − φ1 ρ k −1 = 0, k ≥ 2 وﻳﻤﻜﻦ ﺣﻞ هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﺗﻜﺮارﻳﺎ ﻟﺠﻤﻴﻊ ﻗﻴﻢ k ≥ 2ﺑﺈﺳﺘﺨﺪام اﻟﻘﻴﻢ اﻷوﻟﻴﺔ ρ0 = 1و ) (1 − φ1θ1 )(φ1 − θ1 = ρ1ﻓﻤﺜﻼ 1 + θ12 − 2φ1θ1 ρ2 = φ1 ρ1 ) (1 − φ1θ1 )(φ1 − θ1 ρ 2 = φ1 1 + θ12 − 2φ1θ1 ρ3 = φ1 ρ 2 ) (1 − φ1θ1 )(φ1 − θ1 ρ3 = φ12 1 + θ12 − 2φ1θ1
وهﻜﺬا. ﻧﻜﺘﺐ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﻋﻠﻰ اﻟﺸﻜﻞ 1, k =0 ⎧ ⎪ ) ⎪ (1 − φ1θ1 )(φ1 − θ1 , k =1 ⎨ = ρk 2 ⎪ 1 + θ1 − 2φ1θ1 ⎩⎪ k≥2 φ1 ρ k −1
ﺷﻜﻞ ١٣ﻳﻌﻄﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻘﻴﻢ φ1 = 0.9,θ1 = −0.5
ﺷﻜﻞ)(١٣
) A C F o f A R M A (1 ,1 1 .0 0 .9 0 .8 0 .7 0 .6 0 .4 0 .3 0 .2 0 .1 0 .0
15
5
10
Lag
ﺷﻜﻞ ١٤ﻳﻌﻄﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻘﻴﻢ φ1 = −0.9,θ1 = −0.5
٤٢
0
C1
0 .5
ﺷﻜﻞ)(١٤ ) A C F o f A R M A (1 ,1 0 .5
C1
0 .0
-0 .5
10
15
0
5
Lag
ﻧﻼﺣﻆ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ أو ﻣﺘﺮدد ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ وهﻲ ﻓﻲ هﺬا ﺗﺸﺒﻪ ﺗﻤﺎﻣﺎ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) AR (1ﻣﺎﻋﺪى ان اﻟﺘﺨﺎﻣﺪ ﻳﺒﺪأ ﻣﻦ ) ρ1ﺑﺮهﻦ أن ( ρ k = φ1k −1 ρ1 , k ≥ 2 داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ φkkﺗﺤﺴﺐ ﻣﻦ ﺗﻌﺮﻳﻒ ١١أو ١١ب آﺎﻟﺘﺎﻟﻲ: ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب ﻧﻮﺟﺪ φkkﺗﻜﺮارﻳﺎ φ00 = 1, by definition
) (1 − φ1θ1 )(φ1 − θ1 1 + θ12 − 2φ1θ1
= φ11 = ρ1
ρ 2 − φ11 ρ1 1 − φ11 ρ1 ρ −φ ρ −φ ρ φ33 = 3 21 2 22 1 , φ21 = φ11 − φ22φ11 1 − φ21 ρ1 − φ22 ρ 2
= φ22
وهﻜﺬا ﺗﺤﺴﺐ ﺑﻘﻴﺔ اﻟﻘﻴﻢ ﺗﻜﺮارﻳﺎ. ﻓﻤﺜﻼ ﻟﻠﻘﻴﻢ φ1 = 0.9,θ1 = −0.5ﻧﺠﺪ
φ11 = 0.944186 φ22 = -0.384471 φ33 = 0.183710 φ44 = -0.908462 φ55 = 0.452979 φ66 = -0.226337 φ77 = 0.113154 φ88 = -0.565702 φ99 = 0.282834
وﻧﺮﺳﻢ هﺬﻩ اﻟﻘﻴﻢ ﻓﻲ ﺷﻜﻞ ١٥ ﺷﻜﻞ ١٥ ) P A C F o f A R M A (1 ,1 1 .0
0 .5
C2 0 .0
15
5
10
Lag
٤٣
0
ﺷﻜﻞ ١٦ﻳﺒﻴﻦ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻘﻴﻢ φ1 = −0.9,θ1 = −0.5
ﺷﻜﻞ ١٦ ) P A C F o f A R M A (1 ,1 0 .3 0 .2 0 .1 0 .0
-0 .2
C2
-0 .1
-0 .3 -0 .4 -0 .5 -0 .6
15
5
10
0
Lag
ﻧﻼﺣﻆ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ أو ﻣﺘﺮدد ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ وهﻲ ﻓﻲ هﺬا ﺗﺸﺒﻪ ﺗﻤﺎﻣﺎ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج )MA (1 ﻣﺎﻋﺪى ان اﻟﺘﺨﺎﻣﺪ ﻳﺒﺪأ ﺑﻌﺪ اﻟﻘﻴﻤﺔ اﻷوﻟﻴﺔ . φ11 = ρ1
ﺧﻮاص ﻧﻤﺎذج ): ARMA(p,q أوﻻ :ﻧﻤﻮذج )AR(p وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: -١داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ. -٢داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ ﺗﺘﻜﻮن ﻣﻦ أﺻﻔﺎر ﻟﻘﻴﻢ اﻟﺘﺨﻠﻔﺎت k > pأي φ11 = φ22 = φ33 = L = φ pp ≠ 0 φ p +1, p +1 = φ p +2, p + 2 = L = 0
وﻳﺴﻤﻰ هﺬا ﻗﻄﻌﺎ ﻓﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ . k > p ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج ): MA(q وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: -١داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺗﺘﻜﻮن ﻣﻦ أﺻﻔﺎر ﻟﻘﻴﻢ اﻟﺘﺨﻠﻔﺎت k > qأي ρ1 = ρ 2 = ρ3 = L = ρ q ≠ 0 ρ q+1,q+1 = ρ q+2,q +2 = L = 0
وﻳﺴﻤﻰ هﺬا ﻗﻄﻌﺎ ﻓﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ . k > q -٢داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ. ﻻﺣﻆ اﻹزدواﺟﻴﺔ Dualityﺑﻴﻦ ﻧﻤﻮذﺟﻲ ARو .MA ﺛﺎﻟﺜﺎ :اﻟﻨﻤﻮذج اﻟﻤﺨﺘﻠﻂ ):ARMA(p,q وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﺨﺘﻠﻂ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ اﻟﺘﻲ ﺗﻨﺘﻬﻲ إﻟﻰ اﻟﺼﻔﺮ آﻠﻤﺎ زاد اﻟﺘﺨﻠﻒ . kﻋﻨﺪﻣﺎ ﺗﻜﻮن k > q − pﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﺘﺤﺪد ﻣﻦ ﺟﺰء اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻟﻠﻨﻤﻮذج و ﻋﻨﺪﻣﺎ ﺗﻜﻮن k > p − qﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺤﺪد ﻣﻦ ﺟﺰء اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻟﻠﻨﻤﻮذج. ٤٤
اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة Nonstationar Time :Series Models اوﻻ :ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ: ﻣﻦ ﺗﻌﺮﻳﻒ ٦ﻹﺳﺘﻘﺮار ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻧﺮى ان اﻟﺸﺮط اﻷول ﻟﻺﺳﺘﻘﺮار E ( zt ) = µ = constant ∀tﻳﺘﻄﻠﺐ أن ﻳﻜﻮن ﻣﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ ﺛﺎﺑﺖ ﻋﻠﻰ ﻃﻮل اﻟﺰﻣﻦ ،ﻓﻤﺜﻼ ﻟﻨﻤﻮذج اﻹﻧﺠﺮاف اﻟﺨﻄﻲ ) ∞ WN ( 0, σ 2 ) , b0 , b1 ∈ ( −∞,
zt = b0 + b1t + at , at
ﻧﺠﺪ ان اﻟﻤﺘﻮﺳﻂ هﻮ
E ( z )t = b0 + b1t
وهﻮ ﻏﻴﺮ ﺛﺎﺑﺖ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﺰﻣﻦ ،اي ان ﺷﺮط اﻹﺳﺘﻘﺮار اﻷول ﻏﻴﺮ ﻣﺘﺤﻘﻖ ﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ. ﻟﻨﺤﺎول اﻟﺘﺤﻮﻳﻞ ∇ztوذﻟﻚ ﺑﺘﻄﺒﻴﻖ ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ ﻋﻠﻰ اﻟﻨﻤﻮذج ﻧﺠﺪ
wt = ∇zt = zt − zt −1 = b0 + b1t + at − b0 − b1 ( t − 1) − at −1 =b1 + at − at −1 = b1 + ct
) WN ( 0,ν 2
) ﺗﻤﺮﻳﻦ :أوﺟﺪ اﻟﻌﻼﻗﺔ ﺑﻴﻦ ν 2و ( σ 2 اﻵن ﻧﺠﺪ ﻣﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺠﺪﻳﺪة wt
∴ wt = b1 + ct , ct
E ( wt ) = b1 = constant ∀t
أي ان ﺗﻄﺒﻴﻖ اﻟﺘﺤﻮﻳﻞ ) ∇ = (1 − Bﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ) أي أﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول ﻟﻠﻤﺘﺴﻠﺴﻠﺔ( ﺣﻮﻟﻬﺎ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. آﻤﺜﺎل ﺁﺧﺮ ﻧﻤﻮذج اﻹﻧﺠﺮاف اﻟﺘﺮﺑﻴﻌﻲ ) ∞ WN ( 0, σ 2 ) , b0 , b1 , b2 ∈ ( −∞,
zt = b0 + b1t + b2t 2 + at , at
ﺑﺈﻳﺠﺎد اﻟﻤﺘﻮﺳﻂ 2
E ( zt ) = b0 + b1t + b2t
وهﻮ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ ،أي ان اﻟﻨﻤﻮذج ﻏﻴﺮ ﻣﺴﺘﻘﺮ .ﺑﺄﺧﺬ اﻟﺘﺤﻮﻳﻞ ) ∇ 2 ztأﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻟﺜﺎﻧﻲ( ﻧﺠﺪ
) ∇2 zt = ∇ 2 ( b0 + b1t + b2t 2 + at
) + b1t + b2t 2 + at
(1 − 2 B + B ) z = (1 − 2 B + B )( b 2
2
0
t
wt = {b0 − 2b0 + b0 } + {b1t − 2b1 ( t − 1) + b1 ( t − 2 )} +
}
− 2b2 ( t − 1) + b2 ( t − 2 ) + 2
2
2
{b t 2
} {at − 2at −1 + at −2 } = 2b2 + {at − 2at −1 + at −2 ) WN ( 0,τ 2
وهﻜﺬا
٤٥
=b′ + ht , ht
) WN ( 0,τ 2
wt = ∇2 zt = b′ + ht , ht
E ( wt ) = b′ = constant ∀t
أي ان ﺗﻄﺒﻴﻖ اﻟﺘﺤﻮﻳﻞ ) ∇2أي اﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻟﺜﺎﻧﻲ( ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﺣﻮﻟﻬﺎ اﻟﻰ ﻣﺴﺘﻘﺮة. 2 2 ) ﺗﻤﺮﻳﻦ :أوﺟﺪ اﻟﻌﻼﻗﺔ ﺑﻴﻦ τو .( σ ﺑﺸﻜﻞ ﻋﺎم إذا آﺎن اﻟﻨﻤﻮذج ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮ ﻋﻠﻰ اﻟﺸﻜﻞ
) ∞ WN ( 0, σ 2 ) , b0 , b1 ,L , bd ∈ ( −∞,
zt = b0 + b1t + L + bd t + at , at d
ﻓﺈن اﻟﺘﺤﻮﻳﻞ ∇d ztﻳﺤﻮﻟﻪ إﻟﻰ ﻧﻤﻮذج ﻣﺴﺘﻘﺮ ،أي ان wt = ∇ d ztهﻮ ﻧﻤﻮذج ﻣﺴﺘﻘﺮ.
ﺗﻌﺮﻳﻒ :١٦ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ) ∞ zt = b0 + b1t + L + bd t + at , at WN ( 0, σ 2 ) , b0 , b1 ,L , bd ∈ ( −∞, اﻟﺘﺤﻮﻳﻞ ∇d ztوهﻮ اﻟﺘﻔﺮﻳﻖ ﻟﻠﺪرﺟﺔ dﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻳﺤﻮﻟﻬﺎ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. d
ﺛﺎﻧﻴ ًﺎ :ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﺘﺒﺎﻳﻦ: ﻣﻦ ﺗﻌﺮﻳﻒ ٦ﻹﺳﺘﻘﺮار ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ،اﻟﺸﺮط اﻟﺜﺎﻧﻲ
ﻳﺘﻄﻠﺐ أن ﻳﻜﻮن اﻟﺘﺒﺎﻳﻦ ﺛﺎﺑﺖ ﻟﺠﻤﻴﻊ ﻗﻴﻢ . t ﻓﻤﺜﻼ ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ
V ( zt ) = γ 0 = constant ∀t
) WN ( 0, σ 2
zt = zt −1 + at , at
ﻧﺠﺪ ﻣﻦ اﻟﺘﻌﻮﻳﺾ اﻟﻤﺘﻜﺮر zt = a1 + a2 + L + at
وﺑﺈﺧﺬ اﻟﺘﻮﻗﻊ واﻟﺘﺒﺎﻳﻦ
E ( zt ) = 0 = constant ∀t V ( zt ) = tσ 2
وﻧﻼﺣﻆ أن اﻟﺘﺒﺎﻳﻦ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ . t ﺑﺄﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول wt = ∇zt = zt − zt −1 = at
وﺑﺈﺧﺬ اﻟﺘﻮﻗﻊ واﻟﺘﺒﺎﻳﻦ E ( wt ) = 0 = constant ∀t V ( wt ) = σ 2 = constant ∀t
إذًا اﻟﺘﻔﺮﻳﻖ اﻷول ﺣﻮل اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. ﺑﺸﻜﻞ ﻋﺎم ﻟﻮ آﺎن اﻟﺘﺒﺎﻳﻦ داﻟﺔ ﻟﻤﺴﺘﻮى )ﻣﺘﻮﺳﻂ( ﻣﺘﻐﻴﺮ ﻋﻠﻰ اﻟﺸﻜﻞ
) V ( zt ) = cf ( µt
ﺣﻴﺚ c > 0ﺛﺎﺑﺖ و )⋅( fداﻟﺔ ﻣﻌﺮوﻓﺔ ﺗﻌﻄﻰ ﻗﻴﻤﺔ ﻏﻴﺮ ﺳﺎﻟﺒﺔ و µtﻣﺴﺘﻮى أو ﻣﺘﻮﺳﻂ ﻳﺘﻐﻴﺮ ﻣﻊ اﻟﺰﻣﻦ و ﺑﺎﻟﺘﺎﻟﻲ ﻓﺈن اﻟﺘﺒﺎﻳﻦ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ وهﻨﺎ ﻧﺤﺎول إﻳﺠﺎد ﺗﺤﻮﻳﻞ ) T ( ztأي إﻳﺠﺎد داﻟﺔ )⋅ ( Tﻹﺳﺘﻘﺮار اﻟﺘﺒﺎﻳﻦ. ٤٦
اﻟﺘﺤﻮﻳﻞ λ
zt − 1
λ
= ) yt = T ( zt
ﻳﻌﻄﻲ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ ﺣﻴﺚ ) ∞ λ ∈ ( −∞,هﻮ ﻣﻌﻠﻢ اﻟﺘﺤﻮﻳﻞ .اﻟﺠﺪول اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﻘﻴﻢ اﻷآﺜﺮ إﺳﺘﺨﺪاﻣﺎ ﻟﻠﻤﻌﻠﻢ λﻣﻊ اﻟﺘﺤﻮﻳﻼت اﻟﻤﻘﺎﺑﻠﺔ ﻟﻬﺎ: 1.0 zt
0.5
0.0
-0.5
zt
ln zt
1 zt
-0.1
λ
1 zt
yt
ﻣﺜﺎل: اﻟﺸﻜﻞ)ا( ﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ واﻟﺘﺒﺎﻳﻦ zt O r ig in a l S e r ie s 400
300
)z(t 200
100 90
80
70
60
50
40
30
20
10
In d e x
اﻟﺸﻜﻞ)ب( اﻟﻤﺘﺴﻠﺴﻠﺔ ﺑﻌﺪ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ ﺑﺈﺟﺮاء اﻟﺘﺤﻮﻳﻞ yt = ln zt T r a n s f o r m e d S e r ie s 6 .0
5 .5
)ln z(t 5 .0
90
80
70
60
50
40
30
20
اﻟﺸﻜﻞ)ج( اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺤﻮﻟﺔ ytﺑﻌﺪ إﺟﺮاء اﻟﺘﻔﺮﻳﻖ اﻷول ∇yt = yt − yt −1
٤٧
10
In d e x
D if f e re n c e d a n d T ra n s f o rm e d S e rie s 0 .2
0 .1 )y(t)-y(t-1
0 .0
-0 .1
-0 .2
90
80
70
60
50
40
30
20
10
In d e x
ﻻﺣﻆ آﻴﻒ اﺻﺒﺤﺖ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﻤﺘﻮﺳﻂ واﻟﺘﺒﺎﻳﻦ.
ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )(p,d,q Autoregressive-Integrated-Moving Average Models )ARIMA(p,d,q ﻳﻤﻜﻦ ﻧﻤﺬﺟﺔ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺴﺘﻘﺮة wt = ∇ d ztﻋﻠﻰ ﺷﻜﻞ ﻧﻤﻮذج أﻧﺤﺪار ذاﺗﻲ-ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qآﺎﻟﺘﺎﻟﻲ: ) WN ( 0, σ 2
أو
)
2
φ p ( B ) wt = φ p ( B ) ∇ d zt = δ + θ q ( B ) at , at
φ p ( B )(1 − B ) zt = δ + θ q ( B ) at , at WN ( 0,σ d
وهﺬا اﻟﻨﻤﻮذج ﻳﺴﻤﻰ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, d , q ﺣﻴﺚ ) ∞ δ ∈ ( −∞,ﻣﻌﻠﻢ اﻹﻧﺠﺮاف. أﻣﺜﻠﺔ ﻋﻠﻰ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) : ( p, d , q اوﻻ :ﻧﻤﻮذج اﻹﻧﺤﺪاراﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ ﻣﻦ اﻟﺪرﺟﺔ ) (١،١أو )ARIMA(١،١،٠ ): =ARI(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ1 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at WN ( 0, σ 2
(1 − φ1B )(1 − B ) zt = δ + at {1 − (φ1 + 1) B + φ1B 2 } zt = δ + at أي φ1 < 1
),
2
WN ( 0, σ
zt = δ + (φ1 + 1) zt −1 − φ1 zt −2 + at , at
ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )(١،١ أو ): ARIMA(٠،١،١) = IMA(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ0 ( B )(1 − B ) zt = δ + θ1 ( B ) at , at WN ( 0, σ 2
WN ( 0,σ 2 ) , θ1 < 1
at
(1 − B ) zt = δ + (1 − θ1B ) at ,
WN ( 0, σ 2 ) , θ1 < 1
zt − zt −1 = δ + at − θ1at , at
أي ٤٨
WN ( 0, σ 2 ) ,
θ1 < 1
zt = δ + zt −1 + at − θ1at , at
ﺛﺎﻟﺜﺎ :ﻧﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ﺑﺈﻧﺠﺮاف Random Walk with Trend Modelأو ): ARIMA(٠،١،٠ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
φ0 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at
) WN ( 0, σ 2
أي
)
2
WN ( 0, σ
at
(1 − B ) zt = δ + at ,
zt = δ + zt −1 + at , at
داﻟﺔ اﻷوزان ) ψ ( Bوﺗﻤﺜﻴﻞ ﻧﻤﺎذج ):ARMA(p,q ﺳﺒﻖ أن آﺘﺒﻨﺎ ﻧﻤﺎذج ) ARMA ( p, qﻋﻠﻰ اﻟﺸﻜﻞ
) WN ( 0, σ 2
أو ﺑﺸﻜﻞ اﻹﻧﺤﺮاف ﻋﻦ اﻟﻤﺘﻮﺳﻂ
)
2
WN ( 0, σ
φ p ( B ) zt = δ + θ q ( B ) a t , a t φ p ( B )( zt − µ ) = θ q ( B ) at , at
ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ) φ p ( Bﻧﺠﺪ ) WN ( 0, σ 2
) θq ( B a , at φp (B) t
) WN ( 0, σ 2
+
δ
)φ p (1
) θq ( B a , at φp (B) t
= zt
= zt − µ
) θq ( B ﻻﺣﻆ ان ﻟﻠﻨﻤﺎذج اﻟﻤﺴﺘﻘﺮة اﻟﻨﺴﺒﺔ )φp (B δ ﺗﻘﻊ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة اﻳﻀﺎ = µ وﻟﻬﺬا ﺳﻮف ﻧﻜﺘﻔﻲ ﺑﺸﻜﻞ اﻹﻧﺤﺮاف ﻋﻦ اﻟﻤﺘﻮﺳﻂ ﻓﻲ )φ p (1
ﺗﺸﻜﻞ ﻣﺴﻠﺴﻠﺔ ﻣﺘﻘﺎرﺑﺔ وذﻟﻚ ﻷن ﺟﺬور φ p ( B ) = 0
ﻣﻨﺎﻗﺸﺘﻨﺎ اﻟﺘﺎﻟﻴﺔ ) WN ( 0, σ 2
اﻟﻤﺴﻠﺴﻠﺔ اﻟﻤﺘﻘﺎرﺑﺔ
واﻟﺘﻲ ﺗﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
)θ (B zt − µ = q a , at φp (B) t ) θq ( B )φp (B
)θq ( B = ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + L , ψ 0 = 1 )φp (B
ﺗﺴﻤﻰ داﻟﺔ اﻷوزان.
٤٩
= )ψ (B
= )ψ (B
ﺗﻌﺮﻳﻒ :١٧داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ) ARMA ( p, qاﻟﻤﺴﺘﻘﺮ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ )θq ( B = ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + L , ψ 0 = 1 )φp (B
= )ψ (B
∞ )θq ( B = ∑ψ B j , ψ 0 = 1 φ p ( B ) j =0 j
= )ψ (B
ﺣﻴﺚ اﻷوزان هﻲ ψ 0 = 1,ψ 1 ,ψ 2 ,ψ 3 ,L ) θq ( B a , at φp (B) t
ﻣﻼﺣﻈﺔ :اﻟﻨﻤﻮذج اﻟﺬي ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
= zt − µﻳﺴﻤﻰ ﺗﻤﺜﻴﻞ
اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻼﻧﻬﺎﺋﻲ ﻟﻨﻤﺎذج ) . ARMA ( p, q
أﻣﺜﻠﺔ ﻟﺪاﻟﺔ اﻷوزان ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ): AR(١ ﻧﻤﻮذج ) AR(١ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ1 ( B )( zt − µ ) = θ 0 ( B ) at , at WN ( 0, σ 2
(1 − φ1B )( zt − µ ) = at 1 a (1 − φ1B ) t
= zt − µ
zt − µ = ψ ( B ) at
ﺣﻴﺚ 1 ) (1 − φ1B
= )ψ (B
ﺳﻮف ﻧﻮﺟﺪ اﻷوزان ψ 1 ,ψ 2 ,ψ 3 ,Lﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺘﺎﻟﻴﺔ 1 ) (1 − φ1B
= )ψ (B
ψ ( B )(1 − φ1B ) ≡ 1
B 2 + ψ 3 B 3 + L) (1 − φ1B ) ≡ 1
2
(1 + ψ B + ψ 1
ﻻﺣﻆ أن اﻟﻌﻼﻗﺔ اﻷﺧﻴﺮة هﻲ ﻋﻼﻗﺔ ﺗﻜﺎﻓﺆ أي ان ﻣﻌﺎﻣﻼت B jﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻣﺘﺴﺎوﻳﺔ. وﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت B jﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪ φ1 < 1
+ ψ 3 B 3 + L) (1 − φ1B ) ≡ 1,
2
(1 + ψ B + ψ B 2
1
B 0 : (1)(1) ≡ 1 B1 : ψ 1 − φ1 ≡ 0 ⇒ ψ 1 = φ1 B 2 : ψ 2 − ψ 1φ1 ≡ 0 ⇒ ψ 2 = ψ 1φ1 = φ12 B 3 : ψ 3 − ψ 2φ1 ≡ 0 ⇒ ψ 3 = ψ 2φ1 = φ13 M B j : ψ j − ψ j −1φ1 ≡ 0 ⇒ ψ j = ψ j −1φ1 = φ1j
أي ان اﻷوزان ﻟﻨﻤﻮذج ) AR(١هﻲ ٥٠
ψ j = φ1j , φ1 < 1
: MA(١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞMA(١) ﻧﻤﻮذج
φ0 ( B )( zt − µ ) = θ1 ( B ) at , at WN ( 0, σ 2 )
( zt − µ ) = (1 − θ1B ) at zt − µ = ψ ( B ) at ﺣﻴﺚ
ψ ( B ) = (1 − θ1B )
ﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪB j ﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت
ψ 1 = −θ1 , ψ 2 = ψ 3 = L = 0
أي j=0 j =1
⎧ 1, ⎪ ψ j = ⎨ −θ1 , ⎪ 0, ⎩
j≥2
φ2 ( B )( zt − µ ) = θ 0 ( B ) at , at WN ( 0,σ 2 )
(1 − φ B − φ B ) ( z − µ ) = a
: AR(٢) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞAR(٢) ﻧﻤﻮذج
2
1
zt − µ =
2
t
t
1 a (1 − φ1B − φ2 B 2 ) t
zt − µ = ψ ( B ) at
ﺣﻴﺚ ψ (B) =
1 (1 − φ1B − φ2 B 2 )
ψ ( B ) (1 − φ1B − φ2 B ) ≡ 1
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
2
(1 + ψ B + ψ B 1
2
2
+ ψ 3 B 3 + L)(1 − φ1B − φ2 B 2 ) ≡ 1
B1 : ψ 1 − φ1 = 0 ⇒ ψ 1 = φ1 2 B 2 : ψ 2 − φψ 1 1 − φ2 = 0 ⇒ ψ 2 = φψ 1 1 + φ2 = φ1 + φ 2
B 3 : ψ 3 − φψ 1 2 − φ2ψ 1 = 0 ⇒ ψ 3 = φψ 1 2 + φ2ψ 1 M B j : ψ j − φψ 1 j −1 − φ2ψ j − 2 = 0 ⇒ ψ j = φψ 1 j −1 + φ 2ψ j − 2
هﻲAR(٢) أي ان اﻷوزان ﻟﻨﻤﻮذج
٥١
j=0
⎧ 1, ⎪φ , ⎪ 1 ψj =⎨ 2 ⎪ φ1 + φ2 , ⎪⎩φψ 1 j −1 + φ2ψ j − 2 ,
j =1 j=2 j≥3
: MA(٢) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞMA(٢) ﻧﻤﻮذج
φ0 ( B )( zt − µ ) = θ 2 ( B ) at , at WN ( 0,σ 2 )
( zt − µ ) = (1 − θ1B − θ 2 B 2 ) at zt − µ = ψ ( B ) at ψ ( B ) = (1 − θ1B − θ 2 B
2
ﺣﻴﺚ
)
ﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪB j ﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت
ψ 1 = −θ1 , ψ 2 = −θ 2 , ψ 3 = ψ 4 = ψ 5 L = 0
أي j=0 j =1 j=2 j≥2
⎧ 1, ⎪ −θ , ⎪ ψj =⎨ 1 ⎪ −θ 2 , ⎪⎩ 0,
φ1 ( B )( zt − µ ) = θ1 ( B ) at , at
: ARMA(١،١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞARMA(١،١) ﻧﻤﻮذج
WN ( 0, σ 2 )
(1 − φ1B )( zt − µ ) = (1 − θ1B ) at (1 − θ1B ) a zt − µ = (1 − φ1B ) t zt − µ = ψ ( B ) a t ψ (B) =
(1 − θ1B ) (1 − φ1B )
ψ ( B )(1 − φ1B ) ≡ (1 − θ1B )
(1 + ψ B + ψ 1
ﺣﻴﺚ
2
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
B 2 + ψ 3 B 3 + L) (1 − φ1B ) ≡ (1 − θ1B )
B1 : ψ 1 − φ1 = −θ1 ⇒ ψ 1 = φ1 − θ1 B 2 : ψ 2 − φψ 1 1 = 0 ⇒ ψ 2 = φψ 1 1 = φ1 (φ1 − θ1 ) 2 B 3 : ψ 3 − φψ 1 2 = 0 ⇒ ψ 3 = φψ 1 2 = φ1 (φ1 − θ1 )
M
j −1 B j : ψ j − φψ (φ1 − θ1 ) 1 j −1 = 0 ⇒ ψ j = φψ 1 j −1 = φ1
هﻲARMA(١،١) أي ان اﻷوزان ﻟﻨﻤﻮذج ٥٢
j −1 ψ j = φψ (φ1 − θ1 ) , 1 j −1 = φ1
j ≥ 1,
φ1 < 1, φ1 ≠ θ1
: ARI(١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞARI(١) ﻧﻤﻮذج
φ1 ( B )(1 − B )( zt − µ ) = at , at WN ( 0,σ 2 ) zt − µ =
1 a (1 − φ1B )(1 − B ) t
zt − µ = ψ ( B ) at
ﺣﻴﺚ ψ ( B) =
1 (1 − φ1B )(1 − B )
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
ψ ( B )(1 − φ1B )(1 − B ) ≡ 1
(1 +ψ B +ψ (1 +ψ B +ψ 1
2
1
2
B 2 + ψ 3 B 3 + L) (1 − φ1B )(1 − B ) ≡ 1
B 2 + ψ 3 B 3 + L) (1 − (φ1 + 1) B + φ1B 2 ) ≡ 1
B1 : ψ 1 − (φ1 + 1) = 0 ⇒ ψ 1 = φ1 + 1 B 2 : ψ 2 − (φ1 + 1)ψ 1 + φ1 = 0 ⇒ ψ 2 = (φ1 + 1)ψ 1 + φ1 = (φ1 + 1) + φ1 2
B 3 : ψ 3 − (φ1 + 1)ψ 2 + φψ 1 1 = 0 ⇒ ψ 3 = (φ1 + 1)ψ 2 − φψ 1 1 M
B j : ψ j − (φ1 + 1)ψ j −1 + φψ 1 j −2 1 j − 2 = 0 ⇒ ψ j = (φ1 + 1)ψ j −1 − φψ
هﻲARI(١) أي ان اﻷوزان ﻟﻨﻤﻮذج j=0 j =1
⎧ 1, ⎪ φ + 1, ⎪ 1 ψj =⎨ 2 ⎪ (φ1 + 1) + φ1 , ⎪(φ1 + 1)ψ j −1 − φψ 1 j −2 , ⎩
j=2 j≥3
وأﺧﻴﺮا ﻧﻮﺟﺪ ARIMA(١،٠،١) أوRandom Walk Mdel داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ zt = zt −1 + at , at
WN ( 0, σ 2 )
zt − zt −1 = at , at
WN ( 0, σ
2
أي
)
(1 − B ) zt = at zt =
1 a (1 − B ) t
ﺣﻴﺚ 1 ψ ( B) = (1 − B )
٥٣
و ﻧﻮﺟﺪ اﻷوزان ψ 1 ,ψ 2 ,ψ 3 ,Lﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔ
ψ ( B )(1 − B ) ≡ 1 + ψ 3 B 3 + L) (1 − B ) ≡ 1
2
(1 + ψ B + ψ B 1
2
B1 : ψ 1 − 1 = 0 ⇒ ψ 1 = 1 B 2 :ψ 2 −ψ 1 = 0 ⇒ ψ 2 = ψ 1 = 1 B 3 :ψ 3 −ψ 2 = 0 ⇒ ψ 3 = ψ 2 = 1 M B j : ψ j − ψ j −1 = 0 ⇒ ψ j = ψ j −1 = 1
أي ان اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ) ARIMA ( 0,1,0هﻲ ψ j = 1,
j ≥1
ﺑﻌﺾ ﺧﻮاص داﻟﺔ اﻷوزان ) : ψ ( B ﺳﺒﻖ أن آﺘﺒﻨﺎ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ARMA(p,qﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
وﺑﻜﺘﺎﺑﺔ هﺬﻩ اﻟﻌﻼﻗﻪ ﻋﻠﻰ اﻟﺸﻜﻞ
zt − µ = ψ ( B ) a t , a t
zt − µ = at + ψ 1at −1 + ψ 2 at −2 + ψ 3at −3 + L ∞
= ∑ψ j at − j , ψ 0 = 1 j =0
وإذا اﻓﺘﺮﺿﻨﺎ ان اﻷوزان ﺗﺘﻘﺎرب اي ∞ < ψ 2j
∞
j =0
∑
ﻓﺈﻧﻪ ﻳﻤﻜﻦ إﺛﺒﺎت اﻟﻨﻈﺮﻳﺔ اﻟﺘﺎﻟﻴﺔ:
ﻧﻈﺮﻳﺔ :١ ﻟﻨﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ARMA(p,qاﻟﻤﺴﺘﻘﺮ واﻟﺬي ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ∞ < WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j ∞
∞
zt − µ =∑ψ j at − j , at j =0
-١اﻟﻤﺘﻮﺳﻂ هﻮ
E ( zt ) = µ , ∀t
-٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ∞
j +k
, k = 0,1, 2,L
∑ψ ψ j
j =0
2 j
∞
∑ψ
= ρk
j=0
ﻣﺜﺎل :ﻟﻨﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ ) AR(١وﺟﺪﻧﺎ ﺳﺎﺑﻘﺎ داﻟﺔ اﻷوزان ψ j = φ , φ1 < 1 j 1
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
٥٤
= φ1k , k = 0,1, 2,L
φ1k 1 − φ12 1 1 − φ12
∞
=
∑φ1jφ1j+k j =0 ∞
∑φ
2j 1
j =0
∞
=
∑ψ jψ j+k j =0
2 j
∑ψ
وهﻲ ﻧﻔﺲ اﻟﻨﺘﻴﺠﺔ اﻟﺴﺎﺑﻘﺔ ﺗﻤﺮﻳﻦ :ﺑﺄﺳﺘﺨﺪام ﻧﻈﺮﻳﺔ (٢) ٢أوﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻨﻤﺎذج اﻟﺘﺎﻟﻴﺔ )AR(٢), MA(١), MA(٢), ARMA(١،١), ARMA(٢،١), ARMA(١،٢
٥٥
∞
j=0
= ρk
اﻟﻔﺼﻞ اﻟﺮاﺑﻊ اﻟﺘﻨﺒﺆات ذات ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ﻟﻨﻤﺎذج )ARMA(p,q Minimum Mean Square Error Forecasts for ARMA(p,q) Models ﻓﻲ ﺍﻟﻔﻘﺭﺓ ﺍﻟﺴﺎﺒﻘﺔ ﻜﺘﺒﻨﺎ ﻨﻤﻭﺫﺝ ﺍﻹﻨﺤﺩﺍﺭ ﺍﻟﺫﺍﺘﻲ-ﺍﻟﻤﺘﻭﺴﻁ ﺍﻟﻤﺘﺤﺭﻙ ﻤﻥ ﺍﻟﺩﺭﺠﺔ )ARMA(p,q ﺍﻟﻤﺴﺘﻘﺭ ﻋﻠﻰ ﺍﻟﺸﻜل ∞ < WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j ∞
∞
zt − µ =∑ψ j at − j , at j =0
ﺃﻭ
zt − µ = at + ψ 1at −1 + ψ 2at −2 + ψ 3at −3 +L ∞
=∑ψ j at − j , ψ 0 = 1 j =0
ﻤﻼﺤﻅﺔ :ﻫﺫﺍ ﻴﻨﻁﺒﻕ ﺃﻴﻀﺎ ﻋﻠﻰ ﻨﻤﺎﺫﺝ ) ARIMA(p,d,qﺒﺸﻜل ﻋﺎﻡ. ﻟﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﺓ } {z1 , z2 ,L, zn −1 , znﺍﻟﺘﻨﺒﺅﺍﺕ zn ( l ) , l ≥ 1ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﻴﻤﻜﻥ ﺍﻥ ﺘﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل zn ( l ) = ξ 0an + ξ1an −1 + ξ 2an −2 + L, l ≥ 1
ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﺘﻜﺘﺏ ﺒﺩﻻﻟﺔ ﺍﻟﻨﻤﻭﺫﺝ ﻜﺎﻟﺘﺎﻟﻲ
zn + l − µ = an +l + ψ 1an + l−1 + L + ψ l−1an +1 + ψ l an + ψ l+1an −1 + L, l ≥ 1
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﻴﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ )ﺃﻨﻅﺭ ﺘﻌﺭﻴﻑ ( ٥
2
⎦⎤E ⎡⎣ zn + l − zn ( l )⎤⎦ = E ⎡⎣ an + l + ψ 1an + l−1 + L + ψ l−1an +1 + (ψ l − ξ 0 ) an + (ψ l+1 − ξ1 ) an −1 + L 2
∞
= (1 + ψ 12 + L + ψ l2−1 ) σ 2 + ∑ (ψ l+ j − ξ j ) σ 2 2
j =0
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ ﻴﻨﺘﺞ ﻤﻥ ﺘﺼﻐﻴﺭ ﺍﻟﻌﻼﻗﺔ ﺍﻟﺴﺎﺒﻘﺔ ﺒﺎﻟﻨﺴﺒﺔ ﻟﻸﻭﺯﺍﻥ ξ jﻟﺠﻤﻴﻊ ﻗﻴﻡ j ﻭﻫﺫﺍ ﻴﻤﻜﻥ ﺇﺫﺍ ﻭﻓﻘﻁ ﺇﺫﺍ ﺤﻘﻘﺕ ﺍﻷﻭﺯﺍﻥ ξ jﺍﻟﻌﻼﻗﺔ ﺍﻟﺘﺎﻟﻴﺔ j = 0,1, 2,L , l ≥ 1
ξ j = ψ l+ j ,
ﻭﻋﻠﻴﻪ ﻓﺈﻥ ﺍﻟﺘﻨﺒﺅﺍﺕ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ zn ( l ) = ψ l an + ψ l+1an −1 + ψ l+2an −2 + L, l ≥ 1
ﻨﻅﺭﻴﺔ :٢ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ:
en ( l ) = zn +l − zn ( l ) = an +l + ψ 1an +l−1 + ψ 2an +l−2 + L + ψ l−1an+1 , l ≥ 1
ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ: V ⎡⎣ en ( l )⎤⎦ = σ 2 (1 + ψ 12 + ψ 22 + L + ψ l2−1 ) , l ≥ 1
٥٦
ﺍﻟﺼﻴﻐﺔ zn ( l ) = ψ lan + ψ l+1an−1 + ψ l+2an −2 + L, l ≥ 1ﻏﻴﺭ ﻋﻤﻠﻴﺔ ﻹﻴﺠﺎﺩ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn+l , l ≥ 1ﻭﺫﻟﻙ ﻷﻨﻨﺎ ﻨﺤﺘﺎﺝ ﺇﻟﻰ ﻤﻌﺭﻓﺔ ﺍﻟﻘﻴﻡ } . {a1 , a2 ,L, an−1 , an
ﺘﻌﺭﻴﻑ : ١٨ﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ )} I ({z1 , z2 ,L , zn −1 , znﺘﻜﺎﻓﺊ ﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ )} I ({a1 , a2 ,L , an −1 , anﻭﺫﻟﻙ ﺒﺎﻟﻤﻌﻨﻰ ﺃﻥ ﺍﻟﻤﺠﻤﻭﻋﺔ } {a1 , a2 ,L, an−1 , anﺘﺤﺘﻭﻯ ﻋﻠﻰ ﻨﻔﺱ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ ﻋﻥ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } . {z1 , z2 ,L, zn −1 , zn ﻤﻼﺤﻅﺔ :ﺍﻟﻤﺘﺴﻠﺴﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﻭﻗﻴﺎﺴﻬﺎ ﻭﻟﻜﻥ ﺍﻟﻤﺘﻠﺴﻠﺔ } {a1 , a2 ,L, an−1 , anﻻﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﺃﻭ ﻗﻴﺎﺴﻬﺎ.
ﻨﻅﺭﻴﺔ : ٣ﺍﻟﻤﺘﻨﺒﺊ ﺫﺍ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﻴﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1 ﺃﻱ ﻫﻭ ﺍﻟﺘﻭﻗﻊ ﺍﻟﺸﺭﻁﻲ ﻟﻠﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﻤﻌﻁﻰ } . {z1 , z2 ,L, zn −1 , zn ﺘﺴﺘﺨﺩﻡ ﻨﻅﺭﻴﺔ ٢ﻋﻤﻠﻴﺎ ﻹﻴﺠﺎﺩ ﻗﻴﻡ ﺍﻟﺘﻨﺒﺅﺍﺕ ﺒﺩﻻ ﻤﻥ ﺍﻟﺼﻴﻐﺔ zn ( l ) = ψ l an + ψ l+1an −1 + ψ l+2an −2 + L, l ≥ 1
ﻭﺫﻟﻙ ﺘﺒﻌﺎ ﻟﻠﻤﻼﺤﻅﺔ ﺍﻟﺴﺎﺒﻘﺔ.
ﻗﺎﻋﺩﺓ :٢ ⎧a , j ≤ 0 1 − E ( an + j zn , zn −1 ,L) = ⎨ n + j j>0 ⎩ 0, j≤0 ⎧ zn + j , ⎨ = )2 − E ( zn + j zn , zn −1 ,L ⎩ zn ( j ) , j > 0
ﻨﻅﺭﻴﺔ ٣ﻤﻊ ﺍﻟﻘﺎﻋﺩﺓ ٢ﺘﻌﻁﻲ ﻁﺭﻴﻘﺔ ﻋﻤﻠﻴﺔ ﻭﺴﻬﻠﺔ ﻹﻴﺠﺎﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn + l , l ≥ 1
ﺍﻟﺩﺍﻟﺔ zn ( l ) , l ≥ 1ﻜﺩﺍﻟﺔ ﻟﺯﻤﻥ ﺍﻟﺘﻘﺩﻡ
l ≥1
ﺘﻌﺭﻴﻑ :١٩ ﻋﻨﺩ ﻨﻘﻁﺔ ﺍﻻﺼل ﻟﻠﺯﻤﻥ nﺘﺴﻤﻰ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ.
دوال اﻟﺘﻨﺒﺆ ﻟﻨﻤﺎذج ): ARIMA(p,d,q ﺍﻭﻻ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): AR(١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل
٥٧
) ∞ WN ( 0, σ 2 ) , φ1 < 1, µ ∈ ( −∞,
zt − µ = φ1 ( zt −1 − µ ) + at , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ +E ⎡⎣ ⎡⎣φ1 ( zn + l−1 − µ ) + an +l ⎤⎦ zn , zn −1 ,L⎤⎦ , l ≥ 1
=µ +E ⎡⎣φ1 ( zn + l−1 − µ ) zn , zn −1 ,L + an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
= µ +φ1E ⎡⎣( zn + l−1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﺃﻱ zn ( l ) = µ +φ1E ⎣⎡ ( zn + l−1 zn , zn −1 ,L) − µ ⎦⎤ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
⎦⎤l = 1: zn (1) = µ +φ1E ⎣⎡ ( zn zn , zn −1 ,L) − µ ⎦⎤ + E ⎡⎣ an +1 zn , zn −1 ,L ) = µ +φ1 ( zn − µ
⎦⎤l = 2 : zn ( 2 ) = µ +φ1E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + 2 zn , zn −1 ,L ⎦⎤ = µ +φ1 ⎡⎣ zn (1) − µ
⎦⎤l = 3 : zn ( 3) = µ +φ1E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +3 zn , zn −1 ,L ⎦⎤ = µ +φ1 ⎡⎣ zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 1
ﻭﻫﻲ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ )AR(١
ﺘﻌﺭﻴﻑ :٢٠ ﺸﺭﻁ ﺍﻹﺴﺘﻤﺭﺍﺭ Continuity Conditionﻴﺘﻁﻠﺏ ﺃﻨﻪ ﻋﻨﺩﻤﺎ ﺘﻜﻭﻥ l = 1ﻓﺈﻥ zn ( l − 1) = zn ( 0 ) = zn
ﻤﻥ ﻨﻅﺭﻴﺔ ٢ﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﻤﻥ ﺍﻟﻌﻼﻗﺔ V ⎡⎣ en ( l )⎤⎦ = σ 2 (1 + ψ 12 + ψ 22 + L + ψ l2−1 ) , l ≥ 1
ﺴﺒﻕ ﺃﻥ ﺍﺸﺘﻘﻘﻨﺎ ﺩﺍﻟﺔ ﺍﻷﻭﺯﺍﻥ ﻟﻨﻤﻭﺫﺝ ) AR(١ﻭﻫﻲ ψ j = φ1j , φ1 < 1
), l ≥1
ﻭﺒﺎﻟﺘﻌﻭﻴﺽ ﻓﻲ ﺼﻴﻐﺔ ﺍﻟﺘﺒﺎﻴﻥ ﻨﺠﺩ
(
( V ⎡⎣ en ( l )⎤⎦ = σ 2 1 + φ12 + φ14 + L + φ1
)2 l −1
1 − φ12 l , l ≥1 1 − φ12
=σ2
ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻟﻨﻤﻭﺫﺝ ٥٨
) WN ( 0,0.024
zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻬﺎ. ﺍﻟﺤل :ﻤﻥ ﺍﻟﺼﻴﻐﺔ zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 1ﻨﺠﺩ ) z156 (1) = 0.97+0.85 ( z156 − 0.97
= 0.97+0.85 ( 0.49 − 0.97 ) = 0.56 ) z156 ( 2 ) = 0.97+0.85 ( z156 (1) − 0.97 = 0.97+0.85 ( 0.56 − 0.97 ) = 0.62 ) z156 ( 3) = 0.97+0.85 ( z156 ( 2 ) − 0.97 = 0.97+0.85 ( 0.62 − 0.97 ) = 0.68
ﻭﺍﻟﺘﺒﺎﻴﻨﺎﺕ 1 − φ12 l , l ≥1 1 − φ12
V ⎡⎣ en ( l )⎤⎦ = σ 2
V ⎡⎣ e156 (1)⎤⎦ = 0.024 = 0.041 = 0.054
4
)1 − ( 0.85
2
)1 − ( 0.85
6
)1 − ( 0.85
2
)1 − ( 0.85
V ⎡⎣ e156 ( 2 )⎤⎦ = 0.024 V ⎡⎣ e156 ( 2 )⎤⎦ = 0.024
ﺜﺎﻨﻴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): AR(٢
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(٢ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل WN ( 0, σ 2 ) , µ ∈ ( −∞, ∞ ) ,
zt = µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at , at
φ2 − φ1 < 1, φ2 + φ1 < 1, φ2 < 1
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ +E ⎡⎣ ⎡⎣φ1 ( zn + l−1 − µ ) + φ2 ( zn + l−2 − µ ) + an + l ⎤⎦ zn , zn −1 ,L⎤⎦ , l ≥ 1
=µ +E ⎡⎣φ1 ( zn + l−1 − µ ) zn , zn −1 ,L + φ2 ( zn + l−2 − µ ) zn , zn −1 ,L + an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
= µ +φ1E ⎡⎣( zn +l−1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn + l−2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﺃﻱ
٥٩
zn ( l ) = µ +φ1E ⎡⎣ ( zn + l −1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣ ( zn + l −2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
٢ ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ
l = 1: zn (1) = µ +φ1E ⎡⎣( zn zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn −1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +1 zn , zn −1 ,L⎤⎦ = µ +φ1 ( zn − µ ) + φ2 ( zn −1 − µ )
l = 2 : zn ( 2 ) = µ +φ1E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣ ( zn zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +2 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn (1) − µ ⎤⎦ + φ2 ( zn − µ )
l = 3 : zn ( 3) = µ +φ1E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +3 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn ( 2 ) − µ ⎤⎦ +φ2 ⎡⎣ zn (1) − µ ⎤⎦
l = 4 : zn ( 4 ) = µ +φ1E ⎡⎣( zn +3 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + 4 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn ( 3) − µ ⎤⎦ +φ2 ⎡⎣ zn ( 2 ) − µ ⎤⎦
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ + φ2 ⎡⎣ zn ( l − 2 ) − µ ⎤⎦ , l ≥ 1
AR(٢) ﻭﻫﻲ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ﻭﺩﺍﻟﺔ ﺍﻷﻭﺯﺍﻥ٢ ﻭﻴﻤﻜﻥ ﺤﺴﺎﺏ ﺘﺒﺎﻴﻨﺎﺕ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻤﻥ ﻨﻅﺭﻴﺔ ⎧ 1, ⎪φ , ⎪ 1 ψj =⎨ 2 ⎪ φ1 + φ2 , ⎪⎩φψ 1 j −1 + φ2ψ j − 2 ,
j=0 j =1 j=2 j≥3
: ARIMA(٠،١،١) ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ:ﺜﺎﻟﺜﺎ
ﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎn { ﺤﺘﻰ ﺍﻟﺯﻤﻥz1 , z2 ,L, zn −1 , zn } ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜلARIMA(٠،١،١) ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ zt = zt −1 + at − θ1at −1 , at
WN ( 0, σ 2 )
. zn+l , l ≥ 1 ﺃﻭ ﺒﺸﻜل ﻋﺎﻡzn +1 , zn +2 , zn+3 ,L ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻨﺠﺩ٣ ﻤﻥ ﻨﻅﺭﻴﺔ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=E ( zn +l−1 zn , zn −1 ,L) + E ( an +l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) , l ≥ 1
٢ ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ
٦٠
zn ( l ) =E ( zn + l−1 zn , zn −1 ,L) + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =E ( zn zn , zn −1 ,L) + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = zn − θ1an
)l = 2 : zn ( 2 ) =E ( zn +1 zn , zn −1 ,L) + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L )= zn (1 )l = 3 : zn ( 3) =E ( zn + 2 zn , zn −1 ,L) + E ( an +3 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L ) = zn ( 2
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = zn ( l − 1) , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) ARIMA(٠،١،١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ zn − θ1an , l = 1 ⎨ = ) zn ( l ⎩ zn ( l − 1) , l > 1
ﺭﺍﺒﻌﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): MA(١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) MA(١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل ) WN ( 0, σ 2
zt = µ + at − θ1at −1 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + E ( an + l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
zn ( l ) =µ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = µ − θ1an
)l = 2 : zn ( 2 ) =µ + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L =µ )l = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L =µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) MA(١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ µ − θ1an , l = 1 ⎨ = ) zn ( l l≥2 ⎩ µ,
ﺨﺎﻤﺴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): MA(٢
٦١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) MA(٢ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل ) WN ( 0, σ 2
zt = µ + at − θ1at −1 − θ 2 at − 2 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn+l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + E ( an + l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) − θ 2 E ( an +l−2 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
zn ( l ) =µ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) − θ 2 E ( an + l−2 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L) − θ 2 E ( an −1 zn , zn −1 ,L = µ − θ1an − θ 2an −1
)l = 2 : zn ( 2 ) =µ + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L) − θ 2 E ( an zn , zn −1 ,L = µ − θ 2 an
)l = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L) − θ 2 E ( an +1 zn , zn −1 ,L =µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ , l ≥ 3
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) MA(٢ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ µ − θ1an − θ 2an −1 , l = 1 ⎪ zn ( l ) = ⎨ µ − θ 2 an , l=2 ⎪ µ, l≥3 ⎩
ﺴﺎﺩﺴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): ARMA(١،١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) ARMA(١،١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل WN ( 0, σ 2 ) , φ1 ≠ θ1 , φ1 < 1
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + φ1E ⎡⎣( zn + l−1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
٦٢
zn ( l ) =µ + φ1 E ⎡⎣( zn + l−1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + φ1E ⎣⎡( zn − µ ) zn , zn −1 ,L⎦⎤ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = µ + φ1 ( zn − µ ) − θ1an
)l = 2 : zn ( 2 ) =µ + φ1 E ⎡⎣( zn +1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L ⎦⎤ = µ + φ1 ⎡⎣ zn (1) − µ
)l = 3 : zn ( 3) =µ + φ1 E ⎡⎣( zn + 2 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L ⎦⎤ = µ + φ1 ⎡⎣ zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ + φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) ARMA(١،١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ l =1 l≥2
⎧⎪ µ + φ1 ( zn − µ ) − θ1an , ⎨ = ) zn ( l ⎪⎩ µ + φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ ,
ﺘﻤﺭﻴﻥ: ﻟﻨﻤﻭﺫﺝ ) ARMA(١،١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل
WN ( 0, σ 2 ) , φ1 ≠ θ1 , φ1 < 1
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﺒﺭﻫﻥ ﺍﻥ ﻋﻨﺩﻤﺎ ﺘﺅﻭل φ1 → 1ﻓﺈﻥ ) ARMA(1,1) → IMA(1,1ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ). IMA(١،١ ﺘﻤﺭﻴﻥ: ﺃﻭﺠﺩ ﺩﻭﺍل ﺍﻟﺘﻨﺒﺅ ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻜل ﻤﻥ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﺘﺎﻟﻴﺔ: ARIMA(١،١،١), ARIMA(٢،١،٠), ARIMA(٠،١،٢), ARIMA(١،٢،٠), ARIMA(٠،٢،١), ARIMA(٠،٢،٠). ﺤﺩﻭﺩ ﺍﻟﺘﻨﺒﺅ : Forecasting Limits ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ zn ( l ) , l ≥ 1ﻋﻨﺩ ﻗﻴﻤﺔ ﻤﻌﻴﻨﺔ ﺘﻌﻁﻲ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻟﻨﻘﻁﺔ Point Forecastﻭﺍﻟﺫﻱ ﻻﻴﻜﻔﻲ ﺍﻭ ﻴﻔﻴﺩ ﻓﻲ ﺇﺘﺨﺎﺫ ﻗﺭﺍﺭﺍﺕ ﺇﺤﺼﺎﺌﻴﺔ ﻋﻥ ﺍﻟﻅﺎﻫﺭﺓ ﺍﻟﻌﺸﻭﺍﺌﻴﺔ ﺍﻟﻤﺩﺭﻭﺴﺔ ﻷﻥ P ( Z n + m = zn ( m ) ) = 0, for some m > 0
ﺃﻱ ﺃﻥ ﻤﻘﺩﺍﺭ ﺘﺄﻜﺩﻨﺎ ) ﺃﻭ ﺇﺤﺘﻤﺎل( ﻤﻥ ﺃﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻟﻤﺭﺍﺩ ﺍﻟﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﺴﺎﻭﻱ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻁﺎﺓ ﻤﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺘﺴﺎﻭﻱ ﺍﻟﺼﻔﺭ ﺃﻱ ﺍﻨﻨﺎ ﻏﻴﺭ ﻤﺘﺄﻜﺩﻴﻥ ﺇﻁﻼﻗﺎ ﻭﺒﺎﻟﺘﺎﻟﻲ ﻻﻓﺎﺌﺩﺓ ﻤﻥ ﺍﻟﺘﻨﺒﺅ. ﻟﻠﺘﻐﻠﺏ ﻋﻠﻰ ﺫﻟﻙ ﻭﺃﻹﺴﺘﻔﺎﺩﺓ ﻤﻥ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻨﺴﺘﺨﺩﻡ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻟﻔﺘﺭﺓ Interval Forecast ﻭﻫﻲ ﻋﺒﺎﺭﺓ ﻋﻥ ﻓﺘﺭﺓ ﻤﺜل ] [a, bﻋﻠﻰ ﺨﻁ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ ﺒﺤﻴﺙ ﻴﻜﻭﻥ ) P ( a ≤ Z n +m ≤ b ) = (1 − α
ﻭﺒﻬﺫﺍ ﻨﺴﺘﻁﻴﻊ ﺃﻥ ﻨﺤﺩﺩ ﺩﺭﺠﺔ ﺘﺄﻜﺩﻨﺎ ﻤﻥ ﺃﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻟﻤﺭﺍﺩ ﺍﻟﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﻘﻊ ﺒﻴﻥ ﺍﻟﻘﻴﻡ a ﻭ bﺒﺩﺭﺠﺔ ﺘﺄﻜﺩ ﺃﻭ ﺇﺤﺘﻤﺎل ) 1 − αﺃﻭ ( 100 × (1 − α ) %ﻓﻤﺜﻼ ﻟﻭ ﻜﺎﻨﺕ α = 0.05ﻓﺈﻨﻨﺎ ﻨﻜﻭﻥ ﻤﺘﺄﻜﺩﻴﻥ ﻭﺒﺈﺤﺘﻤﺎل ٩٥٪ﺍﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺘﻘﻊ ﺒﻴﻥ ﺍﻟﻘﻴﻡ aﻭ . b
٦٣
ﺘﻌﺭﻴﻑ :٢١ﻋﻠﻰ ﺇﻓﺘﺭﺍﺽ ﺃﻥ ) at N ( 0,σ 2ﻓﺈﻥ ﺤﺩﻭﺩ 100 × (1 − α ) %ﻓﺘﺭﺓ ﺘﻨﺒﺅ ﻟﻠﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn+l , l ≥ 1ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ
{
}
12
⎦⎤) zn ( l ) ± uα 2 V ⎡⎣ en ( l α ⎟⎞ 100 ⎛⎜ 1 −ﻟﻠﺘﻭﺯﻴﻊ ). N ( 0,1 ⎠2 ⎝
ﺤﻴﺙ uα 2ﺍﻟﻤﺌﻴﻥ
ﻓﻤﺜﻼ ﻋﻨﺩﻤﺎ α = 0.05ﻓﺈﻥ . u0.025 = 1.96 ﻤﻼﺤﻅﺔ :ﻓﻲ ﺍﻟﺘﻌﺭﻴﻑ ﺇﻓﺘﺭﻀﻨﺎ ﺃﻥ ﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﻀﺠﺔ ﺍﻟﺒﻴﻀﺎﺀ ) at N ( 0, σ 2ﻭﻫﺫﺍ ﻤﻤﻜﻥ ﺇﻋﺘﻤﺎﺩﺍ ﻋﻠﻰ ﻨﻅﺭﻴﺔ ﻨﻬﺎﻴﺔ ﻤﺭﻜﺯﻴﺔ. ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻟﻨﻤﻭﺫﺝ ) N ( 0,0.024
zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻬﺎ ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ ٩٥٪ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ. ﺍﻟﺤل :ﺴﺒﻕ ﺃﻥ ﺤﺴﺒﻨﺎ ﻓﻲ ﻤﺜﺎل ﺴﺎﺒﻕ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻭ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻜﺎﻟﺘﺎﻟﻲ: ﺍﻟﺘﻨﺒﺅﺍﺕ z156 (1) = 0.56, z156 ( 2 ) = 0.62, z156 ( 3) = 0.68
ﻭﺍﻟﺘﺒﺎﻴﻨﺎﺕ V ⎡⎣ e156 (1)⎤⎦ = 0.024, V ⎡⎣ e156 ( 2 )⎤⎦ = 0.041, V ⎡⎣ e156 ( 2 )⎤⎦ = 0.054 ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ ٩٥٪ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159ﻨﻭﺠﺩﻫﺎ ﻤﻥ ﺼﻴﻐﺔ ﺘﻌﺭﻴﻑ ٢١
}
12
= 0.56 ± 1.96 0.024 = 0.56 ± 0.304
}
12
{
⎦⎤) zn ( l ) ± uα 2 V ⎡⎣ en ( l
{ {V ⎡⎣e {V ⎡⎣e
⎦⎤)1 − z156 (1) ± u0.025 V ⎡⎣ e156 (1
= 0.62 ± 1.96 0.041 = 0.62 ± 0.397
}⎦⎤) ( 2
156
2 − z156 ( 2 ) ± u0.025
= 0.68 ± 1.96 0.054 = 0.68 ± 0.455
}⎦⎤)( 3
156
3 − z156 ( 3) ± u0.025
12
12
ﺃﻱ ﺃﻥ ) z157 ∈ ( 0.256,0.864ﺒﺈﺤﺘﻤﺎل ٠,٩٥ﻭ ﻭﻜﺫﻟﻙ ) z158 ∈ ( 0.223,1.017ﻭ ﻜﺫﻟﻙ ﺃﻴﻀﺎ ). z159 ∈ ( 0.225,1.135
٦٤
٦٥
اﻟﻔﺼﻞ اﻟﺨﺎﻣﺲ ﺗﺼﻤﻴﻢ وﺑﻨﺎء ﻧﻈﺎم ﺗﻨﺒﺆ إﺣﺼﺎﺋﻲ Designing and Building : Statistical Forecasting System ﺳﺒﻖ أن ذآﺮﻧﺎ ان اﻟﺨﻄﻮة اﻷوﻟﻰ ﻟﺘﺼﻤﻴﻢ ﻧﻈﺎم ﺗﻨﺒﺆ هﻲ ﺑﻨﺎء ﻧﻤﻮذج .إن ﻋﻤﻠﻴﺔ ﺑﻨﺎء ﻧﻤﻮذج إﺣﺼﺎﺋﻲ هﻲ ﻋﻤﻠﻴﺔ ﺗﻜﺮارﻳﺔ Iterativeﺗﺘﻜﻮن ﻣﻦ ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ،ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج )وﻧﻘﺼﺪ ﺑﻬﺎ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج( و إﺧﺘﺒﺎر اﻟﻨﻤﻮذج. ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ): Model Identification (Specification ﻓﻲ ﻣﺮﺣﻠﺔ ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ﻧﺴﺘﺨﺪم اﻟﺒﻴﺎﻧﺎت أو اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ) اﻟﺘﺎرﻳﺦ( واي ﻣﻌﻠﻮﻣﺎت اﺧﺮى ﻋﻦ اﻟﻜﻴﻔﻴﺔ اﻟﺘﻲ ﺗﻮﻟﺪت ﺑﻬﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ وذﻟﻚ ﻹﻗﺘﺮاح ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻨﻤﺎذج اﻟﻤﻨﺎﺳﺒﺔ .وﻳﺘﻢ ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ﺣﺴﺐ اﻟﺨﻄﻮات اﻟﻌﺮﻳﻀﺔ اﻟﺘﺎﻟﻴﺔ: اﻟﺨﻄﻮة اﻻوﻟﻰ :ﺗﺤﻮﻳﻞ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ : Variance-stabilizing Transformation ﺑﻌﺪ رﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotوإﺟﺮاء ﺑﻌﺾ اﻹﺧﺘﺒﺎرات اﻹﺣﺼﺎﺋﻴﺔ ﻟﻤﻌﺮﻓﺔ ﻓﻴﻤﺎ إذا آﺎن اﻟﺘﺒﺎﻳﻦ ﺛﺎﺑﺖ ،وﻓﻲ ﺣﺎﻟﺔ ﻋﺪم ﺛﺒﺎت اﻟﺘﺒﺎﻳﻦ او إذا آﺎن اﻟﺘﺒﺎﻳﻦ ﻳﺘﻐﻴﺮ ﻣﻊ ﻣﺴﺘﻮى اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﺈﻧﻨﺎ ﻧﻄﺒﻖ اﻟﺘﺤﻮﻳﻞ اﻟﻠﻮﻏﺎرﺗﻤﻲ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﻔﺤﺼﻬﺎ ﻣﻦ ﺟﺪﻳﺪ ﻓﺈذا ﺗﻢ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ وإﻻ ﻧﻠﺠﺄ إﻟﻲ ﺗﻄﺒﻴﻖ أﺣﺪ اﻟﺘﺤﻮﻳﻼت اﻟﺘﻲ ذآﺮﻧﺎهﺎ ﻓﻲ ﺟﺪول ﺻﻔﺤﺔ .٤١ اﻟﺨﻄﻮة اﻟﺜﺎﻧﻴﺔ :إﺧﺘﻴﺎر درﺟﺔ اﻟﺘﻔﺮﻳﻖ : d إذا آﺎﻧﺖ اﻟﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ ﺗﺤﺪﻳﺪ درﺟﺔ اﻟﺘﻔﺮﻳﻖ dاﻟﺘﻲ ﺗﺠﻌﻞ اﻟﻤﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ وﻧﻘﻮم ﺑﺄﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول ﺛﻢ ﻧﻔﺤﺺ اﻟﺘﺎﻟﻲ: -١اﻟﻤﺨﻄﻄﺎت اﻟﺰﻣﻨﻴﺔ ﻟﻠﻤﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ. -٢ﻣﺨﻄﻄﺎت داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ SACFو . SPACF -٣إﺟﺮاء ﺗﻔﺮﻳﻖ ﺁﺧﺮ إذا اﺣﺘﺎج اﻷﻣﺮ وإﻋﺎدة اﻟﺨﻄﻮات ١و ٢اﻟﺴﺎﺑﻘﺘﻴﻦ. اﻟﻤﺨﻄﻄﺎت اﻟﺰﻣﻨﻴﺔ ﻟﻠﻤﺘﺴﻠﺴﻼت ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﺗﺒﻴﻦ ﺗﻐﻴﺮ ﻓﻲ اﻟﻤﺴﺘﻮى وداﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﻋﻴﻨﻲ ﻣﺘﺨﺎﻣﺪة ﺑﺒﻂء آﻤﺎ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﺗﻌﻄﻲ ﻗﻴﻤﺔ واﺣﺪة ﻗﺮﻳﺒﺔ ﻣﻦ اﻟﻮاﺣﺪ اﻟﺼﺤﻴﺢ )ﺑﻐﺾ اﻟﻨﻈﺮ ﻋﻦ اﻹﺷﺎرة( وﺑﻘﻴﺔ اﻟﻘﻴﻢ ﻗﺮﻳﺒﺔ ﺟﺪا ﻣﻦ اﻟﺼﻔﺮ. ﻣﻼﺣﻈﺔ :درﺟﺔ اﻟﺘﻔﺮﻳﻖ dﻏﺎﻟﺒﺎ ﻣﺎ ﺗﻜﻮن ٠او ١او . ٢ اﻟﺨﻄﻮة اﻟﺜﺎﻟﺜﺔ :ﺗﺤﺪﻳﺪ pو : q ﺑﻌﺪ ان ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ ﻧﻘﻮم ﺑﺘﺤﺪﻳﺪ درﺟﺔ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ pودرﺟﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك qوذﻟﻚ ﺑﻤﻘﺎرﻧﺔ أﻧﻤﺎط داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻣﻊ اﻷﻧﻤﺎط اﻟﻨﻈﺮﻳﺔ ﻟﺪاﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﺴﺘﺮﺷﺪﻳﻦ ﺑﺨﻮاص ﻧﻤﺎذج ) ARMA(p,qاﻟﻤﺬآﻮرة ﻓﻲ ﺻﻔﺤﺔ ٣٨واﻟﺠﺪول اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ هﺬﻩ اﻟﺨﻮاص ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﺸﺎﺋﻌﺔ:
اﻟﻨﻤﻮذج ) (١,d,٠و)AR(١ ) (٢,d,٠و)AR(٢ ) (p,d,٠و)AR(p ) (٠,d,١و )MA(١ ) (٠,d,٢و )MA(٢
ACF ﺗﺨﺎﻣﺪ اﺳﻲ أواﺳﻲ ﻣﺘﺮدد ﺗﺨﺎﻣﺪ اﺳﻲ او ﺗﺨﺎﻣﺪ ﺟﻴﺒﻲ ﺗﺨﺎﻣﺪ اﺳﻲ و /او ﺗﺨﺎﻣﺪ ﺟﻴﺒﻲ ρ k = 0, k > 1
PACF φkk = 0, k > 1 φkk = 0, k > 2 φkk = 0, k > p
ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ او ﺟﻴﺒﻲ
ρ k = 0, k > 2
٦٦
) (٠,d,qو )MA(q ) (١,d,١و )ARMA(١،١ ) (p,d,qو )ARMA(p,q
ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ و /او ﺟﻴﺒﻲ ρ k = 0, k > q ﺗﺘﻨﺎﻗﺺ وﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ ﺗﺘﻨﺎﻗﺺ وﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻣﻦ اﻟﺘﺨﻠﻒ ١ ﻣﻦ اﻟﺘﺨﻠﻒ ١ ﺗﺘﻨﺎﻗﺺ ﺑﻌﺪ اﻟﺘﺨﻠﻒ ﺗﺘﻨﺎﻗﺺ ﺑﻌﺪ اﻟﺘﺨﻠﻒ q - pوﺗﺘﺨﺎﻣﺪ p – qوﻳﺴﻴﻄﺮ اﺳﻴﺎ و /او ﺟﻴﺒﻴﺎ ﺑﻌﺪ اﻟﺘﺨﻠﻒ q – p ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ و/او ﺟﻴﺒﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ p – q
اﻟﺨﻄﻮة اﻟﺮاﺑﻌﺔ :إﺿﺎﻓﺔ ﻣﻌﻠﻢ إﻧﺠﺮاف: إذا آﺎﻧﺖ اﻟﻤﺘﺴﻠﺴﺔ ﺗﺤﺘﺎج إﻟﻰ ﺗﻔﺮﻳﻖ ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ اﻟﺘﺄآﺪ ﻓﻴﻤﺎ إذا آﺎن ﻋﻠﻴﻨﺎ إﺿﺎﻓﺔ إﻧﺠﺮاف ﻣﻌﻠﻮم δإﻟﻰ اﻟﻨﻤﻮذج وهﺬا ﻳﺘﻢ ﺑﻤﻘﺎرﻧﺔ ﻣﺘﻮﺳﻂ اﻟﻌﻴﻨﺔ wﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ اﻟﻤﺴﺘﻘﺮة ﻣﻊ اﻟﺨﻄﺄ اﻟﻤﻌﻴﺎري ﻟﻬﺬا اﻟﻤﺘﻮﺳﻂ 12
⎡c ⎤ ⎥) s.e ( w ) ≅ ⎢ 0 (1 + 2r1 + 2r2 + L + 2rK ⎣n ⎦
ﺣﻴﺚ c0 = γˆ0و r1 ,L, rKهﻲ اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﻌﻴﻨﻴﺔ اﻟﻤﻌﻨﻮﻳﺔ ﻟﻠﺪرﺟﺔ . Kوﻳﻜﻮن اﻹﺧﺘﺒﺎر
H0 : δ = 0 H1 : δ ± 0
w وﻧﺮﻓﺾ H 0ﻋﻨﺪ α = 0.05إذاآﺎﻧﺖ > 1.96 ) s.e ( w
.
ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج : Model Estimation ﺑﻌﺪ ﺗﺤﺪﻳﺪ ﺷﻜﻞ اﻟﻨﻤﻮذج ﻻﺑﺪ ﻣﻦ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج δو φ1 ,K ,φ pو θ1 ,K ,θ qو σوذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎرﻳﺨﻴﺔ اﻟﻤﺘﻮﻓﺮة ﻟﺪﻳﻨﺎ. ﻟﻨﻔﺘﺮض ان ﻟﺪﻳﻨﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة z1 , z2 ,K, zn −1 , znواﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح 2
أو
) N ( 0, σ 2
φ p ( B ) wt = δ + θ q ( B ) at , at
N ( 0, σ
φ p ( B ) zt = δ + θ q ( B ) at , at
)
2
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻻﻳﻮﺟﺪ ﺑﻴﻨﻬﺎ ﺟﺬور ﻣﺸﺘﺮآﺔ وﺟﺬور اﻟﻤﻌﺎدﻟﺔ φ p ( B ) = 0ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ) ﺷﺮط اﻹﺳﺘﻘﺮار(. هﻨﺎك ﻃﺮق آﺜﻴﺮة ﻟﺘﻘﺪﻳﺮ اﻟﻤﻌﺎﻟﻢ ﺳﻨﺬآﺮ ﻣﻨﻬﺎ هﻨﺎ ﻓﻘﻂ ﻃﺮﻳﻘﺘﻴﻦ ﺗﺪﺧﻞ ﺿﻤﻦ ﻧﻄﺎق هﺬا اﻟﻤﻘﺮر وهﻤﺎ ﻃﺮﻳﻘﺔ اﻟﻌﺰوم وﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ. أوﻻ :ﻃﺮﻳﻘﺔ اﻟﻌﺰوم : The Method of Moments وﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻣﺴﺎوات ﻋﺰوم اﻟﻌﻴﻨﺔ ﻣﺜﻞ ﻣﺘﻮﺳﻂ اﻟﻌﻴﻨﺔ zواﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ ﻟﻠﻌﻴﻨﺔ rkﺑﺎﻟﻌﺰوم اﻟﻨﻈﺮﻳﺔ ﻣﺜﻞ اﻟﻤﺘﻮﺳﻂ µوداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ρ kوﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻨﺎﺗﺠﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﺎﻟﻢ اﻟﻤﺮاد ﺗﻘﺪﻳﺮهﺎ. ﺳﻮف ﻧﺴﺘﻌﺮض اﻟﻄﺮﻳﻘﺔ ﻟﻠﻨﻤﻮذج ) AR(pآﺎﻟﺘﺎﻟﻲ: n -١ﻳﻘﺪر اﻟﻤﺘﻮﺳﻂ µﺑﺎﻟﻤﻘﺪر zاي µˆ = z = ∑i =1 zi n -٢ﻟﺘﻘﺪﻳﺮ φ1 ,K ,φ pﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺔ: ٦٧
ρ k = φ1 ρ k −1 + φ2 ρ k − 2 + L + φ p ρ k − p , k > 1
واﻟﺘﻲ ﺗﻨﺘﺞ ﻣﻦ ﺿﺮب اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻨﻤﻮذج ) AR(pﺑﺎﻟﺤﺪ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ .ﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﺑﻮﺿﻊ k = 1, 2,K , pﻧﺤﺼﻞ ﻋﻠﻰ ﻧﻈﺎم اﻟﻤﻌﺎدﻻت اﻟﻤﺴﻤﻰ ﻣﻌﺎدﻻت ﻳﻮل و ووآﺮ Yule-Walkerاﻟﺘﺎﻟﻲ:
ρ1 = φ1 + φ2 ρ1 + L + φ p ρ p −1
ρ 2 = φ1 ρ1 + φ2 + L + φ p ρ p −2 M
ρ p = φ1 ρ p −1 + φ2 ρ p −2 + L + φ p
و ﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ ρ kﺑﺎﻟﻤﻘﺪر rkﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ φˆ1 ,K,φˆpآﺎﻟﺘﺎﻟﻲ: ﺑﻮﺿﻊ ﻣﻌﺎدﻻت ﻳﻮل و ووآﺮ ﻋﻠﻰ اﻟﺸﻜﻞ اﻟﻤﺼﻔﻮﻓﻲ: ⎞ rp −1 ⎞ ⎛ φˆ1 ⎟ ⎜ ⎟ rp −2 ⎟⎟ ⎜ φˆ2 ⎟ M ⎟⎜ M ⎟ ⎜⎟ ⎟ ˆ1 ⎟⎠ ⎜ φ ⎠⎝ p
r2 L rp −2 r1 L rp −3 M M M rp −3 L r1
r1 ⎛ r1 ⎞ ⎛ 1 ⎜r ⎟ ⎜ r 1 ⎜ 2⎟=⎜ 1 ⎜M⎟ ⎜ M M ⎜⎜ r ⎟⎟ ⎜⎜ r ⎝ p ⎠ ⎝ p −1 rp −2
وﺑﺤﻞ هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻟﻠﻤﻌﺎﻟﻢ ⎞ ⎛ r1 ⎟ ⎜r ⎟⎜ 2 ⎟⎜M ⎟⎟ ⎜⎜ ⎠ ⎝ rp
−1
ﺗﻘﺪر σ 2آﺎﻟﺘﺎﻟﻲ
⎞ rp −1 ⎟⎟ rp −2 ⎟ M ⎟ ⎠⎟ 1
r2 L rp −2 r1 L rp −3 M M M rp −3 L r1
)
r1 1 M rp −2
⎛ φˆ1 ⎞ ⎛ 1 ⎜ ⎟ ⎜ ⎜ φˆ2 ⎟ ⎜ r1 ⎜ M ⎟=⎜ M ⎜ ⎟ ⎜ ⎜ φˆ ⎟ ⎜⎝ rp −1 ⎠⎝ p
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2 −Lφˆp rp
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
هﻮ ﺗﺒﺎﻳﻦ اﻟﻌﻴﻨﺔ. ﺗﻘﺪﻳﺮ اﻟﻌﺰوم ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: -١ﻧﻤﻮذج )AR(١
) N ( 0, σ 2
= γˆ0
zt − µ = φ1 ( zt −1 − µ ) + at , at
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ φ1هﻮ φˆ1 = r1
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ µهﻮ µˆ = z
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ σ 2هﻮ
)
(
σˆ 2 = γˆ0 1 − φˆ1r1
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
٦٨
= γˆ0
-٢ﻧﻤﻮذج )MA(١
)
2
N ( 0, σ
zt − µ = at − θ1at −1 , at
ﻹﻳﺠﺎد ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺔ
−θ1 1 + θ12
= ρ1
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻌﺎﻟﻢ ﺑﻤﻘﺪراﺗﻬﺎ −θˆ1 1 + θˆ12
= r1
وﺑﺤﻞ اﻟﻤﻌﺎدﻟﺔ ﻟﻠﻤﻘﺪر θˆ1ﻧﺠﺪ −1 ± 1 − 4r1 2r1
= θˆ1
هﺬا اﻟﺤﻞ ﻳﻌﻄﻲ ﻗﻴﻤﺘﻴﻦ ﻟﻠﻤﻘﺪر θˆ1ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ اﻟﺘﻲ ﺗﺤﻘﻖ . θˆ1 < 1ﻓﻤﺜﻼ ﻟﻮ آﺎﻧﺖ r1 = −0.4ﻓﺈن (θˆ1 ) = −0.77و (θˆ1 ) = 3.27وﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1هﻮ . θˆ1 = −0.77 1
2
-٣ﻧﻤﻮذج )AR(٢
)
2
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + φ2 ( zt − 2 − µ ) + at , at
ﺑﺈﺳﺘﺨﺪام ﻣﻌﺎدﻻت ﻳﻮل ووآﺮ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ φ1و φ2هﻲ ⎞ ⎛ φˆ1 ⎞ ⎛ 1 r1 ⎞ −1 ⎛ r1 ⎜=⎟ ⎜ ⎠⎟ ⎜ φˆ ⎟ ⎝ r1 1 ⎟⎠ ⎜⎝ r2 ⎠⎝ 2
وﻣﻨﻬﺎ ﻧﺠﺪ r1 − r1r2 r −r , φˆ2 = 2 2 1 − r1 1− r
2 1 2 1
= φˆ1
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ µهﻮ µˆ = z
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ σ 2هﻮ
)
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
-٤ﻧﻤﻮذج )MA(٢
)
2
N ( 0, σ
= γˆ0
zt − µ = at − θ1at −1 − θ 2 at − 2 , at
ﻹﻳﺠﺎد ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و θ2ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺎت
) −θ1 (1 − θ 2 −θ 2 = , ρ2 2 2 1 + θ1 + θ 2 1 + θ12 + θ 22
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻘﺪرات r1و r2ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و θ2
)
(
= ρ1
−θˆ1 1 − θˆ2 −θˆ2 , r = 2 1 + θˆ12 + θˆ22 1 + θˆ12 + θˆ22
٦٩
= r1
وﻧﺤﻞ ﻟﻜﻞ ﻣﻦ θˆ1و θˆ2وﻧﺄﺧﺬ اﻟﺤﻠﻮل اﻟﺘﻲ ﺗﺤﻘﻖ . θ 2 − θ1 < 1, θ 2 + θ1 < 1, θ 2 < 1 -٥ﻧﻤﻮذج )ARMA(١،١
)
2
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﻹﻳﺠﺎد ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و φ1ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺎت
(1 − φ1θ1 )(φ1 − θ1 ) , ρ = (1 − φ1θ1 )(φ1 − θ1 ) φ 2 1 2 2 1 + θ1 − 2φ1θ1
1 + θ1 − 2φ1θ1
= ρ1
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻘﺪرات r1و r2ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و φ1 ) ˆ1 − φˆ θˆ )(φˆ − θ ( ˆφ = r 1
1 1
1
ˆ1 + θˆ 2 − 2φˆ θ
1
1 1
2
1
) ˆ1 − φˆ θˆ )(φˆ − θ ( = r , 1
1
1 1
ˆ1 + θˆ 2 − 2φˆ θ
1 1
1
1
وﺑﻘﺴﻤﺔ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r2ﻋﻠﻰ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r1ﻧﺠﺪ r2 r1
= φˆ1
وهﻮ ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ . φ1ﻹﻳﺠﺎد ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1ﻧﻌﻮض ﻋﻦ φˆ1ﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r1ﻧﺠﺪ ⎞ ˆ ⎛ r2 ˆ ⎞ ⎛ r2 ⎟ ⎜ 1 − r θ1 ⎟ ⎜ r − θ1 1 ⎠⎝ 1 ⎠ ⎝ = r1 r 1 + θˆ12 − 2 2 θˆ1 r1 وﻧﺤﻞ اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ اﻟﻨﺎﺗﺠﺔ ﻟﻠﻤﻘﺪر θˆ1و ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ اﻟﺘﻲ ﺗﺤﻘﻖ . θˆ1 < 1
ﺗﻤﺎرﻳﻦ :أوﺟﺪ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻤﻌﺎﻟﻢ اﻟﻨﻤﺎذج اﻟﺘﺎﻟﻴﺔ ARIMA(١،١،١), ARIMA(٢،١،٠), ARIMA(٠،١،٢), ARIMA(١،٢،٠), ARIMA(٠،٢،١), ARIMA(٠،٢،٠). ﻣﻼﺣﻈﺔ :ﻣﻘﺪرات اﻟﻌﺰوم ﺗﺴﺘﺨﺪم آﻘﻴﻢ أوﻟﻴﺔ ﻹﻳﺠﺎد ﻣﻘﺪرات أآﺜﺮ دﻗﺔ.
ﺛﺎﻧﻴﺎ :ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ Conditional Least Square : Method ﻟﻨﻤﺎذج ) ARMA(p,qواﻟﺘﻲ ﺗﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
)
2
N ( 0, σ
φ p ( B )( zt − µ ) = θ q ( B ) at , at
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻻﻳﻮﺟﺪ ﺑﻴﻨﻬﺎ ﺟﺬور ﻣﺸﺘﺮآﺔ وﺟﺬور اﻟﻤﻌﺎدﻟﺔ θ q ( B ) = 0ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ) ﺷﺮط اﻹﻧﻘﻼب( .ﺑﺈﻋﺎدة آﺘﺎﺑﺔ اﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﻟﻸﺧﻄﺎء atآﺎﻟﺘﺎﻟﻲ: )φp (B )(z − µ θq ( B ) t
= at
اﻟﻄﺮف اﻷﻳﻤﻦ ﻳﻤﻜﻦ إﻋﺘﺒﺎرة آﺪاﻟﺔ ﻓﻲ اﻟﻤﻌﺎﻟﻢ } φ = {φ1 , φ2 ,K , φ pو } θ = {θ1 ,θ 2 ,K ,θ qو µ
و ﻳﻜﺘﺐ
)) (z − µ ) t
٧٠
p
−L − φ p B
2
p
−L −θ pB
2
(1 − φ B − φ B (1 − θ B − θ B 2
2
1
1
= ) at ( φ, θ, µ
ﺗﻌﺘﻤﺪ ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ و ﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة } z = {z1 , z2 ,K, znﻋﻠﻰ ﺗﺼﻐﻴﺮ اﻟﺪاﻟﺔ n
) ∑ a ( φ, θ, µ z 2 t
= ) min Sc ( φ, θ, µ φ,θ , µ
t = p +1
وﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻄﺒﻴﻌﻴﺔ Normal Equationsاﻟﻨﺎﺗﺠﺔ اﻟﺘﺎﻟﻴﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻘﺪرات. ∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑ at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ ∂φ ∂ φ t = p +1 ˆ θ =θ ˆθ =θ ˆµ = µ
ˆµ = µ
∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑p+1 at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ θ ∂θ ∂ t = ˆθ = θ ˆθ =θ ˆµ = µ
ˆµ = µ
∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑ at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ ∂µ ∂µ t = p +1 ˆθ = θ ˆθ =θ ˆµ = µ
ˆµ = µ
هﺬﻩ اﻟﻤﻘﺪرات ﺗﺴﻤﻰ ﺷﺮﻃﻴﺔ ﻷﻧﻨﺎ هﻨﺎ ﻧﺸﺘﺮط ان اﻟﻘﻴﻢ a p = a p −1 = L = a p +1− q = 0أي ﻣﺴﺎوﻳﺔ ﻟﺘﻮﻗﻌﻬﺎ ) .ﻻﺣﻆ أن اﻟﺘﺠﻤﻴﻊ ﻓﻲ اﻟﻤﻌﺎدﻻت اﻟﺴﺎﺑﻘﺔ ﻳﺒﺪأ ﻣﻦ اﻟﻘﻴﻤﺔ .( t = p + 1 ﻳﻘﺪر اﻟﺘﺒﺎﻳﻦ σ 2ﻣﻦ
)
(
Sc φˆ , θˆ , µ
)n − ( p + q + 1
= σˆ 2
ﺗﻘﺪﻳﺮات اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: -١ﻧﻤﻮذج )AR(١
)
ﻟﺘﺒﺴﻴﻂ اﻹﺷﺘﻘﺎﻗﺎت ﺳﻮف ﻧﺴﺘﺒﺪل µﺑﻤﻘﺪرهﺎ z
2
)
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + at , at
N ( 0, σ
zt − z = φ1 ( zt −1 − z ) + at , at
2
ﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة } z = {z1 , z2 ,K, znﻧﻜﺘﺐ اﻷﺧﻄﺎء
at (φ1 ) = ( zt − z ) − φ1 ( zt −1 − z ) , t = 2,3,L, n
وﺗﺮﺑﻴﻊ اﻟﻄﺮﻓﻴﻦ واﻟﺠﻤﻊ ﻋﻠﻰ آﻞ اﻟﻤﺸﺎهﺪات a (φ1 ) = ⎡⎣( zt − z ) − φ1 ( zt −1 − z )⎤⎦ , t = 2,3,L , n 2
2
2 t
n
n
t =2
t =2
⎦⎤) Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ⎡⎣( zt − z ) − φ1 ( zt −1 − z
وهﺬﻩ داﻟﺔ ﻟﻠﻤﻌﻠﻢ φ1ﻓﻘﻂ ،ﻧﺸﺘﻖ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﻠﻢ φ1وﺗﻜﻮن اﻟﻨﺘﻴﺠﺔ ﻣﺴﺎوﻳﺔ ﻟﻠﺼﻔﺮ ﻋﻨﺪﻣﺎ φ1 = φˆ1أي
٧١
2
n
n
⎦⎤) Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ⎡⎣( zt − z ) − φ1 ( zt −1 − z t =2
t =2
2 ∂ ∂ = ) Sc (φ1 ⎦⎤) ⎡⎣( zt − z ) − φ1 ( zt −1 − z ∑ ∂φ1 ∂φ1 t =2 n
n
⎦⎤) = ∑ −2 ( zt −1 − z ) ⎡⎣ ( zt − z ) − φ1 ( zt −1 − z t= 2
n ∂ Sc (φ1 ) ˆ = ∑ −2 ( zt −1 − z ) ⎡⎣( zt − z ) − φˆ1 ( zt −1 − z )⎦⎤ = 0 φ1 =φ1 ∂φ1 t =2 n
∴ ∑ ( zt −1 − z ) ⎡⎣( zt − z ) − φˆ1 ( zt −1 − z )⎤⎦ = 0 t =2
n
n
t =2
t= 2
2 ∑ ( zt −1 − z )( zt − z ) − φˆ1 ∑ ( zt −1 − z ) = 0
أي ) − z )( zt − z
n
t −1
∑( z t =2
2
)−z
n
t −1
∑( z
= φˆ1
t =2
وهﻮ ﻣﻘﺪر اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﻠﻤﻌﻠﻢ . φ1 ﺗﻤﺮﻳﻦ :ﻗﺎرن ﺑﻴﻦ هﺬا اﻟﻤﻘﺪر وﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ . φ1 -٢ﻧﻤﻮذج )MA(١
) N ( 0,σ 2
zt − µ = at − θ1at −1 , at
ﻟﺘﺒﺴﻴﻂ اﻹﺷﺘﻘﺎﻗﺎت ﺳﻮف ﻧﺴﺘﺒﺪل µﺑﻤﻘﺪرهﺎ zوﻧﻌﻤﻞ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻌﺪﻟﺔ ﻟﻠﻤﺘﻮﺳﻂ xt = zt − zﻓﻴﺼﺒﺢ اﻟﻨﻤﻮذج ) N ( 0,σ 2
وﺑﻜﺘﺎﺑﺔ اﻟﻤﻌﺎدﻟﺔ اﻷﺧﻴﺮة ﻋﻠﻰ اﻟﺸﻜﻞ
xt = at − θ1at −1 , at at = xt − θ1at −1
وﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة x1 , x2 ,K , xnو ﺑﻮﺿﻊ a0 = 0ﺷﺮﻃﻴﺎ ﻧﻜﺘﺐ اﻷﺧﻄﺎء
a1 = x1 a2 = x2 − θ1a1 a3 = x3 − θ1a2 M an = xn − θ1an −1
وﺑﺎﻟﺘﺎﻟﻲ n
Sc (θ1 ) = ∑ at2 t =1
اﻟﺪاﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻏﻴﺮ ﺧﻄﻴﺔ ﻓﻲ اﻟﻤﻌﻠﻢ θ1و ﻳﻤﻜﻦ إﻳﺠﺎد ﻗﻴﻤﺔ θ1واﻟﺘﻲ ﺗﺼﻐﺮ ) Sc (θ1ﺑﻄﺮق اﻟﺒﺤﺚ اﻟﻌﺪدﻳﺔ ﻣﺜﻞ اﻟﺒﺤﺚ اﻟﺸﺒﻜﻲ ﻓﻲ اﻟﻤﺠﺎل ) (-١،١أو إﺳﺘﺨﺪام ﻃﺮﻳﻘﺔ ﺟﺎوس-ﻧﻴﻮﺗﻦ واﻟﺘﻲ ﺗﺘﻠﺨﺺ ﻓﻲ ﺗﻘﺮﻳﺐ ) at = at (θ1ﺑﺪاﻟﺔ ﺧﻄﻴﺔ ﻟﻠﻤﻌﻠﻢ θ1ﺣﻮل ﻗﻴﻤﺔ أوﻟﻴﺔ * θﻣﺜﻼ أي
٧٢
) * dat (θ
اﻟﻤﺸﺘﻘﺔ
) * dat (θ d θ1
dθ1
) * at (θ1 ) ≈ at (θ * ) + (θ1 − θ
ﻳﻤﻜﻦ ﺣﺴﺎﺑﻬﺎ ﺗﻜﺮارﻳﺎ وذﻟﻚ ﺑﺈﺷﺘﻘﺎق ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ at = xt − θ1at −1ﺑﺎﻟﻨﺴﺒﺔ
ﻟﻠﻤﻌﻠﻢ θ1ﻟﻨﺤﺼﻞ ﻋﻠﻰ
) da0 (θ1 و ﺑﻘﻴﻤﺔ أوﻟﻴﺔ = 0 dθ1
) dat (θ1 ) θ1dat −1 (θ1 = ) + at −1 (θ1 dθ1 dθ1
.اﻟﻤﻌﺎدﻟﺔ ) * dat (θ dθ1
)
*
−θ
1
) + (θ
*
at (θ1 ) ≈ at (θ
ﺧﻄﻴﺔ ﻓﻲ اﻟﻤﻌﻠﻢ θ1وﺑﺎﻟﺘﺎﻟﻲ ﺑﺎﻻﻣﻜﺎن ﺗﺼﻐﻴﺮ ﻣﺠﻤﻮع اﻟﻤﺮﺑﻌﺎت n
Sc (θ1 ) = ∑ at2 t=1
ﺗﺤﻠﻴﻠﻴﺎ ﻟﻨﺤﺼﻞ ﻋﻠﻲ ﻣﻘﺪر ﺟﺪﻳﺪ وأﻓﻀﻞ ﻟﻠﻤﻌﻠﻢ θ1وﻧﻜﺮر هﺬﻩ اﻟﻌﻤﻠﻴﺔ ﺑﺈﺳﺘﺒﺪال * θﺑﺎﻟﻤﻘﺪر اﻟﺠﺪﻳﺪ وﻧﺴﺘﻤﺮ ﺣﺘﻰ ﻳﺼﺒﺢ اﻟﻔﺮق ﺑﻴﻦ ﻣﻘﺪرﻳﻦ ﺗﺎﻟﻴﻴﻦ ﺻﻐﻴﺮ ﺟﺪا أو اﻟﻨﻘﺺ ﻓﻲ ﻣﺠﻤﻮع اﻟﻤﺮﺑﻌﺎت ﺻﻐﻴﺮ ﺟﺪا .ﻣﻤﻜﻦ إﺳﺘﺨﺪام ﻃﺮﻳﻘﺔ اﻟﻌﺰوم ﻹﻳﺠﺎد اﻟﻘﻴﻤﺔ أﻷوﻟﻴﺔ * θﻟﻜﻲ ﻧﺤﺼﻞ ﻋﻠﻰ ﺗﻘﺎرب ﺳﺮﻳﻊ .ﻃﺒﻌﺎ اﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔ ﻻﺗﺘﻢ ﻳﺪوﻳﺎ ﺑﻞ ﺗﺤﺘﺎج إﻟﻰ ﺣﺎﺳﺐ ﻟﺬﻟﻚ. ﻳﻼﺣﻆ أن ﺗﻘﺪﻳﺮ اﻟﻤﻌﺎﻟﻢ ﻟﻠﻨﻤﻮذج ﻓﻲ ﺣﺎﻟﺔ ﻧﻤﺎذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك او اﻟﻨﻤﺎذج اﻟﻤﺨﺘﻠﻄﺔ اﻟﺘﻲ ﺗﺤﻮي ﻋﻠﻰ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﺗﺸﻜﻞ ﺗﻌﻘﻴﺪا ﻷﻧﻬﺎ ﺗﺤﻮى ﻣﻌﺎﻟﻢ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺑﺸﻜﻞ ﻏﻴﺮ ﺧﻄﻲ وﻟﻬﺬا ﺗﺤﺘﺎج إﻟﻰ ﻃﺮق ﻋﺪدﻳﺔ ﻟﺤﻠﻬﺎ آﻤﺎ ﺷﺎهﺪﻧﺎ ﻓﻲ ﺣﺎﻟﺔ اﻟﻨﻤﻮذج ) MA(١وهﻮ أﺑﺴﻄﻬﺎ ﺟﻤﻴﻌﺎ. ﺳﻮف ﻧﻜﺘﻔﻲ ﻓﻲ ﻣﻘﺮرﻧﺎ هﺬا ﻋﻠﻰ اﻟﻄﺮﻳﻘﺘﻴﻦ اﻟﺴﺎﺑﻘﺔ وﻟﻜﻦ ﻧﺬآﺮ ﺑﻌﺾ اﻟﻄﺮق اﻻﺧﺮى اﻟﻤﺴﺘﺨﺪﻣﺔ ﻓﻲ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج ﻣﺜﻞ: -١ﻃﺮﻳﻘﺔ اﻷرﺟﺤﻴﺔ اﻟﻌﻈﻤﻰ Maximum Likelihood Method -٢ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ ﻏﻴﺮ اﻟﺸﺮﻃﻴﺔ Unconditional Least Squares Method -٣ﻃﺮق اﻟﺘﻘﺪﻳﺮﻏﻴﺮ اﻟﺨﻄﻴﺔ Nonlinear Estimation Methods ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج : Model Checking and Diagnostics ﺑﻌﺪ اﻟﺘﻌﺮف ﻋﻠﻰ ﻧﻤﻮذج ﻣﺒﺪﺋﻲ وﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ هﺬا اﻟﻨﻤﻮذج ﻧﺠﺮي ﺑﻌﺾ اﻟﺘﺸﺨﻴﺼﺎت ﻋﻠﻰ اﻟﺒﻮاﻗﻲ أو أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ )اﻧﻈﺮ ﺗﻌﺮﻳﻒ (٤ﻟﻨﺮى ﻣﺪى ﻣﻄﺎﺑﻘﺔ اﻟﻨﻤﻮذج ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ،وﻳﻔﺘﺮض أن اﻟﺒﻮاﻗﻲ هﻲ ﻣﻘﺪرات ﻟﻤﺘﺴﻠﺴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء atواﻟﺘﻲ ﻧﻔﺘﺮض اﻧﻬﺎ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ . σ 2اﻟﺒﻮاﻗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ et = zt − zˆt = aˆt , t = 1, 2,..., n
أي ان اﻟﺒﻮاﻗﻲ هﻲ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻧﺎﻗﺺ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ. ﻳﻘﻮم اﻟﺘﺸﺨﻴﺺ واﻹﺧﺘﺒﺎرات ﻋﻠﻰ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ واﻟﻨﻈﺮ ﻓﻲ ﻣﺪى ﺗﺤﻘﻴﻘﻬﺎ ﻟﻔﺮﺿﻴﺎت اﻟﻨﻤﻮذج واﻟﺘﻲ هﻲ: -١ﻣﺘﻮﺳﻂ ﺻﻔﺮي -٢اﻟﻌﺸﻮاﺋﻴﺔ ٧٣
-٣ﻋﺪم اﻟﺘﺮاﺑﻂ -٤ﻣﻮزﻋﺔ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ )ﻣﺴﺘﻘﻞ وﻣﺘﻄﺎﺑﻖ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ σ أي ) ( at IIDN ( 0, σ 2 2
ﻟﻬﺬا ﻓﺈﻧﻨﺎ ﻧﺠﺮي ﺗﺸﺨﻴﺺ وهﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ ﻟﻨﺮي ﻓﻴﻤﺎ إذا آﺎﻧﺖ ﺗﺤﻘﻖ هﺬﻩ اﻟﺸﺮوط وﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ ﻧﻌﺘﺒﺮ اﻟﻨﻤﻮذج اﻟﻤﻄﺒﻖ ﻣﻘﺒﻮﻻ أﻣﺎ إذا ﻓﺸﻞ اﺣﺪ هﺬﻩ اﻹﺧﺘﺒﺎرات ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ إﻋﺎدة اﻟﻨﻈﺮ وإﻗﺘﺮاح ﻧﻤﻮذج ﺁﺧﺮ أوﻻ :إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ: H 0 : E ( at ) = 0 H 1 : E ( at ) ≠ 0
وهﻮ إﺧﺘﺒﺎر ﺑﺬﻳﻠﻴﻦ وﻧﺴﺘﺨﺪم ﻓﻴﺔ اﻹﺣﺼﺎﺋﺔ
e
) se ( e
= uواﻟﺘﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ
ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ ان E ( at ) = 0إذا آﺎﻧﺖ ) u < 1.96هﺬا ﻋﻠﻲ إﻋﺘﺒﺎر ان ﺣﺠﻢ اﻟﻌﻴﻨﺔ اآﺒﺮ ﻣﻦ ٣٠وﺣﺪة وهﺬا داﺋﻤﺎ ﻣﺘﺤﻘﻖ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﻧﺪرﺳﻬﺎ ( ﺛﺎﻧﻴﺎ :إﺧﺘﺒﺎر اﻟﻌﺸﻮاﺋﻴﺔ: ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺠﺮي Runs testﺣﻮل اﻟﻤﺘﻮﺳﻂ وﺣﻮل اﻟﺼﻔﺮ وهﻮ اﺣﺪ اﻹﺧﺘﺒﺎرات اﻟﻼﻣﻌﻠﻤﻴﺔ ) ﻳﻮﺟﺪ آﺜﻴﺮ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻟﻠﻌﺸﻮاﺋﻴﺔ ﻳﺪرﺳﻬﺎ اﻟﻄﺎﻟﺐ ﻓﻲ اﻟﻤﻘﺮر ٢٤١ﺑﺤﺚ وﻟﻜﻦ ﻧﻜﺘﻔﻲ هﻨﺎ ﺑﻬﺬا اﻹﺧﺘﺒﺎر(. ﺛﺎﻟﺜﺎ :إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ أو اﻹﺳﺘﻘﻼل: ﻳﺨﺘﺒﺮ ﺗﺮاﺑﻂ أو إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation testوذﻟﻚ ﺑﺤﺴﺎب ورﺳﻢ اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﻌﻴﻨﻴﺔ SACFﻟﻠﺒﻮاﻗﻲ وﻣﻘﺎرﻧﺘﻬﺎ ﻣﻊ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻤﺘﺴﻠﺴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء. اﻹﺧﺘﺒﺎر H 0 : ρ1 = 0
H1 : ρ1 ≠ 0 r1 ﺣﻴﺚ اﻹﺣﺼﺎﺋﺔ ) se ( r1
= uﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ
ان ρ1 = 0إذا آﺎﻧﺖ . u < 1.96 راﺑﻌﺎ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ: ﻧﺨﺘﺒﺮ ﻓﻲ ﻣﺎ إذا آﺎﻧﺖ اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ وذﻟﻚ ﺑﻌﺪة ﻃﺮق ﻣﺜﻞ: -١إﺧﺘﺒﺎر ﺣﺴﻦ اﻟﺘﻄﺎﺑﻖ Goodness of Fit Testوﻧﺴﺘﺨﺪم اﻹﺧﺘﺒﺎر اﻟﻼﻣﻌﻠﻤﻲ آﻮﻟﻤﻮﺟﻮروف -ﺳﻤﻴﺮﻧﻮف . Kolmogorov-Smirnov Test -٢ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ . Normal Probability Plot -٣ﻣﺨﻄﻂ اﻟﺮﺑﻴﻌﺎت-اﻟﺮﺑﻴﻌﺎت . Q-Q Plot ﺑﻌﺾ اﻟﻤﻌﺎﻳﻴﺮ اﻻﺧﺮى ﻹﺧﺘﻴﺎر ﻧﻤﻮذج اﻟﻤﻨﺎﺳﺐ: (١إﺣﺼﺎﺋﻴﺔ آﻴﻮ ﻟـ ﻟﺠﻨﻖ-ﺑﻮآﺲ Ljung-Box Q statistcوﺗﺨﺘﺼﺮ LBQوﺗﺴﺘﺨﺪم ﻹﺧﺘﺒﺎر اﻟﻔﺮﺿﻴﺔ: H 0 : ρ1 = ρ 2 = L = ρ K = 0 وﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ:
٧٤
)χ 2 ( K − m
rk2 k =1 n − k K
∑ )Q = n (n + 2
ﺣﻴﺚ mﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج. (٢ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ Automatic Information Criteriaوﺗﺨﺘﺼﺮ AICوﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: 2 AIC ( m ) = n ln σ a + 2m ﺣﻴﺚ mﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲ ) min AIC ( m m
أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ : Examples and Case Studies -١اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة )z(t
59.9315 61.0344 60.6643 60.5777 59.9086 62.2658 61.9013 54.5045 55.8451 59.3572 61.8956 59.7252 58.4651 61.2797 58.0143 56.6085 56.8697 57.2961 59.6266 58.1105 61.9152 59.0778 58.2759 59.5015 63.4124
60.7196 62.8556 59.3191 59.5353 59.6004 59.1455 60.9492 58.1690 56.7437 63.8189 59.3402 59.9246 57.9624 57.8803 54.2185 57.0642 57.3940 59.3236 61.1107 58.7336 62.1957 56.9972 61.8685 56.6666 60.7356
57.2318 64.6886 60.5820 58.3755 60.1325 58.0151 59.5333 55.7339 58.9585 61.1520 59.0087 60.5289 59.1567 60.3373 55.4219 62.3728 60.1458 57.5307 61.5614 60.0377 60.8256 59.0780 63.1777 56.0309 59.2298
56.1346 63.5049 62.4654 59.3054 58.4174 59.0903 58.8802 63.7261 61.8370 61.6337 58.7564 60.8942 56.5413 61.4310 59.2086 60.8605 61.4451 53.8560 59.4119 59.3488 59.3839 59.8597 58.3583 56.1494 61.7218
56.4828 63.9622 59.1721 60.9225 58.1483 59.4554 61.7122 61.6708 58.5870 60.2990 58.2273 63.6776 54.6083 62.3827 57.8763 60.3843 63.5907 58.1711 59.9346 58.0423 55.4010 59.0997 59.5097 59.2927 61.1168
58.9275 63.1547 57.9813 61.1856 61.8108 59.1609 65.1325 62.6899 57.9363 59.9443 54.7163 60.0538 54.5550 63.3933 59.9569 60.0855 60.6919 61.1852 60.6201 60.7227 58.6501 59.0970 60.3563 60.2513 61.2179
59.5257 61.4230 58.7108 61.4761 62.1789 61.7008 60.5918 59.4444 57.3334 62.1017 57.3292 63.1070 58.1895 61.9205 56.2599 62.1362 58.4221 60.8962 60.3030 60.7021 58.5790 57.1459 60.9815 60.2052 60.9013
60.1815 61.0640 58.0059 61.2223 61.9753 60.4833 63.4411 59.3478 56.7241 58.1281 61.7840 60.9021 60.7001 61.9462 61.9448 60.9805 57.5151 59.2145 58.5278 60.3550 59.4242 60.3319 61.5555 59.7755 59.4755
اوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ MINITAB آﺎﻟﺘﺎﻟﻲ: ;')MTB > TSPlot 'z(t >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC ;Connect >SUBC Title "An obseved Time Series".
٧٥
A n o b s e v e d T im e S e r ie s
z(t)
65
60
55
50
In d e x
100
150
200
ﺛﺎﻧﻴﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮ MTB > %ACF 'z(t)'; SUBC> MAXLAG 20; SUBC> TITLE"SACF of observed Time Series". Executing from file: H:\MTBWIN\MACROS\ACF.MAC
Autocorrelation
S A C F o f o b s e rv e d T im e S e rie s 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
L a g C o rr
T
LBQ
0 .5 1 0 .2 0 -0 .0 0 -0 .0 5 -0 .0 8 -0 .1 8 -0 .1 9
7 .1 9 2 .3 2 - 0 .0 1 - 0 .5 9 - 0 .9 5 - 2 .0 5 - 2 .0 9
5 2 .4 8 6 0 .7 8 6 0 .7 8 6 1 .3 4 6 2 .8 2 6 9 .9 2 7 7 .5 8
1 2 3 4 5 6 7
10
15
L a g C o rr
T
LBQ
- 0 .1 4 - 0 .1 4 - 0 .0 9 - 0 .0 7 - 0 .0 8 - 0 .0 2 0 .0 3
-1 .5 0 -1 .5 2 -0 .9 0 -0 .7 1 -0 .7 9 -0 .2 1 0 .3 2
8 1 .7 6 8 6 .1 4 8 7 .7 3 8 8 .7 3 8 9 .9 7 9 0 .0 5 9 0 .2 7
8 9 10 11 12 13 14
20
L a g C o rr
T
LBQ
0 .0 7 0 .1 3 0 .1 7 0 .2 0 0 .1 2 0 .0 6
0 .6 8 1 .3 3 1 .7 5 2 .0 6 1 .2 1 0 .6 1
9 1 .2 3 9 4 .8 6 1 0 1 .3 3 1 1 0 .6 3 1 1 3 .9 8 1 1 4 .8 6
15 16 17 18 19 20
ﺛﺎﻟﺜﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮ MTB > %PACF 'z(t)'; SUBC> MAXLAG 20; SUBC> TITLE"SPACF of obseved Time Series". Executing from file: H:\MTBWIN\MACROS\PACF.MAC
٧٦
Partial Autocorrelation
S P A C F o f o b se ve d T im e S e rie s 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
10
L a g P AC
T
0 .5 1 -0 .0 8 -0 .1 0 0 .0 0 -0 .0 5 -0 .1 6 -0 .0 4
7 .1 9 -1 .0 7 -1 .3 8 0 .0 4 -0 .7 3 -2 .3 4 -0 .5 0
1 2 3 4 5 6 7
15
L a g P AC
T
-0 .0 1 -0 .1 2 0 .0 1 -0 .0 4 -0 .0 9 0 .0 3 0 .0 2
-0 .1 2 -1 .6 3 0 .1 6 -0 .6 0 -1 .3 4 0 .3 9 0 .3 2
8 9 10 11 12 13 14
20
L a g P AC
T
-0 .0 2 0 .0 9 0 .0 8 0 .0 6 -0 .0 3 0 .0 4
-0 .2 3 1 .2 8 1 .2 0 0 .8 6 -0 .4 5 0 .5 2
15 16 17 18 19 20
ﻣﻦ أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ و اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﺗﺘﺒﻊ ﻧﻤﻮذج وﻟﻬﺬا ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻋﻠﻲ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮAR(١) MTB > Name c7 = 'RESI1' MTB > ARIMA 1 0 0 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 839.667 0.100 53.870 1 746.819 0.250 44.876 2 695.840 0.400 35.883 3 685.086 0.502 29.769 4 685.054 0.507 29.458 5 685.054 0.507 29.443 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.5073 0.0611 Constant 29.4429 0.1309 Mean 59.7571 0.2656
0.0010
T 8.30 224.98
Number of observations: 201 Residuals: SS = 685.020 (backforecasts excluded) MS = 3.442 DF = 199 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 10.8(DF=11) 27.6(DF=23) 35.9(DF=35)
48 45.0(DF=47)
Forecasts from period 201 Period 202 203
Forecast 59.7079 59.7322
95 Percent Limits Lower Upper 56.0707 63.3451 55.6537 63.8106
٧٧
Actual
204 205 206
59.7445 59.7507 59.7539
55.5600 55.5394 55.5357
zt = 59.76 + 0.51( zt −1 − 59.76) + at , at
63.9290 63.9620 63.9721
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
WN ( 0,3.44 )
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
( )
φˆ1 = 0.51, s.e. φˆ1 = 0.061, t = 8.3 µˆ = 59.76, s.e. ( µˆ ) = 0.66
( )
δˆ = 29.44, s.e. δˆ = 0.131, t = 224.98 σˆ 2 = 3.44, with d . f . = 199
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 1.855 'RESI1'; SUBC> Alternative 0; SUBC> GHistogram; SUBC> GDotplot; SUBC> GBoxplot.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 1.85 Variable RESI1
N 201
Mean -0.002
StDev 1.851
SE Mean 0.131
Z -0.01
P 0.99
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 94 The expected number of runs = 101.0796 107 Observations above K 94 below The test is significant at 0.3149 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣
٧٨
ACF of Residuals for z(t) (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
50
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ PACF of Residuals for z(t) (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
50
Lag
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء : إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ-٥ ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ-ا Histogram of the Residuals (response is z(t))
Frequency
30
20
10
0 -5
0
Residual
٧٩
5
: وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ.ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ Normal Probability Plot رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ-ب Normal Probability Plot for RESI1
99
Mean:
-1.6E-03
StDev:
1.85070
95 90
Percent
80 70 60 50 40 30 20 10 5
1
-5.0
-2.5
0.0
2.5
5.0
Data
:واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم ﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲK-S Test ﺑﺈﺧﺘﺒﺎر-ج MTB > %NormPlot 'RESI1'; SUBC> Kstest; SUBC> Title "Normal Test for Residuals". Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
Normal Test for Residuals
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -5
0
5
RESI1 Average: -0.0016272 StDev: 1.85070 N: 201
H 0 : Residuals
Kolmogorov-Smirnov Normality Test D+: 0.045 D-: 0.060 D : 0.060 Approximate P-Value: 0.074
:وﻧﻼﺣﻆ اﻟﺘﺎﻟﻲ اﻹﺧﺘﺒﺎر هﻮ
N ( 0,3.44 )
H1 : Residuals§ N ( 0,3.44 )
ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف +
−
D = 0.045, D = 0.06, D = 0.06 أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔα = 0.05 وهﻲ أآﺒﺮ ﻣﻦ٠٫٠٧٤ ﻟﻺﺧﺘﺒﺎر هﻲP-Value اﻟـ
.اﻟﺼﻔﺮﻳﺔ :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ٨٠
ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ٥ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 201 Period 202 203 204 205 206
Forecast 59.7079 59.7322 59.7445 59.7507 59.7539
95 Percent Limits Lower Upper 56.0707 63.3451 55.6537 63.8106 55.5600 63.9290 55.5394 63.9620 55.5357 63.9721
Actual
وﻧﺮﺳﻤﻬﺎ ﺑﺎﻷﻣﺮ اﻟﺘﺎﻟﻲ Plot C4*C8 C5*C8 C6*C8; SUBC> Connect; SUBC> Type 1; SUBC> Color 1; SUBC> Size 1; SUBC> Title "Forecast of 5 future value with 95% limits"; SUBC> Overlay.
Forecast of 5 future value with 95% limits 64 63 62
C4
61 60 59 58 57 56 55 1
2
3
4
5
C8
.واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺠﺰء اﻷﺧﻴﺮ ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﻊ اﻟﺘﻨﺒﺆات وﻓﺘﺮات اﻟﺘﻨﺒﺆ Forecast of 5 future value with 95% limits 64 63 62
C9
61 60 59 58 57 56 55 180
190
200
C8
اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة-٢ ٨١
494.948 503.170 496.806 504.340 496.372 503.506 496.062 501.521 507.886 506.345 507.735 497.295 497.642 510.287 496.029 503.134 501.634 499.611 500.388 505.987 493.858 494.857 497.861 493.017 498.931 497.794 513.900 504.717 502.209 497.992 501.387
496.208 498.429 496.227 491.923 499.260 500.356 494.033 497.393 495.431 496.285 507.673 503.415 499.783 496.181 497.133 494.878 497.943 499.299 498.787 499.467 502.933 495.430 511.741 501.999 496.078 497.413 495.353 499.134 503.130 504.174 501.701
507.382 502.233 505.884 496.665 500.074 492.286 508.489 504.965 500.664 507.072 485.991 500.921 503.852 498.380 498.060 500.199 504.785 505.517 505.475 497.462 504.817 504.336 490.887 503.465 506.649 505.218 498.477 496.680 491.202 495.699 492.716
498.440 496.678 493.371 506.329 498.598 516.373 499.217 495.000 492.352 495.423 505.577 501.819 501.175 504.666 509.407 504.408 501.417 492.318 496.757 498.403 491.707 505.429 514.220 502.414 491.418 496.150 498.016 501.542 507.590 497.647 504.640
488.539 506.040 501.605 497.785 502.891 491.981 493.161 505.581 501.064 497.883 500.744 493.866 495.868 494.885 494.814 495.954 493.552 497.130 497.626 506.259 504.085 490.683 487.344 494.614 507.438 512.122 505.367 504.012 497.231 505.234 496.877
511.026 489.348 498.229 502.545 503.107 496.830 507.020 495.355 504.712 504.072 495.737 503.580 501.700 507.582 501.928 503.325 500.484 502.066 501.082 493.843 498.571 501.703 498.599 500.256 497.643 490.619 500.324 499.779 497.956 502.681 502.823
496.650 501.649 502.969 506.459 494.559 500.508 498.319 498.304 501.527 494.833 504.129 507.594 498.294 496.932 504.394 499.982 498.640 502.173 504.346 504.913 500.712 504.204 506.403 502.721 503.746 506.001 501.827 496.063 505.709 493.502 497.421 497.048
)z(t 499.148 503.975 498.758 493.057 499.890 507.416 498.090 504.877 494.918 499.173 496.765 482.567 500.989 501.331 489.314 502.720 502.489 495.691 496.252 499.279 498.169 497.015 503.195 500.252 498.158 500.409 496.225 491.726 489.032 500.024 505.194 499.574
ﺍﻭﻻ :ﺴﻭﻑ ﻨﺭﺴﻡ ﻓﻘﻁ ٥٠ﻤﺸﺎﻫﺩﺓ ﻤﻥ ﻫﺫﻩ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ
510
)z(t
500
490
50
40
30
ﺛﺎﻧﻴﺎ :ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ:
٨٢
20
10
In d e x
Autocorrelation
Autocorrelation Function for z(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag Corr
10
T
LBQ
1 -0.53 -8.38 2 -0.05 -0.64 3 0.12 1.52 4 0.04 0.52 5 -0.13 -1.61 6 0.11 1.39 7 -0.09 -1.13
71.03 71.68 75.39 75.83 80.08 83.33 85.52
Lag Corr
T
15
LBQ
8 0.10 1.19 87.98 9 -0.11 -1.31 91.01 10 0.15 1.77 96.64 11 -0.18 -2.16 105.32 12 0.11 1.25 108.35 13 0.05 0.56 108.97 14 -0.12 -1.36 112.61
Lag Corr
20
T
LBQ
15 0.05 0.60 113.33 16 -0.02 -0.26 113.47 17 0.07 0.82 114.85 18 -0.10 -1.16 117.65 19 0.07 0.75 118.81 20 -0.06 -0.73 119.93
: ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ:ﺛﺎﻟﺜﺎ Partial Autocorrelation
Partial Autocorrelation Function for z(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
10
15
Lag PAC
T
Lag PAC
T
1 -0.53 2 -0.46 3 -0.29 4 -0.07 5 -0.11 6 0.02 7 -0.09
-8.38 -7.28 -4.52 -1.08 -1.71 0.30 -1.36
8 0.04 9 -0.07 10 0.12 11 -0.07 12 -0.04 13 0.08 14 -0.04
0.65 -1.09 1.94 -1.15 -0.71 1.19 -0.59
Lag PAC
20
T
15 0.03 0.49 16 -0.13 -2.09 17 0.06 1.01 18 -0.08 -1.32 19 0.03 0.44 20 -0.13 -2.07
وﺑﺘﻄﺒﻴﻖ هﺬا اﻟﻨﻤﻮذج ﻧﺠﺪMA(١) ﻣﻦ اﻷﻧﻤﺎط اﻟﻤﺸﺎهﺪة ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﺔ ﻗﺪ ﺗﺘﺒﻊ ﻧﻤﻮذج MTB > Name c7 = 'RESI1' MTB > ARIMA 0 0 1 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 6081.19 0.100 500.046 1 5265.34 0.250 500.004 2 4615.22 0.400 499.980 3 4109.70 0.550 499.967 4 3766.60 0.700 499.960 5 3727.32 0.841 499.959 6 3687.70 0.797 499.963
٨٣
7 3687.08 0.790 499.962 8 3687.07 0.791 499.962 9 3687.07 0.790 499.962 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev MA 1 0.7905 0.0386 Constant 499.962 0.051 Mean 499.962 0.051
0.0010
T 20.50 9708.40
Number of observations: 250 Residuals: SS = 3684.13 (backforecasts excluded) MS = 14.86 DF = 248 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 26.7(DF=11) 35.9(DF=23) 63.1(DF=35)
48 82.8(DF=47)
Forecasts from period 250 Period 251 252 253 254 255
Forecast 502.256 499.962 499.962 499.962 499.962
95 Percent Limits Lower Upper 494.700 509.812 490.330 509.593 490.330 509.593 490.330 509.593 490.330 509.593
zt = 499.962 + at − 0.7905at −1 , at
WN ( 0,14.86 )
Actual
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
( )
θˆ1 = 0.7905, s.e. θˆ1 = 0.0386, t = 20.50
( )
µˆ = δˆ = 499.962, s.e. δˆ = 0.051, t = 9708.40 σˆ 2 = 14.86, with d . f . = 248
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 3.847 'RESI1'; SUBC> Alternative 0.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 3.85 Variable RESI1
N 250
Mean -0.007
StDev 3.847
SE Mean 0.243
Z -0.03
P 0.98
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1
٨٤
K =
0.0000
The observed number of runs = 134 The expected number of runs = 125.9920 126 Observations above K 124 below The test is significant at 0.3103 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣ A C F o f R e s id u a ls fo r z (t) ( w ith 9 5 % c o n f id e n c e l im it s f o r th e a u to c o r r e l a tio n s ) 1 .0
0 .8
Autocorrelation
0 .6 0 .4
0 .2 0 .0 -0 .2 -0 .4
-0 .6 -0 .8
-1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ P A C F o f R e s id u a ls f o r z (t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء : إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ-٥ ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ-ا
٨٥
H istogram of the R esiduals (resp on se is z(t))
Frequency
30
20
10
0 -10
0
10
R esidual
: وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ.ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ Normal Probability Plot رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ-ب Normal Probability Plot for RESI1
99
Mean:
-6.9E-03
StDev:
3.84651
95 90
Percent
80 70 60 50 40 30 20 10 5
1
-10
-5
0
5
10
Data
:واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم ﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲK-S Test ﺑﺈﺧﺘﺒﺎر-ج MTB > %Qqplot 'RESI1'; SUBC> Conf 95; SUBC> Ci. Executing from file: H:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis Normal Dist. Parameter Estimates Data Mean: StDev:
: RESI1 -6.9E-03 3.84651
MTB > %NormPlot 'RESI1'; SUBC> Kstest. Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
٨٦
Normal Probability Plot
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -10
0
10
RESI1 Average: -0.0069004 StDev: 3.84651 N: 250
Kolmogorov-Smirnov Normality Test D+: 0.034 D-: 0.051 D : 0.051 Approximate P-Value: 0.105
ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف +
−
D = 0.034, D = 0.051, D = 0.051 أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ انα = 0.05 وهﻲ أآﺒﺮ ﻣﻦ٠٫١٠٥ ﻟﻺﺧﺘﺒﺎر هﻲP-Value اﻟـ
.اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ٥ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 250 Period 251 252 253 254 255
95 Percent Limits Lower Upper 494.700 509.812 490.330 509.593 490.330 509.593 490.330 509.593 490.330 509.593
Forecast 502.256 499.962 499.962 499.962 499.962
Actual
ﺗﻨﺒﺆ٩٥% واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات ﻣﻊ ﻓﺘﺮات Forecast of 5 future values with 95% limits
C4
510
500
490 1
2
3
4
5
C8
اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة-٣ ٨٧
224.678 227.500 221.632 234.178 233.005 232.432 228.189 231.718 222.468 231.163 226.479 234.207 230.455 233.640 233.725 233.089 228.655 220.958 231.531 231.363 228.285 229.320 235.587 228.729 232.700 240.884 236.017 229.321 228.141 228.958 226.114
226.760 224.063 225.663 238.463 232.209 230.102 231.121 229.848 223.115 228.418 224.483 232.738 228.163 232.232 236.562 233.427 225.484 220.782 232.030 230.938 229.757 230.636 230.713 230.360 234.625 241.821 230.865 226.222 225.985 229.352 228.837
226.641 222.515 229.136 239.577 227.621 229.564 231.633 227.982 227.859 228.618 228.989 229.792 225.447 234.825 235.224 233.044 224.207 222.819 232.315 233.127 229.733 227.948 227.046 230.662 234.107 238.762 230.499 225.771 225.358 229.369 232.881
226.778 221.562 232.308 232.653 222.156 229.331 231.319 225.734 230.122 228.225 235.122 228.198 224.928 234.707 231.665 229.575 224.747 226.865 232.027 233.852 232.160 224.258 228.606 234.243 231.265 233.628 226.994 225.616 225.291 230.296 236.029
229.903 222.482 236.488 223.408 218.067 229.359 234.668 225.721 230.888 229.851 235.024 226.724 227.812 233.841 228.922 227.089 228.954 232.152 232.135 232.684 235.038 223.994 228.203 239.883 229.486 230.342 223.795 226.271 223.250 228.920 235.339
230.260 223.390 236.033 217.433 217.123 235.744 235.767 224.927 228.472 228.227 236.659 225.196 231.266 233.891 230.327 227.032 232.820 232.461 228.727 231.411 233.799 225.273 231.898 243.963 225.070 227.643 223.381 230.088 225.447 227.934 235.210
227.346 225.772 234.323 213.619 221.484 236.419 233.918 226.765 228.200 225.799 236.399 222.523 232.976 232.067 230.148 227.077 233.256 231.062 224.050 232.032 233.622 223.599 229.100 242.860 225.123 230.082 228.468 235.623 226.745 229.727 232.891 224.096
)z(t 229.574 224.077 230.713 215.405 228.758 234.678 234.155 227.075 230.421 224.663 233.335 223.571 234.561 232.473 231.653 230.146 233.444 230.076 221.171 230.582 232.344 224.880 227.449 239.660 225.860 229.792 235.112 238.292 227.805 230.794 231.092 225.020
اوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ MINITAB آﺎﻟﺘﺎﻟﻲ ٥٠) :ﻣﺸﺎهﺪة ﻓﻘﻂ(
2 4 2
)z(t
2 3 2
2 2 2 5 0
4 0
3 0
ﺛﺎﻧﻴﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ
٨٨
2 0
1 0
In d e x
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
Lag 1 2 3 4 5 6 7
C o rr 0 0 0 -0 -0 -0 -0
.8 .5 .1 .1 .3 .3 .2
4 1 5 6 3 3 0
T 13 5 1 -1 -2 -2 -1
.2 .2 .3 .4 .9 .8 .7
7 0 5 6 8 7 3
10
LBQ 1 2 2 2 2 3 3
7 4 4 5 8 1 2
8 4 9 6 3 1 1
.2 .2 .7 .2 .7 .2 .9
1 9 1 1 5 3 5
Lag
1 1 1 1 1
C o rr
8 -0 .0 2 -0 9 0 .1 4 1 0 0 .2 3 1 1 0 .2 3 1 2 0 .1 5 1 3 0 .0 2 0 4 -0 .1 2 -0
T .1 .1 .9 .8 .2 .1 .9
7 8 5 8 2 5 6
15
LBQ 3 3 3 3 3 3 3
2 2 4 5 6 6 6
2 7 1 5 1 1 5
.0 .2 .4 .1 .1 .2 .0
Lag
5 4 8 6 7 5 4
1 1 1 1 1 2
5 6 7 8 9 0
C o rr -0 -0 -0 -0 0 0
.2 .2 .1 .1 .0 .0
1 2 8 0 0 8
20
T -1 -1 -1 -0 0 0
.6 .7 .4 .7 .0 .6
5 9 2 6 2 1
LBQ 3 3 3 4 4 4
7 8 9 0 0 0
6 9 8 1 1 3
.3 .9 .8 .3 .3 .0
2 3 1 7 7 5
ﺛﺎﻟﺜﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
Lag 1 2 3 4 5 6 7
PAC
10
T
0 .8 4 1 3 .2 7 -0 .6 6 -1 0 .3 9 -1 .1 3 -0 .0 7 -1 .0 0 -0 .0 6 1 .7 8 0 .1 1 2 .1 8 0 .1 4 -0 .2 2 -0 .0 1
Lag
15
PAC
T
8 0 .0 7 9 -0 .0 7 1 0 0 .0 5 1 1 -0 .0 7 1 2 0 .0 6 1 3 -0 .1 5 1 4 0 .0 1
1 .0 6 -1 .1 2 0 .7 3 -1 .1 8 0 .9 0 -2 .4 1 0 .1 8
Lag
20
PAC
T
1 5 0 .0 4 1 6 -0 .0 6 1 7 0 .0 4 1 8 -0 .0 7 1 9 0 .0 9 2 0 -0 .0 9
0 .6 7 -0 .9 3 0 .6 0 -1 .1 4 1 .4 4 -1 .3 9
ﻣﻦ أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ و اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﺗﺘﺒﻊ ﻧﻤﻮذج وﻟﻬﺬا ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻋﻠﻲ اﻟﻤﺸﺎهﺪاتAR(٢) MTB > Name c7 = 'RESI1' MTB > ARIMA 2 0 0 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 10 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 4257.23 0.100 0.100 1 3528.31 0.250 0.012 2 2889.23 0.400 -0.076 3 2338.97 0.550 -0.165 4 1877.39 0.700 -0.253 5 1504.46 0.850 -0.342 6 1220.13 1.000 -0.430
٨٩
183.784 169.535 155.360 141.201 127.051 112.913 98.789
7 1024.34 1.150 8 916.97 1.300 9 894.38 1.402 10 894.31 1.408 11 894.31 1.408 Relative change in each estimate
-0.519 84.690 -0.608 70.623 -0.668 61.154 -0.672 60.670 -0.672 60.646 less than 0.0010
Final Estimates of Parameters Type Coef StDev AR 1 1.4079 0.0473 AR 2 -0.6720 0.0474 Constant 60.6458 0.1203 Mean 229.638 0.456
T 29.78 -14.19 504.11
Number of observations: 250 Residuals: SS = 893.567 (backforecasts excluded) MS = 3.618 DF = 247 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 17.5(DF=10) 27.2(DF=22) 49.7(DF=34)
48 67.7(DF=46)
Forecasts from period 250 Period 251 252 253 254 255 256 257 258 259 260
Forecast 224.939 226.747 228.725 230.296 231.177 231.363 231.033 230.442 229.833 229.372
95 Percent Limits Lower Upper 221.211 228.668 220.308 233.186 220.642 236.808 221.546 239.045 222.311 240.044 222.494 240.233 222.070 239.996 221.327 239.558 220.600 239.067 220.090 238.655
zt = 60.6458 + 1.4079 zt −1 − 0.672 zt −2 + at , at
( ) s.e. (φˆ ) = 0.0474,
Actual
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
WN ( 0,3.618)
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
φˆ1 = 1.4079, s.e. φˆ1 = 0.0473, t = 29.78 φˆ2 = −0.672,
2
t = −14.19
µˆ = 229.638, s.e. ( µˆ ) = 0.456
( )
δˆ = 60.6458, s.e. δˆ = 0.1203, t = 504.11 σˆ 2 = 3.618, with d . f . = 247
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 3.618 'RESI1'; SUBC> Alternative 0.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 3.62 Variable RESI1
N 250
Mean -0.005
StDev 1.894
SE Mean 0.229
٩٠
Z -0.02
P 0.98
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 125 The expected number of runs = 125.8720 129 Observations above K 121 below The test is significant at 0.9119 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣ A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ P A C F o f R e s id u a ls f o r z (t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0 5
10
15
20
25
30
Lag
٩١
35
40
45
50
55
60
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء -٥إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ : ا -ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ H istogram of the R esiduals ))(res p on s e is z(t 30
10
0 5
-5
0
R es idual
ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ .وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ: ب -رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ Normal Probability Plot N orm al P rob ab ility P lot for R E S I1
-4 .6 E -0 3
M ean:
1 .8 9 4 3 6
S tD e v:
99
95 90 80
20 10 5
1
4
2
0
D a ta
واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم: ج -ﺑﺈﺧﺘﺒﺎر K-S Testﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ
٩٢
-2
-4
-6
Percent
70 60 50 40 30
Frequency
20
Normal Probability Plot
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -5
0
5
RESI1 Average: -0.0046305 StDev: 1.89436 N: 250
Kolmogorov-Smirnov Normality Test D+: 0.020 D-: 0.029 D : 0.029 Approximate P-Value > 0.15
وهﻲ أآﺒﺮ ﻣﻦ٠٫١٥ ﻟﻺﺧﺘﺒﺎر هﻲP-Value ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ اﻟـ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف . أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲα = 0.05 :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ١٠ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 250 Period 251 252 253 254 255 256 257 258 259 260
95 Percent Limits Lower Upper 221.211 228.668 220.308 233.186 220.642 236.808 221.546 239.045 222.311 240.044 222.494 240.233 222.070 239.996 221.327 239.558 220.600 239.067 220.090 238.655
Forecast 224.939 226.747 228.725 230.296 231.177 231.363 231.033 230.442 229.833 229.372
Actual
ﻓﺘﺮات ﺗﻨﺒﺆ٩٥% واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات و Forecast of 10 future values with 95% limits
C4
240
230
220 0
1
2
3
4
5
6
7
8
9
C8
٩٣
10
Forecast of 10 future values with 95% limits 245
235
C9 225
215 200
0
100
C8
Forecast of 10 future values with 95% limits
240
C9
230
220 50
60
30
40
20
10
0
C8
اﻟﺸﻜﻞ اﻷول ﻳﺒﻴﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﺑﻜﺎﻣﻠﻬﺎ ﻣﻊ اﻟﺘﻨﺒﺆات واﻟﺸﻜﻞ اﻟﺜﺎﻧﻲ ﻟﻠﺨﻤﺴﻴﻦ ﻗﻴﻤﺔ اﻷﺧﻴﺮة ﻣﻊ اﻟﺘﻨﺒﺆات ﻟﺘﻮﺿﻴﺢ ﺷﻜﻞ داﻟﺔ اﻟﺘﻨﺒﺆ. ﺣﺎﻟﺔ دراﺳﺔ: اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺘﺎﻟﻴﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻴﻮﻣﻲ ﻟﻌﺪد اﻟﺘﻠﻔﺰﻳﻮﻧﺎت اﻟﻤﻌﻴﺒﺔ ﻓﻲ ﺧﻂ إﻧﺘﺎج ﻣﺼﻨﻊ ﻣﺎ )إﻗﺮأ ﺳﻄﺮا ﺑﺴﻄﺮ( 2.09 1.57 2.07 1.78
2.00 1.42 1.82 1.68
1.76 1.54 1.85 1.79
2.83 1.46 2.08 1.37
3.44 2.05 1.42 1.15
2.40 2.50 1.39 1.25
اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
٩٤
1.95 2.25 1.18 1.61
2.70 1.58 1.27 1.77
1.54 1.25 1.08 2.91
Defects 1.20 1.50 1.89 1.80 1.40 1.51 2.32 1.23 1.84
3 .5
3 .0
Defects
2 .5 2 .0 1 .5 1 .0
In d e x
10
20
30
40
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Autocorrelation
Autocorrelation Function for Defects 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
Lag
Corr
4
5
6
7
8
T
LBQ
Lag
Corr
T
LBQ
1 0.43 2.88 2 0.26 1.49 3 0.14 0.77 4 0.08 0.43 5 -0.09 -0.46 6 -0.07 -0.39 7 -0.21 -1.10
8.84 12.18 13.18 13.50 13.89 14.18 16.57
8 9 10 11
-0.11 -0.05 -0.01 -0.04
-0.57 -0.27 -0.04 -0.19
17.25 17.41 17.41 17.50
9
10
11
Partial Autocorrelation
Partial Autocorrelation Function for Defects 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
Lag PAC
4
5
T
1 0.43 2.88 2 0.09 0.63 3 -0.00 -0.01 4 0.00 0.00 5 -0.16 -1.07 6 0.00 0.02 7 -0.18 -1.19
6
Lag PAC
7
8
T
8 0.07 0.44 9 0.05 0.35 10 0.01 0.09 11 -0.03 -0.23
٩٥
9
10
11
: واﻟﺬي ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔAIC ﻹﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ ﺳﻮف ﻧﺴﺘﺨﺪم ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ AIC ( m ) = n ln σ a2 + 2m ﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲm ﺣﻴﺚ min AIC ( m ) m
:ﺳﻮف ﻧﻄﺒﻖ اﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺘﻮاﻟﻲ MTB > ARIMA 1 0 0 'Defects' 'RESI1'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.4421 0.1365 Constant 0.99280 0.06999 Mean 1.7795 0.1254
T 3.24 14.19
Number of observations: 45 Residuals: SS = 9.47811 (backforecasts excluded) MS = 0.22042 DF = 43 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.9(DF=11) 8.9(DF=23) 30.9(DF=35)
48 * (DF= *)
MTB > ARIMA 2 0 0 'Defects' 'RESI2'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.3999 0.1533 AR 2 0.0989 0.1531 Constant 0.89019 0.07047 Mean 1.7762 0.1406
T 2.61 0.65 12.63
Number of observations: 45 Residuals: SS = 9.38567 (backforecasts excluded) MS = 0.22347 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.0(DF=10) 8.1(DF=22) 28.8(DF=34) MTB > ARIMA 1 0 1 'Defects' 'RESI3'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.5983 0.2691 MA 1 0.1926 0.3294 Constant 0.71334 0.05693
T 2.22 0.58 12.53
٩٦
48 * (DF= *)
Mean
1.7759
0.1417
Number of observations: 45 Residuals: SS = 9.39423 (backforecasts excluded) MS = 0.22367 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.1(DF=10) 8.3(DF=22) 29.1(DF=34)
48 * (DF= *)
MTB > ARIMA 0 0 1 'Defects' 'RESI4'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev MA 1 -0.3409 0.1431 Constant 1.78480 0.09651 Mean 1.78480 0.09651
T -2.38 18.49
Number of observations: 45 Residuals: SS = 10.0362 (backforecasts excluded) MS = 0.2334 DF = 43 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 8.0(DF=11) 13.2(DF=23) 35.7(DF=35)
48 * (DF= *)
MTB > ARIMA 0 0 2 'Defects' 'RESI5'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev MA 1 -0.3869 0.1516 MA 2 -0.1816 0.1516 Constant 1.7839 0.1118 Mean 1.7839 0.1118
T -2.55 -1.20 15.96
Number of observations: 45 Residuals: SS = 9.61059 (backforecasts excluded) MS = 0.22882 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.6(DF=10) 9.2(DF=22) 31.0(DF=34) MTB > ARIMA 2 0 1 'Defects' 'RESI6'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.4134 1.5680
T 0.26
٩٧
48 * (DF= *)
AR 2 MA 1 Constant Mean
0.0929 0.0136 0.87675 1.7761
0.7113 1.5749 0.07036 0.1425
0.13 0.01 12.46
Number of observations: 45 Residuals: SS = 9.38561 (backforecasts excluded) MS = 0.22892 DF = 41 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.0(DF= 9) 8.1(DF=21) 28.8(DF=33)
48 * (DF= *)
MTB > ARIMA 1 0 2 'Defects' 'RESI7'; SUBC> Constant;
ARIMA Model ARIMA model for Defects * ERROR * Model cannot be estimated with these data MTB > ARIMA 2 0 2 'Defects' 'RESI8'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 1.6720 0.1165 AR 2 -0.7263 0.1251 MA 1 1.3199 0.0184 MA 2 -0.3196 0.0731 Constant 0.096224 0.003323 Mean 1.77238 0.06121
T 14.35 -5.80 71.63 -4.37 28.95
Number of observations: 45 Residuals: SS = 8.33225 (backforecasts excluded) MS = 0.20831 DF = 40 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.7(DF= 8) 8.9(DF=20) 29.7(DF=32)
48 * (DF= *)
:وﻧﻠﺨﺺ ذﻟﻚ ﺑﺎﻟﺠﺪول اﻟﺘﺎﻟﻲ Model __________ AR (1) AR ( 2 ) MA (1) MA ( 2 ) ARMA (1,1) ARMA ( 2,1) ARMA (1, 2 ) ARMA ( 2, 2 )
σˆ
2
________ 0.22042 0.22347 0.23340 0.22882 0.22367 0.22892 − 0.20831
m ___ 3 4 3 4 4 5 − 6
AIC _________ −62.0499 −59.4315 −59.4751 −58.3669 −59.3913 −56.3472 − −58.5928
٩٨
min AIC ( m ) = −62.0499 m
أي ان أﻓﻀﻞ ﻧﻤﻮذج هﻮ ). AR(١ ﻳﺘﺮك ﻟﻠﻄﺎﻟﺐ آﺘﻤﺮﻳﻦ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ وﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات.
ﺣﺎﻟﺔ دراﺳﺔ: اﻟﻤﺘﺴﻠﺴﺔ اﻟﺘﺎﻟﻴﺔ هﻲ دﺧﻞ اﻟﻤﺒﻴﻌﺎت اﻟﺴﻨﻮﻳﺔ ﺑﻤﻼﻳﻴﻦ اﻟﺮﻳﺎﻻت ﻟﺸﺮآﺔ ﻣﺎ 5.43 3.88 3.57 2.75 6.06 5.80 5.16 6.64 6.43 7.86
3.80 4.30 3.45 4.80 6.12 6.08 5.71 7.49 7.53 7.50
4.14 5.42 1.22 3.08 5.65 4.78 5.61 6.09 5.62 8.27
4.60 3.91 3.98 5.43 5.52 5.67 5.63 6.64 7.59 8.75
4.77 5.07 2.65 4.40 4.79 4.89 5.70 4.72 7.27 8.50
3.45 3.78 3.28 3.84 6.11 4.99 5.75 6.08 9.01 7.23
3.99 6.16 4.05 5.14 6.46 6.12 6.36 6.57 7.49 7.53
5.51 4.05 4.08 4.00 4.31 6.23 8.02 7.56 6.69 7.67
5.74 2.54 4.61 1.58 4.99 5.05 7.07 6.87 7.22 8.22
ﻣﺨﻄﻂ زﻣﻨﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
9 8 7 6
Sales
5 4 3 2 1 100
90
80
70
60
50
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻴﺔ
٩٩
40
30
20
10
In d e x
Sales 3.49 3.96 2.89 2.52 5.77 3.20 5.13 7.20 7.26 6.42
Autocorrelation
Autocorrelation Function for Sales 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag Corr 1 2 3 4 5 6 7
0.71 0.60 0.65 0.64 0.59 0.59 0.51
T
LBQ
7.10 51.97 4.26 89.85 3.94 134.64 3.36 177.75 2.83 215.72 2.63 253.97 2.12 282.61
15
Lag Corr 8 9 10 11 12 13 14
0.56 0.49 0.49 0.51 0.42 0.38 0.45
T
LBQ
2.22 317.32 1.87 344.49 1.79 371.67 1.82 401.71 1.46 422.60 1.29 439.91 1.50 464.06
Lag Corr 15 16 17 18 19 20 21
0.41 0.35 0.31 0.30 0.36 0.31 0.26
T
25
LBQ
1.32 483.85 1.13 499.04 0.97 510.68 0.92 521.52 1.11 537.81 0.95 550.11 0.77 558.52
Lag Corr 22 23 24 25
0.22 0.17 0.21 0.25
T
LBQ
0.67 565.04 0.50 568.77 0.64 574.90 0.75 583.43
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag PAC
T
1 0.71 7.10 2 0.20 1.99 3 0.35 3.53 4 0.15 1.49 5 0.09 0.92 6 0.10 1.03 7 -0.13 -1.27
15
Lag PAC
25
T
Lag PAC
T
8 0.19 1.93 9 -0.17 -1.70 10 0.14 1.44 11 0.03 0.34 12 -0.15 -1.52 13 0.02 0.25 14 0.04 0.39
15 0.03 16 -0.10 17 -0.08 18 -0.01 19 0.14 20 -0.04 21 0.00
0.32 -0.96 -0.84 -0.10 1.43 -0.44 0.04
Lag PAC
T
22 -0.17 -1.68 23 -0.10 -1.00 24 0.15 1.55 25 0.03 0.34
.داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ ﺗﺪل ﻋﻠﻰ ﺗﺨﺎﻣﺪ ﺑﻄﻴﺊ ﻣﻤﺎ ﻗﺪ ﻳﺪل ﻋﻠﻰ ﻋﺪم إﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ وﻧﺮﺳﻤﻬﺎwt = ∇zt ﻟﻨﺠﺮب اﻟﺘﻔﺮﻳﻖ اﻷول ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
3 2 1
w(t)
Partial Autocorrelation
Partial Autocorrelation Function for Sales
0 -1 -2 -3
In d e x
10
20
30
40
50
60
70
80
90
100
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺴﺘﻘﺮة.ﺗﺒﺪو اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة اﻵن
١٠٠
Autocorrelation
Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
Lag Corr
12
T
LBQ
1 -0.30 -3.00 2 -0.31 -2.86 3 0.11 0.90 4 0.04 0.31 5 -0.04 -0.33 6 0.14 1.15 7 -0.24 -1.98
9.26 19.32 20.49 20.63 20.80 22.80 29.02
Lag Corr
T
LBQ
8 0.19 1.53 9 -0.07 -0.53 10 -0.07 -0.52 11 0.16 1.27 12 -0.07 -0.53 13 -0.14 -1.04 14 0.15 1.10
33.07 33.58 34.08 37.12 37.67 39.85 42.37
Lag Corr
22
T
LBQ
15 0.00 0.00 16 0.02 0.13 17 -0.07 -0.52 18 -0.15 -1.09 19 0.20 1.48 20 0.01 0.05 21 -0.04 -0.25
42.37 42.40 42.99 45.65 50.73 50.74 50.90
Lag Corr
T
LBQ
22 0.05 0.35 51.22 23 -0.18 -1.28 55.44 24 0.03 0.20 55.55
Partial Autocorrelation
Partial Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
22
Lag PAC
T
Lag PAC
T
Lag PAC
T
1 -0.30 2 -0.44 3 -0.22 4 -0.21 5 -0.17 6 0.06 7 -0.26
-3.00 -4.41 -2.23 -2.05 -1.72 0.59 -2.55
8 0.12 9 -0.16 10 -0.06 11 0.09 12 -0.06 13 -0.02 14 -0.07
1.17 -1.59 -0.60 0.93 -0.59 -0.17 -0.71
15 0.06 16 0.07 17 -0.03 18 -0.17 19 -0.03 20 -0.10 21 0.09
0.59 0.70 -0.32 -1.65 -0.31 -0.95 0.85
Lag PAC
T
22 0.08 0.80 23 -0.15 -1.53 24 -0.05 -0.55
: واﻟﺬي ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔAIC ﻹﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ ﺳﻮف ﻧﺴﺘﺨﺪم ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ AIC ( m ) = n ln σ a2 + 2m ﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲm ﺣﻴﺚ min AIC ( m ) m
:ﺳﻮف ﻧﻄﺒﻖ اﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺘﻮاﻟﻲ MTB > ARIMA 1 1 0 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.3114 0.0959
T -3.25
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 133.134 (backforecasts excluded)
١٠١
MS =
1.359
DF = 98
Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 31.9(DF=11) 51.2(DF=23) 62.8(DF=35)
48 81.0(DF=47)
MTB > ARIMA 2 1 0 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.4532 0.0897 AR 2 -0.4656 0.0901
T -5.05 -5.17
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 104.715 (backforecasts excluded) MS = 1.080 DF = 97 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 21.8(DF=10) 40.9(DF=22) 49.4(DF=34)
48 59.9(DF=46)
MTB > ARIMA 0 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev MA 1 0.7636 0.0648
T 11.78
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 101.411 (backforecasts excluded) MS = 1.035 DF = 98 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 12.6(DF=11) 27.8(DF=23) 35.9(DF=35)
48 48.5(DF=47)
MTB > ARIMA 0 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev MA 1 0.5756 0.0990 MA 2 0.2029 0.0998
T 5.81 2.03
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 99.2463 (backforecasts excluded) MS = 1.0232 DF = 97
١٠٢
Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 14.3(DF=10) 28.3(DF=22) 36.5(DF=34)
48 47.0(DF=46)
MTB > ARIMA 1 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 0.1283 0.1334 MA 1 0.8027 0.0799
T 0.96 10.04
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 100.421 (backforecasts excluded) MS = 1.035 DF = 97 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 13.3(DF=10) 27.9(DF=22) 36.1(DF=34)
48 48.2(DF=46)
MTB > ARIMA 2 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales * WARNING * Back forecasts not dying out rapidly Final Estimates of Parameters Type Coef StDev AR 1 -1.1389 0.0987 AR 2 -0.1440 0.0983 MA 1 -0.9889 0.0002
T -11.53 -1.47 -3987.49
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 134.250 (backforecasts excluded) MS = 1.398 DF = 96 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 35.1(DF= 9) 53.5(DF=21) 66.6(DF=33) MTB > ARIMA 1 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.3476 0.4077 MA 1 0.2422 0.3771 MA 2 0.4506 0.2656
T -0.85 0.64 1.70
Differencing: 1 regular difference
١٠٣
48 83.2(DF=45)
Number of observations: Original series 100, after differencing 99 Residuals: SS = 97.2357 (backforecasts excluded) MS = 1.0129 DF = 96 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 11.9(DF= 9) 25.0(DF=21) 32.1(DF=33)
48 41.8(DF=45)
MTB > ARIMA 2 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.0691 0.3618 AR 2 -0.2941 0.1450 MA 1 0.5637 0.3737 MA 2 0.0840 0.3266
T -0.19 -2.03 1.51 0.26
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 93.6368 (backforecasts excluded) MS = 0.9857 DF = 95 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 11.0(DF= 8) 23.4(DF=20) 30.1(DF=32)
48 36.5(DF=44)
:وﻧﻠﺨﺺ ذﻟﻚ ﺑﺎﻟﺠﺪول اﻟﺘﺎﻟﻲ Model σˆ __________ ________ ARI (1,1) 1.359 ARI ( 2,1) 1.080 IMA (1,1) 1.035 IMA (1, 2 ) 1.023 ARIMA (1,1,1) 1.035 ARIMA ( 2,1,1) 1.398 ARIMA (1,1, 2 ) 1.013 ARIMA ( 2,1, 2 ) 0.986 2
m ___ 2 3 2 3 3 4 4 5
AIC _________ 34.368 13.619 7.4057 8.2706 9.4057 41.169 9.2689 8.5741
min AIC ( m ) = 7.406 m
. ﻳﺘﺮك ﻟﻠﻄﺎﻟﺐ آﺘﻤﺮﻳﻦ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ وﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات. IMA(١،١) أي ان أﻓﻀﻞ ﻧﻤﻮذج هﻮ
١٠٤
١٠٥
اﻟﻔﺼﻞ اﻟﺴﺎدس ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻴﺔ
Seasonal Autoregressive Integrated Moving Average Models اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ﺗﻌﻄﻲ اﻧﻤﺎط ﻣﺘﺸﺎﺑﻬﺔ ﺗﺘﻜﺮر ﻋﻠﻰ ﻓﺘﺮات زﻣﻨﻴﺔ ﻣﺘﺴﺎوﻳﺔ اﻟﺒﻌﺪ ﻣﺜﻞ ان ﻳﺘﻜﺮر اﻟﻨﻤﻂ آﻞ ارﺑﻌﺔ وﻋﺸﺮون ﺳﺎﻋﺔ او آﻞ ﺳﺒﻌﺔ اﻳﺎم او آﻞ ﺷﻬﺮ او ﺛﻼﺛﺔ اﺷﻬﺮ او ﺳﻨﺔ. اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﺗﺒﻴﻦ ﻣﺜﻞ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻼت S e a s o n a l T im e S e r ie s
70
)z(t 60
50
150
50
100
In d e x
S e a s o n a l T im e S e r ie s 1000
900
)z(t
800
700
600
150
100
In d e x
50
ﻓﻲ هﺬا اﻟﻔﺼﻞ ﺳﻮف ﻧﺴﺘﻌﺮض ﺧﻮاص اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ وﻃﺮق ﻧﻤﺬﺟﺘﻬﺎ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك SARIMA(p,d,q)(P,D,Q)sﻓﻤﺜﻼ اﻟﻨﻤﻮذج SARIMA(٠،١،١)(١،١،٠)١٢ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
١٠٦
at
(1 − Φ B ) (1 − B ) z = (1 − θ B ) a , s
t
1
t
1
وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺑﺎﻟﺪرﺟﺔ (p,d,q)(P,D,Q)s SARIMA(p,d,q)(P,D,Q)sﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ p ( B ) Φ P ( B s ) (1 − B ) (1 − B s ) zt = δ + θ q ( B ) ΘQ ( B s ) at , at WN ( 0, σ 2 D
d
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻋﻤﺎل اﻹﻧﺤﺪار اﻟﺬاﺗﻲ واﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ واﻟﺘﻲ ﻣﺮت ﻋﻠﻴﻨﺎ ﺳﺎﺑﻘﺎ و Φ P ( B s ) = 1 + Φ1B s + Φ 2 B 2 s + L + Φ P B Psﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﻤﻮﺳﻤﻲ و ΘQ ( B s ) = 1 + Θ1B s + Θ2 B 2 s + L + ΘQ B Qsﻋﺎﻣﻞ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻲ وﻳﺴﻤﻰ هﺬا ﺑﺎﻟﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ .Multiplicative Seasonal Models ﻣﻼﺣﻈﺔ :ﻓﻲ ﺟﻤﻴﻊ اﻟﻨﻤﺎذج اﻟﻘﺎدﻣﺔ ﺳﻴﻜﻮن ﻣﻔﻬﻮﻣﺎ ﺿﻤﻨﻴﺎ أن ) at WN ( 0, σ 2
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ: D d ﻓﻲ اﻹﺷﺘﻘﺎﻗﺎت اﻟﺘﺎﻟﻴﺔ ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﺴﺘﻘﺮة wt = (1 − B ) (1 − B s ) zt واﻟﺘﻲ ﺗﺘﺒﻊ اﻟﻨﻤﻮذج SARMA(p,q)(P,Q)s ) WN ( 0, σ 2
φ p ( B ) Φ P ( B ) wt = δ + θ q ( B ) ΘQ ( B ) at , at s
s
ﺳﻮف ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ SARMA(٠،١)(١،١)١٢ ) WN ( 0, σ 2
wt = Φ wt −12 + at − θ at −1 − Θat −12 + θ Θat −13 , at
ﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻓﻲ wtوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ γ 0 = Φ γ 12 + σ + θ σ − Θ ( Φ − Θ ) σ 2 + θ Θ ( −Φ θ + θ Θ ) σ 2 2
2
2
⎦⎤) =Φ γ 12 + σ 2 ⎡⎣(1 + θ 2 ) + Θ ( Φ − Θ ) (1 + θ 2 ⎦⎤) =Φ γ 12 + σ 2 (1 + θ 2 ) ⎡⎣1 + Θ ( Φ − Θ
وﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻓﻲ wt −12وأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ 2
γ 12 = Φ γ 0 − Θσ + θ Θ ( −θ ) σ 2
) =Φ γ 0 − Θσ 2 (1 + θ 2
وﺑﺤﻞ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ 1 + Θ − 2ΦΘ 1 − Φ2 2 ⎡ ⎤ ) Φ (Θ − Φ 2 2 = σ (1 + θ ) ⎢Φ − Θ + ⎥ ⎦⎥ 1 − Φ 2 ⎣⎢ 2
أﻳﻀﺎ
) γ 0 = σ 2 (1 + θ 2 γ 12
) γ 1 = E ( wt wt −1 ) =Φ γ 11 − θσ 2 − ΘE ( at −12 wt −1 ) + θ ΘE ( at −13wt −1 =Φ γ 11 − θσ 2 + θ Θ ( Φ − Θ ) σ 2
و 2
وﺑﺤﻞ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ
١٠٧
γ 11 = E ( wt wt −11 ) = Φγ 1 + Θθσ
⎡ ( Θ − Φ )2 ⎤ γ 1 = −θσ ⎢1 + ⎥ 1 − Φ 2 ⎦⎥ ⎣⎢ 2
⎡
γ 11 = θσ 2 ⎢ Θ − Φ − ⎣⎢
2 Φ (Θ − Φ ) ⎤ ⎥ 1 − Φ 2 ⎦⎥
وﺑﻨﻔﺲ اﻟﻄﺮﻳﻘﺔ ﻳﻤﻜﻦ إﺛﺒﺎت أن
γ 2 = γ 3 = L = γ 10 = 0 γ 13 = γ 11 γ k = Φγ k −12 , k > 13
وﻣﻦ ﺟﻤﻴﻊ اﻟﻌﻼﻗﺎت اﻟﺴﺎﺑﻘﺔ ﻧﻮﺟﺪ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ k =0 ⎧ 1, ⎪ θ ⎪ − , k =1 2 ⎪ 1+θ ⎪ 0, k = 2,...,10 ⎪ γ ⎪ θ ( Θ − Φ )(1 − ΦΘ ) ρk = k = ⎨ , k = 11 γ 0 ⎪1 + θ 2 1 + Θ2 − 2ΦΘ ⎪ ( Θ − Φ )(1 − ΦΘ ) k = 12 , ⎪− 2 1 + Θ − 2 ΦΘ ⎪ k = 13 ⎪ ρ11 , ⎪ Φρ , k > 13 k −12 ⎩
:دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ
wt = (1 − ΘB s ) at SARIMA(٠,d,٠)(٠,D,١)s ﻧﻤﻮذج-١
k =0 ⎧ 1, ⎪⎪ Θ , k=s ρk = ⎨− 2 1 + Θ ⎪ otherwise ⎪⎩ 0,
(1 − Φ B ) w s
t
= at
SARIMA(٠,d,٠)(١,D,١)s ﻧﻤﻮذج-٢
k =0 ⎧1, ⎪ ks ρk = ⎨Φ , k = s, 2 s,... ⎪ 0, otherwise ⎩
wt = (1 − θ B ) (1 − ΘB s ) at
١٠٨
SARIMA(٠,d,١)(٠,D,١)s ﻧﻤﻮذج-٣
k =0 ⎧ 1, ⎪ θ ⎪− , k =1 2 ⎪ 1+θ ⎪ θΘ , k = s −1 ⎪⎪ 2 2 ρ k = ⎨ (1 + θ )(1 + Θ ) ⎪ Θ ⎪− , k=s 2 ⎪ 1+ Θ ⎪ ρ s−1 , k = s +1 ⎪ otherwise ⎪⎩ 0,
(1 − ΦB ) w = (1 − ΘB ) a s
s
t
t
k =0 ⎧ 1, ⎪ ⎪ ( Θ − Φ )(1 − ΦΘ ) k s−1 ρk = ⎨− Φ , k = s, 2 s,... 2 ⎪ 1 + Θ − 2ΦΘ otherwise ⎪⎩ 0, (1 − ΦB s ) wt = (1 − θ B ) at
SARIMA(٠,d,٠)(١,D,١)s ﻧﻤﻮذج-٤
SARIMA(٠,d,١)(١,D,٠)s ﻧﻤﻮذج-٥
k =0 ⎧ 1, ⎪ θ ⎪− , k =1 2 ⎪ 1+θ ⎪ 0, k = 2,..., s − 2 ⎪ ρk = ⎨ θ Φ − , k = s −1 ⎪ 1+θ 2 ⎪ k=s ⎪ Φ, ⎪ ρ s−1 , k = s +1 ⎪ k > s +1 ⎪⎩ Φ ρ k − s ,
wt = (1 − θ1B − θ 2 B 2 )(1 − ΘB12 ) at
١٠٩
SARIMA(٠,d,٢)(٠,D,١)s ﻧﻤﻮذج-٦
k =0 k =1 k =2 k = s−2 k = s −1 k=s k = s +1 k = s+2 otherwise
⎧ 1, ⎪ ⎪ − θ1 (1 − θ 2 ) , ⎪ 1 + θ12 + θ 22 ⎪ θ2 ⎪− , ⎪ 1 + θ12 + θ 22 ⎪ θ 2Θ ⎪ , 2 ) ⎪⎪ (1 + θ1 + θ 22 )(1 + Θ2 ⎨ = ρk ) θ1Θ (1 − θ 2 ⎪ , ) ⎪ (1 + θ 2 + θ 2 )(1 + Θ2 1 2 ⎪ ⎪ Θ , ⎪− 2 + Θ 1 ⎪ ⎪ ρ s −1 , ⎪ ρ , ⎪ s −2 ⎪⎩ 0,
ﺳﻮف ﻧﺴﺘﻌﺮض ﺑﻌﺾ اﻟﺮﺳﻮﻣﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻷﻋﻄﺎء ﻓﻜﺮة ﻋﻦ أﺷﻜﺎﻟﻬﺎ. اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﻟﻨﻤﻮذج : SARIMA(٠,d,١)(١,D,٠)١٢ ﺷﻜﻞ )(١
Φ = 0.6, θ = 0.5 A ( 0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
o f S A R I M
A C F 0 .5
C1
0 .0
-0 .5
5 0
4 0
2 0
3 0
0
1 0
L a g
ﺷﻜﻞ )( ٢
Φ = 0.6, θ = −0.5 A ( 0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
o f S A R I M
A C F 0 .7 0 .6 0 .5
0 .3 0 .2 0 .1 0 .0
5 0
4 0
2 0
3 0
L a g
١١٠
1 0
0
C1
0 .4
ﺷﻜﻞ )(٣
Φ = −0.6, θ = 0.5 A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2 0 .5
C1
0 .0
-0 .5
50
40
20
30
10
0
Lag
ﺷﻜﻞ )(٤
Φ = −0.6, θ = −0.5 A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2
0 .5
C1
0 .0
-0 .5
5 0
4 0
2 0
3 0
1 0
0
L ag
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ: ﻣﻦ اﻟﺼﻌﻮﺑﺔ إﺷﺘﻘﺎق وﺗﻔﺴﻴﺮ أﻧﻤﺎط داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ وﻟﻜﻨﻬﺎ وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن أﺟﺰاء اﻟﻨﻤﻮذج اﻟﻤﻮﺳﻤﻴﺔ وﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ واﻟﺘﻲ ﺗﻨﻤﺬج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺗﻌﻄﻲ ﺗﺨﺎﻣﺪات اﺳﻴﺔ وﺗﺨﺎﻣﺪات ﺟﻴﺒﻴﺔ ﻋﻨﺪ اﻟﺘﺨﻠﻔﺎت اﻟﻤﻮﺳﻤﻴﺔ وﻏﻴﺮاﻟﻤﻮﺳﻤﻴﺔ وﻓﻲ اﻟﻨﻤﺎذج اﻟﺘﻲ ﺗﺤﻮي إﻧﺤﺪار ذاﺗﻲ ﻓﺈن اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﺗﻌﻄﻲ ﻗﻄﻌﺎ . cut off اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﻹﻋﻄﺎء ﻓﻜﺮة ﻋﻦ ﺑﻌﺾ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج : -١ﺷﻜﻞ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج wt = (1 − ΘB12 ) at ا( Θ = 0.6
١١١
A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2 0 .0 -0 .1
C1
-0 .2 -0 .3 -0 .4 -0 .5 -0 .6
0
1 0
2 0
3 0
4 0
5 0
L ag
Θ = −0.6 (ب A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2 0 .6 0 .5 0 .4
C1
0 .3 0 .2 0 .1 0 .0 -0 .1 -0 .2 0
1 0
2 0
3 0
4 0
5 0
L ag
(1 − Φ B ) w 12
t
= at ﺷﻜﻞ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج-٢
Φ = 0.6 (ا A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2 0 .6 0 .5
C1
0 .4 0 .3 0 .2 0 .1 0 .0
0
1 0
2 0
3 0
4 0
5 0
L a g
Φ = −0.6 (ب
١١٢
A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2 0 .0 -0 .1 -0 .2
C1
-0 .3 -0 .4 -0 .5 -0 .6
5 0
4 0
2 0
3 0
0
1 0
L ag
ﺃﻤﺜﻠﺔ :ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ ﺍﻟﻤﻭﺴﻤﻴﺔ )ﻓﻲ ﺠﻤﻴﻊ ﺍﻷﻤﺜﻠﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺇﻗﺭﺃ ﺴﻁﺭﺍ ﺒﺴﻁﺭ( 54.9 58.2 68.2 54.9 53.2 57.3 70.4 53.4 51.7 57.1 75.5
54.9 54.3 70.1 57.5 52.8 52.7 69.9 55.3 51.5 53.9 73.3
55.3 53.4 67.5 61.2 52.8 51.6 61.0 58.2 52.3 53.5 68.1 62.2
56.9 53.0 58.0 69.3 54.5 52.4 52.7 66.9 53.6 53.5 58.1 74.8
57.4 52.8 54.2 69.8 56.4 52.1 53.9 70.7 55.3 53.1 54.8 76.4
61.5 53.3 53.2 66.1 59.3 52.6 53.3 65.3 58.5 53.3 54.3 70.8
72.7 54.4 53.0 56.1 68.7 53.9 53.5 56.5 69.3 53.9 54.6 60.6
72.2 56.0 53.0 53.6 70.0 56.4 53.4 53.4 69.6 55.6 54.2 56.4
71.5 60.0 53.4 53.0 67.9 61.7 53.6 52.5 64.2 60.1 54.8 55.6
59.1 71.0 54.6 52.8 58.7 68.3 55.1 53.2 55.5 68.9 55.8 55.0
57.2 70.6 55.6 52.6 55.4 67.9 57.3 53.0 53.3 68.8 57.9 54.7
56.3 68.2 59.4 52.9 52.9 65.3 61.9 53.5 52.4 63.6 62.6 55.8
55.8 57.7 69.8 54.0 53.0 58.2 69.9 54.3 51.5 57.1 70.3 57.7
55.7 54.6 71.0 54.9 52.7 55.8 71.5 56.3 51.5 52.2 69.4 60.5
ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
7 0
)z(t
6 0
5 0
1 5 0
1 0 0
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
١١٣
5 0
In d e x
)z(t 56.3 54.9 67.4 56.6 53.4 55.3 65.1 59.4 52.1 51.5 64.7 66.4
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
2 Lag 1 2 3 4 5 6 7 8 9 10 11 12
C o rr
12
T
LBQ
Lag
C o rr
0 .7 7 1 0 .3 1 2 .9 3 0 .3 3 -0 .1 1 -0 .9 9 -0 .3 8 -3 .2 8 -0 .5 2 -4 .2 2 -0 .5 6 -4 .1 6 -0 .5 2 -3 .5 0 -0 .3 8 -2 .3 9 -0 .1 2 -0 .7 1 1 .7 6 0 .2 9 4 .1 5 0 .6 9 4 .8 6 0 .8 8
1 0 8 .1 0 1 2 7 .3 4 1 2 9 .7 4 1 5 6 .8 3 2 0 7 .2 4 2 6 6 .5 5 3 1 6 .8 0 3 4 3 .7 1 3 4 6 .2 5 3 6 2 .0 8 4 5 3 .5 8 6 0 3 .8 7
13 14 15 16 17 18 19 20 21 22 23 24
0 .6 8 0 .2 8 -0 .1 2 -0 .3 6 -0 .4 9 -0 .5 3 -0 .4 9 -0 .3 7 -0 .1 4 0 .2 3 0 .6 1 0 .7 9
22 T
32
42
LBQ
Lag
C o rr
T
LBQ
Lag
C o rr
T
LBQ
3 .3 4 6 9 4 .2 8 1 .3 0 7 0 9 .8 2 -0 .5 3 7 1 2 .4 3 -1 .6 5 7 3 8 .3 0 -2 .2 1 7 8 6 .2 8 -2 .3 3 8 4 3 .0 4 -2 .1 0 8 9 2 .3 6 -1 .5 4 9 2 0 .3 9 -0 .5 6 9 2 4 .2 4 0 .9 5 9 3 5 .3 7 2 .4 6 1 0 1 1 .3 1 3 .1 0 1 1 4 0 .9 0
25 26 27 28 29 30 31 32 33 34 35 36
0 .6 1 0 .2 5 -0 .1 1 -0 .3 4 -0 .4 7 -0 .5 1 -0 .4 8 -0 .3 6 -0 .1 4 0 .1 9 0 .5 4 0 .7 2
2 .3 0 0 .9 1 -0 .4 2 -1 .2 4 -1 .6 7 -1 .8 0 -1 .6 5 -1 .2 3 -0 .4 9 0 .6 6 1 .8 2 2 .3 7
1 2 2 0 .0 7 1 2 3 3 .1 8 1 2 3 5 .9 8 1 2 6 1 .2 9 1 3 0 8 .3 2 1 3 6 4 .7 2 1 4 1 4 .0 8 1 4 4 2 .7 1 1 4 4 7 .3 4 1 4 5 5 .7 9 1 5 2 0 .8 9 1 6 3 6 .4 9
37 38 39 40 41 42 43 44
0 .5 7 0 .2 3 -0 .1 1 -0 .3 3 -0 .4 4 -0 .4 8 -0 .4 5 -0 .3 4
1 .8 2 0 .7 3 -0 .3 4 -1 .0 2 -1 .3 8 -1 .4 8 -1 .3 7 -1 .0 4
1 7 0 9 .3 6 1 7 2 1 .7 0 1 7 2 4 .4 3 1 7 4 9 .0 6 1 7 9 5 .0 0 1 8 4 9 .4 1 1 8 9 7 .0 5 1 9 2 5 .3 0
واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
2
12
22
32
42
Lag
P AC
T
Lag
P AC
T
Lag
P AC
T
Lag
P AC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .7 7 -0 .6 8 -0 .0 6 0 .0 1 -0 .4 2 -0 .1 8 -0 .1 1 -0 .2 2 0 .1 8 0 .5 7 0 .2 6 0 .1 6
1 0 .3 1 - 9 .0 1 - 0 .8 2 0 .0 7 - 5 .5 5 - 2 .4 5 - 1 .4 8 - 2 .9 6 2 .3 9 7 .6 5 3 .5 0 2 .1 6
13 14 15 16 17 18 19 20 21 22 23 24
- 0 .4 2 0 .2 9 0 .0 1 - 0 .0 7 0 .0 7 - 0 .0 1 - 0 .0 9 - 0 .0 5 - 0 .0 8 0 .1 0 - 0 .0 3 0 .0 3
- 5 .6 2 3 .8 9 0 .1 6 - 0 .9 2 0 .9 5 - 0 .1 1 - 1 .1 5 - 0 .7 2 - 1 .1 1 1 .3 8 - 0 .3 5 0 .3 7
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 3 - 0 .0 0 0 .0 4 - 0 .1 0 - 0 .0 3 0 .0 1 0 .0 1 - 0 .0 4 - 0 .0 4 0 .0 1 - 0 .0 2 0 .0 7
-1 .7 5 -0 .0 5 0 .5 0 -1 .3 0 -0 .3 4 0 .1 9 0 .0 7 -0 .5 5 -0 .5 2 0 .1 5 -0 .3 2 0 .9 0
37 38 39 40 41 42 43 44
-0 .0 6 -0 .0 5 0 .0 1 0 .0 0 -0 .0 5 0 .0 5 -0 .0 5 -0 .0 3
-0 .7 4 -0 .6 2 0 .1 7 0 .0 0 -0 .6 3 0 .6 6 -0 .6 4 -0 .4 2
.ﻧﻼﺣﻆ اﻷﻧﻤﺎط اﻟﻤﻮﺳﻤﻴﺔ واﺿﺤﺔ ﻓﻲ اﻷﺷﻜﺎل اﻟﺴﺎﺑﻘﺔ ﻤﺜﺎل ﺁﺨﺭ z(t) 589 673 678 621 713 796 801 747 826 898 908 827
561 742 639 602 667 858 764 711 799 957 867 797
640 716 604 635 762 826 725 751 890 924 815 843
656 660 611 677 784 783 723 804 900 881 812
727 617 594 635 837 740 690 756 961 837 773
697 583 634 736 817 701 734 860 935 784 813
640 587 658 755 767 706 750 878 894 791 834
599 565 622 811 722 677 707 942 855 760 782
568 598 709 798 681 711 807 913 809 802 892
577 628 722 735 687 734 824 869 810 828 903
553 618 782 697 660 690 886 834 766 778 966
582 688 756 661 698 785 859 790 805 889 937
600 705 702 667 717 805 819 800 821 902 896
566 770 653 645 696 871 783 763 773 969 858
653 736 615 688 775 845 740 800 883 947 817
ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ
١١٤
1000
z(t)
900
800
700
600
50
In d e x
100
150
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
2 Lag 1 2 3 4 5 6 7 8 9 10 11 12
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.8 .7 .6 .4 .4 .3 .4 .4 .5 .6 .7 .8
T
12 LB Q
9 1 1 .5 6 1 3 5 .9 4 6 .2 7 2 4 0 .1 3 8 4 .1 2 3 0 6 .7 2 2 2 .9 5 3 4 7 .9 7 9 2 .4 7 3 8 0 .0 9 3 2 .1 0 4 0 5 .0 2 8 2 .2 5 4 3 5 .5 4 1 2 .4 0 4 7 2 .3 7 5 2 .8 7 5 2 9 .0 7 6 3 .3 4 6 1 4 .2 7 9 3 .5 2 7 2 1 .7 2 7 3 .6 1 8 5 2 .4 1 4
L ag 1 1 1 1 1 1 1 2 2 2 2 2
3 4 5 6 7 8 9 0 1 2 3 4
22
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.7 .6 .4 .3 .3 .2 .2 .3 .4 .5 .6 .6
4 4 9 6 1 5 9 2 2 3 0 7
T 2 2 1 1 1 0 1 1 1 1 2 2
.9 .4 .7 .3 .0 .9 .0 .1 .4 .8 .0 .2
6 11 91 11 91 01 11 21 41 11 31 11
LB Q 9 0 0 0 1 1 1 1 1 2 3 4
5 3 7 9 1 2 4 6 9 5 2 1
4 0 4 9 7 9 5 5 9 3 5 5
.6 .0 .8 .6 .3 .7 .5 .4 .1 .8 .5 .2
8 9 5 8 9 6 9 2 3 1 1 9
L ag 2 2 2 2 2 3 3 3 3 3 3 3
5 6 7 8 9 0 1 2 3 4 5 6
32
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.5 .4 .3 .2 .1 .1 .1 .2 .2 .3 .4 .5
8 9 5 4 9 4 7 0 8 8 5 2
T 1 1 1 0 0 0 0 0 0 1 1 1
.8 .5 .0 .7 .5 .4 .5 .6 .8 .1 .3 .5
61 21 91 31 71 31 11 01 51 51 51 31
LB Q 4 5 5 5 5 5 5 5 6 6 6 7
8 3 5 6 7 7 8 9 1 4 8 4
3 0 6 8 5 9 5 3 0 1 5 2
.1 .9 .4 .2 .4 .5 .5 .6 .3 .5 .2 .9
1 9 5 0 4 5 2 2 5 7 7 5
L ag 3 3 3 4 4 4
7 8 9 0 1 2
42 C o rr 0 0 0 0 0 0
.4 .3 .2 .1 .0 .0
3 5 2 2 6 2
T 1 1 0 0 0 0
.2 .0 .6 .3 .1 .0
71 01 41 31 81 51
LB Q 7 8 8 8 8 8
8 1 2 2 2 2
4 0 1 4 5 5
.1 .6 .6 .6 .5 .6
7 0 8 7 6 2
واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
2
12
22
32
42
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .8 9 -0 .0 8 -0 .2 8 0 .0 3 0 .3 5 -0 .0 8 0 .2 8 0 .0 9 0 .4 0 0 .3 0 0 .0 6 0 .2 2
1 1 .5 6 -1 .0 6 -3 .6 5 0 .4 2 4 .5 4 -1 .0 7 3 .6 7 1 .1 9 5 .1 7 3 .9 5 0 .8 1 2 .8 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .6 3 -0 .0 2 0 .0 7 -0 .0 4 -0 .0 9 -0 .0 4 -0 .0 5 0 .0 3 0 .0 4 0 .0 5 0 .0 5 0 .0 5
-8 .1 9 -0 .2 1 0 .9 5 -0 .5 2 -1 .1 2 -0 .4 9 -0 .6 0 0 .3 8 0 .4 6 0 .6 7 0 .6 0 0 .5 9
25 26 27 28 29 30 31 32 33 34 35 36
-0 .1 8 0 .0 8 0 .0 6 -0 .0 3 -0 .0 4 0 .0 0 -0 .0 6 -0 .0 1 -0 .0 1 0 .0 3 0 .0 0 0 .0 1
-2 .3 6 1 .0 6 0 .7 3 -0 .4 4 -0 .4 7 0 .0 2 -0 .7 2 -0 .1 1 -0 .1 8 0 .3 8 0 .0 3 0 .0 9
37 38 39 40 41 42
-0 .1 1 -0 .0 2 0 .0 4 -0 .0 3 -0 .0 8 0 .0 1
-1 .3 7 -0 .2 2 0 .5 1 -0 .4 2 -1 .0 6 0 .0 8
١١٥
ﻣﺜﺎل ﺁﺧﺮ z(t) 302 107 055 237 079 035 256
262 056 048 247 045 056 250
218 049 115 215 040 097 198
175 047 185 182 038 210 136
100 047 276 080 041 260 073
077 071 220 046 069 257 039
043 151 181 065 152 210 032
047 244 151 040 232 125 030
049 280 083 044 282 080 031
069 230 055 063 255 042 045
152 185 049 085 161 035
205 148 042 185 107 031
246 098 046 247 053 032
294 061 074 231 040 050
242 046 103 167 039 092
181 045 200 117 034 189
وﻟﻬﺎ اﻟﺸﻜﻞ
3 0 0
z(t)
2 0 0
1 0 0
0
1 0
In d e x
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
وداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
5
1 2 3 4 5 6 7
C o rr 0 0 -0 -0 -0 -0 -0
.8 .4 .0 .4 .6 .7 .6
1 3 3 3 9 8 8
T 8 2 -0 -2 -4 -3 -3
.3 .8 .1 .7 .0 .9 .0
8 9 7 0 3 9 9
L B Q
1 1 2 2
7 9 9 1 6 3 8
2 2 2 3 6 5 9
.2 .3 .4 .1 .3 .1 .3
L a g
2 2 0 3 7 7 3
1 1 1 1 1
C o rr
T
8 -0 .4 3 -1 .7 8 3 1 9 -0 .0 4 -0 .1 8 3 1 0 0 .3 8 1 .5 4 3 2 1 0 .7 1 2 .8 2 3 8 2 0 .8 4 3 .1 0 4 7 3 0 .7 1 2 .4 2 5 3 4 0 .3 7 1 .2 0 5 5
L B Q 0 0 8 9 6 8 6
.6 .8 .2 .7 .2 .8 .1
L a g
6 8 2 2 2 4 0
1 1 1 1 1 2 2
5 6 7 8 9 0 1
2 5
C o rr -0 -0 -0 -0 -0 -0 -0
.0 .3 .6 .6 .5 .3 .0
3 8 0 7 9 7 4
T -0 -1 -1 -2 -1 -1 -0
.0 .1 .8 .0 .7 .0 .1
8 9 8 3 2 5 0
L B Q 5 5 6 6 7 7 7
5 7 2 7 2 4 4
6 4 0 8 4 2 2
.1 .0 .2 .3 .1 .2 .3
8 7 2 6 7 1 8
L a g 2 2 2 2 2
2 3 4 5 6
C o rr 0 0 0 0 0
.3 .6 .7 .5 .3
3 1 1 9 1
T 0 1 1 1 0
.9 .7 .9 .5 .8
2 1 3 5 0
L B Q 7 8 8 9 9
5 0 7 2 4
6 8 9 8 2
.9 .9 .4 .4 .4
8 7 2 1 6
وداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) Partial Autocorrelation
L a g
1 5
1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
5
L a g 1 2 3 4 5 6 7
P A C 0 -0 -0 -0 -0 -0 -0
.8 .7 .2 .2 .1 .3 .1
1 0 9 2 6 2 6
T 8 -7 -2 -2 -1 -3 -1
.3 .1 .9 .2 .6 .2 .6
8 7 8 8 1 9 0
1 5
L a g
1 1 1 1 1
P A C
8 -0 .0 3 9 0 .1 9 0 0 .2 7 1 0 .1 8 2 0 .0 7 3 -0 .1 0 4 -0 .1 5
-0 1 2 1 0 -0 -1
.3 .9 .7 .8 .7 .9 .5
T
L a g
1 1 6 2 4 9 2
1 1 1 1 1 2 2
١١٦
2 5
P A C
0 .1 5 0 .0 6 0 .1 7 0 .0 8 9 -0 .0 0 -0 .0 0 .0 1
9 8 1 4 2 3 4
1 0 1 0 -0 -0 0
.9 .8 .1 .3 .2 .2 .4
T
L a g
3 4 0 9 2 9 6
2 2 2 2 2
P A C
2 0 .0 3 0 .0 4 -0 .0 5 -0 .0 6 0 .1
1 7 7 2 0
T 0 0 -0 -0 1
.1 .7 .7 .2 .0
5 0 2 3 1
وآﻞ هﺬﻩ اﻟﻤﺘﻠﺴﻼت ﺗﺒﺪي اﻧﻤﺎط ﻣﻮﺳﻤﻴﺔ واﺿﺤﺔ.
إﺷﺘﻘﺎق دوال ﺗﻨﺒﺆ ﻟﺒﻌﺾ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ: ﺑﻤﺎ ان ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ هﻲ ﺣﺎﻟﺔ ﺧﺎﺻﺔ ﻣﻦ ﻧﻤﺎذج ARIMAﻓﺈن ﻃﺮق اﻟﺘﻌﺎﻣﻞ ﻣﻌﻬﺎ هﻲ ﻧﻔﺲ اﻟﻄﺮق اﻟﺴﺎﺑﻘﺔ ﻣﻦ ﺣﻴﺚ اﻟﺘﻌﺮف ﻋﻠﻲ ﺷﻜﻞ اﻟﻨﻤﻮذج وﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج واﻹﺧﺘﺒﺎرات اﻟﺘﻔﺤﺼﻴﺔ وﻣﻦ ﺛﻢ اﻟﺘﻨﺒﺆ .ﺟﻤﻴﻊ اﻟﻄﺮق واﻟﻤﻌﺎدﻻت اﻟﺘﻲ درﺳﻨﺎهﺎ ﺳﺎﺑﻘﺎ ﻟﻠﻨﻤﺎذج ﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ ﺗﻨﻄﺒﻖ هﻨﺎ .ﺳﻮف ﻧﺸﺘﻖ دوال اﻟﺘﻨﺒﺆ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ﻟﻠﺘﻮﺿﻴﺢ ﻓﻘﻂ. -١داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج : SARIMA(٠،٠،٠)(٠،١،١)١٢ وﻳﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞ
(1 − B ) z = (1 − ΘB ) a 12
t
12
t
ﻣﻦ اﻟﻤﻌﺎدﻟﺔ اﻟﻔﺮوﻗﻴﺔ zn + l = zn +l−12 + an + l − Θan + l−12
ﻳﻤﻜﻦ اﻟﺤﺼﻮل ﻋﻠﻰ اﻟﺘﻨﺒﺆات آﺎﻟﺘﺎﻟﻲ
zn (1) = zn −11 − Θan −11 zn ( 2 ) = zn −10 − Θan −10
M
zn (12 ) = zn − Θan zn ( l ) = zn ( l − 12 ) , l ≥ 12
أو l = 1, 2,...,12
l > 12
واﺿﺢ أن
⎧ zn + l−12 − Θan + l−12 , ⎨ = ) zn ( l ⎩ zn ( l − 12 ) , zn (1) = zn (13) = zn ( 25) = L
zn ( 2 ) = zn (14 ) = zn ( 26 ) = L
M
zn (12 ) = zn ( 24 ) = zn ( 36 ) = L
ﺗﺒﺎﻳﻦ أﺧﻄﺎء اﻟﺘﻨﺒﺆ
)
2 l −1
+ L +ψ
2 1
(1 + ψ
2
V ⎣⎡ en ( l )⎦⎤ = σ
وداﻟﺔ اﻷوزان ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ )ﺑﺮهﻦ ذﻟﻚ( j = 12, 24,... otherwise
⎧1 − Θ, ⎩ 0,
⎨= ψj
وﺑﺘﻌﻮﻳﺾ اﻷوزان ﻓﻲ ﺻﻴﻐﺔ ﺗﺒﺎﻳﻦ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﻧﺠﺪ ⎥⎧ ⎢ l − 1 ⎫2 ⎢ V ⎡⎣ en ( l )⎤⎦ = σ 2 ⎨1 + ⎬ ) (1 − Θ ⎥ ⎦ ⎩ ⎣ 12 ⎭
ﺣﻴﺚ ⎦⎥ ⎢⎣ xﺗﻌﻨﻲ اﻟﺠﺰء اﻟﺼﺤﻴﺢ ﻣﻦ . x -٢داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج : SARIMA(٠،١،١)(٠،١،١)١٢ وﻳﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞ
(1 − B ) (1 − B12 ) zt = (1 − θ B ) (1 − ΘB12 ) at
١١٧
ﻣﻦ اﻟﻤﻌﺎدﻟﺔ اﻟﻔﺮوﻗﻴﺔ
zn + l = zn +l−1 + zn + l−12 − zn + l−13 + an + l − θ an +l−1 − Θan + l−12 + θ Θan+ l−13
ﻳﻤﻜﻦ اﻟﺤﺼﻮل ﻋﻠﻰ اﻟﺘﻨﺒﺆات آﺎﻟﺘﺎﻟﻲ
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12 zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11 M
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1 zn (13) = zn (12 ) + zn (1) − zn + θ Θan )zn ( l ) = zn ( l − 1) + zn ( l − 12 ) − zn ( l − 13
وهﻜﺬا ﺑﻘﻴﻢ أوﻟﻴﺔ
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12 zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11
M
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1 zn (13) = zn (12 ) + zn (1) − zn + θ Θan
وﻋﻼﻗﺔ ﺗﻜﺮارﻳﺔ
zn ( l ) = zn ( l − 1) + zn ( l − 12 ) − zn ( l − 13) , l > 13
ﻳﻤﻜﻦ ﺗﻮﻟﻴﺪ اﻟﻌﺪد اﻟﻤﻄﻠﻮب ﻣﻦ اﻟﺘﻨﺒﺆات. أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ﻟﺒﻌﺾ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ: ﻣﺜﺎل ) : (١ﺳﻮف ﻧﺤﺎول إﻳﺠﺎد ﻧﻤﻮذج ﻣﻦ ﻋﺎﺋﻠﺔ SARIMAﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ: 653 736 615 688 775 845 740 800 883 947 817
566 770 653 645 696 871 783 763 773 969 858
600 705 702 667 717 805 819 800 821 902 896
582 688 756 661 698 785 859 790 805 889 937
553 618 782 697 660 690 886 834 766 778 966
577 628 722 735 687 734 824 869 810 828 903
568 598 709 798 681 711 807 913 809 802 892
599 565 622 811 722 677 707 942 855 760 782
واﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻠﻤﺸﺎهﺪات
١١٨
640 587 658 755 767 706 750 878 894 791 834
697 583 634 736 817 701 734 860 935 784 813
727 617 594 635 837 740 690 756 961 837 773
656 660 611 677 784 783 723 804 900 881 812
640 716 604 635 762 826 725 751 890 924 815 843
561 742 639 602 667 858 764 711 799 957 867 797
)z(t 589 673 678 621 713 796 801 747 826 898 908 827
1000
900
)z(t
800
700
600
150
50
100
In d e x
ﻳﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ ﻟﺬﻟﻚ ﻧﺜﺒﺖ اﻟﺘﺒﺎﻳﻦ أوﻻ ﺑﺘﺤﻮﻳﻞ ﻟﻮﻏﺎرﺛﻤﻲ أي ) yt = ln ( ztوﻧﺮﺳﻢ اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻬﺎ 6 .9 6 .8 6 .7
)y(t
6 .6 6 .5 6 .4 6 .3
1 5 0
5 0
1 0 0
In d e x
ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﺳﺘﻘﺮت ﻓﻲ اﻟﺘﺒﺎﻳﻦ وﻟﻜﻦ ﻻﺗﺰال ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ﻟﺬﻟﻚ ﻧﺄﺧﺬ اﻟﻔﺮق اﻷول ) xt = (1 − B ) yt = (1 − B ) ln ( ztوﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ 0 .1 5 0 .1 0
0 .0 0
)y(t)-y(t-1
0 .0 5
-0 .0 5 -0 .1 0 1 5 0
1 0 0
5 0
In d e x
اﻟﻤﺘﺴﻠﺴﻠﺔ اﻵن ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ .ﻟﻨﻨﻈﺮ إﻟﻰ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ
١١٩
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
10
20
30
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
0 .0 1 0 .2 5 - 0 .0 8 - 0 .3 7 - 0 .0 6 - 0 .5 0 - 0 .0 4 - 0 .3 5 - 0 .0 5 0 .2 3 0 .0 1 0 .9 0
0 .1 2 3 .2 3 - 1 .0 1 - 4 .4 8 - 0 .6 8 - 5 .4 1 - 0 .3 3 - 3 .3 0 - 0 .4 3 2 .0 1 0 .1 1 7 .7 4
0 .0 2 1 0 .7 0 1 1 .8 8 3 5 .6 0 3 6 .2 8 7 9 .9 7 8 0 .1 9 1 0 2 .3 7 1 0 2 .8 0 1 1 2 .2 9 1 1 2 .3 2 2 6 1 .4 1
13 14 15 16 17 18 19 20 21 22 23 24
0 .0 2 0 .2 3 - 0 .0 7 - 0 .3 4 - 0 .0 6 - 0 .4 6 - 0 .0 3 - 0 .3 2 - 0 .0 5 0 .2 1 0 .0 1 0 .8 2
0 .1 2 1 .5 1 - 0 .4 7 - 2 .1 8 - 0 .3 9 - 2 .8 8 - 0 .2 0 - 1 .9 3 - 0 .2 6 1 .2 0 0 .0 8 4 .7 6
2 6 1 .4 7 2 7 1 .2 9 2 7 2 .3 0 2 9 3 .7 3 2 9 4 .4 5 3 3 4 .6 7 3 3 4 .8 8 3 5 4 .9 6 3 5 5 .3 6 3 6 3 .5 5 3 6 3 .5 8 4 9 7 .3 0
25 26 27 28 29 30 31 32 33 34 35 36
0 .0 2 0 .2 1 - 0 .0 6 - 0 .3 1 - 0 .0 6 - 0 .4 2 - 0 .0 2 - 0 .3 0 - 0 .0 5 0 .1 8 0 .0 0 0 .7 6
0 .0 8 1 .1 0 - 0 .2 9 - 1 .5 7 - 0 .2 9 - 2 .0 9 - 0 .1 2 - 1 .4 7 - 0 .2 4 0 .8 9 0 .0 2 3 .6 3
4 9 7 .3 5 5 0 6 .5 0 5 0 7 .1 6 5 2 6 .4 5 5 2 7 .1 4 5 6 3 .0 5 5 6 3 .1 6 5 8 2 .0 5 5 8 2 .5 8 5 8 9 .8 4 5 8 9 .8 5 7 1 3 .1 4
40
L ag
T
LBQ
3 7 0 .0 1 0 .0 5 3 8 0 .2 0 0 .9 0 3 9 - 0 .0 4 - 0 .1 9 4 0 - 0 .2 7 - 1 .2 1 4 1 - 0 .0 5 - 0 .2 4
C o rr
7 1 3 .1 6 7 2 2 .0 6 7 2 2 .4 5 7 3 8 .8 6 7 3 9 .5 1
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t i o n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
10
20
30
40
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .0 1 0 .2 5 - 0 .0 9 - 0 .4 6 - 0 .0 2 - 0 .3 5 - 0 .1 4 - 0 .4 8 - 0 .4 1 - 0 .2 3 - 0 .5 8 0 .6 3
0 .1 2 3 .2 3 - 1 .2 0 - 5 .9 7 - 0 .2 2 - 4 .5 2 - 1 .8 2 - 6 .2 6 - 5 .3 6 - 2 .9 9 - 7 .5 5 8 .0 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 4 -0 .3 3 0 .0 0 0 .1 8 0 .0 1 0 .0 8 -0 .0 8 0 .0 8 0 .0 2 -0 .0 4 0 .0 2 0 .0 8
- 0 .4 8 - 4 .2 6 0 .0 6 2 .2 9 0 .0 7 1 .0 6 - 1 .0 4 0 .9 8 0 .2 9 - 0 .5 3 0 .2 9 1 .0 4
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 3 - 0 .0 5 0 .0 6 0 .0 0 - 0 .0 7 0 .0 3 0 .0 7 - 0 .0 1 - 0 .1 0 0 .0 6 0 .0 3 0 .0 1
-1 .6 7 -0 .6 1 0 .8 4 0 .0 5 -0 .9 4 0 .3 4 0 .9 3 -0 .0 9 -1 .3 1 0 .8 0 0 .3 9 0 .1 3
37 38 39 40 41
- 0 .0 4 0 .0 2 0 .0 4 0 .0 6 - 0 .0 2
- 0 .5 1 0 .3 0 0 .4 8 0 .8 1 - 0 .2 0
و١٢ ﻳﻼﺣﻆ ﻣﻦ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻣﻮﺳﻤﻴﺎ ﻷن ﻗﻴﻤﻬﺎ ﻋﻨﺪ اﻟﺘﺨﻠﻔﺎت wt = (1 − B12 ) (1 − B ) ln ( zt ) ﺗﺘﺨﺎﻣﺪ ﺑﺒﻂء ﻟﺬﻟﻚ ﻧﺄﺧﺬ اﻟﻔﺮق اﻟﻤﻮﺳﻤﻲ اﻷول٣٦ و٢٤ وﻧﺮﺳﻤﻬﺎ ﺑﻌﺪ هﺬا اﻟﺘﻔﺮﻳﻖ
y(t)-y(t-1)12
0 .0 5
0 .0 0
-0 .0 5 In d e x
50
100
150
ﻧﻮﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ
١٢٠
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
5
15
25
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .2 1 - 0 .0 1 0 .1 0 - 0 .1 3 - 0 .1 0 - 0 .0 2 0 .1 2 0 .0 5 - 0 .0 5 0 .1 3 - 0 .0 1 - 0 .4 4
- 2 .6 5 - 0 .1 3 1 .1 4 - 1 .4 9 - 1 .1 1 - 0 .2 8 1 .3 3 0 .5 4 - 0 .5 8 1 .4 5 - 0 .0 9 - 4 .9 0
7 .1 5 7 .1 7 8 .6 3 1 1 .2 0 1 2 .6 7 1 2 .7 7 1 4 .9 5 1 5 .3 1 1 5 .7 5 1 8 .4 9 1 8 .5 0 5 0 .8 8
13 14 15 16 17 18 19 20 21 22 23 24
0 .1 8 - 0 .0 7 - 0 .0 5 0 .0 3 0 .1 2 - 0 .0 0 - 0 .1 1 0 .0 3 - 0 .0 2 - 0 .0 9 0 .1 1 - 0 .0 4
1 .8 0 - 0 .7 0 - 0 .4 8 0 .2 6 1 .1 3 - 0 .0 2 - 1 .0 8 0 .2 5 - 0 .2 0 - 0 .8 0 0 .9 9 - 0 .3 8
5 6 .6 4 5 7 .5 4 5 7 .9 7 5 8 .1 0 6 0 .5 8 6 0 .5 8 6 2 .9 1 6 3 .0 4 6 3 .1 2 6 4 .4 5 6 6 .5 2 6 6 .8 2
25 26 27 28 29 30 31 32 33 34 35 36
0 .0 7 - 0 .0 0 - 0 .0 6 0 .0 3 - 0 .1 0 0 .0 1 0 .0 4 0 .0 0 0 .0 2 0 .0 0 0 .0 9 - 0 .0 6
0 .6 2 - 0 .0 2 - 0 .5 7 0 .3 0 - 0 .9 0 0 .0 8 0 .3 8 0 .0 4 0 .1 9 0 .0 3 0 .8 0 - 0 .5 1
6 7 .6 7 6 7 .6 7 6 8 .3 8 6 8 .5 8 7 0 .4 0 7 0 .4 1 7 0 .7 4 7 0 .7 4 7 0 .8 3 7 0 .8 3 7 2 .3 6 7 2 .9 9
35 L ag
T
LBQ
3 7 - 0 .0 6 - 0 .5 4 3 8 0 .0 3 0 .2 6
C o rr
7 3 .7 1 7 3 .8 8
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
15
25
35
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .2 1 - 0 .0 6 0 .0 8 - 0 .0 9 - 0 .1 5 - 0 .1 0 0 .1 1 0 .1 1 - 0 .0 4 0 .0 6 0 .0 5 - 0 .4 2
- 2 .6 5 - 0 .7 3 1 .0 6 - 1 .1 6 - 1 .8 2 - 1 .1 9 1 .4 1 1 .4 2 - 0 .5 1 0 .8 0 0 .5 9 - 5 .2 3
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 0 -0 .0 2 0 .0 0 -0 .1 1 0 .0 4 0 .0 1 -0 .0 4 -0 .0 3 -0 .0 3 0 .0 3 0 .0 7 -0 .2 7
- 0 .0 3 - 0 .2 5 0 .0 0 - 1 .4 0 0 .4 7 0 .1 2 - 0 .5 2 - 0 .3 5 - 0 .3 1 0 .3 2 0 .9 1 - 3 .4 1
25 26 27 28 29 30 31 32 33 34 35 36
0 .1 2 - 0 .0 1 - 0 .1 2 - 0 .0 7 0 .0 4 - 0 .0 4 - 0 .0 5 0 .0 2 - 0 .0 1 - 0 .0 2 0 .1 8 - 0 .2 1
1 .5 1 -0 .1 5 -1 .4 4 -0 .9 1 0 .4 4 -0 .5 3 -0 .6 0 0 .1 9 -0 .1 2 -0 .1 9 2 .2 1 -2 .5 7
37 38
0 .1 0 - 0 .0 6
1 .2 3 - 0 .7 9
wt = (1 − B12 ) (1 − B ) ln ( zt ) ﻣﻦ اﻧﻤﺎط دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
اي ﻧﻄﺒﻖ اﻟﻨﻤﻮذج٠ و٠ اﻟﻤﻤﻜﻨﺔ هﻲq وp ﻧﺠﺪ ان ﻗﻴﻢ
(1 − B ) (1 − B ) ln ( z ) = (1 − ΘB ) a 12
12
t
t
ﻳﻄﺒﻖ هﺬا اﻟﻨﻤﻮذجMINITAB اﻷﻣﺮ اﻟﺘﺎﻟﻲ ﻓﻲSARIMA(٠،١،٠)(٠،١،١)١٢ هﻮ ARIMA 0 1 ٠ 0 1 1 12 'y(t)' ; NoConstant. zt = e y وﻟﻠﺤﺼﻮل ﻋﻠﻰ اﻟﻨﺘﺎﺋﺞ اﻟﻨﻬﺎﺋﻴﺔ ﻧﺠﺮي اﻟﺘﺤﻮﻳﻞyt = ln ( zt ) ﻻﺣﻆ اﻧﻨﺎ اﺳﺘﺨﺪﻣﻨﺎ :اﻟﻨﺘﺎﺋﺞ t
MTB > Name c14 = 'RESI3' c15 = 'FITS3' MTB > ARIMA 0 1 0 0 1 1 12 'y(t)' 'RESI3' 'FITS3'; SUBC> NoConstant; SUBC> Forecast 24 c7 c8 c9; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model
١٢١
ARIMA model for y(t) Estimates at each iteration Iteration SSE Parameters 0 0.0228597 0.100 1 0.0204943 0.250 2 0.0187066 0.400 3 0.0174234 0.550 4 0.0169841 0.684 5 0.0169841 0.683 6 0.0169841 0.683 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev SMA 12 0.6831 0.0610
0.0010
T 11.20
Differencing: 1 regular, 1 seasonal of order 12 Number of observations: Original series 168, after differencing 155 Residuals: SS = 0.0165799 (backforecasts excluded) MS = 0.0001077 DF = 154 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 9.0(DF=11) 29.9(DF=23) 44.5(DF=35)
48 59.4(DF=47)
Forecasts from period 168 Period 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192
Forecast 6.76750 6.70901 6.83815 6.85381 6.92288 6.89349 6.84654 6.80008 6.74395 6.75028 6.70664 6.75999 6.79052 6.73203 6.86117 6.87684 6.94590 6.91651 6.86956 6.82310 6.76697 6.77330 6.72966 6.78301
95 Percent Limits Lower Upper 6.74716 6.78784 6.68024 6.73778 6.80292 6.87338 6.81313 6.89450 6.87739 6.96836 6.84366 6.94331 6.79272 6.90035 6.74255 6.85761 6.68293 6.80497 6.68596 6.81461 6.63918 6.77410 6.68952 6.83045 6.71514 6.86590 6.65203 6.81203 6.77680 6.94554 6.78832 6.96535 6.85342 7.03838 6.82023 7.01279 6.76962 6.96950 6.71963 6.92657 6.66009 6.87385 6.66312 6.88349 6.61627 6.84305 6.66649 6.89952
(1 − B ) (1 − B ) ln ( z ) = (1 − 0.683B ) a , 12
12
t
t
Actual
أي أن اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
at
N ( 0, 0.0001077 )
ﻻﺣﻆ ان
Θ = 0.683, s.e. ( Θ ) = 0.061, t = 11.2
.أي ان اﻟﻤﻌﻠﻢ ﻋﺎﻟﻲ اﻟﻤﻌﻨﻮﻳﺔ :ﻓﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ MTB > ZTest 0.0 0.0103778 'RESI3'; SUBC> Alternative 0.
١٢٢
Z-Test Test of mu = 0.000000 vs mu not = 0.000000 The assumed sigma = 0.0104 Variable RESI3
N Mean StDev 155 -0.000111 0.010375
SE Mean 0.000834
Z -0.13
P 0.89
اي ﻻﻧﺮﻓﺾ أن ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﺻﻔﺮا٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦP-value=٠٫٨٩ ﻻﺣﻆ ان اﻟـ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 0 'RESI3'.
Runs Test RESI3 K =
0.0000
The observed number of runs = 70 The expected number of runs = 78.1097 72 Observations above K 83 below The test is significant at 0.1893 Cannot reject at alpha = 0.05
اي اﻧﻨﺎ ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ٠٫١٨٩٣ اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ :إﺧﺘﺒﺎر إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ A C F o f R e s id u a ls f o r y ( t ) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 3
6
9
12
15
18
21
Lag
١٢٣
24
27
30
33
36
39
P A C F o f R e s id u a ls f o r y ( t ) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
.ﻧﻼﺣﻆ اﻧﻬﺎ ﺗﻌﻄﻲ اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء أي اﻧﻬﺎ ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ وإذا آﺎﻧﺖ ﻃﺒﻴﻌﻴﺔ ﻓﻬﻲ ﻣﺴﺘﻘﻠﺔ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Histogram of the Residuals (response is y(t))
Frequency
30
20
10
0 -0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
Residual
Normal Probability Plot of the Residuals (response is y(t)) 0.04 0.03 0.02
Residual
Partial Autocorrelation
0 .8
0.01 0.00 -0.01 -0.02 -0.03 -3
-2
-1
0
1
2
Normal Score
١٢٤
3
K-S test for Residuals
.999 .99 .95
.50 .20
Probability
.80
.05 .01 .001 0.03
0.02
0.01
-0.01
0.00
-0.02
-0.03
RESI3 Average: -0.0001115 StDev: 0.0103754 N: 155
Kolmogorov-Smirnov Normality Test D+: 0.074 D-: 0.045 D : 0.074 Approximate P-Value: 0.041
ﻻﺣﻆ ان اﻟـ P-valueﻹﺧﺘﺒﺎر K-Sﻳﻌﻄﻲ ٠٫٠٤١وهﻲ اﻗﻞ ﻣﻦ ٠٫٠٥اذا اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ α = 0.05اي ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ. اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻨﻤﻮذج: ﻓﻲ اﻟﻤﺨﺮﺟﺎت اﻟﺴﺎﺑﻘﺔ ﻗﻤﻨﺎ ﺑﺎﻟﺘﻨﺒﺆ ﻋﻦ ٢٤ﻗﻴﻤﺔ ﻣﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ ٩٥%ﻓﺘﺮات ﺗﻨﺒﺆ وﻧﺮﺳﻤﻬﺎ ﺑﺎﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ: 1150
1050
850
750
25
20
10
15
T im e
واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻣﻊ اﻟﺘﻨﺒﺆات وﺣﺪود اﻟﺘﻨﺒﺆ
١٢٥
5
0
Forecast
950
1150
1050
850 750
Forecast
950
650 550 200
0
100
Time
ﺣﺎﻟﺔ دراﺳﺔ : ١ ﺳﻮف ﻧﺤﺎول إﻳﺠﺎد ﻧﻤﻮذج ﻣﻦ ﻋﺎﺋﻠﺔ SARIMAﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ: 57.4 60.0 69.8 70.1 66.1 58.7 55.8 51.6 53.5 53.0 52.1 53.6 55.6 62.6 75.5 76.4
61.5 71.0 71.0 67.5 56.1 55.4 55.3 52.4 53.4 53.5 53.4 55.3 60.1 70.3 73.3 70.8
72.7 70.6 67.4 58.0 53.6 52.9 53.2 52.1 53.6 54.3 55.3 58.5 68.9 69.4 68.1 60.6
72.2 68.2 58.2 54.2 53.0 53.0 52.8 52.6 55.1 56.3 58.2 69.3 68.8 64.7 58.1 56.4
71.5 57.7 54.3 53.2 52.8 52.7 52.8 53.9 57.3 59.4 66.9 69.6 63.6 57.1 54.8 55.6
59.1 54.6 53.4 53.0 52.6 53.4 54.5 56.4 61.9 70.4 70.7 64.2 57.1 53.9 54.3 55.0
57.2 54.9 53.0 53.0 52.9 54.9 56.4 61.7 69.9 69.9 65.3 55.5 52.2 53.5 54.6 54.7
56.3 54.9 52.8 53.4 54.0 57.5 59.3 68.3 71.5 61.0 56.5 53.3 51.5 53.5 54.2 55.8
55.7 55.3 54.4 55.6 56.6 69.3 70.0 65.3 57.3 53.9 52.5 51.5 51.5 53.3 55.8 60.5 62.2
55.8 54.9 53.3 54.6 54.9 61.2 68.7 67.9 65.1 52.7 53.4 52.4 51.7 53.1 54.8 57.7
اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ
7 0
)z(t 6 0
5 0 1 5 0
5 0
1 0 0
١٢٦
In d e x
)z(t 56.3 56.9 56.0 59.4 68.2 69.8 67.9 58.2 52.7 53.3 53.2 51.5 52.3 53.9 57.9 66.4 74.8
ﻧﺠﺪwt = (1 − B ) zt ﺑﺄﺧﺬ اﻟﻔﺮق اﻷول ﻹﺳﺘﻘﺮار اﻟﻤﺘﻮﺳﻂ
y(t)
1 0
0
-1 0
5 0
In d e x
1 0 0
1 5 0
وﻟﻬﺎ دوال ﺗﺮاﺑﻂ ذاﺗﻲ وﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ Autocorrelation
Autocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
0.47 -0.02 -0.37 -0.28 -0.21 -0.19 -0.21 -0.27 -0.32 -0.00 0.46 0.86
6.32 -0.22 -4.10 -2.82 -2.00 -1.77 -1.92 -2.46 -2.85 -0.02 3.91 6.71
40.60 40.67 65.83 80.10 87.92 94.41 102.29 115.89 135.47 135.48 176.37 318.37
13 14 15 16 17 18 19 20 21 22 23 24
0.43 -0.01 -0.33 -0.25 -0.19 -0.17 -0.19 -0.25 -0.31 -0.01 0.42 0.79
2.76 -0.03 -2.00 -1.49 -1.12 -0.97 -1.09 -1.42 -1.76 -0.06 2.35 4.28
354.69 354.69 375.77 388.04 395.15 400.68 407.71 419.89 439.21 439.24 475.10 602.75
25 26 27 28 29 30 31 32 33 34 35 36
0.41 -0.00 -0.29 -0.23 -0.17 -0.16 -0.17 -0.23 -0.28 -0.01 0.37 0.72
2.06 -0.01 -1.43 -1.08 -0.82 -0.78 -0.83 -1.06 -1.31 -0.06 1.70 3.27
638.60 638.60 656.89 667.77 674.14 679.93 686.56 697.63 714.89 714.93 745.11 860.42
37 38 39 40 41 42 43 44
0.40 0.02 -0.27 -0.21 -0.17 -0.15 -0.16 -0.20
1.72 0.07 -1.14 -0.89 -0.71 -0.61 -0.65 -0.83
896.48 896.54 913.29 923.75 930.47 935.55 941.37 950.82
2
12
22
32
42
Partial Autocorrelation
Partial Autocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
Lag PAC
T
0.47 -0.32 -0.30 0.09 -0.23 -0.25 -0.16 -0.43 -0.64 -0.31 -0.16 0.41
6.32 -4.21 -4.00 1.22 -3.05 -3.33 -2.09 -5.67 -8.56 -4.11 -2.18 5.42
1 2 3 4 5 6 7 8 9 10 11 12
Lag PAC 13 14 15 16 17 18 19 20 21 22 23 24
-0.27 -0.01 0.01 -0.07 -0.01 0.02 0.03 0.07 -0.12 -0.02 -0.06 0.13
22 T -3.62 -0.17 0.13 -0.95 -0.16 0.24 0.44 0.89 -1.62 -0.26 -0.86 1.74
32
Lag PAC 25 26 27 28 29 30 31 32 33 34 35 36
-0.02 -0.06 0.07 -0.00 -0.02 -0.04 0.01 0.03 -0.02 0.00 -0.09 0.05
T -0.33 -0.74 0.99 -0.05 -0.26 -0.60 0.20 0.37 -0.25 0.06 -1.17 0.61
42
Lag PAC 37 38 39 40 41 42 43 44
0.03 -0.03 -0.01 0.02 -0.04 0.03 -0.00 0.03
T 0.38 -0.35 -0.17 0.23 -0.56 0.35 -0.00 0.43
وﻳﻨﺘﺞ اﻟﻤﺘﺴﻠﺴﻠﺔwt = (1 − B12 ) (1 − B ) zt أي١٢ ﻧﺮى اﻧﻬﺎ ﺗﺤﺘﺎج إﻟﻰ ﺗﻔﺮﻳﻖ ﻣﻮﺳﻤﻲ ﻣﻦ اﻟﺮﺗﺒﺔ اﻟﺘﺎﻟﻴﺔ
١٢٧
w(t)
5
0
-5 In d e x
50
100
150
وﻟﻬﺎ دوال ﺗﺮاﺑﻂ ذاﺗﻲ وﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r w ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
10
20
30
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
-0 .0 1 -0 .1 9 -0 .2 4 -0 .0 0 0 .0 0 -0 .0 4 -0 .0 4 -0 .0 3 0 .2 2 0 .1 0 0 .0 8 -0 .4 2
- 0 .1 9 - 2 .4 8 - 2 .9 7 - 0 .0 3 0 .0 3 - 0 .5 0 - 0 .4 9 - 0 .3 1 2 .5 6 1 .1 6 0 .9 3 - 4 .6 3
0 .0 4 6 .3 1 1 6 .1 0 1 6 .1 0 1 6 .1 0 1 6 .4 1 1 6 .7 1 1 6 .8 3 2 5 .2 6 2 7 .1 3 2 8 .3 7 5 9 .4 0
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 8 0 .1 0 0 .1 2 -0 .0 2 -0 .0 3 0 .1 0 0 .0 7 -0 .0 1 -0 .2 0 -0 .0 2 0 .1 0 0 .1 4
-0 .8 4 0 .9 8 1 .1 5 -0 .1 7 -0 .2 8 1 .0 1 0 .7 0 -0 .0 5 -1 .9 4 -0 .2 3 0 .9 5 1 .3 0
6 0 .6 9 6 2 .5 0 6 5 .0 1 6 5 .0 7 6 5 .2 2 6 7 .2 4 6 8 .2 3 6 8 .2 4 7 5 .9 4 7 6 .0 5 7 8 .0 2 8 1 .7 8
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .0 8 - 0 .1 3 - 0 .0 1 0 .0 6 0 .1 2 - 0 .1 5 - 0 .0 8 - 0 .0 3 0 .1 9 0 .1 7 - 0 .1 0 - 0 .2 2
-0 .7 6 -1 .1 8 -0 .1 3 0 .5 6 1 .0 9 -1 .3 6 -0 .7 2 -0 .2 8 1 .6 6 1 .4 9 -0 .8 6 -1 .9 2
8 3 .0 9 8 6 .2 8 8 6 .3 3 8 7 .0 7 8 9 .9 4 9 4 .4 7 9 5 .8 0 9 6 .0 1 1 0 3 .2 0 1 0 9 .1 9 1 1 1 .2 4 1 2 1 .7 7
40
L ag
T
LBQ
3 7 0 .0 5 0 .4 6 3 8 0 .1 0 0 .8 4 3 9 0 .0 2 0 .1 5 4 0 -0 .0 2 - 0 .1 6 4 1 -0 .1 2 - 1 .0 1
C o rr
1 2 2 .4 0 1 2 4 .5 2 1 2 4 .5 9 1 2 4 .6 7 1 2 7 .8 4
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r w ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
10
20
30
40
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .0 1 - 0 .1 9 - 0 .2 6 - 0 .0 7 - 0 .1 1 - 0 .1 4 - 0 .1 1 - 0 .1 3 0 .1 4 0 .0 7 0 .1 7 - 0 .3 2
- 0 .1 9 - 2 .4 8 - 3 .2 8 - 0 .8 5 - 1 .4 3 - 1 .8 4 - 1 .4 6 - 1 .6 6 1 .7 6 0 .9 5 2 .1 9 - 4 .0 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 5 0 .0 1 -0 .0 4 -0 .0 3 -0 .0 2 0 .0 6 0 .0 5 -0 .0 3 -0 .0 5 0 .0 5 0 .1 9 0 .0 1
- 0 .6 9 0 .1 5 - 0 .5 0 - 0 .3 3 - 0 .2 3 0 .7 8 0 .6 2 - 0 .4 2 - 0 .6 8 0 .6 3 2 .5 0 0 .0 9
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 2 - 0 .0 8 - 0 .0 3 - 0 .0 3 0 .0 7 - 0 .1 1 - 0 .0 2 - 0 .1 0 - 0 .0 1 0 .1 8 0 .0 7 - 0 .0 9
-1 .4 9 -0 .9 7 -0 .3 2 -0 .3 6 0 .8 7 -1 .3 9 -0 .2 9 -1 .2 9 -0 .0 9 2 .3 1 0 .9 5 -1 .1 8
37 38 39 40 41
0 .0 2 - 0 .0 8 0 .0 3 0 .0 5 - 0 .0 3
0 .2 4 - 1 .0 6 0 .4 4 0 .6 3 - 0 .3 6
ﻣﻦ اﻷﻧﻤﺎط اﻟﻤﺸﺎهﺪة ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻗﺪ ﻳﻜﻮن اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ هﻮ أيSARIMA(١،١،١)(٠،١،١)١٢
١٢٨
(1 − φ B ) (1 − B12 ) (1 − B ) zt = (1 − θ B ) (1 − ΘB12 ) at :ﻧﻄﺒﻖ هﺬا اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎﻟﺘﺎﻟﻲ MTB > ARIMA 1 1 1 0 1 1 12 'z(t)' 'RESI2'; SUBC> NoConstant; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 307.653 0.100 0.100 1 281.217 0.100 0.100 2 262.275 0.226 0.231 3 262.027 0.376 0.381 4 261.770 0.526 0.531 5 261.426 0.675 0.681 6 260.905 0.824 0.831 7 260.036 0.970 0.981 8 227.926 0.835 0.980 9 221.838 0.748 0.980 10 221.665 0.738 0.980 11 221.637 0.738 0.980 12 221.610 0.737 0.980 13 221.585 0.737 0.980 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.7374 0.0620 MA 1 0.9796 0.0017 SMA 12 0.5898 0.0736
0.100 0.250 0.400 0.401 0.401 0.402 0.403 0.405 0.536 0.576 0.586 0.589 0.589 0.590 0.0010
T 11.89 582.86 8.01
Differencing: 1 regular, 1 seasonal of order 12 Number of observations: Original series 178, after differencing 165 Residuals: SS = 214.393 (backforecasts excluded) MS = 1.323 DF = 162 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 15.7(DF= 9) 30.9(DF=21) 61.6(DF=33)
48 67.1(DF=45)
أي أن اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
(1 − 0.74 B ) (1 − B ) (1 − B ) zt = (1 − 0.98B ) (1 − 0.59 B12 ) at , 12
at
N ( 0,1.323)
ﻻﺣﻆ ان
φ = 0.74, s.e. (φ ) = 0.062, t = 11.89 θ = 0.96, s.e. (θ ) = 0.0017, t = 582.86 Θ = 0.59, s.e. ( Θ ) = 0.074, t = 8.01
.أي ان اﻟﻤﻌﺎﻟﻢ ﻋﺎﻟﻴﺔ اﻟﻤﻌﻨﻮﻳﺔ :ﻓﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ MTB > ZTest 0.0 1.15 'RESI1'; SUBC> Alternative 0.
Z-Test
١٢٩
Test of mu = 0.0000 vs mu not = 0.0000 The assumed sigma = 1.15 Variable RESI1
N 165
Mean -0.0144
StDev 1.1433
SE Mean 0.0895
Z -0.16
P 0.87
اي ﻻﻧﺮﻓﺾ أن ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﺻﻔﺮا٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦP-value=٠٫٨٧ ﻻﺣﻆ ان اﻟـ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 67 The expected number of runs = 82.4061 73 Observations above K 92 below The test is significant at 0.0149
اي اﻧﻨﺎ ﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ وهﺬا ﻳﺤﺘﺎج إﻟﻰ إﺟﺮاء٠٫٠٥ اﻹﺧﺘﺒﺎر ﻏﻴﺮﻣﻌﻨﻮي ﻋﻨﺪ : ﻋﻠﻰ اﻟﻮﺳﻴﻂ اﻟﺘﺎﻟﻲSign Test إﺧﺘﺒﺎر ﺁﺧﺮ أآﺜﺮ ﻗﻮة ﻣﻦ إﺧﺘﺒﺎر اﻟﺠﺮي ﻣﺜﻞ إﺧﺘﺒﺎر اﻹﺷﺎرة MTB > STest 0.0 'RESI1'; SUBC> Alternative 0.
Sign Test for Median Sign test of median = 0.00000 versus RESI1
N 165
N* 13
Below 92
Equal 0
not = Above 73
0.00000 P 0.1611
Median -0.08139
وﻟﻠﺘﺄآﺪ ﻧﺠﺮي إﺧﺘﺒﺎر وﻟﻜﻮآﺴﻮن ﻹﺷﺎرات اﻟﺮﺗﺐ ﻋﻠﻰ اﻟﻮﺳﻴﻂ٠٫١٦١١ واﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ اﻟﺘﺎﻟﻲ MTB > WTest 0.0 'RESI1'; SUBC> Alternative 0.
Wilcoxon Signed Rank Test Test of median = 0.000000 versus median not = 0.000000
RESI1
N 165
Number Missing 13
N for Test 165
Wilcoxon Statistic 6321.0
P 0.392
Estimated Median -0.05940
٠٫٣٩٢ واﻹﺧﺘﺒﺎر اﻳﻀﺎ ﻣﻌﻨﻮي ﻋﻨﺪ :إﺧﺘﺒﺎر إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
١٣٠
A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
36
39
Lag
P A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
3
6
9
12
15
18
21
24
27
30
33
Lag
.ﻧﻼﺣﻆ اﻧﻬﺎ ﺗﻌﻄﻲ اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء أي اﻧﻬﺎ ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ وإذا آﺎﻧﺖ ﻃﺒﻴﻌﻴﺔ ﻓﻬﻲ ﻣﺴﺘﻘﻠﺔ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Histogram of the Residuals (response is z(t)) 50
Frequency
40
30
20
10
0 -6
-5
-4
-3
-2
-1
0
1
2
Residual
١٣١
3
4
Normal Probability Plot of the Residuals (response is z(t)) 4 3 2
Residual
1 0 -1 -2 -3 -4 -5 -3
-2
-1
0
1
2
3
Normal Score
K-S Test for Residuals
.999
Probability
.99 .95 .80 .50 .20 .05 .01 .001 -5
-4
-3
-2
-1
0
1
2
3
RESI1 Average: -0.0144171 StDev: 1.14327 N: 165
Kolmogorov-Smirnov Normality Test D+: 0.117 D-: 0.140 D : 0.140 Approximate P-Value < 0.01
ايα = 0.05 اذا اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ٠٫٠١ أﻗﻞ ﻣﻦK-S ﻹﺧﺘﺒﺎرP-value ﻻﺣﻆ ان اﻟـ .ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ :اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻨﻤﻮذج ﻓﺘﺮات ﺗﻨﺒﺆ٩٥% ﻗﻴﻤﺔ ﻣﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ٣٦ ﺳﻨﻘﻮم ﺑﺎﻟﺘﻨﺒﺆ ﻋﻦ Forecasts from period 178 Period 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193
Forecast 57.7885 55.8516 54.7429 54.1820 54.6298 54.9152 55.6640 59.3778 68.6924 74.0698 73.9650 63.5866 58.9095 56.7768 55.5237
95 Percent Limits Lower Upper 55.5332 60.0437 53.0220 58.6812 51.6264 57.8594 50.9063 57.4578 51.2601 57.9994 51.4875 58.3430 52.1986 59.1294 55.8869 62.8688 65.1833 72.2015 70.5472 77.5925 70.4317 77.4983 60.0447 67.1286 55.1843 62.6347 52.9407 60.6128 51.6169 59.4305
١٣٢
Actual
50.9022 51.2380 51.4409 52.1278 55.7947 65.0729 70.4217 70.2940 59.8968 55.0418 52.7962 51.4676 50.7472 51.0774 51.2749 51.9568 55.6189 64.8927 70.2374 70.1057 59.7046
58.8105 59.2131 59.4653 60.1905 63.8885 73.1929 78.5647 78.4575 68.0793 63.5664 61.5364 60.3514 59.7315 60.1357 60.3903 61.1184 64.8194 74.1270 79.5021 79.3983 69.0235
194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214
54.8564 55.2256 55.4531 56.1592 59.8416 69.1329 74.4932 74.3758 63.9881 59.3041 57.1663 55.9095 55.2394 55.6066 55.8326 56.5376 60.2191 69.5099 74.8697 74.7520 64.3640
وﻧﺮﺳﻤﻬﺎ ﺑﺎﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ: 8 0
Forecast
7 0
6 0
5 0
3 0
4 0
2 0
0
1 0
T im e
اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﺑﻜﺎﻣﻠﻬﺎ ﻣﻊ اﻟﺘﻨﺒﺆات وﻓﺘﺮات اﻟﺘﻨﺒﺆ
80
60
50
200
100
T im e
١٣٣
0
Forecast
70
١٣٤
اﻟﻔﺼﻞ اﻟﺴﺎﺑﻊ ورﻗﺔ ﺗﺪرﻳﺐ ﻋﻤﻠﻲ ﻋﻠﻲ اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك-اﻹﻧﺤﺪار اﻟﺬاﺗﻲ Forecasting By ARMA Models اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺴﺠﻠﺔ ﻋﻠﻲ ﺷﻜﻞ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ 28.0 28.0 29.5
23.5 30.5 32.3
21.3 29.6 30.6
24.5 36.5 26.5
23.5 36.5 20.7 16.4
15.3 25.0 16.0 23.4
15.5 25.3 19.0 26.4
21.0 17.3 19.7 32.2
12.0 24.0 26.0 28.3
20.5 15.5 21.5 31.3
وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: ;MTB > TSPlot C1 >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
30
C 1 20
10 30
10
20
اوﻻ ﻧﻮﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
١٣٥
Index
MTB > %ACF C1.
Autocorrelation
Autocorrelation Function for C1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag
Corr
T
LBQ
1 2 3 4 5 6 7
0.63 0.30 0.14 -0.05 -0.24 -0.30 -0.24
3.79 1.35 0.59 -0.20 -1.01 -1.22 -0.94
15.62 19.27 20.07 20.17 22.71 26.71 29.40
5
Lag
6
Corr
7
T
LBQ
8 -0.20 -0.78 9 -0.12 -0.44
31.42 32.10
8
9
MTB > %PACF C1. Executing from file: E:\MTBWIN\MACROS\PACF.MAC
Partial Autocorrelation
Partial Autocorrelation Function for C1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag PAC 1 2 3 4 5 6 7
0.63 -0.16 0.04 -0.20 -0.18 -0.04 0.02
5
6
T
Lag PAC
T
3.79 -0.98 0.22 -1.20 -1.09 -0.23 0.12
8 -0.07 9 0.05
-0.39 0.29
7
8
9
ARMA (1,1) ﻧﻼﺣﻆ ﻣﻦ اﻧﻤﺎط اﻟﺪاﻟﺘﻴﻦ ان اﻟﻤﺸﺎهﺪات ﻗﺪ ﺗﻜﻮن ﻣﻦ ﻧﻤﻮذج ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح MTB > Name c17 = 'RESI1' MTB > ARIMA 1 0 1 C1 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c14 c15 c16; SUBC> GACF; SUBC> GPACF; SUBC> GNormalplot.
١٣٦
ARIMA Model ARIMA model for C1 Estimates at each iteration Iteration SSE Parameters 0 1337.71 0.100 0.100 21.918 1 936.95 0.250 -0.049 18.193 2 849.78 0.211 -0.199 19.106 3 751.53 0.215 -0.349 18.941 4 658.66 0.266 -0.499 17.594 5 592.30 0.372 -0.649 14.890 6 580.80 0.433 -0.699 13.314 7 579.30 0.455 -0.714 12.698 8 579.11 0.464 -0.719 12.470 9 579.08 0.467 -0.721 12.386 10 579.08 0.468 -0.722 12.356 11 579.08 0.468 -0.722 12.345 Relative change in each estimate less than 0.0010 Final Estimates of Parameters Type Coef StDev AR 1 0.4684 0.1755 MA 1 -0.7221 0.1380 Constant 12.345 1.154 Mean 23.221 2.170
T 2.67 -5.23 10.70
Number of observations: 36 Residuals: SS = 523.365 (backforecasts excluded) MS = 15.860 DF = 33 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 7.2(DF=10) 15.9(DF=22) * (DF= *) * (DF= *) Forecasts from period 36 Period 37 38 39 40 41
Forecast 14.7649 19.2606 21.3663 22.3524 22.8143
95 Percent Limits Lower Upper 6.9578 22.5720 7.1228 31.3985 8.4715 34.2610 9.2975 35.4074 9.7245 35.9041
zt = 12.345 + 0.4684 zt −1 + at − 0.7221at −1 , at ﺣﻴﺚ φˆ1 = 0.4684 se φˆ1 = 0.1755 t = 2.67
Actual
:اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ WN ( 0,15.86 ) ∀t
( ) θˆ = −0.7221 se (θˆ ) = 0.1380 t = −5.23 δˆ = 12.345 se (δˆ ) = 1.154 t = 10.70 1
1
σˆ 2 = 15.86 df = 33 ﻓﻤﺜﻼ اﻟﻔﺮﺿﻴﺔα = 0.05 وﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻘﺪرات ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪ
١٣٧
H 0 : φ1 = 0 H1 : φ1 ≠ 0 أيα = 0.05 وهﻲ ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪt =
0.4684 φˆ1 = = 2.6689 ﻧﺨﺘﺒﺮهﺎ ﺑﺎﻹﺣﺼﺎﺋﺔ 0.1755 se φˆ1
( )
وﺑﺎﻟﻤﺜﻞ ﻟﺠﻤﻴﻊ اﻟﻘﺪرات اﻻﺧﺮىφ1 = 0 اﻧﻨﺎ ﻧﺮﻓﺾ ان
ﺛﺎﻧﻴﺎ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ :إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ اﻹﺧﺘﺒﺎر هﻮ
H 0 : µa = 0, H1 : µa ≠ 0
MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean Test of mu = 0.000 vs mu not = 0.000 Variable RESI1
N 36
Mean 0.344
StDev 3.851
SE Mean 0.642
T 0.54
P 0.60
أي ان اﻹﺧﺘﺒﺎرα = 0.05 وهﻲ اآﺒﺮ ﻣﻦ0.6 ﻟﻬﺎ هﻲP-Value واﻟـt = 0.54 ﻻﺣﻆ ان ﻏﻴﺮ ﻣﻌﻨﻮي أي ﻳﻤﻜﻦ إﻋﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﻳﺴﺎوي اﻟﺼﻔﺮ :إﺧﺘﺒﺎرﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ وﻧﺴﺘﺨﺪم ﻟﺬﻟﻚ إﺧﺘﺒﺎر اﻟﺠﺮيRuns Test MTB > Runs 'RESI1'.
Runs Test RESI1 K =
0.3443
The observed number of runs = 21 The expected number of runs = 19.0000 18 Observations above K 18 below The test is significant at 0.4989 Cannot reject at alpha = 0.05
α = 0.05 ﻻﻳﻤﻜﻨﻨﺎ رﻓﺾ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﻋﻨﺪ :إﺧﺘﺒﺎر ﺗﺮاﺑﻂ اﻟﺒﻮاﻗﻲ وﻧﺴﺘﺨﺪم ﻟﺬﻟﻚ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
ACF of Residuals for C1 (with 95% confidence limits for the autocorrelations) 1.0 0.8
orrelation
0.6 0.4 0.2 0.0
١٣٨
PACF of Residuals for C1 (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1
2
3
4
5
6
7
8
9
Lag
ﻧﻼﺣﻆ اﻧﻪ ﻻﻳﻮﺟﺪ أي ﺗﺮاﺑﻂ ﻣﻦ أي درﺟﺔ ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﺨﺘﻠﻔﺔ ﻟﻠﺒﻮاﻗﻲ أي اﻧﻬﺎ ﺗﻈﻬﺮ اﻧﻤﺎط ﺗﺘﻤﺸﻲ ﻣﻊ آﻮﻧﻬﺎ ﻣﺘﺴﻠﺴﺔ ﺿﺠﺔ ﺑﻴﻀﺎء Normal Probability Plot واﺧﻴﺮا ﻧﺨﺘﺒﺮ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﺎﻟـ
Normal Probability Plot of the Residuals (response is C1)
Residual
10
0
-10
-2
-1
0
Normal Score
١٣٩
1
2
( وهﻮ ﻣﻘﺒﻮل ) ﻧﻮﻋﺎ إذا ﻳﻤﻜﻦ إﻋﺘﺒﺎر اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻣﻨﺎﺳﺒﺎ ﺗﻨﺒﺆ ﻟﻬﺎ95% اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﺘﻨﺒﺆات ﻟﺨﻤﺴﺔ ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ وﻓﺘﺮات
MTB > TSPlot C14 C15 C16; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect; SUBC> overlay.
Time Series Plot for C1 (with forecasts and their 95% confidence limits)
36
C1
26
16
6
5
10
15
20
25
30
35
Time
:ﻣﺜﺎل ﺁﺧﺮ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺴﺠﻠﺔ ﻋﻠﻲ ﺷﻜﻞ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ 10.38 11.68 9.10 10.01 9.51 9.48 7.23 10.85 6.89
11.86 11.17 9.09 9.37 9.24 7.38 8.42 10.41 5.96
10.97 10.53 9.35 8.69 8.66 6.90 9.61 9.96 6.80
10.80 10.01 8.82 8.19 8.86 6.94 9.05 9.61 7.68
9.79 9.91 9.32 8.67 8.05 6.24 9.26 8.76 8.38
10.39 9.14 9.01 9.55 7.79 6.84 9.22 8.18 8.52
10.42 9.16 9.00 8.92 6.75 6.85 9.38 7.21 9.74
10.82 9.55 9.80 8.09 6.75 6.90 9.10 7.13 9.31
11.40 9.67 9.83 9.37 7.82 7.79 7.95 9.10 9.89
11.32 11.44 8.44 8.24 9.72 9.89 10.13 10.14 8.64 10.58 8.18 7.51 8.12 9.75 8.25 7.91 9.96
MTB > TSPlot C10; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
12 11
C 1 0
10 9 8
١٤٠
7 6 Index
10
20
30
40
50
60
70
80
90
ﻧﻔﺤﺺ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
Autocorrelation
Autocorrelation Function for C10 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
Lag Corr 1 2 3 4 5 6 7
0.83 0.61 0.46 0.37 0.33 0.28 0.26
T
12
LBQ
8.24 69.92 3.91 107.90 2.56 129.56 1.95 143.87 1.65 155.04 1.40 163.68 1.28 171.23
Lag Corr 8 9 10 11 12 13 14
0.26 0.26 0.18 0.09 0.04 0.03 0.04
T
LBQ
22
Lag Corr
1.26 178.83 1.21 186.14 0.84 189.86 0.43 190.87 0.20 191.09 0.13 191.19 0.19 191.39
T
LBQ
15 0.05 0.21 191.63 16 0.04 0.16 191.78 17 0.00 0.02 191.78 18 -0.03 -0.15 191.91 19 -0.05 -0.24 192.26 20 -0.05 -0.24 192.60 21 0.01 0.07 192.63
Lag Corr
T
LBQ
22 0.10 0.47 194.02 23 0.18 0.81 198.12 24 0.20 0.89 203.24
Partial Autocorrelation
Partial Autocorrelation Function for C10 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
Lag PAC
T
Lag PAC
T
1 0.83 2 -0.27 3 0.13 4 0.03 5 0.06 6 -0.02 7 0.09
8.24 -2.64 1.29 0.34 0.61 -0.21 0.91
8 0.05 9 0.00 10 -0.20 11 0.02 12 0.01 13 0.01 14 0.03
0.45 0.03 -1.98 0.19 0.09 0.12 0.34
22
Lag PAC 15 16 17 18 19 20 21
-0.01 -0.03 -0.07 -0.03 0.06 0.02 0.21
T
Lag PAC
T
-0.15 -0.25 -0.73 -0.26 0.60 0.20 2.03
22 0.05 23 0.06 24 -0.07
0.51 0.59 -0.65
AR ( 2 ) ﻧﻼﺣﻆ ان اﻷﻧﻤﺎط ﺗﻘﺘﺮح ﻧﻤﻮذج إﻧﺤﺪار ذاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح MTB > Name c17 = 'RESI1' MTB > ARIMA 2 0 0 C10 'RESI1'; SUBC> Constant;
١٤١
SUBC> SUBC> SUBC> SUBC> SUBC>
Forecast 5 c14 c15 c16; GSeries; GACF; GPACF; GNormalplot.
ARIMA Model ARIMA model for C10 Estimates at each iteration Iteration SSE Parameters 0 126.398 0.100 0.100 1 103.515 0.250 0.043 2 84.535 0.400 -0.014 3 69.407 0.550 -0.071 4 58.132 0.700 -0.128 5 50.724 0.850 -0.184 6 47.212 1.000 -0.239 7 46.918 1.053 -0.256 8 46.916 1.054 -0.255 9 46.916 1.054 -0.255 10 46.916 1.054 -0.255 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 1.0542 0.0992 AR 2 -0.2547 0.0993 Constant 1.81360 0.07092 Mean 9.0480 0.3538
7.283 6.434 5.586 4.738 3.887 3.030 2.163 1.838 1.816 1.814 1.814 0.0010
T 10.63 -2.56 25.57
Number of observations: 98 Residuals: SS = 46.7518 (backforecasts excluded) MS = 0.4921 DF = 95 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 7.2(DF=10) 13.7(DF=22) 21.3(DF=34) 28.8(DF=46) Forecasts from period 98 Period 99 100 101 102 103
Forecast 9.7950 9.6033 9.4432 9.3232 9.2375
95 Percent Limits Lower Upper 8.4198 11.1703 7.6050 11.6016 7.1234 11.7629 6.8446 11.8018 6.6825 11.7925
Actual
اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ zt = 1.8136 + 1.0542 zt −1 − 0.2547 zt −2 + at , at WN ( 0,0.4921) ∀t ﻟﻬﺎ هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻢ φˆ1 = 1.0542 se φˆ1 = 0.0992 t = 10.63
( ) φˆ = −0.2547 se (φˆ ) = 0.0993 t = −2.56 δˆ = 1.8136 se (δˆ ) = 0.07092 t = 25.57 2
2
σˆ 2 = 0.4921 df = 95
α = 0.05 ﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻘﺪرات ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪ ١٤٢
ﻟﻜﻲ ﻧﻘﺒﻞ ﺑﻬﺬا اﻟﻨﻤﻮذج ﻋﻠﻲ اﻧﻪ ﻣﻨﺎﺳﺐ ﻟﻠﺘﻨﺒﺆ ﻧﺠﺮي إﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean Test of mu = 0.0000 vs mu not = 0.0000 Variable RESI1
N 98
Mean StDev -0.0082 0.6942
SE Mean 0.0701
T -0.12
P 0.91
هﻲ إﺣﺘﻤﺎل ان اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔP-Value واﺿﺢ ﺟﺪا ان اﻷﺧﺘﺒﺎر ﻏﻴﺮ ﻣﻌﻨﻮي ) ﻣﻼﺣﻈﺔ اﻟـ ( ﺻﺤﻴﺤﺔ ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 'RESI1'.
Runs Test RESI1 K =
-0.0082
The observed number of runs = 47 The expected number of runs = 49.9184 47 Observations above K 51 below The test is significant at 0.5529 Cannot reject at alpha = 0.05
أي ﻻﻳﻤﻜﻨﻨﺎ رﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﻧﻔﺤﺺ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
ACF of Residuals for C10 (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
4
6
8
10
12
14
Lag
١٤٣
16
18
20
22
24
PACF of Residuals for C10 (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
4
6
8
10
12
14
16
18
20
22
24
Lag
واﺿﺢ ﺟﺪا اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء ﻳﺒﻘﻲ ﻓﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Normal Probability Plot of the Residuals (response is C10) 2
Residual
1
0
-1
-2 -3
( at
-2
-1
0
1
Normal Score
2
3
IIDN ( 0,0.4921) وﻧﺴﺘﻄﻴﻊ ان ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ) أي ﺗﻨﺒﺆ ﻟﻬﺎ95% اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﺘﻨﺒﺆات ﻟﺨﻤﺴﺔ ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ وﻓﺘﺮات
MTB > TSPlot C14 C15 C16; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect; SUBC> overlay.
Time Series Plot for C10 (with forecasts and their 95% confidence limits) 12 11
C10
10 9 8 7 6
١٤٤
ﺗﻤﺮﻳﻦ :ﻃﺒﻖ ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻧﻤﻮذج ) AR (1وﻗﺎرن ﺑﻴﻦ اﻟﻨﺘﺎﺋﺞ
١٤٥
١٤٦
اﻟﻔﺼﻞ اﻟﺜﺎﻣﻦ ﻣﺜﺎل ﻋﻠﻰ ﺗﺤﻠﻴﻞ اﻟﺒﻮاﻗﻲ وﻣﻌﺎﻳﻴﺮ إﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ Example on Residual Analysis and Model Selection : Criteria ﻟﻘﺪ ﻋﺮﻓﻨﺎ ﺳﺎﺑﻘﺎ اﻟﺒﻮاﻗﻲ ﻋﻠﻲ اﻧﻬﺎ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻧﺎﻗﺺ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻓﻤﻦ ﻣﺸﺎهﺪات ﻣﻌﻄﺎة z1 , z2 ,..., znوﻧﻤﻮذج ﻣﻄﺒﻖ ﻳﻨﺘﺞ ﻟﺪﻳﻨﺎ ﻗﻴﻢ ﻣﻄﺒﻘﺔ zˆ1 , zˆ2 ,..., zˆnوﺗﻜﺘﺐ اﻟﺒﻮاﻗﻲ: ei = zi − zˆi , i = 1, 2,..., n واﻟﺒﻮاﻗﻲ هﻲ ﻣﻘﺪرات اﻷﺧﻄﺎء ﻓﻲ اﻟﻨﻤﻮذج أي aˆi = ei , i = 1, 2,..., nوﻟﻬﺬا ﻳﺠﺐ ان ﺗﺤﻘﻖ اﻟﺸﺮوط اﻟﻤﻔﺮوﺿﺔ ﻋﻠﻲ اﻷﺧﻄﺎء ﻓﻲ هﺬا اﻟﻨﻤﻮذج واﻟﺘﻲ ﻣﻨﻬﺎ: -١ﻣﺘﻮﺳﻂ اﻷﺧﻄﺎء ﻳﺴﺎوي اﻟﺼﻔﺮ -٢اﻷﺧﻄﺎء ﻋﺸﻮاﺋﻴﺔ و ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ أو ﻣﺴﺘﻘﻠﺔ ) وﻓﻲ آﺜﻴﺮ ﻣﻦ اﻟﻨﻤﺎذج ﻧﻔﺘﺮض ان اﻷﺧﻄﺎء ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻣﺴﺘﻘﻞ وﻣﺘﻄﺎﺑﻖ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ σ 2أي ) ( at IIDN ( 0, σ 2 ﻟﻬﺬا ﻓﺈﻧﻨﺎ ﻧﺠﺮي ﺗﺤﻠﻴﻼ وهﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ ﻟﻨﺮي ﻓﻴﻤﺎ إذا آﺎﻧﺖ ﺗﺤﻘﻖ هﺬﻩ اﻟﺸﺮوط وﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ ﻧﻌﺘﺒﺮ اﻟﻨﻤﻮذج اﻟﻤﻄﺒﻖ ﻣﻘﺒﻮﻻ أﻣﺎ إذا ﻓﺸﻞ اﺣﺪ هﺬﻩ اﻹﺧﺘﺒﺎرات ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ إﻋﺎدة اﻟﻨﻈﺮ وإﻗﺘﺮاح ﻧﻤﻮذج ﺁﺧﺮ أوﻻ :إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ H 0 : E ( at ) = 0 H 1 : E ( at ) ≠ 0
وهﻮ إﺧﺘﺒﺎر ﺑﺬﻳﻠﻴﻦ وﻧﺴﺘﺨﺪم ﻓﻴﺔ اﻹﺣﺼﺎﺋﺔ
e
) se ( e
= uواﻟﺘﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ
ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ ان E ( at ) = 0إذا آﺎﻧﺖ ) u < 1.96هﺬا ﻋﻠﻲ إﻋﺘﺒﺎر ان ﺣﺠﻢ اﻟﻌﻴﻨﺔ اآﺒﺮ ﻣﻦ ٣٠وﺣﺪة وهﺬا داﺋﻤﺎ ﻣﺘﺤﻘﻖ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﻧﺪرﺳﻬﺎ (
ﻣﺜﺎل: ﺳﻮف ﻧﻌﻮد اﻟﻲ ﻣﺜﺎل ﺗﻄﺒﻴﻖ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ ﻋﻠﻲ اﻟﻤﺘﻐﻴﺮ Metals MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'. Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW Worksheet was saved on 6/ 5/1996 ;'MTB > TSPlot 'Metals >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
50
M e ta ls
45
40 60
50
40
30
20
10
Index
'MTB > Name c4 = 'AVER1' c5 = 'FITS1' c6 = 'RESI1
١٤٧
MTB > %MA 'Metals' 3; SUBC> Averages 'AVER1'; SUBC> Fits 'FITS1'; SUBC> Residuals 'RESI1'. Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average Data Length NMissing
Metals 60.0000 0
Moving Average Length: 3 Accuracy Measures MAPE: 1.55036 MAD: 0.70292 MSD: 0.76433
Moving Average
Actual
Predicted
M etals
50
Actual Predicted
45 Moving Average Length:
40 0
10
20
30
40
50
3
MAPE:
1.55036
MAD:
0.70292
MSD:
0.76433
60
Time
ﻻﺣﻆ اﻧﻨﺎ ﺧﺰﻧﺎ اﻟﺒﻮاﻗﻲ ﻓﻲ اﻟﻌﻤﻮد اﻟﺴﺎدس واﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻓﻲ اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ MTB > print c3 c6 c5
Data Display
Row
Metals
RESI1
FITS1
1 2 3 4 5 6 7 8 9
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4
* * * -0.90000 -1.23333 0.76667 0.90000 0.96667 -0.10000
* * * 44.3000 44.0333 43.5333 43.5000 43.8333 44.5000
١٤٨
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
-1.43333 -1.50000 -0.96667 -0.50000 -0.60000 -2.03333 0.63333 1.10000 1.60000 -0.10000 0.46667 0.33333 -0.10000 -0.03333 -0.20000 -0.36667 -0.16667 -0.33333 0.43333 1.00000 1.76667 0.76667 0.73333 0.06667 0.13333 0.30000 0.23333 -0.10000 0.36667 0.96667 1.23333 1.33333 1.46667 0.76667 1.06667 0.33333 0.63333 0.86667 0.56667 -0.20000 0.03333 -0.23333 1.00000 0.63333 0.56667 -0.26667 0.50000 -0.16667 -0.50000 -1.53333 -1.96667
44.5333 44.1000 43.3667 42.7000 42.4000 42.1333 41.3667 41.3000 41.5000 42.5000 42.6333 42.8667 42.9000 43.0333 43.0000 42.8667 42.7667 42.6333 42.4667 42.6000 42.9333 43.7333 44.2667 44.7333 44.7667 44.9000 44.9667 45.1000 45.1333 45.2333 45.5667 46.1667 46.8333 47.5333 48.0333 48.5667 48.7667 49.1333 49.4333 49.8000 49.8667 49.8333 49.7000 50.0667 50.3333 50.7667 50.7000 50.8667 50.8000 50.7333 50.0667
اﻵن ﻧﺨﺘﺒﺮ ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean
١٤٩
Test of mu = 0.000 vs mu not = 0.000 Variable RESI1
N 57
Mean 0.158
StDev 0.868
SE Mean 0.115
T 1.37
P 0.17
ﻳﺴﺘﺨﺪم ﺑﺮﻧﺎﻣﺞ ﻋﺎم ﻋﻨﺪﻣﺎ ﻳﻜﻮن اﻹﻧﺤﺮاف اﻟﻤﻌﻴﺎري ) او اﻟﺘﺒﺎﻳﻦ ( ﻏﻴﺮMinitab ﻓﻲ:ﻣﻼﺣﻈﺔ أي١٫٩٦ وهﻲ اﻗﻞ ﻣﻦT=١٫٣٧ ﻻﺣﻆ ان ﻗﻴﻤﺔ اﻹﺣﺼﺎﺋﺔ هﻲ. Ttest ﻣﻌﺮوف وﻳﻄﻠﻖ ﻋﻠﻴﻪ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺣﻮل اﻟﻤﺘﻮﺳﻂ وﺣﻮل اﻟﺼﻔﺮRuns test ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺠﺮي:ﺛﺎﻧﻴﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > Runs 'RESI1'.
Runs Test
RESI1 K =
0.1579
The observed number of runs = 17 The expected number of runs = 29.4211 30 Observations above K 27 below The test is significant at 0.0009 MTB > Runs 0 'RESI1'.
Runs Test
RESI1 K =
0.0000
The observed number of runs = 17 The expected number of runs = 28.7895 33 Observations above K 24 below The test is significant at 0.0013
ﻧﻼﺣﻆ اﻧﻪ ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﻻﻧﺮﻓﺾ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ Autocorrelation test ﻧﺨﺘﺒﺮ ﺗﺮاﺑﻂ أو إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ:ﺛﺎﻟﺜﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > %ACF 'RESI1'. Executing from file: E:\MTBWIN\MACROS\ACF.MAC
١٥٠
Autocorrelation
Autocorrelation Function for RESI1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
4
9
14
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1 2 3 4 5 6 7
0.56 0.24 -0.01 0.04 0.02 -0.07 -0.14
4.24 1.39 -0.06 0.21 0.11 -0.38 -0.81
18.93 22.31 22.32 22.40 22.42 22.72 24.07
8 9 10 11 12 13 14
-0.04 0.12 0.29 0.24 0.20 0.05 0.03
-0.22 0.68 1.64 1.30 1.05 0.25 0.17
24.17 25.21 31.36 35.67 38.71 38.90 38.99
أيρ1 = 0 أي اﻧﻨﺎ ﻧﺮﻓﺾ ان٤٫٢٤ ﻟﻠﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻋﻨﺪ اﻟﺘﺨﻠﻒ اﻷول ﺗﺴﺎويT ﻻﺣﻆ ان اﻟـ ﻳﻮﺟﺪ ﺗﺮاﺑﻂ ﺑﻴﻦ اﻟﺒﻮاﻗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻰ ﻓﻲ اﻹﺧﺘﺒﺎر H 0 : ρ1 = 0
H1 : ρ1 ± 0 r1 = 4.24 ﺣﻴﺚ اﻹﺣﺼﺎﺋﺔ هﻲ se ( r1 ) ﻧﺨﺘﺒﺮ ﻓﻲ ﻣﺎ إذا آﺎﻧﺖ اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ:راﺑﻌﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > %NormPlot 'RESI1'; SUBC> Kstest. Executing from file: E:\MTBWIN\MACROS\NormPlot.MAC
Normal Probability Plot
.999
P ro b a b ility
.99 .95 .80 .50 .20 .05 .01 .001 -2
-1
0
1
RESI1 Average: 0.157895 StDev: 0.867525 N: 57
Kolmogorov-Smirnov Normality Test D+: 0.054 D-: 0.084 D : 0.084 Approximate P-Value > 0.15
أي ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ اﻟﺘﻮزﻳﻊ٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦ٠٫١٥ اﻟﻨﺎﺗﺠﺔ ﺗﺴﺎويP-Value ﻻﺣﻆ اﻟـ واﻟﺬي ﻳﺒﻴﻦ ﻣﺪيQ-Q Plot هﻨﺎك اﻳﻀﺎ إﺧﺘﺒﺎر ﺁﺧﺮ ﻟﻠﻄﺒﻴﻌﻴﺔ هﻮ اﻟـα = 0.05 اﻟﻄﺒﻴﻌﻲ ﻋﻨﺪ ﺗﻄﺎﺑﻖ ﻣﺸﺎهﺪات ﻣﺎ ﻣﻊ ﺗﻮزﻳﻊ ﻣﻌﻴﻦ MTB > %Qqplot 'RESI1'; SUBC> Table; SUBC> Conf 95; SUBC> Ci. Executing from file: E:\MTBWIN\MACROS\Qqplot.MAC
١٥١
Distribution Function Analysis
Normal Dist. Parameter Estimates Data
: RESI1
Mean: StDev:
0.157895 0.867525
Normal Probability Plot for RESI1
99
0.157895
Mean:
0.867525
StDev: 95 90 80
60 50 40
Percent
70
30 20 10 5
1
2
0
1
-1
-2
Data
ﻻﺣﻆ ان ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﻓﺈﻧﻨﺎ ﻻﻧﺮﻓﺾ ان اﻟﺒﻮاﻗﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻣﻼﺣﻈﺔ اﺧﻴﺮة :ﻳﺒﺪو ان اﻟﺒﻮاﻗﻲ ﺗﺤﻘﻖ ﻣﻌﻈﻢ اﻟﺸﺮوط ﻓﻴﻤﺎ ﻋﺪي اﻟﺘﺮاﺑﻂ اﻟﺬي ﻳﻮﺟﺪ ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﺘﺘﺎﻟﻴﺔ وهﺬا ﻳﺠﻌﻠﻨﺎ ﻧﺮﻓﺾ ﺟﻮدة اﻟﺘﻄﺒﻴﻖ ﻟﻄﺮﻳﻘﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ ﺣﻴﺚ ادي اﻟﻲ ﺑﻮاﻗﻲ ﻣﺘﺮاﺑﻄﺔ.
١٥٢
اﻟﻔﺼﻞ اﻟﺘﺎﺳﻊ ﺗﺤﻠﻴﻞ او ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻲ ﻣﺮآﺒﺎت :Decomposition Method ﻳﻨﻈﺮ إﻟﻰ اﻟﻤﺘﻠﺴﻠﺴﺔ اﻟﺰﻣﻨﻴﺔ ﻋﻠﻰ اﻧﻬﺎ ﻣﻜﻮﻧﺔ ﻣﻦ ﻋﺪة ﻣﺮآﺒﺎت أو اﺟﺰاء ﻣﺘﺤﺪة ﻣﻊ ﺑﻌﻀﻬﺎ ﻟﺘﻜﻮﻳﻦ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ، ﻟﻨﻔﺘﺮض ان ﻟﺪﻳﻨﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪﻩ . z1 , z2 ,..., znﻟﻘﺪ وﺟﺪ ﺑﺎﻟﺘﺠﺮﺑﺔ أﻧﻪ ﻳﻤﻜﻦ ﻧﻤﺬﺟﺘﻬﺎ ﻋﻠﻰ اﻟﺸﻜﻞ zt = Tt + St + Ct + Et , t = 1, 2,..., n
ﺣﻴﺚ Ztاﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة و Ttﻣﺮآﺒﺔ اﻹﻧﺠﺮاف وهﻲ اﻟﺘﻲ ﺗﻨﻤﺬج اﻹﺗﺠﺎﻩ اﻟﻌﺎم اﻟﺬي ﺗﻨﺤﻲ أو ﺗﻨﺠﺮف اﻟﻴﻪ اﻟﻤﺘﺴﻠﺴﻠﺔ و Stﻣﺮآﺒﺔ ﻣﻮﺳﻤﻴﺔ وﺗﻨﻤﺬج اﻟﺘﺄﺛﻴﺮ اﻟﻤﻮﺳﻤﻲ )إذا وﺟﺪ( وهﻮ اﻟﺘﻐﻴﺮ اﻟﺬي ﻳﺤﺪث ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻧﺘﻴﺠﺔ اﻟﺘﺄﺛﻴﺮات اﻟﻤﻮﺳﻤﻴﺔ ﻣﺜﻞ اﻟﺸﻬﺮﻳﺔ واﻟﺴﻨﻮﻳﺔ و Ctﻣﺮآﺒﺔ دورﻳﺔ )إذا وﺟﺪت( و ﺗﻨﻤﺬج ﻣﻨﺤﻰ أو إﺗﺠﺎة ﻳﺘﻜﺮر ﺑﻌﺪ ﻓﺘﺮات زﻣﻨﻴﺔ ﻃﻮﻳﻠﺔ ﻏﻴﺮ ﻣﻮﺳﻤﻴﺔ و Et ﻣﺮآﺒﺔ اﻟﺨﻄﺄ وﺗﺸﻤﻞ ﺟﻤﻴﻊ اﻟﻌﻮاﻣﻞ اﻻﺧﺮى اﻟﺘﻲ ﺗﺆﺛﺮ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ واﻟﺘﻲ ﻻﻳﻤﻜﻦ ﻧﻤﺬﺟﺘﻬﺎ ﺿﻤﻨﻴﺎ أو اﻟﺘﻲ ﻻﻳﻤﻜﻦ ﻣﺸﺎهﺪﺗﻬﺎ او ﻗﻴﺎﺳﻬﺎ .اﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﻳﺴﻤﻲ ﺑﺎﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ Additive Modelوذﻟﻚ ﻷن آﻞ اﻟﻤﺮآﺒﺎت ﺗﺪﺧﻞ ﺑﺸﻜﻞ إﺿﺎﻓﻲ ﻓﻲ اﻟﻨﻤﻮذج .هﻨﺎك أﺷﻜﺎل اﺧﺮى ﻣﺜﻞ zt = Tt St + Ct + Et , t = 1, 2,..., n zt = Tt St Ct + Et , t = 1, 2,..., n
واﻟﺘﻲ ﺗﺴﻤﻰ ﺑﺎﻟﻨﻤﺎذج اﻟﺘﻀﺎﻋﻔﻴﺔ . Multiplicative Models ﻓﻲ هﺬا اﻟﻤﺴﺘﻮى ﺳﻮف ﻧﻬﻤﻞ اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ Ctوذﻟﻚ ﻷن اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ ﻧﺎدرا ﻣﺎﺗﻜﻮن ﻣﻮﺟﻮدة ﻓﻲ اﻟﻤﺘﺴﻠﺴﻼت اﻟﻘﺼﻴﺮة أو اﻟﻄﻮﻳﻠﺔ ﻧﺴﺒﻴﺎ ﻷﻧﻬﺎ ﺗﺤﺘﺎج اﻟﻰ ﻣﺸﺎهﺪات ﻃﻮﻳﻠﺔ ﺟﺪا ﻋﻠﻰ ﻣﺪي ﻋﺪد آﺒﻴﺮ ﻣﻦ اﻟﻌﻘﻮد. وﻧﻜﺘﻔﻲ ﺑﺎﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺸﻜﻞ zt = Tt + St + Et , t = 1,2,..., n
zt = Tt St + Et , t = 1,2,..., n أﻧﻈﺮ آﺘﺎب FORECASTING: METHODS AND APPLICATIONSﻟﻠﻤﺆﻟﻔﻴﻦ MAKRIDAKIS/ WHEELWRIGHT/ McGEEص ١٤١-١٣١ ﺳﻮف ﻧﺴﺘﻌﺮض ﻓﻲ اﻟﻤﺜﺎل اﻟﺘﺎﻟﻲ ﻃﺮق اﻟﺘﺤﻠﻴﻞ اﻹﺿﺎﻓﻴﺔ واﻟﺘﻀﺎﻋﻔﻴﺔ ﺑﺪون اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ أي اﻟﻨﻤﺎذج zt = Tt + St + Et , t = 1,2,..., n
zt = Tt St + Et , t = 1,2,..., n اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ هﻲ اﻟﻄﻠﺐ ﻋﻠﻲ اﻟﺒﻨﺰﻳﻦ ﺑﻤﻼﻳﻴﻦ اﻟﻠﺘﺮات ﻓﻲ ﻣﺪﻳﻨﺔ اوﻧﺘﺎرﻳﻮ ﺑﻜﻨﺪة ﻣﻦ ﺳﻨﺔ ١٩٦٠ وﺣﺘﻰ ﺳﻨﺔ ١٩٧٥ GasDemand MONTHLY GASOLINE DEMAND ONTARIO GALLON MILLIONS 1960-1975 87695 86890 96442 98133 113615 123924 128924 134775 117357 114626 107677 108087 92188 88591 98683 99207 125485 124677 132543 140735 124008 121194 111634 111565 101007 94228 104255 106922 130621 125251 140318 146174 122318 128770 117518 115492 108497 100482 106140 118581 132371 132042 151938 150997 130931 137018 121271 123548 109894 106061 112539 125745 136251 140892 158390 148314 144148 140138 124075 136485 109895 109044 122499 124264 142296 150693 163331 165837 151731 142491 140229 140463 116963 118049 137869 127392 154166 160227 165869 173522 155828 153771 143963 143898 124046 121260 138870 129782 162312 167211 172897 189689 166496 160754 155582 145936 139625 137361 138963 155301 172026 165004 185861 190270 163903 174270 160272 165614 146182 137728 148932 156751 177998 174559 198079 189073 175702 180097 155202 174508
١٥٣
154277 174176 198688 218099 227443 193522 199024 217775
144998 184416 190474 229001 233038 212870 191813 227621
159644 158167 194502 203200 234119 248565 195997
168646 156261 190755 212557 255133 221532 208684
166273 176353 166286 197095 216478 252642 244113
190176 175720 170699 193693 232868 255007 243108
205541 193939 181468 188992 221616 206826 255918
193657 201269 174241 175347 209893 233231 244642
182617 218960 210802 196265 194784 212678 237579
189614 209861 212262 203526 189756 217173 237579
أوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ MTB > TSPlot 'GasDemand'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
G asD em and
250
200
150
100
100 150 اﻟﻰ اﻻﻋﻠﻰ ﻣﻮﺳﻤﻴﺔ وﻣﻨﺠﺮﻓﺔ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ zt = Tt + St + Et , t = 1, 2,..., n : ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ:اوﻻ 50
Index
SUBC> SUBC> SUBC> SUBC>
MTB > %Decomp 'GasDemand' 12; Additive ; Forecasts 24; Title "Forecast of Gasoline Demand"; Start 1.
Time Series Decomposition Data Length NMissing
GasDeman 192.000 0
Trend Line Equation Yt = 96.4074 + 0.680579*t Seasonal Indices Period 1 2 3 4 5 6 7 8 9 10 11 12
Index -20.5625 -26.8125 -14.8958 -11.0625 9.89583 11.8958 22.7708 25.1875 5.64583 7.27083 -4.81250 -4.52083
١٥٤
Accuracy of Model MAPE: MAD: MSD:
3.6952 5.6622 52.7851
Forecasts Row
Period
Forecast
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
207.197 201.627 214.225 218.738 240.377 243.058 254.614 257.711 238.850 241.155 229.753 230.725 215.364 209.794 222.391 226.905 248.544 251.225 262.780 265.878 247.017 249.322 237.919 238.892
Forecast of Gasoline Demand Seasonal Indices
Original Data, by Seasonal Period
30
250
20 10
200
0 150
-10 -20
100
-30 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period
Residuals, by Seasonal Period 30 20
10
10 0 5
-10 -20
0
-30 1 2 3 4 5 6 7 8 9 10 11 12
١٥٥
1 2 3 4 5 6 7 8 9 10 11 12
(١) ﺷﻜﻞ
ﺷﻜﻞ )(٢ Forecast of Gasoline Demand Original Data
Detrended Data 50 40 30 20 10 0 -10 -20 -30 -40 100
200
250 200 150 100 200
0
Seasonally Adj. and Detrended Data
0
100
Seasonally Adjusted Data 250
30 20 10
200
-10
150
0 -20 -30 100
200
ﺷﻜﻞ )(٣
100 200
0
0
100
Forecast of Gasoline Demand
Actual Predicted
250
Actual Predicted Forecast
200
Forecast
GasDeman
150
100 3.6952 5.6622 52.7851
MAPE: MAD: MSD:
200
100
0
Time
ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ: ﺷﻜﻞ ) (١ﻳﻮﺿﺢ اﻟﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ ، Seasonal Indicesﻓﺎﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﺴﺎر ﻳﺒﻴﻦ ﺗﺄﺛﺮاﻟﻄﻠﺐ ﻓﻲ اﻷﺷﻬﺮ اﻟﻤﺨﺘﻠﻔﺔ ﻣﻦ اﻟﺴﻨﺔ ﻓﻔﻲ اﻷﺷﻬﺮ ١١و ١٢و ١و ٢و ٣و ٤ﻳﺤﺪث ﻧﻘﺺ ﻓﻲ اﻟﻄﻠﺐ إذ ﻳﺘﻨﺎﻗﺺ ﺗﺪرﻳﺠﻴﺎ ﺣﺘﻰ ﻳﺼﻞ إﻟﻰ أﻗﻞ ﻣﻌﺪل ﻟﻪ ﻓﻲ اﻟﺸﻬﺮ ٢ﺛﻢ ﻳﺘﺰاﻳﺪ ﺣﺘﻰ ﻳﺼﺒﺢ ﻣﻮﺟﺒﺎ ﻓﻲ اﻟﺸﻬﺮ ٥وﻳﺘﺰاﻳﺪ ﺣﺘﻰ ﻳﺼﻞ اﻗﺼﻰ ﻗﻴﻤﺔ ﻣﻮﺟﺒﺔ ﻓﻲ اﻟﺸﻬﺮ ٨ﺛﻢ ﻳﻨﻘﺺ ﺑﺸﻜﻞ آﺒﻴﺮ ﺑﻌﺪﺋﺬ .اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﻤﻴﻦ ﻳﻌﻄﻲ رﺳﻢ اﻟﺼﻨﺪوق Box Plotﻟﻠﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ﻣﻮزﻋﺔ ﻋﻠﻰ اﻷﺷﻬﺮ وهﻮ ﻳﻮﺿﺢ ﺗﻮزﻳﻊ وإﻧﺘﺸﺎر اﻟﻤﺸﺎهﺪات ﻋﻠﻰ آﻞ ﺷﻬﺮ واﻟﻘﻴﻢ اﻟﺨﺎرﺟﺔ . Out Liers اﻟﺸﻜﻞ اﻷﺳﻔﻞ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﺘﻐﻴﺮ اﻟﻨﺴﺒﻲ اﻟﻤﺌﻮي ﻋﻠﻰ اﻟﻔﺘﺮات اﻟﻤﻮﺳﻤﻴﺔ )اﻷﺷﻬﺮ( .اﻟﺸﻜﻞ اﻷﺳﻔﻞ اﻷﻳﻤﻦ ﻳﻌﻄﻲ رﺳﻢ اﻟﺼﻨﺪوق ﻟﻠﺒﻮاﻗﻲ أو اﻷﺧﻄﺎء ﻣﻮزﻋﺔ ﻋﻠﻰ اﻷﺷﻬﺮ. ﺷﻜﻞ ) (٢اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﻤﻴﻦ ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ،اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات ﺑﻌﺪ إزاﺣﺔ ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف أي wt = zt − Tt , t = 1, 2,..., n =St + Et , t = 1, 2,..., n
١٥٦
اﻟﺸﻜﻞ اﻷﺳﻔﻞ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ﺑﻌﺪ إزاﺣﺔ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ أي yt = zt − St , t = 1, 2,..., n =Tt + Et , t = 1, 2,..., n
أو اﻟﺒﻮاﻗﻲ ﺑﻌﺪ إزاﺣﺔ ﻣﺮآﺒﺘﻲ اﻹﻧﺠﺮافEt اﻟﺸﻜﻞ اﻷﺳﻔﻞ اﻷﻳﻤﻦ ﻳﻌﻄﻲ ﻣﺮآﺒﺔ اﻟﺨﻄﺄ واﻟﻤﻮﺳﻤﻴﺔ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ أي et = zt − Tt − St , t = 1, 2,..., n =Et , t = 1, 2,..., n
. اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ ﻣﻘﺎﻳﻴﺲ دﻗﺔ اﻟﺘﻄﺒﻴﻖ٢٤ ( ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ٣) ﺷﻜﻞ zt = Tt St + Et , t = 1, 2,..., n
: ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:ﺛﺎﻧﻴﺎ
MTB > %Decomp 'GasDemand' 12; SUBC> Forecasts 24; SUBC> Title "Forecast of Gasoline Demand"; SUBC> Start 1. Executing from file: D:\MTBWIN\MACROS\Decomp.MAC Macro is running ... please wait
Time Series Decomposition Data Length NMissing
GasDeman 192.000 0
Trend Line Equation Yt = 96.4074 + 0.680579*t Seasonal Indices Period
Index
1 2 3 4 5 6 7 8 9 10 11 12
0.860355 0.828555 0.892431 0.936273 1.06124 1.07274 1.15775 1.17075 1.03409 1.05059 0.966300 0.968923
Accuracy of Model MAPE: MAD: MSD:
3.6338 5.7720 56.8996
Forecasts Row
Period
Forecast
1 2
193 194
195.954 189.275
١٥٧
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
204.474 215.156 244.596 247.977 268.415 272.227 241.154 245.718 226.660 227.935 202.980 196.042 211.762 222.803 253.263 256.738 277.870 281.789 249.599 254.298 234.552 235.848
(٤) ﺷﻜﻞ Forecast of Gasoline Demand Seasonal Indices
Original Data, by Seasonal Period
1.2
250
1.1
200
1.0 150 0.9 100 0.8 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period 14 12 10 8 6 4 2 0
Residuals, by Seasonal Period 20 10 0 -10 -20 -30
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
(٥) ﺷﻜﻞ Forecast of Gasoline Demand Original Data
Detrended Data
250
1.3 1.2
200
1.1 1.0
150
0.9 100
0.8 0
100
200
0
Seasonally Adjusted Data 240 220 200 180 160 140
١٥٨
100
200
Seasonally Adj. and Detrended Data 20 10 0 -10
Forecast of Gasoline Demand
ﺷﻜﻞ )(٦ Actual
280
Predicted Forecast
180
3.6338 5.7720 56.8996
MAPE: MAD: MSD:
GasDeman
Actual Predicted Forecast
80
200
100
0
Time
ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ: اﻷﺷﻜﺎل ) (٤و ) (٥و ) (٦ﻟﻬﺎ ﻧﻔﺲ اﻟﺘﻔﺴﻴﺮ آﻤﺎ ﻓﻲ اﻷﺷﻜﺎل ) (١و ) (٢و ).(٣ ﺑﻤﺎ اﻧﻨﺎ ﻃﺒﻘﻨﺎ ﻧﻤﻮذﺟﻴﻦ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ﻓﻴﺠﺐ أن ﻧﺨﺘﺎر اﻓﻀﻞ ﻧﻤﻮذج ،وهﻨﺎ ﻳﺄﺗﻲ دور ﻣﻘﺎﻳﻴﺲ دﻗﺔ اﻟﺘﻄﺒﻴﻖ واﻟﺘﻲ ﺗﻨﺘﺞ ﻣﻦ اﻟﺒﺮﻧﺎﻣﺞ ،ﻟﺪﻳﻨﺎ ﺛﻼﺛﺔ ﻣﻘﺎﻳﻴﺲ دﻗﺔ: -١ﻣﺘﻮﺳﻂ اﻟﺨﻄﺄ اﻟﻨﺴﺒﻲ اﻟﻤﻄﻠﻖ Mean Absolute Percentage Errorأو MAPE وﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ zt − zˆt zt × 100, zt ≠ 0 n
n
∑ t =1
= MAPE
-٢ﻣﺘﻮﺳﻂ اﻹﻧﺤﺮاف اﻟﻤﻄﻠﻖ Mean Absolute Deviationأو MADوﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ n
− zˆt
t
∑z t =1
n
= MAD
-٣ﻣﺘﻮﺳﻂ أﻹﻧﺤﺮاف اﻟﻤﺮﺑﻊ )أو ﻣﺘﻮﺳﻂ اﻟﺨﻄﺄ اﻟﻤﺮﺑﻊ( ) MSDأو ( MSEوﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ 2
) − zˆt
n
t
∑( z
n
t =1
= MSD
ﺑﺈﺧﺘﻴﺎر أﺣﺪ هﺬة اﻟﻤﻘﺎﻳﻴﺲ ﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲ أﻗﻞ ﻗﻴﻤﺔ ﻟﻬﺬا اﻟﻤﻘﻴﺎس ،اﻟﻤﻘﻴﺎس اﻷآﺜﺮ إﺳﺘﺨﺪاﻣﺎ وﺷﻴﻮﻋﺎ هﻮ MSDأو MSEوهﻮ اﻟﺬي ﺳﻮف هﻨﺎ. ﻟﻠﻨﻤﻮذج اﻹﺿﺎﻓﻲ ﻣﻘﺎﻳﻴﺲ اﻟﺪﻗﺔ هﻲ: 3.6952 ١٥٩
MAPE:
MAD: MSD:
5.6622 52.7851 و ﻟﻠﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:
MAPE: 3.6338 MAD: 5.7720 MSD: 56.8996 ﻧﻼﺣﻆ أن اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ اﻋﻄﻰ اﻗﻞ ﻗﻴﻤﺔ ﻟﻠﻤﻘﻴﺎس MSDوﻟﺬﻟﻚ ﻧﻘﺮر إﺳﺘﺨﺪام هﺬا اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻄﻠﺐ.
ﺗﻮﺿﻴﺢ ﻃﺮﻳﻘﺔ ﺗﺤﻠﻴﻞ او ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻲ ﻣﺮآﺒﺎت Decomposition :Method ﺳﻮف ﻧﻮﺿﺢ ﺑﺎﻟﻤﺜﺎل اﻟﺘﺎﻟﻲ ﻃﺮﻳﻘﺔ ﺗﺤﻠﻴﻞ أو ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ إﻟﻰ ﻣﺮآﺒﺎت .اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ إﻧﺘﺎج ١٦٨ﻳﻮﻣﺎ ﻟﻠﺤﻠﻴﺐ ﻓﻲ أﺣﺪ اﻟﻤﺰارع ﺑﺎﻟﻜﻴﻠﻮ ﺟﺮام MTB > Read "E:\Mtbwin\milk.dat" c1. Entering data from file: E:\Mtbwin\milk.dat 168 rows read. 'MTB > name c1='MilkProd MTB > print c1
Data Display MilkProd 582 598 634 635 688 698 711 734 751 800 805 802 813 843
553 565 594 602 645 660 677 690 711 763 766 760 773 797
577 587 611 621 667 687 706 723 747 800 810 791 812 827
568 583 604 615 661 681 701 725 740 790 809 784 815 817
599 617 639 653 697 722 740 764 783 834 855 837 867 858
640 660 678 702 735 767 783 801 819 869 894 881 908 896
697 716 736 756 798 817 826 845 859 913 935 924 947 937
727 742 770 782 811 837 858 871 886 942 961 957 969 966
640 653 688 709 736 762 775 785 807 860 890 883 889 892
656 673 705 722 755 784 796 805 824 878 900 898 902 903
561 566 618 622 635 667 696 690 707 756 799 773 778 782
589 600 628 658 677 713 717 734 750 804 826 821 828 834
وﻧﺮﺳﻢ اﻟﺒﻴﺎﻧﺎت اﻟﺴﺎﺑﻘﺔ ﺑﺎﻷواﻣﺮ: ;'MTB > TSPlot 'MilkProd >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
1000
900
700
600
150
100
١٦٠
50
Index
MilkProd
800
: ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ:اوﻻ ﻟﻜﻲzt = Tt + St + Et , t = 1, 2,..., n ﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞz1 , z2 ,..., zn ﻟﻤﺸﺎهﺪات :ﻧﻘﻮم ﺑﺘﻔﻜﻴﻚ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻲ اﻟﻤﺮآﺒﺎت اﻟﺴﺎﺑﻘﺔ ﺳﻮف ﻧﺴﺘﻌﺮض ﻃﺮﻳﻘﺘﻴﻦ ﻣﻤﻜﻨﺔ :اﻟﻄﺮﻳﻘﺔ اﻻوﻟﻲ : أيTt ﻧﻄﺒﻖ إﻧﺤﺪار ﺧﻄﻲ ﺑﺴﻴﻂ ﻟﻠﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ ﻟﺘﻘﺪﻳﺮ ﻣﺮآﺒﺔ اﻻﻧﺠﺮاف-١ Tˆt ≡ zˆt = a + bt , t = 1, 2,...,168
:أي MTB > set c2 DATA> 1:168 DATA> end MTB > name c1='MilkProd' c2='Time' c3='Trend' c5='Detrend' c6='Index' c8='Fitted' c9='Resid' MTB > regr c1 1 c2; SUBC> fits c3.
Regression Analysis The regression equation is MilkProd = 612 + 1.69 Time Predictor Constant Time S = 60.74
Coef StDev T P 611.682 9.414 64.97 0.000 1.69262 0.09663 17.52 0.000 R-Sq = 64.9% R-Sq(adj) = 64.7%
Analysis of Variance Source Regression Error Total
DF 1 166 167
SS 1132003 612439 1744443
MS 1132003 3689
F 306.83
P 0.000
وﺷﻜﻞ اﻹﻧﺠﺮاف هﻮ
900
Trend
800
700
600
Index
50
100
150
ﻧﻄﺮح ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف ﻣﻦ اﻟﻤﺸﺎهﺪات اﻻﺻﻠﻴﺔ ﻓﻨﺤﺼﻞ ﻋﻠﻲ ﻣﺎﻳﺴﻤﻲ ﺑﺎﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ-٢ zt − zˆt = zt − Tˆt , t = 1, 2,...,168 أيDetrended Series اﻹﻧﺠﺮاف MTB > let c5=c1-c3
:وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
١٦١
Detrend
100
0
-100 Index
50
100
150
zt − Tˆt = St + Et , t = 1, 2,...,168 ﻧﻼﺣﻆ اﻧﻬﺎ اﺻﺒﺤﺖ ﻣﻮﺳﻤﻴﺔ ﻓﻘﻂ ﻷن : آﺎﻟﺘﺎﻟﻲSeasonal Indices ﻟﺘﻘﺪﻳﺮ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻧﻮﺟﺪ اﻟﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ-٣ اﻟﻤﺆﺷﺮ اﻟﻤﻮﺳﻤﻲ ﻟﻠﺸﻬﺮI1 ﺣﻴﺚI s , s = 1, 2,...,12 ﻟﻨﺮﻣﺰ ﻟﻠﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ ﺑﺎﻟﺮﻣﺰ d t = zt − Tˆt , t = 1, 2,...,168 اﻟﻤﺆﺷﺮ اﻟﻤﻮﺳﻤﻲ ﻟﻠﺸﻬﺮ اﻟﺜﺎﻧﻲ وهﻜﺬا وﻟﻨﺮﻣﺰ ﺑـI 2 اﻷول و :ﺗﻘﺪر هﺬﻩ اﻟﻤﺆﺷﺮات آﺎﻟﺘﺎﻟﻲ
1 ( d1 + d13 + d 25 + L + d157 ) 14 1 I 2 = ( d 2 + d14 + d 26 + L + d158 ) 14 M I1 =
I12 =
1 ( d12 + d 24 + d 36 + L + d168 ) 14 :وﻳﺘﻢ ذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻷواﻣﺮ اﻟﺘﺎﻟﻴﺔ
MTB > DATA> DATA> MTB > SUBC> SUBC> MTB > & CONT> MTB > MTB > MTB > DATA> DATA> MTB >
set c4 14(1:12) end stat c5; by c4; mean c6. Stack 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' c7. let c8=c3+c7 let c9=c1-c8 set c10 1:12 end print c10 c6
Data Display
١٦٢
-٤اﻟﺘﻨﺒﺆات ﺗﻮﻟﺪ آﺎﻟﺘﺎﻟﻲ:
Index
Season
Row
-18.328 -57.806 34.716 49.595 110.616 82.281 32.517 -9.747 -52.297 -48.775 -79.754 -43.018
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
z168 ( l ) = 612 + 1.69 ( l + 168) + I ( l mod 12 ) , l = 1,2,...
ﻓﻤﺜﻼ اﻟﺘﻨﺒﺆ ﻋﻨﺪ اﻟﻴﻮم ١٦٩هﻮ
z168 (1) = 612 + 1.69 (169 ) + I1 =897.61 + ( −18.328 ) = 879.282
اﻟﻄﺮﻳﻘﺔ اﻟﺜﺎﻧﻴﺔ: وهﻲ اﻟﺘﻲ ﻳﺴﺘﺨﺪﻣﻬﺎ ﺑﺮﻧﺎﻣﺞ : Minitab -١آﺎﻟﻄﺮﻳﻘﺔ اﻻوﻟﻰ ﻧﻄﺒﻖ إﻧﺤﺪار ﺧﻄﻲ ﺑﺴﻴﻂ ﻟﻠﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ ﻟﺘﻘﺪﻳﺮ ﻣﺮآﺒﺔ اﻻﻧﺠﺮاف Ttﻓﻨﺤﺼﻞ ﻋﻠﻰ ﻧﻔﺲ اﻟﻨﺘﻴﺠﺔ آﻤﺎ ﻓﻲ اﻟﻄﺮﻳﻘﺔ اﻻوﻟﻰ )(١ -٢اﻳﻀﺎ هﻨﺎ ﻧﻄﺮح ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف ﻣﻦ اﻟﻤﺸﺎهﺪات اﻻﺻﻠﻴﺔ ﻓﻨﺤﺼﻞ ﻋﻠﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ اﻹﻧﺠﺮاف Detrended Series -٣ﻧﻄﺒﻖ اﻵن ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ درﺟﺔ اﻟﻤﻮﺳﻢ وﻧﻮﺳﻄﻪ اذا اﺣﺘﺎج اﻻﻣﺮ -٤ﻧﻄﺮح اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ ﻣﻦ ﻧﻈﻴﺮاﺗﻬﺎ ﻓﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ اﻹﻧﺠﺮاف ﻓﻨﺤﺼﻞ ﻋﻠﻲ ﻣﺘﺴﻠﺴﻠﺔ ﺗﺤﻮي اﻟﻤﺮآﺒﺎت اﻟﻤﻮﺳﻤﻴﺔ -٥ﺗﻘﺪر اﻟﻤﺮآﺒﺎت اﻟﻤﻮﺳﻤﻴﺔ آﺎﻟﺘﺎﻟﻲ: ) I1 = Median ( d1 , d13 , d 25 ,L, d157
) I 2 = Median ( d 2 , d14 , d 26 ,L, d158
M
) I12 = Median ( d12 , d 24 , d 36 ,L, d168 -٦ﺗﻮﻟﺪ اﻟﺘﻨﺒﺆات آﺎﻟﺴﺎﺑﻖ وﺳﻮف ﻧﺴﺘﻌﺮض هﺬا آﺎﻟﺘﺎﻟﻲ:
MTB > Read "E:\Mtbwin\milk.dat" c1. Entering data from file: E:\Mtbwin\milk.dat 168 rows read. 'MTB > name c1='MilkProd MTB > set c2 DATA> 1:168 DATA> end 'MTB > name c2='Time ;MTB > regr c1 1 c2 SUBC> fits c3.
Regression Analysis
١٦٣
The regression equation is MilkProd = 612 + 1.69 Time Predictor Constant Time
Coef 611.682 1.69262
S = 60.74
StDev 9.414 0.09663
R-Sq = 64.9%
T 64.97 17.52
P 0.000 0.000
R-Sq(adj) = 64.7%
Analysis of Variance Source Regression Error Total
DF 1 166 167
SS 1132003 612439 1744443
Unusual Observations Obs Time MilkProd 113 113 942.00 125 125 961.00
MS 1132003 3689
Fit 802.95 823.26
F 306.83
StDev Fit 5.44 6.11
P 0.000
Residual 139.05 137.74
St Resid 2.30R 2.28R
R denotes an observation with a large standardized residual MTB MTB MTB MTB
> > > >
name c3='Trend' let c4=c1-c3 name c4='Detrend' Name c5 = 'AVER1'
: وﻧﻮﺳﻄﻪ١٢ ﻓﻲ اﻟﺨﻄﻮة اﻟﺘﺎﻟﻴﺔ ﻧﻄﺒﻖ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ MTB > %MA 'Detrend' 12; SUBC> Center; SUBC> Averages 'AVER1'. Executing from file: E:\MTBWIN\MACROS\MA.MAC Macro is running ... please wait
Moving average Data Length NMissing
Detrend 168.000 0
Moving Average Length: 12 Accuracy Measures MAPE: 111.68 MAD: 52.36 MSD: 3564.77
ﻧﻄﺮح اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ اﻟﻤﻮﺳﻄﺔ ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺰال إﻧﺠﺮاﻓﻬﺎ MTB > MTB > MTB > DATA> DATA> MTB > SUBC> SUBC> MTB >
let c6=c4-c5 name c6='DeSeason' set c2 14(1:12) end stat c6; by c2; median c7. name c7='SeasInx'
Data Display Row
Season
SeasInx
١٦٤
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
-20.750 -58.958 35.625 50.083 109.542 81.292 33.917 -10.000 -52.792 -50.250 -79.958 -44.375
:ﻧﻘﺎرن اﻟﺤﺴﺎﺑﺎت اﻟﺘﻲ اﺟﺮﻳﻨﺎهﺎ ﻣﻊ اﻟﺒﺮﻧﺎﻣﺞ اﻷﺻﻠﻲ MTB > %Decomp 'MilkProd' 12; SUBC> Additive ; SUBC> Start 1. Executing from file: E:\MTBWIN\MACROS\Decomp.MAC Macro is running ... please wait
Time Series Decomposition Data Length NMissing
MilkProd 168.000 0
Trend Line Equation Yt = 611.682 + 1.69262*t Seasonal Indices Period 1 2 3 4 5 6 7 8 9 10 11 12
Index -20.1979 -58.4062 36.1771 50.6354 110.094 81.8437 34.4687 -9.44792 -52.2396 -49.6979 -79.4063 -43.8229
Accuracy of Model MAPE: MAD: MSD:
1.583 12.088 244.406
.وﺑﻤﻘﺎرﻧﺔ اﻟﻨﺘﻴﺠﺘﻴﻦ ﻧﺠﺪ اﻧﻬﻤﺎ ﺗﻘﺮﻳﺒﺎ ﻣﺘﺴﺎوﻳﺘﺎن : آﺎﻟﺘﺎﻟﻲ%Decomp ﺳﻮف ﻧﻮﻟﺪ ﺗﻨﺒﺆات ﺑﺎﺳﺘﺨﺪام اﻟﺒﺮﻧﺎﻣﺞ MTB > %Decomp 'MilkProd' 12; SUBC> Additive ; SUBC> Forecasts 12; SUBC> Start 1.
:واﻟﺘﻲ ﺗﻌﻄﻲ اﻟﺘﻨﺒﺆات Forecasts
١٦٥
Forecast
Period
Row
877.54 841.02 937.30 953.45 1014.60 988.04 942.36 900.13 859.04 863.27 835.25 872.53
169 170 171 172 173 174 175 176 177 178 179 180
1 2 3 4 5 6 7 8 9 10 11 12
ﺗﻤﺮﻳﻦ :وﻟﺪ ﺗﻨﺒﺆات ﻹﻧﺘﺎج اﻟﺤﻠﻴﺐ اﻟﻴﻮﻣﻲ ﺑﺈﺳﺘﺨﺪام اﻟﻄﺮﻳﻘﺘﻴﻦ اﻟﻤﻌﻄﺎة وﻗﺎرﻧﻬﺎ ﺑﺎﻟﻘﻴﻢ اﻷﺧﻴﺮة اﻟﻨﺎﺗﺠﺔ ﻣﻦ اﻟﺒﺮﻧﺎﻣﺞ
١٦٦
اﻟﻔﺼﻞ اﻟﻌﺎﺷﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Using Moving Average Smoothing for Forecasting ﻳﺴﺘﺨﺪم اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻟﺘﻤﻬﻴﺪ اﻟﻤﺸﺎهﺪات وذﻟﻚ ﺑﺘﻘﻠﻴﻞ ﺗﺒﺎﻳﻦ اﻷﺧﻄﺎء ﻓﻤﺜﻼ ﻟﻮ آﺎن ﻟﺪﻳﻨﺎ ﻣﺸﺎهﺪات ﻣﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻓﺎﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ m ﻟﻠﻤﺸﺎهﺪات ﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ 1 zˆt = ( zt + zt −1 + zt −2 + L + zt −m+1 ) , t = m, m + 1,..., n m أو 1 zˆt = zˆt −1 + ( zt − zt −m ) , t = m, m + 1,..., n m ﻻﺣﻆ ان ﻋﺪد اﻟﻤﺸﺎهﺪات اﺻﺒﺢ ﺑﻌﺪ اﻟﺘﻤﻬﻴﺪ . n − m + 1 ﻓﻤﺜﻼ ﻟﻮ آﺎﻧﺖ m=٣ﻓﺈن اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ هﻮ 1 ) ( z3 + z2 + z1 3 1 1 ) zˆ4 = ( z4 + z3 + z2 ) or zˆ4 = zˆ3 + ( z4 − z1 3 3 M 1 1 ) zˆn = ( zn + zn −1 + zn −2 ) or zˆn = zˆn −1 + ( zn − zn −3 3 3 = zˆ3
وﻟﻜﻲ ﻧﺮى آﻴﻒ ﻳﻌﻤﻞ اﻟﺘﻤﻬﻴﺪ ﻟﺘﻘﻠﻴﻞ ﺗﺒﺎﻳﻦ اﻷﺧﻄﺎء ﻟﻨﻔﺘﺮض ان اﻟﻤﺸﺎهﺪات ﺗﺘﺒﻊ اﻟﻨﻤﻮذج WN ( 0, σ 2 ) , t = 1, 2,..., n
ﻓﻴﻜﻮن
zt = µ + at , at
V ( zt ) = σ , ∀t 2
وﺑﺎﻟﺘﺎﻟﻲ 2
σ
= ) V ( zˆt
, t = m, m + 1,..., n m أي ان اﻟﻤﺸﺎهﺪات اﻟﻤﻤﻬﺪة اﺻﺒﺢ ﺗﺒﺎﻳﻨﻬﺎ أﺻﻐﺮ ﺑـ mﺿﻌﻒ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ وهﺬا اﻟﺘﻤﻬﻴﺪ ﻟﻸﺧﻄﺎء ﻳﻈﻬﺮ أي ﻧﻤﻂ ﻓﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎن ﻣﺪﻓﻮﻧﺎ او ﻣﻐﻄﻰ ﻣﻦ ﺗﺄﺛﻴﺮ اﻷﺧﻄﺎء. ﻣﻼﺣﻈﺔ :ﺗﺆﺧﺬ mداﺋﻤﺎ ﻓﺮدﻳﺔ وذﻟﻚ ﻟﻨﺘﺠﻨﺐ ﺗﻮﺳﻴﻂ اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة.
اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك: ﻳﺆﺧﺬ آﻤﺘﻨﺒﺊ ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك:
zn ( l ) = zˆn −1 , l > 0
ﻣﺜﺎل: ﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻷﺣﺼﺎﺋﻴﺔ MINITABﻧﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﻧﻨﻈﺮ ﻣﺎذا ﺗﺤﻮي ﻣﻦ ﻣﺘﻐﻴﺮات MTB > info
١٦٧
Information on the Worksheet Column C1 C2 C3
Count 60 60 60
Name Trade Food Metals
Metals ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻟﻠﻤﺘﻐﻴﺮ Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
44.3 42.6 43.1 43.6 45.5 50.0 51.2
44.4 42.4 43.2 44.7 46.2 50.0 50.7
43.4 42.2 42.8 44.5 46.8 49.6 50.3
42.8 41.8 43.0 45.0 47.5 49.9 49.2
44.3 40.1 42.8 44.8 48.3 49.6 48.1
44.4 42.0 42.5 44.9 48.3 50.7
44.8 42.4 42.6 45.2 49.1 50.7
44.4 43.1 42.3 45.2 48.9 50.9
:ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات MTB > TSPlot 'Metals'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
Metals
50
45
40 Index
10
20
30
40
50
60
وﻧﻮﺟﺪ ﺗﻨﺒﺆاتm=٣ ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ : ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﻪ٦ ﻟـ MTB > %MA 'Metals' 3; SUBC> Forecasts 6; SUBC> Title "Smoothing and Forecasting Metals". Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average Data Length NMissing
Metals 60.0000 0
Moving Average Length: 3 Accuracy Measures MAPE: 1.55036
١٦٨
0.70292 0.76433 Lower
Upper
50.9135 50.9135 50.9135 50.9135 50.9135 50.9135
Forecast
Period
49.2 49.2 49.2 49.2 49.2 49.2
61 62 63 64 65 66
47.4865 47.4865 47.4865 47.4865 47.4865 47.4865
MAD: MSD:
Row 1 2 3 4 5 6
Smoothing and Forecasting Metals
Actual Predicted
50
Forecast
45 Moving Average 3
Metals
Actual Predicted Forecast
Length:
MAPE: 1.55036 0.70292
MAD:
0.76433
MSD:
40 60
ﺛﺎﻧﻴﺎ :ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ ﻓﻲ اﻟﻤﺜﺎل اﻟﺤﺎﻟﻲ
50
40
30
20
10
0
Time
50.3 + 49.2 + 48.1 147.6 = = 49.2 3 3 ﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟـ ٦اﻟﻤﺴﺘﻘﺒﻠﻴﺔ أي ﻟﻠﻘﻴﻢ zn +1 , zn +2 ,..., zn+6أو ﻓﻲ هﺬا اﻟﻤﺜﺎل z61 , z62 ,..., z66آﺎﻟﺘﺎﻟﻲ: z60 (1) = z60 ( 2 ) = L = z60 ( 6 ) = 49.2
= zˆ59
ﻟﺤﺴﺎب ﻓﺘﺮات ﺗﻨﺒﺆ 95%ﻧﺤﺴﺐ اﻟﻜﻤﻴﺎت ⎡⎣ zn ( l ) ± 1.96σˆ ⎤⎦ , l > 0أي ] ˆ [ 49.2 ± 1.96σﻟﺠﻤﻴﻊ ﻗﻴﻢ اﻟﺘﻨﺒﺆات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ،ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ MSD = 0.76433آﻤﻘﺪر ﻟـ
σ 2أي σˆ 2 = 0.76433ﻓﻴﻜﻮن σˆ = 0.8743وﻋﻠﻴﻪ ﺗﻜﻮن ﻓﺘﺮة ﺗﻨﺒﺆ 95%ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ هﻲ: ]⎡⎣ 49.2 ± 1.96 ( 0.8743) ⎤⎦ = [ 49.2 ± 1.7135] = [ 47.4865,50.9135 أي: z60+l ∈ [ 47.4865,50.9135] , l > 0 with probability 0.95 ﻣﻼﺣﻈﺔ :ﺗﺤﺴﺐ MSDآﺎﻵﺗﻲ
١٦٩
) − zˆi
n −1
∑( z
i
i =2
n−2
= ˆMSD = σ 2
ﺗﻤﺮﻳﻦ: ﻃﺒﻖ ﻣﺘﻮﺳﻄﺎت ﻣﺘﺤﺮآﺔ ﻣﻦ اﻟﺪرﺟﺎت ٥و ٧ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ وﻗﺮر اﻳﻬﺎ اﻓﻀﻞ ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ؟. اﻟﻮﺳﻴﻂ اﻟﺠﺎري Running Median اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﺘﺄﺛﺮ آﺜﻴﺮا ﺑﺎﻟﻤﺸﺎهﺪات اﻟﺨﺎرﺟﺔ Outliersاو اﻟﻤﺘﻄﺮﻓﺔ ﻓﺎﻟﻘﻴﻤﺔ اﻟﻤﺘﻄﺮﻓﺔ اﻟﻮاﺣﺪة ﺗﺆﺛﺮ ﻋﻠﻲ mﻣﻦ اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ﻓﻤﺜﻼ ﻟﻮآﺎﻧﺖ ﻟﺪﻳﻦ اﻟﻤﺸﺎهﺪات 18
13
15
11
12
1500
10
6
8
9
)z(t 5 7 20
3
وﻟﻬﺎ اﻟﺸﻜﻞ 1500
1000
)z(t 500
0 10
In d e x
5
ﺑﺄﺧﺬ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ٣ﻧﺠﺪ M o v in g A v e r a g e
A c tu a l
1500
P re d ic te d A c tu a l P re d ic te d
1000
)z(t
M o v in g A v e ra g e 3 1081
500
L e n g th : M APE :
273
M AD :
268811
MSD:
0
15
5
10
0
T im e
ﻻﺣﻆ ان اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻨﺎﺗﺞ ﺗﺄﺛﺮت ﻓﻴﺔ ﺛﻼﺛﺔ ﻗﻴﻢ ﺑﺎﻟﻘﻴﻤﺔ اﻟﻤﺘﻄﺮﻓﺔ. ﻟﻠﺘﻐﻠﺐ ﻋﻠﻰ ﻣﺜﻞ هﺬﻩ اﻟﺼﻌﻮﺑﺎت ﻳﺴﺘﺨﺪم اﻟﻮﺳﻴﻂ اﻟﺠﺎري ذا اﻟﻄﻮل اﻟﻔﺮدي آﻤﻤﻬﺪ ﻏﻴﺮ ﺧﻄﻲ واﻟﺬي ﻻﻳﺘﺄﺛﺮ ﺑﺎﻗﻴﻢ اﻟﻤﺘﻄﺮﻓﺔ. اﻟﻮﺳﻴﻂ اﻟﺠﺎري ذا اﻟﻄﻮل اﻟﻔﺮدي j = 2i + 1ﻟﻤﺸﺎهﺪات z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ
z%t = med ( zt −i ,..., zt ,..., zt +i ) , j = 2i + 1
١٧٠
ﻓﻤﺜﻼ ﻟﻘﻴﻤﺔ j = 3ﺗﺼﺒﺢ اﻟﻌﻼﻗﺔ ) z%t = med ( zt −1 , zt , zt +1وﺑﺄﺧﺬ وﺳﻴﻂ ﺟﺎري ذا اﻟﻄﻮل ٣ ﻟﻠﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ
15
)smoothz(t
10
5
12
8
10
4
6
2
In d e x
واذا آﺎﻧﺖ اﻟﻘﻴﻤﺔ اﻟﺤﻘﻴﻘﻴﺔ ﻟـ z9هﻲ ١٥وﻟﻴﺲ ١٥٠٠ﻓﺈن اﻟﻤﺸﺎهﺪات اﻟﺤﻘﻴﻘﻴﺔ ﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
20
15
)z(t 10
5
10
ﻗﺎرن ﺑﻴﻦ اﻟﻨﺘﻴﺠﺘﻴﻦ.
١٧١
5
Index
اﻟﻔﺼﻞ اﻟﺤﺎدي ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ Using Single : Exponential Smoothing for Forecasting اﻟﺘﻤﻬﻴﺪ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﻌﻄﻲ ﺟﻤﻴﻊ اﻟﺒﻴﺎﻧﺎت ﻧﻔﺲ اﻷهﻤﻴﺔ وﺑﺎﻟﺘﺎﻟﻲ ﻓﺈن اﻟﻘﻴﻢ اﻟﻘﺪﻳﻤﺔ ﻧﻮﻋﺎ ﺗﺆﺛﺮ ﻧﻔﺲ اﻟﺘﺄﺛﻴﺮ آﺎﻟﻘﻴﻢ اﻟﺤﺪﻳﺜﺔ وهﺬا ﻗﺪ ﻻﻳﻜﻮن ﻣﻦ اﻟﻨﺎﺣﻴﺔ اﻟﻌﻤﻠﻴﺔ ﺻﺤﻴﺤﺎ ،اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ ﻋﻠﻰ اﻟﻌﻜﺲ ﻳﻌﻄﻲ اﻟﻘﻴﻢ اﻷآﺜﺮ ﺣﺪاﺛﺔ أهﻤﻴﺔ أآﺒﺮ واﻟﻘﻴﻢ اﻻﺧﺮي ﺗﻌﻄﻰ اهﻤﻴﺔ ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻣﻊ ﻗﺪﻣﻬﺎ .ﻓﻤﺜﻼ ﻟﻮ آﺎن ﻟﺪﻳﻨﺎ ﻣﺸﺎهﺪات ﻣﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻓﺎﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ m ﻟﻠﻤﺸﺎهﺪات ﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ 1 zˆt = ( zt + zt −1 + zt −2 + L + zt −m+1 ) , t = m, m + 1,..., n m واﻟﺘﻲ ﻳﻤﻜﻦ آﺘﺎﺑﺘﻬﺎ 1 1 1 1 zˆt = zt + zt −1 + zt −2 + L + zt −m+1 , t = m, m + 1,..., n m m m m 1 = zˆt = β zt + β zt −1 + β zt −2 + L + β zt −m+1 , t = m, m + 1,..., n, β m أي ان اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﻌﻄﻲ ﺟﻤﻴﻊ اﻟﺒﻴﺎﻧﺎت ﻧﻔﺲ اﻟﻮزن β اﻵن ﻟﻮ أﻋﻄﻴﻨﺎ اﻟﺒﻴﺎﻧﺎت اوزان ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻣﻊ ﺑُﻌﺪ اﻟﻤﺸﺎهﺪات ﻋﻦ اﻟﻘﻴﻤﺔ اﻟﺤﺎﺿﺮة znآﺎﻟﺘﺎﻟﻲ st = α zt + α (1 − α ) zt −1 + α (1 − α ) zt −2 + L , t = 1, 2,..., n, 0 < α < 1 2
اﻟﻘﻴﻤﺔ stهﻲ ﻣﺘﻮﺳﻂ ﻣﻮزون ﺑﺄوزان ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﺴﺎﺑﻘﺔ وهﺬا ﻣﺎﻳﺴﻤﻰ ﺑﺎﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ وﻳﻜﺘﺐ ﺑﺸﻜﻞ ﺗﻜﺮاري
st = α zt + (1 − α ) st −1 , t = 1, 2,..., n, s0 = z
وﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات
zn ( l ) = sn , l ≥ 1
ﻣﺜﺎل:
ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ
EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Metals 44.4 43.1 42.3 45.2 48.9 50.9
44.8 42.4 42.6 45.2 49.1 50.7
44.4 42.0 42.5 44.9 48.3 50.7
44.3 40.1 42.8 44.8 48.3 49.6 48.1
42.8 41.8 43.0 45.0 47.5 49.9 49.2
43.4 42.2 42.8 44.5 46.8 49.6 50.3
44.4 42.4 43.2 44.7 46.2 50.0 50.7
44.3 42.6 43.1 43.6 45.5 50.0 51.2
Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات: ;'MTB > TSPlot 'Metals >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol
١٧٢
SUBC>
Connect.
M etals
50
45
40 Index
10
20
30
40
50
60
α = 0.2 ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ وﻧﺄﺧﺬ ﺛﺎﺑﺖ ﺗﻤﻬﻴﺪ : ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﻪ٦ وﻧﻮﺟﺪ ﺗﻨﺒﺆات ﻟـ MTB > %SES 'Metals'; SUBC> Weight 0.2; SUBC> Forecasts 6; SUBC> Title "Smoothing and Forecasting Metals"; SUBC> Initial 6. Single Exponential Smoothing Data Metals Length 60.0000 NMissing 0 Smoothing Constant Alpha: 0.2 Accuracy Measures MAPE: 2.17304 MAD: 1.00189 MSD: 1.45392 Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
49.7216 49.7216 49.7216 49.7216 49.7216 49.7216
47.2670 47.2670 47.2670 47.2670 47.2670 47.2670
52.1763 52.1763 52.1763 52.1763 52.1763 52.1763
Smoothing and Forecasting Metals
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constant Alpha:
40 0
10
20
30
Time
١٧٣
40
50
60
0.200
MAPE:
2.17304
MAD:
1.00189
MSD:
1.45392
ﺛﺎﻧﻴﺎ :ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ -١ﻳﺤﺴﺐ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ ﻟﻤﺸﺎهﺪات ﻣﻦ اﻟﻌﻼﻗﺔ اﻟﺘﻜﺮارﻳﺔ:
z1 , z2 ,K , zn −1 , zn −2ﺑﺜﺎﺑﺖ ﺗﻤﻬﻴﺪ α = 0.2
si = α zi + (1 − α ) si −1 , i = 1, 2,..., n ﻟﻜﻲ ﻧﺒﺪأ اﻟﻌﻼﻗﺔ اﻟﺘﻜﺮارﻳﺔ ﻟﺤﺴﺎب اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة اﺳﻴﺎ ﻧﺤﺘﺎج اﻟﻲ اﻟﻘﻴﻤﺔ اﻻوﻟﻴﺔ s0واﻟﺘﻲ ﺗﺤﺴﺐ ﺑﻌﺪة ﻃﺮق ،أﺣﺪ هﺬﻩ اﻟﻄﺮق واﻟﺘﻲ ﺳﻨﺴﺘﺨﺪﻣﻬﺎ هﻲ وﺿﻊ s0ﻣﺴﺎوﻳﺔ ﻟﻠﻤﺘﻮﺳﻂ m
) , m = 6 ( or n, if n<6
∑z
i
i =1
m
= s0ﻓﻔﻲ ﻣﺜﺎﻟﻨﺎ
44.2 + 44.3 + 44.4 + 43.4 + 42.8 + 44.3 = 43.9 6
= s0
وﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن s1 = α z1 + (1 − α ) s0 = 0.2 ( 44.2 ) + 0.8 ( 43.9 ) = 8.84 + 35.12 = 43.96 s2 = α z2 + (1 − α ) s1 = 0.2 ( 44.3) + 0.8 ( 43.96 ) = 8.86 + 35.168 = 44.028 وهﻜﺬا ﻧﺴﺘﻤﺮ ﺣﺘﻲ ﺁﺧﺮ ﻣﺸﺎهﺪة ﻓﻴﻨﺘﺞ اﻟﺘﺎﻟﻲ: RESI1
FITS1
SMOO1
Metals
Time
0.30000 0.34000 0.37200 -0.70240 -1.16192 0.57046 0.55637 0.84510 0.27608 -1.07914 -1.36331 -1.29065 -1.23252 -1.38601 -2.80881 -0.34705 0.12236 0.79789 -0.06169 0.65065 0.62052 0.09642 0.27713 0.02171 -0.28264 -0.12611 -0.40089 0.27929 0.92343 1.83875 1.27100 1.51680 1.01344 0.91075 1.02860
43.9000 43.9600 44.0280 44.1024 43.9619 43.7295 43.8436 43.9549 44.1239 44.1791 43.9633 43.6906 43.4325 43.1860 42.9088 42.3470 42.2776 42.3021 42.4617 42.4494 42.5795 42.7036 42.7229 42.7783 42.7826 42.7261 42.7009 42.6207 42.6766 42.8613 43.2290 43.4832 43.7866 43.9892 44.1714
43.9600 44.0280 44.1024 43.9619 43.7295 43.8436 43.9549 44.1239 44.1791 43.9633 43.6906 43.4325 43.1860 42.9088 42.3470 42.2776 42.3021 42.4617 42.4494 42.5795 42.7036 42.7229 42.7783 42.7826 42.7261 42.7009 42.6207 42.6766 42.8613 43.2290 43.4832 43.7866 43.9892 44.1714 44.3771
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
١٧٤
0.82288 0.45830 0.86664 1.39331 1.71465 2.07172 2.45738 1.96590 2.37272 1.69818 1.85854 2.08683 1.66947 0.93557 1.04846 0.53877 1.53101 1.22481 1.17985 0.54388 1.13510 0.40808 -0.07353 -1.15883 -2.02706
44.3771 44.5417 44.6334 44.8067 45.0853 45.4283 45.8426 46.3341 46.7273 47.2018 47.5415 47.9132 48.3305 48.6644 48.8515 49.0612 49.1690 49.4752 49.7202 49.9561 50.0649 50.2919 50.3735 50.3588 50.1271
44.5417 44.6334 44.8067 45.0853 45.4283 45.8426 46.3341 46.7273 47.2018 47.5415 47.9132 48.3305 48.6644 48.8515 49.0612 49.1690 49.4752 49.7202 49.9561 50.0649 50.2919 50.3735 50.3588 50.1271 49.7216
45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
اﻟﻌﻤﻮد اﻟﺮاﺑﻊ SMOO١ﻳﺤﻮى اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة أي si , i = 1, 2,...,60اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ FITS١ ﻳﺤﻮى اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ أي zˆi = si −1 , i = 1, 2,...,60اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ RESI١ﻳﺤﻮي اﻷﺧﻄﺎء )اﻟﺒﻮاﻗﻲ (Residualsأي ei = zi − zˆi , i = 1, 2,...,60 -٢ﻳﺆﺧﺬ آﻤﺘﻨﺒﺊ ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺁﺧﺮ ﻗﻴﻤﺔ ﻣﻤﻬﺪة أي: z n ( l ) = sn , l > 0 ﻓﻔﻲ اﻟﻤﺜﺎل اﻟﺤﺎﻟﻲ z60 ( l ) = 49.7216, l > 0 ﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟـ ٦اﻟﻤﺴﺘﻘﺒﻠﻴﺔ أي ﻟﻠﻘﻴﻢ zn +1 , zn +2 ,..., zn +6أو ﻓﻲ هﺬا اﻟﻤﺜﺎل z61 , z62 ,..., z66آﺎﻟﺘﺎﻟﻲ: z60 (1) = z60 ( 2 ) = L = z60 ( 6 ) = 49.7216
-٣ﻟﺤﺴﺎب ﻓﺘﺮات ﺗﻨﺒﺆ 95%ﻧﺤﺴﺐ اﻟﻜﻤﻴﺎت ⎡⎣ zn ( l ) ± 1.96σˆ ⎤⎦ , l > 0أي ] ˆ [ 49.7216 ± 1.96σﻟﺠﻤﻴﻊ ﻗﻴﻢ اﻟﺘﻨﺒﺆات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ،ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ MSD = 1.45392آﻤﻘﺪر
ﻟـ σ 2أي σˆ 2 = 1.45392ﻓﻴﻜﻮن σˆ = 1.205786وﻋﻠﻴﻪ ﺗﻜﻮن ﻓﺘﺮة ﺗﻨﺒﺆ 95%ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ هﻲ: ]⎡⎣ 49.7216 ± 1.96 (1.205786 ) ⎤⎦ = [ 49.7216 ± 2.3633] = [ 47.35826,52.08494 أي: z60+l ∈ [ 47.3582,50.0849] , l > 0 with probability 0.95 ﻣﻼﺣﻈﺔ :ﺗﺤﺴﺐ MSDآﺎﻵﺗﻲ
١٧٥
) − zˆi
n
∑( z
i
n −1
i =1
= ˆMSD = σ 2
ﺗﻤﺮﻳﻦ: ﻃﺒﻖ ﺗﻤﻬﻴﺪ اﺳﻲ ﺑﺴﻴﻂ ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻣﺴﺘﺨﺪﻣﺎ α = 0.3,0.4,0.5وﻗﺮر اﻳﻬﺎ اﻓﻀﻞ ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ؟.
١٧٦
اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﻤﺰدوج Using Double : Exponential Smoothing for Forecasting أوﻻ :ﻃﺮﻳﻘﺔ ﺑﺮاون :Brown’s Method
ﻟﻤﺸﺎهﺪات z1 , z2 ,K , zn −1 , zn −2وﻟﺜﺎﺑﺖ ﺗﻤﻬﻴﺪ 0 < α < 1ﻧﻮﺟﺪ اﻟﺘﺎﻟﻲ:
s = α zt + (1 − α ) st(−1) , t = 1,2,..., n )(1 t
1
ﺣﻴﺚ ) ( stﺗﻤﻬﻴﺪ اﺳﻲ ﺑﺴﻴﻂ ) st( ) = stاﻧﻈﺮ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ( و ) (1ﺗﺮﻣﺰ اﻟﻲ درﺟﺔ هﺬا اﻟﺘﻤﻬﻴﺪ )(2 )(1 )(2 st = α st + (1 − α ) st −1 , t = 1,2,..., n 1
1
ﺣﻴﺚ ) ( stﺗﻤﻬﻴﺪ اﺳﻲ ﻣﺰدوج و ) ( 2ﺗﺮﻣﺰ اﻟﻲ درﺟﺔ هﺬا اﻟﺘﻤﻬﻴﺪ 2
at = 2 st( ) − st( ) , t = 1, 2,..., n 1
2
α 1 2 st( ) − st( ) , t = 1, 2,..., n 1−α
)
ﺗﺤﺴﺐ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻣﻦ اﻟﻤﻌﺎدﻟﺔ
t = 1, 2,..., n
وﺗﺤﺴﺐ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l > 0ﻣﻦ
zˆt = at + bt t ,
zn ( l ) = an + bn l, l > 0
ﺣﺴﺎب اﻟﻘﻴﻢ اﻻوﻟﻴﺔ ) ( s0و ) ( : s0ﻣﻦ اﻟﻌﻼﻗﺎت اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ 1
(
= bt
2
b0 b0 ﻧﻮﺟﺪ a0و b0ﺑﺈﻧﺤﺪار اﻟﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ t = 1, 2,..., n ˆ a0 = αو ˆb0 = β
1−α
α
1−α
α
s0( ) = a0 − 1
s0( ) = a0 − 2 2
zt = α + β t + et ,وﻳﻜﻮن
ﺛﺎﻧﻴﺎ :ﻃﺮﻳﻘﺔ هﻮﻟﺖ :Holt’s Method
ﻟﻤﺸﺎهﺪات z1 , z2 ,K , zn −1 , zn −2وﻟﺜﺎﺑﺘﻲ ﺗﻤﻬﻴﺪ 0 < α < 1و 0 < γ < 1ﻧﻮﺟﺪ اﻟﺘﺎﻟﻲ: st = α zt + (1 − α )( st −1 + bt −1 ) , t = 1, 2,..., n t = 1, 2,..., n ﻧﺤﺴﺐ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻣﻦ
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 , t = 1, 2,..., n
واﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻦ
zˆt = st + bt t ,
zn ( l ) = sn + bn l, l > 0
ﻧﺤﺴﺐ اﻟﻘﻴﻢ اﻻوﻟﻴﺔ s0و b0ﻣﻦ ١٧٧
s0 = z1 b0 = z2 − z1
b0 =
or
( z2 − z1 ) + ( z3 − z2 ) = ( z3 − z1 )
or 2 2 ( z − z ) + ( z3 − z2 ) + ( z4 − z3 ) ( z4 − z1 ) b0 = 2 1 = 3 3 :ﻣﺜﺎل EMPLOY.MTW ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ MTB > Retrieve 'E:\Mtbwin\DATA\EMPLOY.MTW'.
Metals ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
44.3 42.6 43.1 43.6 45.5 50.0 51.2
44.4 42.4 43.2 44.7 46.2 50.0 50.7
43.4 42.2 42.8 44.5 46.8 49.6 50.3
42.8 41.8 43.0 45.0 47.5 49.9 49.2
44.3 40.1 42.8 44.8 48.3 49.6 48.1
44.4 42.0 42.5 44.9 48.3 50.7
44.8 42.4 42.6 45.2 49.1 50.7
44.4 43.1 42.3 45.2 48.9 50.9
:ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات MTB > TSPlot 'Metals'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
Metals
50
45
40 Index
10
20
30
١٧٨
40
50
60
ﻧﺴﺘﺨﺪم اﻵن،ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﻤﺰدوج ﺑﻄﺮﻳﻘﺔ ﺑﺮاون ﺑﺄوزان ﻣﺘﺴﺎوﻳﺔ ﺳﻮف ﻧﺄﺧﺬهﺎWEIGHT ﻣﻊ اﻻﻣﺮ اﻟﻔﺮﻋﻲ%DES (Macro) اﻟﺒﺮﻣﺞ 0.2 MTB > %DES 'Metals'; SUBC> Weight 0.2 0.2; SUBC> Forecasts 6; SUBC> Title "Brown's Double Exponential Smoothing"; SUBC> Table.
Double Exponential Smoothing Data Length NMissing
Metals 60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Accuracy Measures MAPE: 2.16187 MAD: 0.97032 MSD: 1.62936
Time Metals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7
Smooth
Predict
Error
41.7739 42.4976 43.1686 43.5546 43.7373 44.1459 44.4990 44.8575 45.0620 44.9391 44.6673 44.3271 43.9378 43.4769 42.7011 42.3565 42.1465 42.1286 42.0132 42.0763 42.1877 42.2374 42.3396 42.4078 42.4181 42.4495 42.4207 42.5129 42.7420 43.1797
41.1674 42.0470 42.8607 43.5933 43.9716 44.1074 44.5238 44.8719 45.2275 45.3989 45.1841 44.8088 44.3723 43.8962 43.3514 42.4456 42.0831 41.8857 41.9164 41.8204 41.9347 42.0967 42.1745 42.3098 42.3976 42.4119 42.4509 42.4161 42.5276 42.7996
3.03257 2.25303 1.53927 -0.19330 -1.17163 0.19257 -0.12377 -0.07189 -0.82751 -2.29891 -2.58407 -2.40884 -2.17229 -2.09617 -3.25142 -0.44557 0.31694 1.21426 0.48355 1.27964 1.26533 0.70327 0.82549 0.49024 0.10244 0.18809 -0.15090 0.48394 1.07245 1.90036
١٧٩
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
43.5507 43.9854 44.3338 44.6511 44.9749 45.2430 45.4157 45.6373 45.9491 46.3285 46.7909 47.3492 47.8339 48.4002 48.8413 49.2966 49.7849 50.1841 50.4162 50.6292 50.7104 50.9509 51.1334 51.3019 51.3407 51.4782 51.4770 51.3649 51.0127 50.4384
43.3133 43.7317 44.2172 44.5889 44.9187 45.2538 45.5197 45.6716 45.8863 46.2106 46.6136 47.1115 47.7173 48.2253 48.8267 49.2707 49.7311 50.2302 50.6202 50.8114 50.9880 51.0137 51.2417 51.4024 51.5509 51.5477 51.6712 51.6311 51.4659 51.0230
1.18668 1.26826 0.58280 0.31112 0.28133 -0.05376 -0.51967 -0.17162 0.31368 0.58938 0.88637 1.18850 0.58266 0.87469 0.07332 0.12930 0.26890 -0.23017 -1.02022 -0.91145 -1.38797 -0.31368 -0.54169 -0.50244 -1.05093 -0.34770 -0.97120 -1.33115 -2.26587 -2.92300
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
50.3318 50.2252 50.1186 50.0120 49.9054 49.7988
47.9545 47.7984 47.6384 47.4749 47.3080 47.1381
52.7091 52.6520 52.5987 52.5490 52.5027 52.4594
Brown's Double Exponential Smoothing
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 MAPE: MAD: MSD:
40 0
10
20
30
Time
١٨٠
40
50
60
2.16187 0.97032 1.62936
: b0 وa0 إﻳﺠﺎد MTB > DATA> DATA> MTB >
set c4 1:60 end regr c3 1 c4
Regression Analysis The regression equation is Metals = 41.0 + 0.152 C4
1−α
وﻣﻨﻬﺎ ﻧﺤﺴﺐb0 = 0.152 وa0 = 41.0 إذا
0.8 ( 0.152 ) = 41.608 α 0.2 1−α ⎛ 0.8 ⎞ 2 s0( ) = a0 − 2 b0 = 41.0 − 2 ⎜ ⎟ ( 0.152 ) = 42.216 α ⎝ 0.2 ⎠ 1 s1( ) = ( 0.2 )( 44.2 ) + ( 0.8 )( 41.608 ) = 42.1264 s0( ) = a0 − 1
b0 = 41.0 −
s1( ) = ( 0.2 )( 42.1264 ) + ( 0.8 )( 42.216 ) = 42.19808 2
a1 = ( 2 )( 42.1264 ) − 42.19808 = 42.05472
( 0.2 ) ( 42.19808 − 42.05472 ) = 0.03584 ( 0.8) zˆ1 = 42.05472 + ( 0.03584 )(1) = 42.09056
b1 =
… وهﻜﺬا اﻟﺦ . آﻤﺎ ﻓﻲ اﻷﻣﺜﻠﺔ اﻟﺴﺎﺑﻘﺔMSD ﺗﺤﺴﺐ ﻓﺘﺮات اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام :ﻣﺜﺎل WEIGHT ﻣﻊ اﻻﻣﺮ اﻟﻔﺮﻋﻲ%DES (Macro) ﻟﺘﻄﺒﻴﻖ ﻃﺮﻳﻘﺔ هﻮﻟﺖ ﻧﺴﺘﺨﺪم اﻵن اﻟﺒﺮﻣﺞ γ = 0.3 وα = 0.2 ﺑﺄوزان ﻣﺨﺘﻠﻔﺔ ﺳﻮف ﻧﺄﺧﺬ MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'. Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW Worksheet was saved on 6/ 5/1996 MTB > %DES 'Metals'; SUBC> Weight 0.2 0.3; SUBC> Forecasts 6; SUBC> Title "Holt's Double Exponential Smoothing"; SUBC> Table.
Double Exponential Smoothing Data Length NMissing
Metals 60.0000 0
١٨١
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.3 Accuracy Measures MAPE: 2.15656 MAD: 0.96328 MSD: 1.56274
Time
Metals
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0
Smooth 41.7739 42.5461 43.2891 43.7501 43.9779 44.3895 44.7334 45.0685 45.2405 45.0676 44.7113 44.2595 43.7466 43.1634 42.2751 41.8139 41.5361 41.5057 41.4371 41.5799 41.8054 41.9895 42.2254 42.4205 42.5395 42.6522 42.6793 42.7982 43.0394 43.4861 43.8762 44.3257 44.6858 45.0007 45.3066 45.5449 45.6749 45.8384 46.0888 46.4159 46.8406 47.3799 47.8665 48.4419 48.9016 49.3693 49.8653 50.2702
Predict 41.1674 42.1076 43.0113 43.8376 44.2724 44.4118 44.8167 45.1356 45.4506 45.5595 45.2391 44.7244 44.1332 43.5043 42.8188 41.7674 41.3202 41.1072 41.1964 41.1999 41.4568 41.7869 42.0317 42.3257 42.5493 42.6653 42.7741 42.7728 42.8993 43.1826 43.7202 44.1571 44.6573 45.0259 45.3333 45.6312 45.8436 45.9230 46.0610 46.3199 46.6757 47.1499 47.7582 48.2774 48.9020 49.3617 49.8317 50.3378
Error 3.03257 2.19238 1.38868 -0.43760 -1.47237 -0.11184 -0.41671 -0.33560 -1.05057 -2.45952 -2.63911 -2.32443 -1.93322 -1.70426 -2.71884 0.23263 1.07985 1.99283 1.20365 1.90008 1.74322 1.01314 0.96829 0.47431 -0.04933 -0.06528 -0.47413 0.12724 0.70070 1.51743 0.77977 0.84285 0.14275 -0.12590 -0.13327 -0.43116 -0.84361 -0.42295 0.13895 0.48014 0.82428 1.15014 0.54181 0.82265 -0.00205 0.03832 0.16832 -0.33779
١٨٢
49 50 51 52 53 54 55 56 57 58 59 60
49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
50.4979 50.6862 50.7296 50.9166 51.0532 51.1813 51.1869 51.2901 51.2673 51.1350 50.7591 50.1448
50.7224 50.8827 51.0121 50.9708 51.1415 51.2516 51.3586 51.3127 51.4092 51.3438 51.1489 50.6560
-1.12240 -0.98275 -1.41206 -0.27079 -0.44152 -0.35162 -0.85860 -0.11267 -0.70916 -1.04381 -1.94889 -2.55603
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
49.8884 49.6319 49.3755 49.1190 48.8626 48.6061
47.5283 47.1597 46.7803 46.3915 45.9946 45.5908
52.2484 52.1041 51.9707 51.8466 51.7306 51.6215
Holt's Double Exponential Smoothing
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.300 MAPE: MAD: MSD:
40 0
10
20
30
40
50
2.15656 0.96328 1.56274
60
Time
ﻳﺘﻮﻗﻊ ﻋﺪم ﺗﻄﺎﺑﻖ: ﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ اﻟﺤﺴﺎﺑﺎت اﻟﺴﺎﺑﻘﺔ ﺑﺘﺘﺒﻊ ﺑﻌﺾ اﻟﻘﻴﻢ ﻳﺪوﻳﺎ ) ﻣﻼﺣﻈﺔ:ﺗﻤﺮﻳﻦ اﻟﺤﺴﺎﺑﺎت ﺗﻤﺎﻣﺎ وذﻟﻚ ﻹﺧﺘﻼف ﻃﺮﻳﻖ ﺗﻤﺜﻴﻞ اﻷﻋﺪاد ﺑﻴﻦ اﻟﺤﺎﺳﺐ واﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ وآﺬﻟﻚ ﻓﻲ (ﺗﺨﺰﻳﻦ اﻷرﻗﺎم ﻓﻲ ذاآﺮات آﻞ ﻣﻨﻬﻤﺎ
١٨٣
اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺜﻼﺛﻲ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ Triple Exponential Smoothing: Winters' ThreeParameter Trend and Seasonality Smoothing Method ﺟﻤﻴﻊ اﻟﻄﺮق اﻟﺴﺎﺑﻘﺔ اﻟﺘﻲ درﺳﻨﺎهﺎ ﻓﻲ هﺬا اﻟﻔﺼﻞ ﻻ ﺗﻨﻔﻊ ﻟﺘﺤﻠﻴﻞ اﻟﻈﻮاهﺮ اﻟﻤﻮﺳﻤﻴﺔ ﻣﺎﻋﺪى ﻃﺮﻳﻘﺔ اﻟﺘﻔﻜﻴﻚ Decomposition Methodوﻃﺮﻳﻘﺔ وﻧﺘﺮز Winters' trend and seasonal smoothingاﻟﺘﻲ ﺳﻮف ﻧﺴﺘﻌﺮﺿﻬﺎ هﻨﺎ
ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ أوﻻ :ﺗﻤﻬﺪ اﻟﻤﺸﺎهﺪات ﺗﻤﻬﻴﺪا آﻠﻴﺎ ﺑﺎﻟﻌﻼﻗﺔ zt + (1 − α )( st −1 + bt −1 ) , t = 1, 2,..., n St − s ﺛﺎﻧﻴﺎ :ﺗﻤﻬﻴﺪ اﻹﻧﺠﺮاف
st = α
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 , t = 1, 2,..., n
ﺛﺎﻟﺜﺎ :ﺗﻤﻬﻴﺪ اﻟﻤﻮﺳﻤﻴﺔ zt + (1 − β ) St − s , t = 1, 2,..., n st ﺣﻴﺚ Siهﻲ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻋﻨﺪ اﻟﺰﻣﻦ iو sهﻲ دورة اﻟﻤﻮﺳﻤﻴﺔ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﺗﻌﻄﻲ ﺑﺎﻟﻌﻼﻗﺔ zˆt = ( st + bt t ) St − s , t = 1, 2,..., n واﻟﺘﻨﺒﺆات ﻣﻦ اﻟﻌﻼﻗﺔ zn ( l ) = ( sn + bn l ) Sn − s+l , l > 0
St = β
ﻣﻦ اﻟﺼﻌﺐ ﺟﺪا ﺗﺘﺒﻊ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﺑﺎﻟﺤﺴﺎﺑﺎت اﻟﻴﺪوﻳﺔ ﺣﻴﺚ ان اﻟﻘﻴﻢ اﻻوﻟﻴﺔ ﺗﺤﺴﺐ ﺑﺨﻮارزﻣﺎت ﻏﻴﺮ ﺧﻄﻴﺔ ﺑﺈﺳﺘﺨﺪام اﻟﺤﺎﺳﺐ وﻟﻬﺬا ﻟﻦ ﻧﺴﺘﻌﺮﺿﻬﺎ هﻨﺎ وﻧﻜﺘﻔﻲ ﺑﺎﻟﻨﺘﺎﺋﺞ اﻟﻤﺨﺮﺟﺔ ﻣﻦ اﻟﺤﺎﺳﺐ. ﻣﺜﺎل: ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Food Food
52.1 51.5 53.5 54.3
53.4 52.3 53.1 54.6
55.3 53.6 53.3 54.2
58.2 55.3 53.9 54.8
66.9 58.5 55.6 55.8
70.7 69.3 60.1 57.9 57.7
65.3 69.6 68.9 62.6 60.5
١٨٤
56.5 64.2 68.8 70.3 66.4
53.4 55.5 63.6 69.4 75.5
52.5 53.3 57.1 64.7 73.3
53.2 52.4 52.2 57.1 68.1
53.0 51.5 51.5 53.9 58.1
53.5 51.5 51.7 53.5 54.8
وﻧﺮﺳﻢ اﻟﻤﺸﺎهﺪات
75
Food
70 65 60 55 50 Index
10
20
30
40
50
60
: ﻧﻄﺒﻖ اﻵن ﻃﺮﻳﻘﺔ وﻧﺘﺮز آﺎﻟﺘﺎﻟﻲ١٢ ﻧﻼﺣﻆ ان اﻟﻈﺎهﺮة ﻣﻮﺳﻤﻴﺔ ﺑﺪورة Additive Model ﻟﻠﻨﻤﻮذج اﻹﺿﺎﻓﻲ:أوﻻ zt = bt + St + et , t = 1, 2,..., n MTB > %Wintadd 'Food' 12; SUBC> Weight 0.2 0.2 0.2; SUBC> Forecasts 12; SUBC> Title "Wintrs' Trend and Seasonal Smoothing"; SUBC> Table.
Winters' additive model Data Length NMissing
Food 60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Delta (seasonal): 0.2 Accuracy Measures MAPE: 1.94769 MAD: 1.15100 MSD: 2.66711
١٨٥
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Food 53.5 53.0 53.2 52.5 53.4 56.5 65.3 70.7 66.9 58.2 55.3 53.4 52.1 51.5 51.5 52.4 53.3 55.5 64.2 69.6 69.3 58.5 55.3 53.6 52.3 51.5 51.7 51.5 52.2 57.1 63.6 68.8 68.9 60.1 55.6 53.9 53.3 53.1 53.5 53.5 53.9 57.1 64.7 69.4 70.3 62.6 57.9 55.8 54.8 54.2 54.6 54.3 54.8 58.1 68.1 73.3 75.5 66.4 60.5
Smooth 48.7755 49.6020 51.0736 52.0733 53.5117 57.4851 66.2299 71.7852 71.8932 62.3206 57.5208 55.1544 55.0393 53.6493 53.1185 52.2661 52.6528 55.7616 63.8483 68.8787 68.0386 59.2825 55.0979 53.0432 53.0377 52.1394 51.9889 51.7214 52.1875 55.0750 63.7515 68.8427 68.0160 58.9061 55.3220 53.4419 53.3084 52.6717 52.9095 52.9745 53.7952 57.2065 65.2747 70.4039 69.6200 60.5655 57.0392 55.3721 55.2403 54.6681 54.8138 54.7366 55.3001 58.5310 66.4168 71.8806 71.8794 63.7240 60.3141
Predict 49.4303 50.4197 51.9944 53.0424 54.4591 58.3901 67.0593 72.5443 72.5785 62.7787 57.7958 55.3296 55.1373 53.6258 53.0100 52.0971 52.4960 55.6369 63.7181 68.7678 67.9610 59.2585 55.0435 52.9991 53.0177 52.0907 51.9165 51.6403 52.1009 54.9923 63.7531 68.8382 68.0099 58.9356 55.3981 53.5262 53.4076 52.7666 53.0177 53.1020 53.9386 57.3484 65.4066 70.5076 69.6794 60.6497 57.2014 55.5623 55.4399 54.8422 54.9622 54.8705 55.4112 58.6176 66.4827 72.0112 72.0616 64.0437 60.7281
Error 4.06965 2.58027 1.20556 -0.54244 -1.05914 -1.89013 -1.75932 -1.84430 -5.67851 -4.57874 -2.49577 -1.92957 -3.03734 -2.12584 -1.50996 0.30287 0.80401 -0.13695 0.48187 0.83218 1.33902 -0.75851 0.25650 0.60092 -0.71765 -0.59067 -0.21651 -0.14031 0.09913 2.10774 -0.15314 -0.03822 0.89007 1.16436 0.20190 0.37384 -0.10757 0.33345 0.48233 0.39803 -0.03858 -0.24838 -0.70661 -1.10758 0.62065 1.95031 0.69858 0.23773 -0.63993 -0.64220 -0.36218 -0.57048 -0.61119 -0.51765 1.61731 1.28878 3.43843 2.35629 -0.22810
١٨٦
60
57.7
58.6397
59.0446
-1.34455
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6 7 8 9 10 11 12
61 62 63 64 65 66 67 68 69 70 71 72
58.6167 58.3236 58.8195 58.9840 59.8723 63.4804 72.0757 77.4486 77.7540 68.9067 64.6434 62.7731
55.7968 55.4449 55.8775 55.9746 56.7913 60.3243 68.8410 74.1321 74.3528 65.4180 61.0647 59.1020
61.4366 61.2023 61.7614 61.9935 62.9532 66.6365 75.3104 80.7651 81.1552 72.3954 68.2221 66.4441
Wintrs' Trend and Seasonal Smoothing
Actual
80
Predicted Forecast Actual Predicted Forecast
Food
70
60
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 Delta (season):0.200
50
MAPE: MAD: MSD:
0
10
20
30
40
50
60
1.94769 1.15100 2.66711
70
Time
zt = bt St + et , t = 1, 2,..., n
Multiplicative Model ﻟﻠﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:ﺛﺎﻧﻴﺎ
MTB > %Wintmult 'Food' 12; SUBC> Weight 0.2 0.2 0.2; SUBC> Forecasts 12; SUBC> Title "Winters' Trend and Seasonal Smoothing"; SUBC> Table.
Winters' multiplicative model Data
Food
١٨٧
Length NMissing
60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Delta (seasonal): 0.2 Accuracy Measures MAPE: 1.88377 MAD: 1.12068 MSD: 2.86696
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Food 53.5 53.0 53.2 52.5 53.4 56.5 65.3 70.7 66.9 58.2 55.3 53.4 52.1 51.5 51.5 52.4 53.3 55.5 64.2 69.6 69.3 58.5 55.3 53.6 52.3 51.5 51.7 51.5 52.2 57.1 63.6 68.8 68.9 60.1 55.6 53.9 53.3 53.1 53.5 53.5 53.9 57.1 64.7 69.4
Smooth
48.7870 49.6755 51.1521 52.1675 53.6181 57.6509 66.6199 72.4105 72.5679 62.7837 57.9154 55.5108 54.4920 53.2117 52.8118 52.0929 52.5894 55.7388 63.7189 68.7087 67.9722 59.4594 55.4037 53.4103 52.6818 51.8659 51.8002 51.6271 52.1643 55.0424 63.6079 68.6702 67.9561 59.1021 55.6210 53.7881 53.0479 52.4502 52.7444 52.8747 53.7689 57.2790 65.4702 70.6713
Predict
49.3853 50.4303 52.0132 53.0746 54.5132 58.5541 67.5607 73.3280 73.3777 63.2634 58.1732 55.6485 54.5392 53.1621 52.6957 51.9302 52.4439 55.6209 63.5782 68.5838 67.8890 59.4361 55.3468 53.3536 52.6356 51.8071 51.7290 51.5549 52.0890 54.9676 63.6199 68.6823 67.9727 59.1487 55.7003 53.8609 53.1211 52.5294 52.8467 53.0029 53.9188 57.4374 65.6357 70.8095
Error
4.11470 2.56966 1.18677 -0.57458 -1.11323 -2.05414 -2.26072 -2.62800 -6.47768 -5.06337 -2.87320 -2.24849 -2.43920 -1.66212 -1.19573 0.46985 0.85611 -0.12087 0.62178 1.01617 1.41104 -0.93606 -0.04680 0.24639 -0.33562 -0.30705 -0.02902 -0.05492 0.11103 2.13244 -0.01988 0.11774 0.92727 0.95133 -0.10032 0.03912 0.17892 0.57055 0.65329 0.49714 -0.01879 -0.33743 -0.93567 -1.40954
١٨٨
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
70.3 62.6 57.9 55.8 54.8 54.2 54.6 54.3 54.8 58.1 68.1 73.3 75.5 66.4 60.5 57.7
69.8908 60.7552 57.1925 55.5181 54.8764 54.3244 54.5372 54.5298 55.1925 58.6054 66.7739 72.4056 72.3385 63.6729 60.0395 58.3023
69.9719 60.8370 57.3348 55.6775 55.0383 54.4749 54.6769 54.6661 55.3155 58.7141 66.8698 72.5622 72.5236 63.9378 60.3781 58.6338
0.32815 1.76302 0.56523 0.12253 -0.23826 -0.27486 -0.07694 -0.36612 -0.51551 -0.61410 1.23016 0.73784 2.97638 2.46217 0.12191 -0.93381
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6 7 8 9 10 11 12
61 62 63 64 65 66 67 68 69 70 71 72
57.8102 57.3892 57.8332 57.9307 58.8311 62.7415 72.1849 78.1507 78.5092 68.6689 63.9258 61.8189
55.0645 54.5864 54.9687 55.0005 55.8313 59.6686 69.0354 74.9215 75.1976 65.2721 60.4414 58.2446
60.5558 60.1921 60.6977 60.8609 61.8309 65.8145 75.3344 81.3798 81.8208 72.0657 67.4103 65.3933
Winters' Trend and Seasonal Smoothing
Actual
80
Predicted Forecast Actual Predicted Forecast
Food
70
60
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 Delta (season):0.200
50
MAPE: MAD: MSD:
0
10
20
30
40
50
60
1.88377 1.12068 2.86696
70
Time
:ﻣﻼﺣﻈﺎت ﺛﺎﺑﺖγ ﺛﺎﺑﺖ اﻟﺘﻤﻬﻴﺪ اﻟﻜﻠﻲ وα ﻟﺘﻄﺒﻴﻖ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻧﺤﺘﺎج اﻟﻲ إﺧﺘﻴﺎر ﻗﻴﻢ ﺛﻼﺛﺔ ﻣﻌﺎﻟﻢ هﻲ ﻓﻲ ﺛﻼﺛﺔOptimization ﺛﺎﺑﺖ ﺗﻤﻬﻴﺪ اﻟﻤﻮﺳﻤﻴﺔ وهﺬﻩ ﻋﻤﻠﻴﺔ أﻓﻀﻠﻴﺔβ ﺗﻤﻬﻴﺪ اﻹﻧﺠﺮاف و ١٨٩
اﺑﻌﺎد ) ﻓﻀﺎء اﻟﻤﻌﺎﻟﻢ ( إﻣﺎ أن ﻧﺘﺮك ﻟﻠﺒﺮﻧﺎﻣﺞ اﻹﺣﺼﺎﺋﻲ اﻟﻤﺴﺘﺨﺪم ﺣﺴﺎﺑﻬﺎ ﺗﻠﻘﺎﺋﻴﺎ ﺑﺈﺳﺘﺨﺪام ﺧﻮارزﻣﺎت ﻏﻴﺮ ﺧﻄﻴﺔ ﻣﺒﻨﻴﺔ داﺧﻞ اﻟﺒﺮﻧﺎﻣﺞ أو ﻧﻘﻮم ﻧﺤﻦ ﺑﺈﻣﺪاد اﻟﺒﺮﻧﺎﻣﺞ ﺑﺘﻠﻚ اﻟﻘﻴﻢ. ﺗﻤﺮﻳﻦ :ﻓﻲ اﻷﻣﺜﻠﺔ اﻟﺴﺎﺑﻘﺔ اﺧﺬﻧﺎ . α = γ = β = 0.2ﺛﺒﺖ ﻓﻲ آﻞ ﻣﺮة ﻣﻌﻠﻤﻴﻦ وﻏﻴﺮ اﻟﺜﺎﻟﺚ ﺣﺘﻲ ﺗﺤﺼﻞ ﻋﻠﻲ أﻗﻞ MSD ﻣﺜﺎل ﻋﻤﻠﻲ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: ﺳﻮف ﻧﺴﺘﺨﺪم ورﻗﺔ اﻟﻌﻤﻞ CPI.MTWﻣﻦ ﻣﺠﻤﻮﻋﺔ اﻟﺒﻴﺎﻧﺎت ﻟﻠﺒﺮﻧﺎﻣﺞ MINITAB MTB > Retrieve
'C:\MTBWIN\STUDENT9\CPI.MTW'.
ﺳﻮف ﻧﺄﺧﺬ اﻟﻤﺘﻐﻴﺮ CPIChange 2.9 6.2 13.5 3.6
1.6 3.2 11.3 1.9
1.3 5.7 6.5 4.3 4.2
1.3 4.4 7.6 3.6 3.0
1.0 4.2 9.1 6.2 4.8
1.0 5.5 5.8 3.2 5.4
CPIChnge 1.7 3.1 11.0 10.3 4.1
ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﻮﺟﺪ اﻟﺘﺮﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻟﻌﻴﻨﻴﺔ MTB > TSPlot
14 12 10
6
CPIChnge
8
4 2 0 30
20
25
10
15
Index
5
MTB > %acf c2 Autocorrelation Function for CPIChnge
8
7
5
6
4
Corr
Lag
LBQ
T
41.23
8 -0.16 -0.51
١٩٠
3
2
Corr
1
Lag
LBQ
T
22.42 30.32 33.59 36.24 38.97 40.05 40.07
1 0.79 4.53 2 0.46 1.77 3 0.29 1.02 4 0.26 0.88 5 0.26 0.86 6 0.16 0.52 7 -0.02 -0.06
Autocorrelation
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
MTB > %pacf c2 Partial Autocorrelation
Partial Autocorrelation Function for CPIChnge 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag PAC 1 2 3 4 5 6 7
0.79 -0.42 0.35 -0.04 0.08 -0.28 -0.05
5
6
T
Lag PAC
T
4.53 -2.44 2.01 -0.24 0.43 -1.62 -0.29
8 -0.09
-0.52
7
8
ARMA(١،١) ﻣﻦ اﻧﻤﺎط اﻟﺘﺮﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻟﻌﻴﻨﻴﺔ ﻗﺪ ﻳﻜﻮن اﻟﻨﻤﻮذج ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ٥ اﻷواﻣﺮ اﻟﺘﺎﻟﻴﺔ ﺗﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح وﺗﻮﻟﺪ،ﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ MTB > SUBC> SUBC> SUBC> SUBC> SUBC> SUBC>
arima 1 0 1 c2; fore 5 c3 c4 c5; gser; gacf; gpacf; ghist; gnormal.
ARIMA Model ARIMA model for CPIChnge Estimates at each iteration Iteration SSE Parameters 0 323.251 0.100 0.100 4.522 1 200.616 0.250 -0.050 3.745 2 182.146 0.184 -0.200 4.067 3 163.067 0.135 -0.350 4.308 4 142.864 0.107 -0.500 4.434 5 121.402 0.111 -0.650 4.407 6 99.668 0.150 -0.800 4.197 7 77.036 0.268 -0.950 3.590 8 67.550 0.418 -0.956 2.828 9 62.802 0.568 -0.964 2.062 10 62.108 0.637 -0.973 1.687 11 62.030 0.644 -0.979 1.619 12 62.003 0.647 -0.982 1.584 13 61.996 0.651 -0.985 1.549 14 61.996 0.651 -0.986 1.539 Unable to reduce sum of squares any further Final Estimates of Parameters
١٩١
Type AR 1 MA 1 Constant Mean
Coef 0.6513 -0.9857 1.5385 4.412
StDev 0.1434 0.0516 0.4894 1.403
T 4.54 -19.11 3.14
Number of observations: 33 Residuals: SS = 61.8375 (backforecasts excluded) MS = 2.0613 DF = 30 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 9.6(DF=10) 17.0(DF=22) * (DF= *)
48 * (DF= *)
Forecasts from period 33 Period 34 35 36 37 38
95 Percent Limits Lower Upper 0.3216 5.9507 -1.8180 8.9801 -2.3061 10.0476 -2.4192 10.5380 -2.4201 10.7848
Forecast 3.1362 3.5810 3.8708 4.0594 4.1823
zt = 1.54 + 0.65zt −1 + at − 0.99at −1 , at
Actual
أي ان اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
N ( 0, 2.06 )
هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر ˆ ˆ φ1 = 0.6513, s.e. φ1 = 0.1434, t = 4.54
( ) θˆ = −0.9857, s.e. (θˆ ) = 0.0516, t = −19.11 δˆ = 1.5385, s.e. (δˆ ) = 0.4894, t = 3.14 1
1
σˆ 2 = 2.0613, with d . f . = 30
.ﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻌﺎﻟﻢ ﻣﻌﻨﻮﻳﺔ
:اﻵن ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ
ACF of Residuals for CPIChnge (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
١٩٢
PACF of Residuals for CPIChnge (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1
2
3
4
5
6
7
8
Lag
أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ ﺗﺪل ﻋﻠﻰ أن اﻟﺒﻮاﻗﻲ ﺗﺘﺒﻊ ﺗﻮزﻳﻊ ﺿﺠﺔ : ﻟﻨﻔﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ،ﺑﻴﻀﺎء أي ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ رﺳﻢ اﻟﻤﺪرج اﻟﺘﻜﺮاري
Histogram of the Residuals (response is CPIChnge) 8 7
Frequency
6 5 4 3 2
١٩٣
.ﻳﺒﺪو ﻣﺘﻨﺎﻇﺮ ﺑﻌﺾ اﻟﺸﻴﺊ :ﻟﻨﻨﻈﺮ إﻟﻰ ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ ﻟﻠﺒﻮاﻗﻲ
Normal Probability Plot of the Residuals (response is CPIChnge) 4 3
Residual
2 1 0 -1 -2 -3 -2
-1
0
1
2
Normal Score
.ﻧﺴﺘﻄﻴﻊ أن ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ ﺗﻘﺮﻳﺒﺎ . ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ٥ اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﺔ ﻣﻊ
Time Series Plot for CPIChnge (with forecasts and their 95% confidence limits)
CPIChnge
10
5
0
١٩٤
: ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎﻟﺘﺎﻟﻲAR(٢) دﻋﻨﺎ ﻧﺤﺎول ﺗﻄﺒﻴﻖ ﻧﻤﻮذج MTB > arima 2 0 0 c2 Type AR 1 AR 2 Constant Mean
Coef 1.1872 -0.4657 1.3270 4.765
StDev 0.1625 0.1624 0.2996 1.076
T 7.31 -2.87 4.43
Number of observations: 33 Residuals: SS = 88.6206 (backforecasts excluded) MS = 2.9540 DF = 30 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 19.8(DF=10) 25.4(DF=22) * (DF= *)
zt = 1.33 + 1.187 zt −1 − 0.4657 zt −1 + at , at
N ( 0, 2.95)
48 * (DF= *)
أي ان اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر
( ) φˆ = −0.4657, s.e. (θˆ ) = 0.1624, t = −2.87 δˆ = 1.327, s.e. (δˆ ) = 0.2996, t = 4.43
φˆ1 = 1.1872, s.e. φˆ1 = 0.1625, t = 7.31 2
1
σˆ 2 = 2.954, with d . f . = 30
ﻟﻨﻨﻈﺮ إﻟﻰ اﻹﺧﺘﺒﺎر
H 0 : φ2 = 0 H 1 : φ2 ± 0
اﻹﺣﺼﺎﺋﺔ t0 =
φˆ
2
( )
s.e. φˆ2
=
−0.4657 = −2.8676 0.1624
ﻟﻬﺎ ﺑﺎﻷﻣﺮP-value ﻧﻮﺟﺪ اﻟـ MTB > cdf -2.8676; SUBC> t 30.
Cumulative Distribution Function
١٩٥
Student's t distribution with 30 DF )P( X <= x 0.0037
x -2.8676
أي اﻟـ P-valueﻟﻬﺎ ﺗﺴﺎوي ٠٫٠٠٣٧وهﻲ أﻗﻞ ﻣﻦ ٠٫٠٥أي ﻻﻧﺮﻓﺾ ان φ2 = 0وﺑﺎﻟﺘﺎﻟﻲ ﻧﺮﻓﺾ اﻟﻨﻤﻮذج ). AR(٢ ﺗﻤﺮﻳﻦ: ﺣﺎول ﺗﻄﺒﻴﻖ ﻧﻤﺎذج اﺧﺮى ﻣﻨﺎﺳﺒﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺴﺎﺑﻘﺔ وإﺧﺘﺎر أﻓﻀﻞ ﻧﻤﻮذج ،أﺟﺮي اﻹﺧﺘﺒﺎرات اﻟﻤﻨﺎﺳﺒﺔ واﺳﺘﺨﺪم اﻳﻀﺎ اﻟﻤﻌﻴﺎر . AIC ﻣﺜﺎل ﻋﻤﻠﻲ ﺁﺧﺮ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: ﺳﻮف ﻧﺤﺎول ﺑﻨﺎء ﻧﻤﻮذج ﻟﻠﻤﺘﺴﻠﺴﻠﺔ )z(t -103.2 -391.4 -836.1 -1307.3 -1736.9 -2097.5 -2363.4 -2721.4 -3353.3 -4153.0 -5068.7 -5848.8 -6387.3 -6834.5 -7259.9 -7891.4 -8791.2 -9878.4 -11071.9 -12157.5 -13190.3 -14196.3 -15335.9 -16697.1 -18155.6 -19547.0 -20827.7 -21908.3
-76.7 -339.9 -766.7 -1242.9 -1679.0 -2055.4 -2328.5 -2651.9 -3253.4 -4028.2 -4937.2 -5760.6 -6317.8 -6773.3 -7193.3 -7784.7 -8649.8 -9713.3 -10903.2 -12011.7 -13039.8 -14051.7 -15158.3 -16492.7 -17951.5 -19356.3 -20655.7 -21762.1
-52.8 -291.3 -698.2 -1177.4 -1620.8 -2010.0 -2290.6 -2589.5 -3156.1 -3906.6 -4805.0 -5663.8 -6244.3 -6711.8 -7131.7 -7683.2 -8512.3 -9552.0 -10734.0 -11863.9 -12889.7 -13910.1 -14986.4 -16290.9 -17745.2 -19161.7 -20478.0 -21614.0
-33.1 -246.0 -631.6 -1111.3 -1561.5 -1960.6 -2250.5 -2533.0 -3060.8 -3788.7 -4673.3 -5558.0 -6167.9 -6649.0 -7073.1 -7586.4 -8379.1 -9394.6 -10564.0 -11713.8 -12740.8 -13769.9 -14819.8 -16091.5 -17535.7 -18965.1 -20294.5 -21463.7 -22474.4
وﻟﻬﺎ اﻟﺮﺳﻢ اﻟﺰﻣﻨﻲ اﻟﺘﺎﻟﻲ:
١٩٦
-19.2 -204.0 -566.7 -1044.2 -1500.0 -1906.9 -2210.8 -2482.0 -2968.6 -3675.0 -4542.3 -5444.4 -6090.4 -6584.6 -7015.6 -7495.3 -8249.2 -9239.6 -10392.1 -11560.2 -12593.4 -13629.3 -14657.5 -15895.8 -17324.4 -18766.2 -20107.3 -21310.9 -22335.1
-9.1 -165.9 -504.8 -975.1 -1436.5 -1851.1 -2173.3 -2437.6 -2880.4 -3564.3 -4412.0 -5323.6 -6011.8 -6518.9 -6957.1 -7411.2 -8124.0 -9086.3 -10219.0 -11402.0 -12447.5 -13486.3 -14499.7 -15705.0 -17113.5 -18564.2 -19919.5 -21154.4 -22195.6
-2.5 -132.4 -446.4 -905.3 -1371.8 -1794.5 -2136.4 -2398.7 -2797.6 -3456.9 -4281.7 -5197.8 -5931.6 -6453.5 -6896.0 -7332.9 -8004.6 -8936.4 -10047.2 -11238.9 -12302.5 -13339.6 -14345.9 -15518.4 -16904.2 -18360.1 -19733.5 -20993.7 -22053.2
O r ig in a l T im e S e r ie s
-1 0 0 0 0
-2 0 0 0 0
In d e x
50
100
150
200
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ هﻲ
Autocorrelation Function for z(t) Autocorrelation
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
10 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9813.93 196.91 0.97 8.00 388.73 0.95 6.15 575.44 0.94 5.15 757.06 0.92 4.50 933.59 0.91 4.041105.06 0.89 3.681271.47 0.88 3.391432.85 0.86 3.151589.23 0.84 2.951740.66 0.83 2.781887.17 0.81 2.622028.81
Lag Corr 13 14 15 16 17 18 19 20 21 22 23 24
20 T
LBQ
2.49 2165.64 2.37 2297.73 2.25 2425.14 2.15 2547.94 2.06 2666.22 1.97 2780.05 1.89 2889.51 1.82 2994.69 1.75 3095.67 1.68 3192.54 1.62 3285.39 1.56 3374.32
0.80 0.78 0.76 0.75 0.73 0.72 0.70 0.68 0.67 0.65 0.64 0.62
30
Lag Corr 25 26 27 28 29 30 31 32 33 34 35 36
0.61 0.59 0.58 0.56 0.55 0.53 0.52 0.50 0.49 0.47 0.46 0.45
T
LBQ
1.50 3459.42 1.45 3540.78 1.40 3618.50 1.35 3692.67 1.30 3763.38 1.26 3830.74 1.21 3894.84 1.17 3955.76 1.13 4013.61 1.09 4068.47 1.05 4120.44 1.01 4169.59
40
Lag Corr 37 38 39 40 41 42 43 44 45 46 47 48
0.43 0.42 0.41 0.39 0.38 0.36 0.35 0.34 0.32 0.31 0.30 0.28
T
LBQ
50
Lag Corr
T
LBQ
49 0.27 0.594589.24 50 0.26 0.564607.24
0.98 4216.02 0.94 4259.81 0.91 4301.04 0.87 4339.79 0.84 4376.14 0.81 4410.16 0.77 4441.93 0.74 4471.52 0.71 4498.99 0.68 4524.43 0.65 4547.90 0.62 4569.48
P artial A utocorrelation Function for z(t) Partial Autocorrelation
z(t)
0
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
10
20
30
40
Lag
PAC
T
Lag
PAC
T
Lag
PAC
T
Lag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0.98 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
13.93 -0.17 -0.17 -0.16 -0.16 -0.16 -0.15 -0.15 -0.15 -0.14 -0.14 -0.13
13 14 15 16 17 18 19 20 21 22 23 24
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.13 -0.12 -0.12 -0.11 -0.11 -0.11 -0.11 -0.10 -0.10 -0.09 -0.09 -0.09
25 26 27 28 29 30 31 32 33 34 35 36
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.09 -0.09 -0.09 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.09 -0.09
37 38 39 40 41 42 43 44 45 46 47 48
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.09 -0.09 -0.10 -0.10 -0.11 -0.11 -0.12 -0.12 -0.12 -0.13 -0.13 -0.13
50 Lag
PAC
T
49 -0.01 50 -0.01
-0.14 -0.14
. ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂzt واﺿﺢ ﺟﺪا ان اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﺮﺳﻤﻬﺎwt = zt − zt −1 ﻧﺄﺧﺬ اﻟﻔﺮوق اﻻوﻟﻰ ١٩٧
F irs t D if f e re n c e s w (t)= z (t)-z (t-1 )
w(t)
0
-1 0 0
-2 0 0 In d e x
50
100
150
200
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ هﻲ Autocorrelation
Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9913.93 197.10 0.97 7.98 389.03 0.95 6.12 574.74 0.93 5.10 753.55 0.91 4.43 924.81 0.89 3.941087.93 0.86 3.561242.44 0.83 3.251388.10 0.81 2.981524.83 0.78 2.761652.58 0.75 2.561771.42 0.72 2.381881.49
15
Lag Corr 13 14 15 16 17 18 19 20 21 22 23 24
0.69 0.66 0.63 0.60 0.57 0.54 0.52 0.49 0.47 0.45 0.43 0.41
T
LBQ
2.21 1983.03 2.07 2076.38 1.93 2161.98 1.81 2240.33 1.70 2311.99 1.60 2377.42 1.50 2437.13 1.41 2491.57 1.33 2541.23 1.26 2586.61 1.19 2628.22 1.13 2666.48
25 Lag Corr 25 26 27 28 29 30 31 32 33 34 35 36
0.39 0.38 0.37 0.36 0.35 0.34 0.34 0.34 0.34 0.34 0.34 0.34
T
LBQ
1.08 2701.91 1.03 2734.99 1.00 2766.16 0.96 2795.84 0.94 2824.46 0.92 2852.35 0.91 2879.79 0.90 2907.03 0.89 2934.32 0.89 2961.85 0.89 2989.82 0.89 3018.34
35 Lag Corr 37 38 39 40 41 42 43 44 45 46 47 48
0.34 0.35 0.35 0.35 0.35 0.35 0.35 0.34 0.34 0.33 0.33 0.32
T
45 LBQ
0.90 3047.50 0.90 3077.32 0.90 3107.68 0.90 3138.42 0.90 3169.33 0.89 3200.21 0.88 3230.81 0.87 3260.99 0.85 3290.64 0.84 3319.59 0.82 3347.55 0.79 3374.17
Lag Corr
T
LBQ
49 0.31 0.763399.16
Partial Autocorrelation
P a rtia l A u to c o rre la tio n F u n c tio n fo r w (t) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5 Lag
P AC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .9 9 -0 .1 4 -0 .1 1 -0 .0 7 -0 .0 6 -0 .0 7 -0 .0 6 -0 .0 3 -0 .0 3 -0 .0 3 -0 .0 2 -0 .0 2
1 3 .9 3 -1 .9 7 -1 .5 1 -0 .9 6 -0 .9 0 -0 .9 7 -0 .7 8 -0 .4 8 -0 .3 7 -0 .3 7 -0 .3 4 -0 .2 2
15
Lag
P AC
T
1 3 -0 .0 1 1 4 -0 .0 0 1 5 0 .0 1 1 6 0 .0 2 1 7 0 .0 2 1 8 0 .0 1 1 9 0 .0 1 2 0 0 .0 1 2 1 0 .0 1 2 2 0 .0 3 2 3 0 .0 2 2 4 0 .0 2
-0 .1 6 -0 .0 2 0 .1 0 0 .2 3 0 .2 5 0 .1 0 0 .1 2 0 .1 1 0 .1 6 0 .3 5 0 .3 1 0 .2 7
25 Lag
35
45
P AC
T
Lag
P AC
T
2 5 0 .0 4 2 6 0 .0 4 2 7 0 .0 3 2 8 0 .0 4 2 9 0 .0 4 3 0 0 .0 2 3 1 0 .0 2 3 2 0 .0 1 3 3 0 .0 1 3 4 0 .0 1 3 5 0 .0 0 3 6 -0 .0 0
0 .5 2 0 .6 2 0 .4 7 0 .5 5 0 .5 6 0 .3 4 0 .2 2 0 .1 6 0 .1 8 0 .1 5 0 .0 4 -0 .0 7
37 38 39 40 41 42 43 44 45 46 47 48
-0 .0 1 -0 .0 2 -0 .0 4 -0 .0 4 -0 .0 4 -0 .0 4 -0 .0 3 -0 .0 0 0 .0 1 -0 .0 1 -0 .0 4 -0 .0 5
-0 .0 7 -0 .2 9 -0 .6 3 -0 .6 0 -0 .5 4 -0 .5 5 -0 .4 2 -0 .0 7 0 .0 8 -0 .1 8 -0 .5 6 -0 .7 6
Lag
P AC
T
4 9 -0 .0 3
-0 .4 8
. ﻻﺗﺰال ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂwt واﺿﺢ ﺟﺪا ان اﻟﻤﺘﺴﻠﺴﻠﺔ )ﻻﺣﻆ ان هﺬا اﻟﻔﺮق اﻟﺜﺎﻧﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻷﺻﻠﻴﺔ( وﻧﺮﺳﻤﻬﺎyt = wt − wt −1 ﻧﺄﺧﺬ اﻟﻔﺮوق اﻻوﻟﻰ
١٩٨
F ir s t D if f e r e n c e s
y (t)= w (t)-w (t-1 )
1 0
y(t)
5
0
-5
In d e x
5 0
1 0 0
1 5 0
2 0 0
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ هﻲ Autocorrelation
A utocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9313.08 173.62 0.80 6.83 303.36 0.70 4.89 401.81 0.62 3.89 479.62 0.56 3.26 542.93 0.50 2.77 593.77 0.43 2.32 632.35 0.36 1.90 659.84 0.29 1.50 677.66 0.22 1.11 687.71 0.15 0.75 692.41 0.08 0.41 693.82
15
Lag Corr
T
LBQ
25 Lag Corr
13 0.01 0.06 693.85 14 -0.05 -0.23 694.31 15 -0.09 -0.46 696.12 16 -0.14 -0.70 700.34 17 -0.19 -0.97 708.51 18 -0.25 -1.24 722.16 19 -0.31 -1.53 743.30 20 -0.37 -1.83 774.40 21 -0.44 -2.13 818.29 22 -0.51 -2.40 876.73 23 -0.56 -2.57 948.00 1030.79 24 -0.60 -2.67
T
LBQ
1124.21 25 -0.64 -2.73 1224.05 26 -0.66 -2.72 1324.28 27 -0.66 -2.62 1418.06 28 -0.63 -2.44 1500.63 29 -0.59 -2.22 1572.45 30 -0.55 -2.01 1632.42 31 -0.50 -1.80 1677.44 32 -0.43 -1.53 1708.46 33 -0.36 -1.25 1728.63 34 -0.29 -1.00 1740.51 35 -0.22 -0.76 1746.42 36 -0.16 -0.53
35 Lag Corr
T
45 LBQ
1748.28 37 -0.09 -0.30 1748.29 38 -0.01 -0.03 1749.70 39 0.08 0.26 1755.14 40 0.15 0.50 1765.20 41 0.20 0.68 1778.55 42 0.23 0.78 1793.44 43 0.24 0.82 1810.08 44 0.25 0.86 1831.09 45 0.28 0.96 1860.92 46 0.34 1.13 1903.36 47 0.40 1.34 1956.81 48 0.45 1.49
Lag Corr
T
LBQ
49 0.48 1.562017.77
Partial Autocorrelation
P artial A utocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag PAC
T
0.93 -0.46 0.33 -0.14 0.12 -0.13 0.00 -0.06 -0.08 -0.04 -0.04 -0.09
13.08 -6.44 4.71 -1.97 1.72 -1.78 0.01 -0.81 -1.11 -0.55 -0.58 -1.21
1 2 3 4 5 6 7 8 9 10 11 12
15
Lag PAC 13 14 15 16 17 18 19 20 21 22 23 24
-0.05 0.05 -0.07 -0.09 -0.04 -0.08 -0.12 -0.11 -0.15 -0.07 -0.06 -0.15
25
T
Lag
PAC
T
-0.75 0.67 -0.95 -1.30 -0.55 -1.11 -1.71 -1.50 -2.12 -0.93 -0.87 -2.12
25 26 27 28 29 30 31 32 33 34 35 36
-0.03 -0.03 0.00 0.08 -0.01 -0.04 0.15 0.06 -0.00 0.04 0.03 0.04
-0.46 -0.44 0.04 1.13 -0.14 -0.56 2.17 0.88 -0.05 0.49 0.46 0.50
35 Lag PAC 37 38 39 40 41 42 43 44 45 46 47 48
0.08 0.11 0.04 0.04 -0.05 -0.05 -0.14 -0.01 0.07 0.06 -0.01 -0.09
45 T
1.16 1.56 0.51 0.52 -0.74 -0.66 -1.95 -0.10 0.92 0.79 -0.17 -1.24
Lag PAC 49
0.07
T 1.00
ﻧﻼﺣﻆ ﻣﻦ ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ و اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻧﻬﺎ اﺻﺒﺤﺖ ﻣﺴﺘﻘﺮة . d=٢ ﻓﻲ اﻟﻤﺘﻮﺳﻂ اي ان ﻣﻦ اﻧﻤﺎط اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻧﺮى اﻧﻬﺎ ﺗﺘﺨﺎﻣﺪ ﻣﻦ اﻟﺘﺨﻠﻒ اﻷول ﻣﻤﺎ وﺳﻮف ﻧﻄﺒﻖ هﺬا اﻟﻨﻤﻮذج ﺑﺎﻷﻣﺮzt ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻷﺻﻠﻴﺔARIMA(١،٢،١) ﻳﺮﺷﺢ ﻧﻤﻮذج MTB > ARIMA 1 2 1 'z(t)' 'RESI2' 'FITS2'; SUBC> NoConstant; SUBC> Forecast 10 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram;
١٩٩
SUBC>
GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 2462.77 0.100 0.100 1 1345.58 0.250 -0.050 2 1170.63 0.203 -0.200 3 984.83 0.182 -0.350 4 782.47 0.200 -0.500 5 560.15 0.278 -0.650 6 363.93 0.428 -0.765 7 259.20 0.578 -0.814 8 202.76 0.728 -0.842 9 185.51 0.861 -0.859 10 185.36 0.873 -0.860 11 185.36 0.875 -0.860 12 185.36 0.875 -0.860 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.8749 0.0353 MA 1 -0.8599 0.0357
0.0010
T 24.75 -24.12
Differencing: 2 regular differences Number of observations: Original series 200, after differencing 198 Residuals: SS = 183.717 (backforecasts excluded) MS = 0.937 DF = 196 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 3.9(DF=10) 13.0(DF=22) 33.1(DF=34)
48 46.0(DF=46)
Forecasts from period 200 Period 201 202 203 204 205 206 207 208 209 210
Forecast -22615.2 -22757.2 -22900.4 -23044.5 -23189.5 -23335.2 -23481.5 -23628.4 -23775.8 -23923.7
95 Percent Limits Lower Upper -22617.1 -22613.3 -22764.6 -22749.9 -22917.2 -22883.5 -23075.3 -23013.7 -23238.8 -23140.1 -23407.8 -23262.6 -23582.1 -23380.9 -23761.8 -23495.0 -23946.7 -23605.0 -24136.6 -23710.7
zt = 0.875t −1 z + at − 0.859at −1 , at
N ( 0,0.937 )
Actual
اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
هﻲt وﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر
٢٠٠
( ) s.e. (θˆ ) = 0.0357,
φˆ1 = 0.8749, s.e. φˆ1 = 0.0353, t = 24.75 θˆ1 = −0.8599,
1
t = −24.12
σˆ 2 = 0.937, with d . f . = 196
.ﻧﻼﺣﻆ ان اﻟﻤﻌﺎﻟﻢ ﻣﻌﻨﻮﻳﺔ :اﻵن ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ ACF of Residuals for z(t) (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
Lag
PACF of Residuals for z(t) (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
Lag
أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ ﺗﺪل ﻋﻠﻰ أن اﻟﺒﻮاﻗﻲ ﺗﺘﺒﻊ ﺗﻮزﻳﻊ ﺿﺠﺔ : ﻟﻨﻔﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ،ﺑﻴﻀﺎء أي ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ رﺳﻢ اﻟﻤﺪرج اﻟﺘﻜﺮاري
٢٠١
Histogram of the Residuals (response is z(t))
Frequency
20
10
0 -3
-2
-1
0
1
2
Residual
.ﻳﺒﺪو ﻣﺘﻨﺎﻇﺮ ﺑﻌﺾ اﻟﺸﻴﺊ :ﻟﻨﻨﻈﺮ إﻟﻰ ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ ﻟﻠﺒﻮاﻗﻲ Normal Probability Plot of the Residuals (response is z(t))
2
Residual
1
0
-1
-2
-3 -3
-2
-1
0
1
2
3
Normal Score
.ﻧﺴﺘﻄﻴﻊ أن ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ ﺗﻘﺮﻳﺒﺎ . ﻓﺘﺮات ﺗﻨﺒﺆ٩٥٪ ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ١٠ اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟـ Forecast of 20 Future values with 95% limits -22500
z(t)
-23000
-23500
-24000
0
1
2
3
4
5
6
7
8
9
Time
٢٠٢
10
٢٠٣
:اﻟﻤﺮاﺟﻊ ١-
Abraham, B. and Ledoter, J. (1983). Statistical Methods for Forecasting, John Wiley, New York. ٢Anderson, T. W. (1971). The Statistical Analysis of Time Series, John Wiley, New York. ٣Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis
Forecasting and Control, 2nd ed., Holden-Day, San Francisco. Brillinger, D. R. (1975). Time Series: Data Analysis and Theory, Holt, Rinehart and Winston, New York. Fuller, W. A. (1976). Introduction to Statistical Time Series, John Wiley, New York. Granger, C. W. J. and Newbold, P. (1977). Forecasting Economic Time Series, Academic Press, New York. Hannan, E. J. (1970). Multiple Time Series, John Wiley, New York. Harvey, A. C. (1980). Time Series Models, Halsted Press, New York. P
٤٥٦٧٨-
P
٩-
Montgomery, D. C., Johnson, L. A. and Gardiner, J. S. (1990). Forecasting and Time Series Analysis, 2nd ed., McGraw-Hill International Edition. ١٠Makridakis, S., Wheelwright, S. C. and McGee, V. E. (1983). Forecasting Methods and Applications, 2nd ed., P
P
P
P
John Wiley, New York. Shumway, R. H. (1988). Applied Statistical Time Series ١١Analysis, Prentice-Hall, New York. Wei, W. W. S. (1990). Time Series Analysis Univariate and ١٢Multivariate Methods, Addison Wesley. ١٣Minitab Reference Manual, Release 11 for Windows. (1998).
T
ﻟﻸﺳﻒ اﻟﺸﺪﻳﺪ ﻻﺗﻮﺟﺪ ﺣﺴﺐ ﻋﻠﻤﻲ ﻣﺮاﺟﻊ ﻋﺮﺑﻴﺔ ﺗﻐﻄﻲ آﻞ أو ﺟﺰء ﻣﻦ ﻣﺤﺘﻮى اﻟﻤﺎدة اﻟﻤﻐﻄﺎة ﻓﻲ هﺬا اﻟﻜﺘﺎب وأرﺟﻮا ﻣﻦ أي ﻃﺎﻟﺐ أو ﺑﺎﺣﺚ أو ﻣﺪرس ﻳﻌﻠﻢ ﺑﻤﺜﻞ :هﺬا اﻟﻤﺮﺟﻊ او اﻟﻜﺘﺎب أن ﻳﺮﺳﻞ ﻟﻲ ﻣﻼﺣﻈﺔ ﻋﻠﻰ اﻟﺒﺮﻳﺪ اﻹﻟﻜﺘﺮوﻧﻲ [email protected] أو[email protected] .وﺷﻜﺮا T
T
T
٢٠٤
ﻃﺮق اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ) اﻟﺠﺰء اﻷول(
ﺗﺄﻟﻴﻒ د .ﻋﺪﻧﺎن ﻣﺎﺟﺪ ﻋﺒﺪاﻟﺮﺣﻤﻦ ﺑﺮي أﺳﺘﺎذ اﻹﺣﺼﺎء وﺑﺤﻮث اﻟﻌﻤﻠﻴﺎت اﻟﻤﺸﺎرك
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ﺑﺴﻢ اﷲ اﻟﺮﺣﻤﻦ اﻟﺮﺣﻴﻢ اﻟﺤﻤﺪ ﷲ رب اﻟﻌﺎﻟﻤﻴﻦ واﻟﺼﻼة واﻟﺴﻼم ﻋﻠﻰ اﺷﺮف ﺧﻠﻖ اﷲ ﺳﻴﺪﻧﺎ وﻧﺒﻴﻨﺎ ﻣﺤﻤﺪ وﻋﻠﻰ ﺁﻟﻪ وﺻﺤﺒﻪ وﺳﻠﻢ. أﻣﺎ ﺑﻌﺪ. هﺬﻩ هﻲ اﻟﻤﺴﻮدة اﻷوﻟﻰ ﻟﻜﺘﺎب ﻃﺮق اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﻟﻄﻼب ﻣﺮﺣﻠﺔ اﻟﺒﻜﺎﻟﻮرﻳﻮس. هﺬا اﻟﻜﺘﺎب ﺳﻴﻈﻞ ﻣﺴﻮدة إﻟﻰ ﻣﺎﺷﺎء اﷲ ﻷﻧﻲ وﺑﺈذن اﷲ ﺗﻌﺎﻟﻰ ﺳﻮف أﻗﻮم ﺑﺘﻄﻮﻳﺮﻩ وﺗﺠﺪﻳﺪﻩ وﺗﺤﺴﻴﻨﻪ ﺑﺸﻜﻞ ﻣﺴﺘﻤﺮ وﺳﻴﻈﻞ ﺑﺸﻜﻠﻪ اﻹﻟﻜﺘﺮوﻧﻲ هﺬا ﻷﻧﻲ أﻋﺘﻘﺪ ان اﻟﻌﻠﻮم واﻟﺘﻘﻨﻴﺔ ﺗﺘﻄﻮر ﻳﻮﻣﻴﺎ وﺑﺸﻜﻞ ﻣﺘﺴﺎرع ﺑﺤﻴﺚ ان وﺿﻌﻬﺎ ﻓﻲ آﺘﺎب ﺟﺎﻣﺪ ﺳﺘﺎﺗﻴﻜﻲ ﻻﻳﺘﻨﺎﺳﺐ ﻣﻊ دﻳﻨﺎﻣﻴﻜﻴﺔ اﻟﻤﻮﺿﻮع وﺧﺎﺻﺔ ﻓﻲ ﻋﺼﺮ ﺛﻮرة اﻟﻤﻌﻠﻮﻣﺎت واﻹﻧﺘﺮﻧﺖ. ﻳﻐﻄﻲ اﻟﺠﺰء اﻷول ﻣﻦ اﻟﻜﺘﺎب اﻷﺳﺎﺳﻴﺎت اﻷوﻟﻴﺔ ﻟﻠﻤﻮﺿﻮع وﻳﺘﻄﺮق إﻟﻰ ﻣﻮﺿﻮع اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﺑﺈﺳﺘﺨﺪام ﻧﻤﺎذج ARIMAواﻟﺘﻲ آﺎﻧﺖ اول ﻣﻌﺎﻟﺠﺔ رﻳﺎﺿﻴﺔ ﺟﺎدة وﻣﺤﻜﻤﺔ ﻟﻠﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ Time Seriesآﻤﺎ ﺗﻄﺮﻗﺖ ﻓﻲ ﺁﺧﺮ اﻟﻜﺘﺎب إﻟﻰ ﺑﻌﺾ اﻟﻄﺮق اﻟﺘﻘﻠﻴﺪﻳﺔ اﻟﻬﻮرﺳﺘﻴﻜﻴﺔ ﻟﻠﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ وﻓﻲ ﺟﻤﻴﻊ أﺟﺰاء اﻟﻜﺘﺎب ﻗﻤﺖ ﺑﺘﻮﺿﻴﺢ اﻷﻣﺜﻠﺔ واﻟﺤﺎﻻت اﻟﺪراﺳﻴﺔ ﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ Minitabوهﻲ ﺑﺮاﻣﺞ ﺣﺎﺳﺐ ﻃﻮرت ﺧﺎﺻﺔ ﻟﺘﻌﻠﻴﻢ ﻋﻠﻢ اﻹﺣﺼﺎء ﺑﺠﻤﻴﻊ ﻓﺮوﻋﻪ وهﺬﻩ اﻟﺤﺰﻣﺔ ﻣﺘﻮﻓﺮة ﻟﻠﻄﻼب ﺑﺎﻟﻤﺠﺎن. اﻟﺠﺰء اﻟﺜﺎﻧﻲ ﻣﻦ اﻟﻜﺘﺎب وﻣﻮﺟﻪ ﻟﻄﻼب اﻟﺪراﺳﺎت اﻟﻌﻠﻴﺎ ﺳﻮف ﻳﺘﻄﺮق ﺑﺈذن اﷲ ﻟﻤﻮاﺿﻴﻊ ﻣﺜﻞ ﺗﺤﻠﻴﻞ اﻟﺘﺪﺧﻞ Intervention Analysisوﻧﻤﺎذج داﻟﺔ اﻟﺘﺤﻮﻳﻞ Transfer Function Modelsوﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﺘﻌﺪدة Multivariate Time Series Modelsوﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺟﻬﻪ Vector Time Series Modelsوﻧﻤﺎذج ﻓﻀﺎء اﻟﺤﺎﻟﺔ وﻣﺮﺷﺢ آﺎﻟﻤﻦ State Space Models and Kalman Filteringوﻧﻤﺎذج اﻟﺤﺪ Threshold Time Series Modelsوﻧﻤﺎذج ARCHوﻧﻤﺎذج GARCHوﺗﻄﺒﻴﻘﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ اﻟﻤﺎﻟﻲ Finantial Time Series Forecastingآﻤﺎ ﺳﻨﺘﻄﺮق إﻟﻰ اﻟﺸﺒﻜﺎت اﻟﻌﺼﺒﻴﺔ Neural Networksوإﺳﺘﺨﺪاﻣﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ اﻹﺣﺼﺎﺋﻲ. هﺬا وارﺟﻮا ﻣﻦ اﷲ ان ﻳﻮﻓﻘﻨﻲ ﻓﻲ إﻧﺠﺎز هﺬا اﻟﻌﻤﻞ ﻟﻮﺟﻬﻪ اﻟﻜﺮﻳﻢ وﻹﺛﺮاء اﻟﻤﻜﺘﺒﺔ اﻟﻌﺮﺑﻴﺔ اﻟﻔﻘﻴﺮة إﻟﻰ ﻣﺜﻞ هﺬا اﻟﻜﺘﺎب. ﺳﻴﻜﻮن هﺬا اﻟﻜﺘﺎب ﻣﺠﺎﻧﻲ ﻷي ﻃﺎﻟﺐ ﻋﻠﻢ وهﻮ ﺳﻴﻜﻮن ﻣﺘﻮاﺟﺪ ﻋﻠﻰ ﺷﺒﻜﺔ اﻹﻧﺘﺮﻧﺖ ﻓﻲ اﻟﻤﻮﻗﻊ http://www.abarry.net/or/or٢٢١book١.pdf واﷲ اﻟﻤﻮﻓﻖ. اﻟﻤﺆﻟﻒ د .ﻋﺪﻧﺎن ﻣﺎﺟﺪ ﻋﺒﺪ اﻟﺮﺣﻤﻦ ﺑﺮي ﺟﺎﻣﻌﺔ اﻟﻤﻠﻚ ﺳﻌﻮد ذو اﻟﻘﻌﺪة ١٤٢٢هـ ﻳﻨﺎﻳﺮ ٢٠٠٢م ٣
اﻟﻤﺤﺘﻮﻳﺎت ﻣﻘﺪﻣﺔ -١اﻟﻔﺼﻞ اﻷول :ﻣﻘﺪﻣﺔ وﺗﻌﺎرﻳﻒ١٠.................................................................. ١-١أﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ١٠....................................................... ٢-١اﻟﻐﺮض ﻣﻦ دراﺳﺔ وﺗﺤﻠﻴﻞ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ١٠ ................................. ٣-١اﻟﺨﻄﻮات اﻟﻤﺘﺨﺬة ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ١٠ . ................................................ ١-٣-١ﺗﻌﻴﻴﻦ اﻟﻨﻤﻮذج ١٠ .. ................................................................... ٢-٣-١ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج ١١ ................................................................... ٣-٣-١ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج ١١...................................................... ٤-٣-١ﺗﻮﻟﻴﺪ اﻟﺘﻨﺒﺆات ١١..................................................................... ٥-٣-١إﺳﺘﺨﺪام اﻟﺘﻨﺒﺆات ووﺿﻊ اﻟﻘﺮارات ١١ ........................................... ٤-١ﺗﻌﺎرﻳﻒ وﻣﺒﺎدئ أوﻟﻴﺔ ١١..................................................................... ١-٤-١ﺗﻌﺮﻳﻒ ﻣﺎﺿﻲ أو ﺗﺎرﻳﺦ اﻟﻈﺎهﺮة ١١ ............................................. ٢-٤-١ﺗﻌﺮﻳﻒ اﻟﺤﺎﺿﺮ أو اﻵن ١١ ......................................................... ٣-٤-١ﺗﻌﺮﻳﻒ أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ١١ ......................................................... ٤-٤-١ﺗﻌﺮﻳﻒ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ١٢ ............................................................. ٥-٤-١ﺗﻌﺮﻳﻒ اﻹﺳﺘﻘﺮار ١٢ ................................................................. ٦-٤-١ﺗﻌﺮﻳﻒ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٢ ......................................................... ٧-٤-١ﻣﺜﺎل : ١اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ١٢ ................................................... ٨-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ١٣ .................................................... ٩-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ١٣ ................................................... ١٠-٤-١ﻣﺜﺎل : ٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٣ ......................... ١١-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ١٤ ...................................... ١٢-٤-١ﻣﺜﺎل :٣داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻀﺠﺔ اﻟﺒﻴﻀﺎء ١٥.................. ١٣-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ ١٦......................................... ١٤-٤-١ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ١٧ .............................. ١٥-٤-١ﻣﺜﺎل :٤داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ١٨ ......... -٢اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ :ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ARMAوإﺳﺘﺨﺪاﻣﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ ٢٢................................................................................................ ١-٢ﺗﻌﺮﻳﻒ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )٢٢ ........ (p,q ٢-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ ٢٢......................................................... ٣-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻷﻣﺎﻣﻲ ٢٢ ..................................................... ٤-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ ٢٢.................................................................. ٥-٢ﺗﻌﺮﻳﻒ ﻋﺎﻣﻞ اﻟﺘﺠﻤﻴﻊ ٢٢ ................................................................. ٦-٢أﻣﺜﻠﺔ ٢٣ ..................................................................................... ٧-٢ﺧﺼﺎﺋﺺ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ٢٤ ............................. ١-٧-٢ﻧﻤﻮذج )٢٤ ........................................................... ARMA(٠،٠ ٢-٧-٢ﻧﻤﻮذج )٢٧ .................................................................................. AR(١ ٣-٧-٢ﻧﻤﻮذج )٣١ .................................................................... AR(٢ ٤
٤-٧-٢ﻧﻤﻮذج )٣٦ ..................................................................... MA(١ ٥-٧-٢ﻧﻤﻮذج )٣٩ .................................................................... MA(٢ ٦-٧-٢ﻧﻤﻮذج )٤٠ ............................................................. ARMA(١،١ ٧-٧-٢ﺧﻮاص ﻧﻤﺎذج )٤٧ ................................................ ARMA(p,q -٣اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ :ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ٤٩ ...................................... ١-٣ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ ٤٩ .............................................................. ٢-٣ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﺘﺒﺎﻳﻦ ٥٠ ................................................................ ٣-٣ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )٥٢ ...... (p,d,q ١-٣-٣ﻧﻤﻮذج )٥٢ ....................................................... ARIMA(١،١،٠ ٢-٣-٣ﻧﻤﻮذج )٥٢ ........................................................ ARIMA(٠،١،١ ٣-٣-٣ﻧﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ﺑﺈﻧﺠﺮاف ٥٣ ................................................. ٤-٣داﻟﺔ اﻷوزان ) y (Bوﺗﻤﺜﻴﻞ ﻧﻤﺎذج )٥٣ ............................... ARMA(p,q ٥-٣اﻣﺜﻠﺔ ﻟﺪاﻟﺔ اﻷوزان ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٥٤ ..................................................... ١-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٤ .................................................... AR(١ ٢-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٥ .................................................... MA(١ ٣-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٥ .................................................... AR(٢ ٤-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٦ ................................................... MA(٢ ٥-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٦ ........................................... ARMA(١،١ ٦-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج )٥٧ ................................................... ARI(١ ٧-٥-٣داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ )٥٧ .................. ARIMA(١،٠،١ ٦-٣ﺑﻌﺾ ﺧﻮاص داﻟﺔ اﻷوزان ) ٥٨ .................................................... y (B -٤اﻟﻔﺼﻞ اﻟﺮاﺑﻊ :اﻟﺘﻨﺒﺆات ذات ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ﻟﻨﻤﺎذج )٥٩ ....... ARMA(p,q ١-٤ﻧﻈﺮﻳﺔ :٢أﺧﻄﺎء اﻟﺘﻨﺒﺆ ٦١ ......................................................................... ٢-٤ﻣﺠﻤﻮﻋﺔ اﻟﻤﻌﻠﻮﻣﺎت ٦٢ ................................................. Information Sets ٣-٤ﻧﻈﺮﻳﺔ :٣اﻟﻤﺘﻨﺒﺊ ذا ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ٦٢ ............................................ ٤-٤ﻗﺎﻋﺪة ٦٢ ............................................................................................. ٢ ٥-٤ﺗﻌﺮﻳﻒ داﻟﺔ اﻟﺘﻨﺒﺆ ٦٢ ................................................................................. ٦-٤دوال اﻟﺘﻨﺒﺆ ﻟﻨﻤﺎذج )٦٢ ....................................................... ARIMA(p,d,q ١-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٣ ............................................................... AR(١ ٢-٦-٤ﺷﺮط اﻹﺳﺘﻤﺮار ٦٣ ............................................................................ ٣-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٤ .............................................................. AR(٢ ٤-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٥ ................................................. ARIMA(٠،١،١ ٥-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٦ .............................................................. MA(١ ٦-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٦ .............................................................. MA(٢ ٧-٦-٤داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻨﻤﻮذج )٦٧ ..................................................... ARMA(١،١ ٧-٤ﺣﺪود اﻟﺘﻨﺒﺆ ٦٨ ......................................................................................... ١-٧-٤ﺗﻌﺮﻳﻒ ﻓﺘﺮة ﺗﻨﺒﺆ ﻟﻠﻘﻴﻤﺔ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ٦٨ ..................................................... ٢-٧-٤ﻣﺜﺎل ٦٩ ........................................................................................ -٥اﻟﻔﺼﻞ اﻟﺨﺎﻣﺲ :ﺗﺼﻤﻴﻢ وﺑﻨﺎء ﻧﻈﺎم ﺗﻨﺒﺆ إﺣﺼﺎﺋﻲ ٧١ ............................................ ١-٥ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ٧١ ....................................................................... ١-١-٥ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ ٧١ .............................................................................. ٢-١-٥إﺧﺘﻴﺎر درﺟﺔ اﻟﺘﻔﺮﻳﻖ ٧١ ................................................................ d
٥
٣-١-٥ﺗﺤﺪﻳﺪ ٧١ ............................................................................... p,q ٤-١-٥إﺿﺎﻓﺔ ﻣﻌﻠﻢ إﻧﺠﺮاف ٧١ ................................................................... ٢-٥ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج ٧٢ .............................................................................. ١-٢-٥ﻃﺮﻳﻘﺔ اﻟﻌﺰوم ٧٢ ........................................................................ ٢-٢-٥ﺗﻘﺪﻳﺮ اﻟﻌﺰوم ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٧٣ ....................................................... ١-٢-٢-٥ﻟﻨﻤﻮذج )٧٣ .............................................................. AR(١ ٢-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ ......................................................... MA(١ ٣-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ .......................................................... AR(٢ ٤-٢-٢-٥ﻟﻨﻤﻮذج )٧٤ ......................................................... MA(٢ ٥-٢-٢-٥ﻟﻨﻤﻮذج )٧٥ ................................................ ARMA(١،١ ٣-٢-٥ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ٧٥ .............................................. ٤-٢-٥ﺗﻘﺪﻳﺮات اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ٧٦ ......................... ١-٤-٢-٥ﻟﻨﻤﺎذج )٧٦ .........................................................AR(١ ٢-٤-٢-٥ﻟﻨﻤﺎذج )٧٧ ....................................................... MA(١ ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج٧٨ ......................................................... ٣-٥ ١-٣-٥ﻓﺤﺺ اﻟﺒﻮاﻗﻲ ٧٨ .................................................................... ١-١-٣-٥إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ ﻟﻠﺒﻮاﻗﻲ ٧٩ .................................................. ٢-١-٣-٥إﺧﺘﺒﺎر اﻟﻌﺸﻮاﺋﻴﺔ ﻟﻠﺒﻮاﻗﻲ ٧٩ ................................................. ٣-١-٣-٥إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ أو اﻹﺳﺘﻘﻼل ﻟﻠﺒﻮاﻗﻲ ٧٩ ................................... ٤-١-٣-٥إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ ٧٩ .................................................... ٢-٣-٥ﺑﻌﺾ اﻟﻤﻌﺎﻳﻴﺮ اﻻﺧﺮى ﻹﺧﺘﻴﺎر ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ٧٩ ............................. ١-٢-٣-٥إﺣﺼﺎﺋﻴﺔ آﻴﻮ ﻟﻠﺠﻨﻖ وﺑﻜﺲ ٧٩ ........................................... ٢-٢-٣-٥ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ ٨٠ ......................................... AIC ٣-٣-٥أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ٨٠ .......................................................... -٦اﻟﻔﺼﻞ اﻟﺴﺎدس :ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻴﺔ ١١٣ ...... ١-٦دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ ١١٤ ..... ١-١-٦ﻟﻨﻤﻮذج ١١٤ ......................................... SARMA(٠،١)(١،١)١٢ ٢-٦دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ ١١٥ .................................. ١١٥............................................. SARIMA(٠,d,٠)(٠,D,١)s ١-٢-٦ ١١٥ ........................................... SARIMA(٠,d,٠)(١,D,١)s ٢-٢-٦ ١١٥ ............................................ SARIMA(٠,d,١)(٠,D,١)s ٣-٢-٦ ١١٦ ........................................... SARIMA(٠,d,٠)(١,D,١)s ٤-٢-٦ ١١٦ ........................................... SARIMA(٠,d,١)(١,D,٠)s ٥-٢-٦ ١١٦ ............................................ SARIMA(٠,d,٢)(٠,D,١)s ٦-٢-٦ ٣-٦داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ ١١٨ .................... ٤-٦أﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ١١٩ ....................................... ٥-٦إﺷﺘﻘﺎق دوال ﺗﻨﺒﺆ ﻟﺒﻌﺾ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ ١٢٣ .......... ١-٥-٦داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج ١٢٣ .................... SARIMA(٠،٠،٠)(٠،١،١)١٢ ٢-٥-٦داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج ١٢٤ .................... SARIMA(٠،١،١)(٠،١،١)١٢ ٦-٦أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ١٢٥ .......................... اﻟﺠﺰء اﻟﻌﻤﻠﻲ: -٧اﻟﻔﺼﻞ اﻟﺴﺎﺑﻊ :ورﻗﺔ ﺗﺪرﻳﺐ ﻋﻤﻠﻲ ﻋﻠﻰ اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ١٤٢ ............................................................................................ ٦
-٨اﻟﻔﺼﻞ اﻟﺜﺎﻣﻦ :ﻣﺜﺎل ﺗﺤﻠﻴﻞ اﻟﺒﻮاﻗﻲ وﻣﻌﻴﻴﺮ إﺧﺘﻴﺎر ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ١٥٥ .................. -٩اﻟﻔﺼﻞ اﻟﺘﺎﺳﻊ :ﺗﺤﻠﻴﻞ أو ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ إﻟﻰ ﻣﺮآﺒﺎت ١٦٢ ..................... -١٠اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ١٧٨ ........................................... ١-١٠اﻟﻮﺳﻴﻂ اﻟﺠﺎري ١٨١ ............................................................. -١١اﻟﻔﺼﻞ اﻟﺤﺎدي ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﺒﺴﻴﻂ ١٨٤ ............. -١٢اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﻤﺰدوج ١٩٠ ........... ١-١٢ﻃﺮﻳﻘﺔ ﺑﺮاون ١٩٠ ........................................................................ ٢-١٢ﻃﺮﻳﻘﺔ هﻮﻟﺖ ١٩٠ ........................................................................ ٣-١٢أﻣﺜﻠﺔ ١٩١ .................................................................................. -١٣اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻋﺸﺮ :اﻟﺘﻤﻬﻴﺪ اﻷﺳﻲ اﻟﺜﻼﺛﻲ واﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ ١٩٨ ............................................................................. ١-١٣اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ ١٩٩ ............................................................... ٢-١٣اﻟﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ ٢٠١ .............................................................. ٣-١٣ﻣﺜﺎل ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ٢٠٤ ........................................................... ٤-١٣ﻣﺜﺎل ﺁﺧﺮ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ ٢١٠ ..................................................... اﻟﻤﺮاﺟﻊ ٢١٩ ...........................................................................................
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اﻟﻔﺼﻞ اﻷول ﻣﻘﺪﻣﺔ وﺗﻌﺎرﻳﻒ ﺗﻌﺮﻳﻒ :١اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ Time Series
ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺮﺗﺒﺔ ﻣﻊ اﻟﺰﻣﻦ ) او ﻣﺮﺗﺒﺔ ﻣﻊ اﻟﻤﻜﺎن (
اﻣﺜﻠﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ: -١ﺳﻌﺮ اﻗﻔﺎل ﺳﻬﻢ ﺑﻨﻚ اﻟﺮﻳﺎض ﻳﻮﻣﻴﺎ. -٢ﻋﺪد اﻟﻮﺣﺪات اﻟﻤﻄﻠﻮﺑﺔ اﺳﺒﻮﻋﻴﺎ ﻣﻦ اﻧﺘﺎج ﺳﻠﻌﺔ ﻣﻌﻴﻨﺔ. -٣ﺣﺠﻢ اﻟﻤﺒﻴﻌﺎت ﺷﻬﺮﻳﺎ ﻣﻦ ﺳﻠﻌﺔ ﻣﺎ. -٤ﺣﺠﻢ اﻹﻧﺘﺎج اﻟﻴﻮﻣﻲ ﻟﻠﻨﻔﻂ اﻟﺨﺎم ﺑﺎﻟﻤﻤﻠﻜﺔ. واﻟﻐﺮض ﻣﻦ دراﺳﺔ وﺗﺤﻠﻴﻞ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ هﻮ: -١ﻓﻬﻢ وﻧﻤﺬﺟﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﻈﺎهﺮة اﻟﻤﺸﺎهﺪة. -٢اﻟﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻟﻠﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ. -٣اﻟﺘﺤﻜﻢ ﺑﺎﻟﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ إذا اﻣﻜﻦ ذﻟﻚ. اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة وهﻲ ﻋﺒﺎرة ﻋﻦ اﻹﻧﺘﺎج اﻟﻴﻮﻣﻲ ﻟﻠﺤﻠﻴﺐ ﺑﺎﻟﺮﻃﻞ ﻟﺒﻘﺮة ﻣﺎ
850
750
C1 650
550
70
60
50
30
40
20
10
Index
اﻟﺨﻄﻮات اﻟﻤﺘﺨﺬة ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: إن إﻳﺠﺎد ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ﺗﻨﻄﺒﻖ ﻋﻠﻴﺔ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة ﻳﻌﺘﺒﺮ ﻣﻦ اﻟﻤﻬﺎم اﻟﺼﻌﺒﺔ واﻟﺘﻲ ﺗﺤﺘﺎج اﻟﻰ اﻟﻜﺜﻴﺮ ﻣﻦ اﻟﺒﺤﺚ واﻟﺨﺒﺮة .ﺳﻮف ﻧﺴﺘﻌﺮض ﺑﻌﺾ اﻟﺨﻄﻮات اﻟﻌﺮﻳﻀﺔ ﻟﺒﻨﺎء ﻧﻤﻮذج رﻳﺎﺿﻲ ﻟﻠﺘﻨﺒﺆ ﻋﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺎ: -١ ﺗﻌﻴﻴﻦ اﻟﻨﻤﻮذج أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج :Model Identificationوهﺬا ﻳﺘﻢ ﺑﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﻓﻴﻤﺎ ﻳﺴﻤﻰ Time Plotﺣﻴﺚ ﻳﻜﻮن اﻹﺣﺪاﺛﻲ اﻻﻓﻘﻲ هﻮ اﻟﺰﻣﻦ واﻟﺮأﺳﻲ ﺣﺠﻢ اﻟﻈﺎهﺮة اﻟﻤﺸﺎهﺪة وﻣﻦ ﺛﻢ إﺧﺘﻴﺎر ﻧﻤﻮذج رﻳﺎﺿﻲ ﻣﻌﺘﻤﺪﻳﻦ ﻋﻠﻲ ﺑﻌﺾ اﻟﻤﻘﺎﻳﻴﺲ اﻹﺣﺼﺎﺋﻴﺔ اﻟﺘﻰ ﺗﻤﻴﺰ ﻧﻤﻮذج ﻋﻦ ﺁﺧﺮ وﻋﻠﻰ اﻟﺨﺒﺮة اﻟﻤﺴﺘﻤﺪة ﻣﻦ اﻟﺪراﺳﺎت واﻻﺑﺤﺎث.
٨
-٢
-٣
-٤
-٥
ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج :Model Fittingﺑﻌﺪ ﺗﺮﺷﻴﺢ ﻧﻤﻮذج او اآﺜﺮ آﻨﻤﻮذج ﻣﻨﺎﺳﺐ ﻟﻮﺻﻒ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ﻧﻘﻮم ﺑﺘﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ هﺬا اﻟﻨﻤﻮذج ﻣﻦ اﻟﺒﻴﺎﻧﺎت اﻟﻤﺸﺎهﺪة ﺑﺈﺳﺘﺨﺪام ﻃﺮق اﻟﺘﻘﺪﻳﺮ اﻹﺣﺼﺎﺋﻲ اﻟﺨﺎﺻﺔ ﺑﺎﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ وهﺬا اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ ﻳﺆﺧﺬ آﻨﻤﻮذج اوﻟﻲ ﻗﺎﺑﻞ ﻟﻠﺘﻌﺪﻳﻞ ﻻﺣﻘﺎ. ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج :Model Diagnosticsإﺟﺮاء إﺧﺘﺒﺎرات ﺗﻔﺤﺼﻴﺔ ﻋﻠﻰ أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ Fitting Errorsﻟﻤﻌﺮﻓﺔ ﻣﺪى ﺗﻄﺎﺑﻖ اﻟﻤﺸﺎهﺪات ﻣﻊ اﻟﻘﻴﻢ اﻟﻤﺤﺴﻮﺑﺔ ﻣﻦ اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ وﻣﺪى ﺻﺤﺔ ﻓﺮﺿﻴﺎت اﻟﻨﻤﻮذج .ﻓﻲ ﺣﺎﻟﺔ إﺟﺘﻴﺎز اﻟﻨﻤﻮذج اﻟﻤﺮﺷﺢ ﻟﻬﺬﻩ اﻹﺧﺘﺒﺎرات ﻧﻘﻮم ﺑﺈﻋﺘﻤﺎدة ﻋﻠﻰ اﻧﻪ اﻟﻨﻤﻮذج اﻟﻨﻬﺎﺋﻲ وﻳﺴﺘﺨﺪم ﻟﺘﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ وإﻻ ﻧﻌﻮد ﻟﻠﺨﻄﻮة اﻻوﻟﻰ ﻟﺘﻌﻴﻴﻦ ﻧﻤﻮذج ﺟﺪﻳﺪ. ﺗﻮﻟﻴﺪ اﻟﺘﻨﺒﺆات :Forecast Generationﻳﺴﺘﺨﺪم اﻟﻨﻤﻮذج اﻟﻨﻬﺎﺋﻲ ﻟﺘﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ وﻣﻦ ﺛﻢ ﺣﺴﺎب أﺧﻄﺎء اﻟﺘﻨﺒﺆ Forecast Errorsآﻠﻤﺎ اﺳﺘﺠﺪت ﻗﻴﻢ ﺟﺪﻳﺪة ﻣﺸﺎهﺪة ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ وﻣﺮاﻗﺒﺔ هﺬﻩ اﻷﺧﻄﺎء ﻓﻰ ﻣﺎ ﻳﺴﻤﻰ ﺑﻤﺨﻄﻄﺎت اﻟﻤﺮاﻗﺒﺔ Control Chartsواﻟﺘﻲ ﺗﻮﺿﻊ ﻟﻠﻘﺒﻮل ﺑﻨﺴﺒﺔ ﺧﻄﺄ ﻣﻌﻴﻦ إذا ﺗﺠﺎوزﺗﺔ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﻳﻌﺎد اﻟﻨﻈﺮ ﻓﻲ اﻟﻨﻤﻮذج وﺗﻌﺎد اﻟﺪورة ﻣﻦ ﺟﺪﻳﺪ ﺑﺘﺤﺪﻳﺪ ﻧﻤﻮذج ﻣﺮﺷﺢ ﺁﺧﺮ. إﺳﺘﺨﺪام اﻟﺘﻨﺒﺆات ووﺿﻊ اﻟﻘﺮارات :Implementation and Decision making ﺗﻘﺪم اﻟﺘﻨﺒﺆات ﻓﻰ ﺗﻘﺮﻳﺮ ﻟﺼﺎﻧﻌﻲ اﻟﻘﺮار ﻟﻠﻨﻈﺮ ﻓﻲ إﺳﺘﺨﺪاﻣﻬﺎ ﺑﺎﻟﺸﻜﻞ اﻟﻤﻨﺎﺳﺐ.
ﺗﻌﺎرﻳﻒ وﻣﺒﺎدئ اوﻟﻴﺔ: ﺳﻮف ﻧﺮﻣﺰ ﻟﻠﻈﺎهﺮة اﻟﻌﺸﻮاﺋﻴﺔ أو اﻟﻌﻤﻠﻴﺔ اﻟﻌﺸﻮاﺋﻴﺔ اﻟﺘﻲ ﺗﻮﻟﺪ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﺑﺎﻟﺮﻣﺰ } {L, Z −1 , Z 0 , Z1 , Z 2 ,Lاو اﺧﺘﺼﺎرا }} {Z t , t ∈{L, −1, 0,1, 2,Lاو ﺑﺒﺴﺎﻃﺔ
وﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة ﺑﺎﻟﺮﻣﺰ } {z1 , z2 ,L, zn−1 , zn
} {Z t
ﺗﻌﺮﻳﻒ :٢اﻟﻘﻴﻢ z1 , z2 ,L , zn −1ﺗﺴﻤﻰ ﺑﺎﻟﻤﺎﺿﻰ او ﺗﺎرﻳﺦ اﻟﻈﺎهﺮة History
واﻟﺘﺎرﻳﺦ ﻣﻬﻢ ﺟﺪا ﻓﻲ ﻋﻤﻠﻴﺔ اﻟﻨﻤﺬﺟﺔ
ﺗﻌﺮﻳﻒ : ٣اﻟﻘﻴﻤﺔ znﺗﺴﻤﻰ اﻟﺤﺎﺿﺮ او اﻵن وهﻲ اﻟﻤﺸﺎهﺪة اﻷﺧﻴﺮة .
ﺗﻌﺮﻳﻒ :٤أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ et = zt − zˆt , t = 1, 2,..., nﺣﻴﺚ zˆtهﻲ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ) اﻟﻘﻴﻢ اﻟﺘﻲ ﻧﺘﺤﺼﻞ ﻋﻠﻴﻬﺎ ﻣﻦ اﻟﻨﻤﻮذج( وﺗﺴﻤﻲ أﻳﻀﺎ اﻟﺮواﺳﺐ Residuals وﻳﻼﺣﻆ ان اﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ ﻧﺤﺼﻞ ﻋﻠﻴﻬﺎ دﻓﻌﺔ واﺣﺪة ﺑﻌﺪ ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج. ﻣﻼﺣﻈﺔ :ﺳﻮف ﻧﺮﻣﺰ ﻟﻠﻤﺸﺎهﺪات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺑﺎﻟﺮﻣﻮز zn +1 , zn + 2 , zn +3 ,...او ﺑﺸﻜﻞ ﻋﺎم zn +l , l ≥ 0وﻧﺮﻣﺰ ﻟﺘﻨﺒﺆاﺗﻬﺎ ﺑﺎﻟﺮﻣﺰ zn (1) , zn ( 2 ) , zn ( 3) ,...او ﺑﺸﻜﻞ ﻋﺎم zn ( l ) , l ≥ 0
٩
en ( l ) = zn +l − zn ( l ) , l ≥ 0
ﺗﻌﺮﻳﻒ :٥أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ
وأﺧﻄﺎء اﻟﺘﻨﺒﺆ ﺗﻨﺘﺞ اﻟﻮاﺣﺪة ﺗﻠﻮ اﻻﺧﺮى آﻠﻤﺎ ﺗﻘﺪم اﻟﺰﻣﻦ وﺷﻮهﺪت اﻟﻘﻴﻢ اﻟﺤﻘﻴﻘﻴﺔ ﺗﻌﺮﻳﻒ :٦ﻳﻘﺎل ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة } {z1 , z2 ,L, zn−1 , znﻣﺴﺘﻘﺮة Stationary
إذا ﺣﻘﻘﺖ اﻟﺸﺮوط اﻟﺘﺎﻟﻴﺔ:
1) E ( zt ) = constant = µ , ∀t ⎧constant = γ 0 , ∀t , ∀s, t = s ⎨ = ) 2) cov ( zt , zs ⎩ f ( s − t ) , ∀t , ∀ s , t ≠ s
اﻵن ﺳﻮف ﻧﻌﺮف ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﻬﻤﺔ ﺟﺪا ﻟﻜﻮﻧﻬﺎ ﺣﺠﺮة او ﻃﻮب اﻟﺒﻨﺎء Building Blocks ﻟﺠﻤﻴﻊ اﻟﻨﻤﺎذج اﻟﺘﻲ ﺳﻮف ﻧﺪرﺳﻬﺎ
ﺗﻌﺮﻳﻒ :٧ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء White Noise Seriesاوﻋﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء {at } White Noise Processهﻲ ﻋﺒﺎرة ﻋﻦ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻟﻌﺸﻮاﺋﻴﺔ ﻏﻴﺮ اﻟﻤﺘﺮاﺑﻄﺔ ) واﺣﻴﺎﻧﺎ ﻧﻔﺘﺮض اﻧﻬﺎ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻦ اﻟﻤﺘﻐﻴﺮات اﻟﻌﺸﻮاﺋﻴﺔ اﻟﺘﻲ ﺗﻜﻮن ﻣﺴﺘﻘﻠﺔ وﻟﻬﺎ ﺗﻮزﻳﻌﺎت ﻣﺘﻄﺎﺑﻘﺔ ) ( Independent, Identically Distributed (IIDﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ ﺛﺎﺑﺖ σ 2أي:
1) E ( at ) = 0, ∀t
وﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ ) WN ( 0, σ 2
⎧σ 2 , ∀t , ∀s, t = s ⎨ = ) 2) cov ( at , as ⎩ 0 , ∀t , ∀ s , t ≠ s at
ﻣﺜﺎل :١ﻣﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ : Random Walk ﺳﻮف ﻧﺒﻨﻲ ﻋﻤﻠﻴﺔ ﻋﺸﻮاﺋﻴﺔ } {Z tآﺎﻟﺘﺎﻟﻲ:
Z1 = a1 Z 2 = a1 + a2 M
Z t = a1 + a2 +L + at أو
Z t = Z t −1 + at أي ﻟﻮ اﻋﺘﺒﺮﻧﺎ ان a jهﻮ ﺣﺠﻢ اﻟﺨﻄﻮة اﻟﺘﻲ ﺗﺆﺧﺬ اﻟﻲ اﻻﻣﺎم او اﻟﺨﻠﻒ ﻋﻨﺪ اﻟﺰﻣﻦ jﻓﺎن Z t
هﻲ ﻣﻮﻗﻊ ﻣﺎﺷﻲ ﻋﺸﻮاﺋﻲ ﻋﻨﺪ اﻟﺰﻣﻦ t ﻣﻼﺣﻈﺔ :هﺬﻩ اﻟﻌﻤﻠﻴﺔ او اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﻦ اﻟﻨﻤﺎذج اﻟﻬﺎﻣﺔ ﺟﺪا اﻟﺘﻲ ﺗﺼﻒ اﺳﻮاق اﻟﻤﺎل اﻟﻌﺎﻟﻤﻴﺔ ﺗﻤﺮﻳﻦ :اوﺟﺪ ) E ( Z tو ) cov ( Z t , Z sﻟﺠﻤﻴﻊ ﻗﻴﻢ t , sوهﻞ اﻟﻌﻤﻠﻴﺔ ﻣﺴﺘﻘﺮة؟ ١٠
ﺗﻌﺮﻳﻒ :٨داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ Autocovariance Functionوﺗﻌﺮف آﺎﻟﺘﺎﻟﻲ: = cov ( Z t , Z s ) , ∀t , ∀s
γ t ,s
= E ⎡⎣( Z t − µ )( Z s − µ ) ⎤⎦ , ∀t , ∀s
وإذا ﻋﺮﻓﻨﺎ اﻟﺘﺨﻠﻒ kﻋﻠﻲ اﻧﻪ اﻟﻔﺘﺮة اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﺗﻔﺼﻞ ﺑﻴﻦ Z tوﺑﻴﻦ Z t −kأو Z t +kﻓﺈن داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: γ k = cov ( Z t , Z t −k ) , k = 0, ±1, ±2,L = E ⎡⎣( Z t − µ )( Z t −k − µ ) ⎤⎦ , k = 0, ±1, ±2,L ﻣﻼﺣﻈﺔ :ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﺘﻌﺮﻳﻒ اﻟﺜﺎﻧﻲ داﺋﻤﺎ
ﺗﻌﺮﻳﻒ :٩داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ) Autocorrelation Function (ACFوﺗﻌﺮف آﺎﻟﺘﺎﻟﻲ: γ ρ k = k , k = 0, ±1, ±2,L γ0 وﻟﻬﺎ اﻟﺨﻮاص اﻟﺘﺎﻟﻴﺔ:
1. ρ 0 = 1 2. ρ − k = ρ k
ρk ≤ 1
3.
ﻣﺜﺎل :٢ﺳﻮف ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء داﻟﺔ اﻟﺘﻐﺎﻳﺮ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء هﻲ: 2 ⎧σ , k = 0 ⎨ = ) γ k = cov ( at , at −k ⎩ 0, k ≠0 وذﻟﻚ ﻣﻦ اﻟﺘﻌﺮﻳﻒ ٧وﻣﻨﻬﺎ ﻧﺠﺪ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ: ⎧1 , k = 0 γ ⎨ = ρk = k γ 0 ⎩0 , k ≠ 0 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: Autocorrelation function of White Noise 1.0
0.0 9
8
7
6
5
4
Lag
١١
3
2
1
0
Autocorr
0.5
ﺗﻌﺮﻳﻒ :١٠داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation Function )(PACF وﺗﻌﻄﻲ ﻣﻘﺪار اﻟﺘﺮاﺑﻂ ﺑﻴﻦ Z tو Z t −kﺑﻌﺪ إزاﻟﺖ ﺗﺄﺛﻴﺮ اﻟﺘﺮاﺑﻂ اﻟﻨﺎﺗﺞ ﻣﻦ اﻟﻤﺘﻐﻴﺮات Z t −1 , Z t −2 ,..., Z t −k +1اﻟﻮاﻗﻌﺔ ﺑﻴﻨﻬﻤﺎ وﻳﺮﻣﺰ ﻟﻬﺎ ﻋﻨﺪ اﻟﺘﺨﻠﻒ kﺑﺎﻟﺮﻣﺰ φkkوأﺣﺪ ﻃﺮق ﺣﺴﺎﺑﻬﺎ ﺗﻘﻮم ﻋﻠﻲ ﺣﺴﺎب ﻣﻌﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺠﺰﺋﻲ φkkﻓﻲ اﻟﺘﻤﺜﻴﻞ: Z t = φk 1Z t −1 + φk 2 Z t −2 + L + φkk Z t −k + at
ﺣﺴﺎب : φ11
Z t = φ11Z t −1 + at
ﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﺑـ Z t −1وأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
) E ( Z t −1Z t ) = φ11 E ( Z t −1Z t −1 ) + E ( Z t −1at
أي
γ 1 = φ11γ 0 ﺣﻴﺚ ) E ( Z t −1at ) = 0ﺑﺸﻜﻞ ﻋﺎم E ( Z t −k at ) = 0, k = 1, 2,...آﻤﺎ ﺳﻨﺒﻴﻦ ﻻﺣﻘﺎ ( وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ γ 0ﻧﺠﺪ φ11 = ρ1
ﺗﻌﺮﻳﻒ :١١ﺑﺸﻜﻞ ﻋﺎم ﺗﻌﺮف داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ آﺎﻟﺘﺎﻟﻲ: k =0 k =1
k = 2,3,...
1,
ρ1 , ρ1 ρ2
ρ k −2 ρ k −3
M
M
ρk , ρ k −1 ρ k −2 M 1
ﺣﻴﺚ
ρ1 ρ k −2 ρ k −3 M
ρ1
L L L L L L L L
ρ1
1
1 M
ρ1 M
ρ k −1 ρ k −2 1 ρ1 1 ρ1 M
M
ρ k −1 ρ k −2
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = φkk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪
ﺗﺮﻣﺰ اﻟﻲ ﻣﺤﺪدة ﻣﺼﻔﻮﻓﺔ
اﻟﺘﻌﺮﻳﻒ اﻟﺴﺎﺑﻖ ﺻﻌﺐ اﻹﺳﺘﺨﺪام ﻟﻘﻴﻢ kاﻟﻜﺒﻴﺮة وﻟﻬﺬا ﺳﻮف ﻧﻌﻄﻲ ﺗﻌﺮﻳﻒ ﺁﺧﺮ ﻟﺤﺴﺎب داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﻜﺮارﻳﺎ: ١٢
ﺗﻌﺮﻳﻒ ١١ب :ﺗﺤﺴﺐ φkkﺗﻜﺮارﻳﺎ ﻣﻦ اﻟﻌﻼﻗﺎت
φ00 = 1, by definition φ11 = ρ1 k −1
, k = 2,3,...
ρ k − ∑φk −1, j ρ k − j j =1 k −1
1 − ∑φk −1, j ρ j
= φkk
j =1
ﺣﻴﺚ φkj = φk −1, j − φkkφk −1,k −1 ,
j = 1, 2,..., k − 1 ﺣﺴﺎب : φ22 ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب:
ρ 2 − φ11 ρ1 ρ 2 − ρ12 = φ22 = 1 − φ11 ρ1 1 − ρ12
وذﻟﻚ ﻷن . φ11 = ρ1
ﻣﺜﺎل :٣ﺳﻮف ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء: ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب φ00 = 1, by definition φ11 = ρ1 = 0 وذﻟﻚ ﻣﻦ ﻣﺜﺎل ١اﻟﺴﺎﺑﻖ وﺑﺎﻟﺘﻌﻮﻳﺾ ﻓﻲ ﺗﻌﺮﻳﻒ ١١ب ﻋﻦ φkkﻧﺠﺪ φ22 = φ33 = L = 0 وهﻜﺬا: ⎧1, k = 0 ⎨ = φkk ⎩0, k ≠ 0 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: Partial Autocorrelation function of White Noise 1.0
0.0
PACF
0.5
ﻣﻼﺣﻈﺔ :ﻻﺣﻆ أن آﻞ ﻣﻦ داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء Lag ﺗﺴﺎوي اﻟﺼﻔﺮ ﻣﻦ اﻟﺘﺨﻠﻒ اﻷول .وهﺬﻩ ﺧﺎﺻﻴﺔ ﺟﻤﻴﻊ اﻟﻤﺘﻐﻴﺮات اﻟﻌﺸﻮاﺋﻴﺔ ﻏﻴﺮ اﻟﻤﺘﺮاﺑﻄﺔ او اﻟﻤﺴﺘﻘﻠﺔ .ﻹﺧﺘﺒﺎر ﻋﺪم اﻟﺘﺮاﺑﻂ ﺑﻴﻦ ﻗﻴﻢ ﻣﺸﺎهﺪة ﻟﻤﺘﻐﻴﺮ ﻋﺸﻮاﺋﻲ ﺗﺴﺘﺨﺪم داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺬﻟﻚ. 9
8
7
6
5
4
١٣
3
2
1
0
ﺗﻌﺮﻳﻒ : ١٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ Sample Autocorrelation Function SACF ﻟﻤﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة z1 , z2 ,L , zn −1 , znوﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ rk , k = 0,1, 2,...وﺗﻌﻄﻰ
ﺑﺎﻟﻌﻼﻗﺔ:
, k = 0,1, 2, ...
) − z )( zt +k − z 2
)−z
n −k
t
∑( z t =1
n
t
∑( z
= rk
t =1
1 n ﺣﻴﺚ z = ∑ zt n t =1 وهﻲ ُﻣﻘ ﱢﺪر Estimatorﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ أي ρˆ k = rk , k = 0,1, 2,...وﺑﻤﺎ اﻧﻬﺎ ُﻣﻘ ﱢﺪر ﻓﻬﻲ إذا ﺗﺘﻐﻴﺮ ﻋﺸﻮاﺋﻴﺎ ﻣﻦ ﻋﻴﻨﺔ ﻻﺧﺮى وﻟﻬﺬا ﻓﺈن ﻟﻬﺎ اﻟﺨﻮاص اﻟﻌﻴﻨﻴﺔ اﻟﺘﺎﻟﻴﺔ: -١إذا آﺎﻧﺖ ρ k = 0, k > qﻓﺈن q ⎛1 ⎞ V ( rk ) ≅ ⎜ 1 + 2∑ ρ k2 ⎟ , k > q ⎝n k =1 ⎠ 1 وﻓﻲ اﻟﺤﺎﻟﺔ اﻟﺨﺎﺻﺔ ﻋﻨﺪﻣﺎ ρ k = 0, k > 0ﻓﺈن V ( rk ) ≅ , k > 0 n -٢ﻟﻘﻴﻢ nاﻟﻜﺒﻴﺮة و ρ k = 0ﻓﺈن rkﻳﻜﻮن ﻟﻬﺎ ﺗﻘﺮﻳﺒﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ وﺑﺎﻟﺘﺎﻟﻲ ﻧﺴﺘﻄﻴﻊ اﻟﻘﻴﺎم ﺑﺎﻻﺧﺘﺒﺎر اﻟﺘﺎﻟﻲ:
H 0 : ρk = 0 H1 : ρ k ≠ 0 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻹﺣﺼﺎﺋﺔ: = n rk
rk − 12
n
وذﻟﻚ ﻋﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05وﺗﺮﻓﺾ H 0إذا آﺎﻧﺖ n rk > 1.96 -٣ﺗﺤﺖ اﻟﻔﺮﺿﻴﺔ H 0 : ρ k = 0, ∀kﻓﺈن corr ( rk , rk − s ) ≅ 0, s ≠ 0 ُ -٤ﺗﻘ ﱠﺪر اﻟﺘﺒﺎﻳﻨﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ آﺎﻟﺘﺎﻟﻲ:
⎛1 ⎞ Vˆ ( rk ) ≅ ⎜ 1 + 2∑ rk2 ⎟ , k > q ⎝n k =1 ⎠ ﺗﻌﺮﻳﻒ :١٣داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ Sample Partial Autocorrelation Function SPACFﻟﻤﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة z1 , z2 ,L , zn −1 , znوﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ rkk , k = 0,1, 2,...ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: q
١٤
k =0 k =1 rk −2 r1 rk −3 r2 M M r1 rk , rk −2 rk −1 rk −3 rk −2 M M r1 1
k = 2,3,...
L L L L L L L L
⎧ 1, ⎪ r, ⎪ 1 ⎪ 1 r1 ⎪ 1 ⎪ r1 ⎪ M M ⎪ rkk = ⎨ rk −1 rk −2 ⎪ 1 r1 ⎪ 1 ⎪ r1 ⎪ M M ⎪ ⎪ rk −1 rk −2 ⎪ ⎩⎪
و ﻟﺤﺴﺎب داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ ﺗﻜﺮارﻳﺎ:
ﺗﻌﺮﻳﻒ ١٣ب :ﺗﺤﺴﺐ rkkﺗﻜﺮارﻳﺎ ﻣﻦ اﻟﻌﻼﻗﺎت
r00 = 1, by definition r11 = r1 k −1
, k = 2,3, ...
rk − ∑ rk −1, j rk − j j =1 k −1
1 − ∑ rk −1, j rj
= rkk
j =1
j = 1, 2,..., k − 1
ﺣﻴﺚ
rkj = rk −1, j − rkk rk −1,k −1 ,
وهﻲ اﻳﻀﺎ ﻣﻘﺪﱠر Estimatorﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ أي φˆkk = rkk , k = 0,1, 2,...وﺑﻤﺎ اﻧﻬﺎ ُﻣﻘ ﱢﺪر ﻓﻬﻲ إذا ﺗﺘﻐﻴﺮ ﻋﺸﻮاﺋﻴﺎ ﻣﻦ ﻋﻴﻨﺔ ﻻﺧﺮى وﻟﻬﺬا ﻓﺈن ﻟﻬﺎ اﻟﺨﻮاص اﻟﻌﻴﻨﻴﺔ اﻟﺘﺎﻟﻴﺔ: 1 V ( rkk ) ≅ , k > 0 -١ n -٢ﻟﻘﻴﻢ nاﻟﻜﺒﻴﺮة ﻓﺈن rkkﻳﻜﻮن ﻟﻬﺎ ﺗﻘﺮﻳﺒﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ وﺑﺎﻟﺘﺎﻟﻲ ﻧﺴﺘﻄﻴﻊ اﻟﻘﻴﺎم ﺑﺎﻻﺧﺘﺒﺎر اﻟﺘﺎﻟﻲ:
H 0 : φkk = 0 H1 : φkk ≠ 0 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻹﺣﺼﺎﺋﺔ: ١٥
= n rkk
rkk − 12
n
وذﻟﻚ ﻋﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05وﺗﺮﻓﺾ H 0إذا آﺎﻧﺖ n rkk > 1.96
-٣ﺗﺤﺖ اﻟﻔﺮﺿﻴﺔ H 0 : φkk = 0, ∀kﻓﺈن corr (φkk ,φk − s ,k −s ) ≅ 0, s ≠ 0
ُ -٤ﺗﻘ ّﺪر اﻟﺘﺒﺎﻳﻨﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻌﻴﻨﺔ آﺎﻟﺘﺎﻟﻲ: 1 Vˆ ( rkk ) ≅ , k > 0 n ﻣﺜﺎل :٤اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﺗﻤﺜﻞ اﻟﻄﻠﺐ ﻋﻠﻲ ﻣﻨﺘﺞ ﻣﻌﻴﻦ ﻳﻮﻣﻴﺎ: 158 222 248 216 226 239 206 178 169 أﺣﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ وارﺳﻤﻬﻤﺎ: 1 1 اوﻻ :ﻧﺤﺴﺐ اﻟﻤﺘﻮﺳﻂ zt = (158 + 222 + L + 169 ) = 206.89 ∑ 9 n t =1 ﺛﺎﻧﻴﺎ :ﻧﺤﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻣﻦ اﻟﻌﻼﻗﺔ n
, k = 0,1,2,...
=z n −k
) − z )( zt +k − z 2
)−z
t
∑( z
n
t
∑( z
t =1
= rk
t =1
) (158 × 222 + 222 × 248 + L + 178 × 169 = 0.265116 ) (158 × 158 + 222 × 222 + L + 169 × 169 ) (158 × 248 + 222 × 216 + L + 206 × 169 = r2 = -0.212 ) (158 × 158 + 222 × 222 + L + 169 × 169 ) (158 × 216 + 222 × 226 + L + 239 × 169 = r3 = −0.076 ) (158 × 158 + 222 × 222 + L + 169 × 169 = r1
وهﻜﺬا r8 = 0.230, r7 = 0.104, r6 = −0.242, r5 = −0.387, r4 = −0.183 ﺛﺎﻟﺜﺎ :ﻧﺤﺴﺐ اﻟﺘﺒﺎﻳﻨﺎت ﻣﻦ q ⎛1 ⎞ Vˆ ( rk ) ≅ ⎜ 1 + 2∑ rk2 ⎟ , k > q ⎝n k =1 ⎠ 1 ≅ ) Vˆ ( r1 9 1 1 2 Vˆ ( r2 ) ≅ (1 + 2 r12 ) = 1 + 2 ( 0.265) = 0.1267 n 9 1 1 2 2 Vˆ ( r3 ) ≅ 1 + 2 ( r12 + r22 ) = 1 + 2 ( 0.265) + ( −0.212 ) = 0.1367 n 9
)
))
(
(
١٦
( )
(
Vˆ ( r4 ) ≅ 0.138 Vˆ ( r5 ) ≅ 0.1454 Vˆ ( r6 ) ≅ 0.1787 اﻟﺦ… راﺑﻌﺎ :ﻧﺤﺴﺐ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ:
r00 = 1, by definition r11 = r1 = 0.265
ﺛﻢ ﻧﺤﺴﺐ ﺑﺎﻗﻲ اﻟﺘﺮاﺑﻄﺎت ﻣﻦ اﻟﻌﻼﻗﺎت اﻟﺘﻜﺮارﻳﺔ k −1
, k = 2,3, ...
rk − ∑ rk −1, j rk − j j =1 k −1
1 − ∑ rk −1, j rj
= rkk
j =1
ﺣﻴﺚ
j = 1, 2,..., k − 1
rkj = rk −1, j − rkk rk −1,k −1 , 1
r2 − r11r1 ( −0.212 ) − ( 0.265)( 0.265) −0.282225 = = 1 − r11r1 )1 − ( 0.265)( 0.265 0.929775
=
r2 − ∑ r1, j r2− j j =1 1
1 − ∑ r1, j rj
= r22
j =1
= −0.30354 ﻟﺤﺴﺎب r33ﻧﺤﺘﺎج اﻟﻰ r21وﺗﺤﺴﺐ ﻣﻦ r21 = r11 − r22 r11 = 0.265 − ( −0.303)( 0.265) = 0.345295
2
) r3 − ( r21r2 + r22 r1 ) 1 − ( r21r1 + r22 r2
=
r3 − ∑ r2, j r3− j j =1 k −1
1 − ∑ r2, j rj
= r33
j=1
) )( −0.076 ) − ( ( 0.345)( −0.212 ) + ( −0.303)( 0.265 ) ) 1 − ( ( 0.345)( 0.265) + ( −0.303)( −0.212
=
= 0.092
وهﻜﺬا ﻧﺤﺴﺐ ﺑﺎﻗﻲ اﻟﺘﺮاﺑﻄﺎت اﻟﺠﺰﺋﻴﺔ ﻟﻠﻌﻴﻨﺔ r88 = 0.042, r77 = 0.013, r66 = −0.207, r55 = −0.294, r44 = −0.298 وﻟﻬﺎ ﺟﻤﻴﻌﺎ اﻟﺘﺒﺎﻳﻨﺎت ﺗﺴﺎوي ﺗﻘﺮﻳﺒﺎ 1 = 0.1111 9 ﺧﺎﻣﺴﺎ :رﺳﻢ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻌﻴﻨﺔ Autocorrelation Function for Demand
6
١٧
3
4
2
Corr
1
Lag
LBQ
T
Corr
Lag
LBQ
T
13.66
0.52
0.23
8
0.87 1.50
1 0.27 0.80 2 -0.21 -0.59
Autocorrelation
8
7
5
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
Partial Autocorrelation
Partial Autocorrelation Function for Demand 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
5
6
Lag PAC
T
Lag PAC
T
0.27 -0.30 0.09 -0.30 -0.29 -0.21 0.01
0.80 -0.91 0.27 -0.89 -0.88 -0.62 0.04
8 0.04
0.13
1 2 3 4 5 6 7
١٨
7
8
اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Autoregressive-Moving Average Modelsوإﺳﺘﺨﺪاﻣﺎﺗﻬﺎ ﻓﻲ اﻟﺘﻨﺒﺆ: هﻨﺎك ﻋﺎﺋﻠﺔ آﺒﻴﺮة ﻣﻦ اﻟﻨﻤﺎذج اﻟﺘﻲ ﻳﻄﻠﻖ ﻋﻠﻴﻬﺎ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Autoregressive-Moving Average Modelsواﻟﺘﻲ اﺛﺒﺘﺖ اﻷﺑﺤﺎث اﻟﻜﺜﻴﺮة ﻓﻲ ﻣﺨﺘﻠﻒ اﻟﻤﻴﺎدﻳﻦ اﻟﺘﻄﺒﻴﻘﻴﺔ ﻋﻠﻲ ﺗﻔﻮﻗﻬﺎ اﻟﻬﺎﺋﻞ ﻋﻠﻲ اﻟﻄﺮق اﻟﺘﻘﻠﻴﺪﻳﺔ ﻓﻲ اﻟﺘﻨﺒﺆ.
ﺗﻌﺮﻳﻒ :١٤ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qوﻳﺮﻣﺰ ﻟﻪ ) ARMA ( p, qﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة } {z1 , z2 ,K, zn−1 , znﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: zt = δ + φ1 zt −1 + φ2 zt −2 + L + φ p zt − p + at − θ1at −1 − θ 2 at −2 − L − θ q at −q
ﺣﻴﺚ ) at WN ( 0,σ 2ﻣﺘﺴﻠﺴﻠﺔ ﺿﺠﺔ ﺑﻴﻀﺎء و ∞ < −∞ < δﻣﻌﻠﻢ ﺛﺎﺑﺖ ﻳﻤﺜﻞ اﻟﻤﺴﺘﻮي و φ1 ,φ2 ,K,φ pهﻲ ﻣﻌﺎﻟﻢ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ Autoregressive Parametersو θ1 ,θ 2 ,K,θ qهﻲ ﻣﻌﺎﻟﻢ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Moving Average Operators
ﺳﻮف ﻧﺴﺘﻌﻴﻦ ﺑﺠﺒﺮ اﻟﻌﻤﺎل Operators Algebraﻟﺘﺒﺴﻴﻂ هﺬﻩ اﻟﻨﻤﺎذج ﻟﻜﻲ ﻳﺴﻬﻞ اﻟﺘﻌﺎﻣﻞ ﻣﻌﻬﺎ
ﺗﻌﺮﻳﻒ :١٥ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ Backshift Operatorوﻳﺮﻣﺰ ﻟﻪ Bوﻟﻪ اﻟﺨﻮاص اﻟﺘﺎﻟﻴﺔ: 1 − Bzt = zt −1 2 − B m zt = B m−1 ( Bzt ) = B m−2 ( B ( Bzt ) ) = L = zt −m
3 − Bc = c, c is a constant ﺑﺎﻹﺿﺎﻓﺔ اﻟﻲ ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻟﺨﻠﻔﻲ ﺗﻮﺟﺪ ﻋﻤﺎل اﺧﺮي ﻧﺤﺘﺎج اﻟﻴﻬﺎ ﻻﺣﻘﺎ هﻲ:
ﺗﻌﺮﻳﻒ ١٥ب: -١ﻋﺎﻣﻞ اﻹزاﺣﺔ اﻷﻣﺎﻣﻲ Forewardshift Operatorوﻳﺮﻣﺰ ﻟﻪ Fوﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: F = B −1 -٢ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ Difference Operatorوﻳﺮﻣﺰ ﻟﻪ ∇ وﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: ) ∇ = (1 − B -٣ﻋﺎﻣﻞ اﻟﺘﺠﻤﻴﻊ Sum Operatorوﻳﺮﻣﺰ ﻟﻪ Sوﻳﻌﺮف آﺎﻟﺘﺎﻟﻲ: −1 ) S = ∇ −1 = (1 − B اﻵن ﻧﻌﻮد اﻟﻲ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qوﻧﻜﺘﺒﻪ ﻋﻠﻲ اﻟﺸﻜﻞ:
١٩
zt − φ1 zt −1 − φ2 zt −2 − L − φ p zt − p = δ + at − θ1at −1 − θ 2 at −2 − L − θ q at −q zt − φ1 Bzt − φ2 B 2 zt − L − φ p B p zt = δ + at − θ1 Bat − θ 2 B 2 at − L − θ q B q at
(1 − φ B − φ B 1
2
2
− L − φ p B p ) zt = δ + (1 − θ1 B − θ 2 B 2 − L − θ q B q ) at
أو
φ p ( B ) zt = δ + θ q ( B ) at
Autoregressive هﻮ ﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲφ p ( B ) = 1 − φ1B − φ2 B 2 − L − φ p B p ﺣﻴﺚ هﻮ ﻋﺎﻣﻞ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮكθ q ( B ) = 1 − θ1 B − θ 2 B 2 − L − θ q B q وOperator Moving Average Operator
:أﻣﺜﻠﺔ وﻳﻜﺘﺐARMA ( 0,0 ) وﻳﺮﻣﺰ ﻟﻪConstant Mean Model ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ-١ :ﻋﻠﻲ اﻟﺸﻜﻞ φ0 ( B ) zt = δ + θ 0 ( B ) at او (1) zt = δ + (1) at WN ( 0,σ 2 )
z t = δ + a t , at
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA (1,0 ) ≡ AR (1) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ-٢ φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
zt = δ + φ1 zt −1 + at , at
WN ( 0,σ 2 )
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA ( 0,1) ≡ MA (1) ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ-٣ φ0 ( B ) zt = δ + θ1 ( B ) at zt = δ + (1 − θ1 B ) at
zt = δ + at − θ1at −1 , at
WN ( 0,σ 2 )
: وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞARMA ( 2,0 ) ≡ AR ( 2 ) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ-٤ φ 2 ( B ) z t = δ + θ 0 ( B ) at
(1 − φ B − φ B ) z 2
1
2
t
= δ + at
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at
WN ( 0,σ 2 )
: وﻧﻜﺘﺒﻪ ﻋﻠﻲ اﻟﺸﻜﻞARMA (1,1) (١و١) ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ-٥
φ1 ( B ) zt = δ + θ1 ( B ) at (1 − φ1B ) zt = δ + (1 − θ1B ) at
zt = δ + φ1 zt −1 + at − θ1at −1 , at
WN ( 0,σ 2 )
٢٠
ﺧﺼﺎﺋﺺ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك: ﺳﻮف ﻧﺪرس اﻟﺨﺼﺎﺋﺺ اﻹﺣﺼﺎﺋﻴﺔ اﻟﺘﻲ ﺗﻤﻴﺰ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ_اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك وﻣﻌﺮﻓﺔ آﻴﻔﻴﺔ اﻟﺘﻌﺮف ﻋﻠﻲ اﺣﺪ هﺬﻩ اﻟﻨﻤﺎذج ﻣﻦ ﻋﻴﻨﺔ ﻣﺸﺎهﺪة وذﻟﻚ ﻟﺘﻌﻴﻴﻦ او ﺗﺤﺪﻳﺪ ﻧﻤﻮذج ﻣﻨﺎﺳﺐ ﻳﺼﻒ اﻟﻤﺸﺎهﺪات.
أوﻻ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ):ARMA(٠،٠ وﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ
φ0 ( B ) zt = δ + θ 0 ( B ) at
او
) WN ( 0, σ 2
z t = δ + a t , at
ﺳﻮف ﻧﺸﺘﻖ اﻟﺨﻮاص اﻹﺣﺼﺎﺋﻴﺔ ﻟﻬﺬا اﻟﻨﻤﻮذج وذﻟﻚ ﺑﺈﻳﺠﺎد اﻟﺘﻮﻗﻊ )اﻟﻤﺘﻮﺳﻂ( وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ آﺎﻟﺘﺎﻟﻲ: ) E ( z t ) = δ + E ( at وذﻟﻚ ﻷن ) WN ( 0, σ 2
=δ at
ﺳﻮف ﻧﺮﻣﺰ ﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ ) E ( ztﺑﺎﻟﺮﻣﺰ µأي ) µ = E ( ztوﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن δ = µ
وﻳﻜﺘﺐ اﻟﻨﻤﻮذج:
zt − µ = at ﻹﺷﺘﻘﺎق داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻧﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي
⎦⎤ E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at
وﻟﻜﻦ E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ = γ kﻣﻦ ﺗﻌﺮﻳﻒ ٨إذا γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L وﻧﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ: ⎦⎤ k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ﻹﻳﺠﺎد اﻟﻄﺮف اﻷﻳﻤﻦ ﻧﻀﺮب ﻃﺮﻓﻲ zt − µ = atﻓﻲ atوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي وذﻟﻚ ﻷن ) WN ( 0, σ 2
E ⎡⎣( zt − µ ) at ⎤⎦ = E ( at at ) = σ 2
atإذا
k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ⎤⎦ = σ 2 k = 1: γ 1 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ = 0
وذﻟﻚ ﻷن
zt −1 − µ = at −1 E ⎡⎣( zt −1 − µ ) at ⎤⎦ = E ( at −1at ) = 0
ﻓﻲ اﻟﺤﻘﻴﻘﺔ ﻓﺈن
٢١
zt −k − µ = at −k , k = 1, 2,K
E ⎡⎣( zt −k − µ ) at ⎦⎤ = E ( at −k at ) = 0, k = 1, 2,K أي
:١ ﻗﺎﻋﺪة ⎧σ , k = 0 E ⎡⎣ ( zt −k − µ ) at ⎤⎦ = E ( at −k at ) = ⎨ ⎩ 0, k = 1, 2,.. 2
أي
γ0 =σ γ k = 0, k = ±1, ±2,K 2
:وﺗﻮﺿﻊ ﻋﻠﻲ ﺷﻜﻞ داﻟﻲ
⎧σ , k = 0 γk = ⎨ 2
⎩ 0, k ≠ 0
ﻧﺠﺪγ 0 = σ 2 وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ
γ k ⎧1, k = 0 =⎨ γ 0 ⎩0, k ≠ 0 :وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ Autocorrelation function of Constant Mean Model 1.0
Autocorr
ρk =
0.5
0.0 0
1
2
3
4
5
6
7
8
9
Lag
ﻧﺠﺪ١١ ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﻦ اﻟﺘﻌﺮﻳﻒ
٢٢
φ00 = 1, by definition φ11 = ρ1 , by definition φ11 = 0
1
ρ1 1 0 ρ2 0 0 = =0 ρ1 1 0
ρ1
1
1
ρ1
ρ1 ρ2
1
1
φ22 =
φ33 =
ρ1
0 1
1
ρ1 1 0 ρ2 0 1 ρ3 0 0 = ρ2 1 0 ρ1 0 1
ρ1
1
ρ1 ρ1
1
ρ1 ρ2
0 0 0 =0 0 0 0 0 1
M
1
ρ1
ρ1
1 M
M
ρ k −1 ρ k −2 ρ1 1 ρ1 1
φkk =
M
M
ρ k −1 ρ k −2
L L M L L L M L
ρ1 ρ2
1 0 M 0 = 1 0 M 0
M
ρk ρ k −1 ρ k −2 M 1
0 1 M 0 0 1 M 0
L L M L L L M L
0 0 M 0 0 = = 0, k = 2,3,L 0 1 0 M 1 :وﺗﻮﺿﻊ ﻋﻠﻲ ﺷﻜﻞ داﻟﻲ
⎧1, k = 0 ⎩0, k ≠ 0
φkk = ⎨
Partial Autocorrelation function of Constant Mean Model
:وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
PACF
1.0
0.5
0.0 0
1
2
3
4
5
6
7
8
9
Lag اﻟﺒﻴﻀﺎء اﻻ ﻓﻲ ان ﻟﻪ ﻣﺘﻮﺳﻂ ﻏﻴﺮ ﺻﻔﺮي ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﺜﺎﺑﺖ ﻻﻳﻔﺘﺮق ﻋﻦ ﻧﻤﻮذج اﻟﻀﺠﺔ:ﻣﻼﺣﻈﺔ
٢٣
ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻲ )ARMA(١،٠) = AR(١ وهﻮ ﻋﻠﻲ اﻟﺸﻜﻞ: φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
) WN ( 0,σ 2
zt = δ + φ1 zt −1 + at , at
آﺎﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﺳﻮف ﻧﻮﺟﺪ اﻟﺘﻮﻗﻊ )اﻟﻤﺘﻮﺳﻂ( وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ:
(1 − φ1B ) zt = δ + at
+ (1 − φ1 B ) at
δ
−1
) (1 − φ1
δ
−1 ⎤ + E ⎡(1 − φ1 B ) at ⎦ ⎣ ) (1 − φ1
= zt
= ) E ( zt
اﻟﺤﺪ اﻟﺜﺎﻧﻲ ﻓﻲ اﻟﻄﺮف اﻷﻳﻤﻦ هﻮ
⎤ ⎞ ⎡⎛ ∞ j j −1 ⎡ ⎤ ⎥ E (1 − φ1 B ) at = E ⎢⎜ ∑φ1 B ⎟ at ⎣ ⎦ ⎢⎣⎝ j =0 ⎦⎥ ⎠ ﻹدﺧﺎل اﻟﺘﻮﻗﻊ ﻋﻠﻲ اﻟﻤﺠﻤﻮع اﻟﻶﻧﻬﺎﺋﻲ ﻳﺠﺐ ان ﺗﻜﻮن اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻶﻧﻬﺎﺋﻴﺔ ∞ < B j
j
∞
∑φ
1
ﻣﺘﻘﺎرﺑﺔ وذﻟﻚ
j =0
ﻳﺘﺤﻘﻖ إذا آﺎﻧﺖ φ1 < 1وذﻟﻚ إذا اﻋﺘﺒﺮﻧﺎ اﻟﻌﺎﻣﻞ Bاﻵن ﻳﻠﻌﺐ دور ﻣﺘﻐﻴﺮ ﻣﺮآﺐ Complex Variableﻟﻪ اﻟﺸﻜﻞ B = a + ibوﻟﻪ اﻟﻘﻴﺎس B = 1ﻓﻲ اﻟﺤﻘﻴﻘﺔ ﻻﺑﺪ ان ﻧﺘﻄﻠﺐ ان ﺗﻜﻮن ﺟﺰور او اﺻﻔﺎر (1 − φ1 B ) = 0ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة أي B > 1أي 1 − φ1 B = 0 1
φ1 > 1 ⇒ φ1 < 1
1
φ1
=B
⇒B >1
وهﺬا هﻮ ﺷﺮط اﻹﺳﺘﻘﺮار .ﻧﻌﻮد اﻟﻲ اﻟﻌﻼﻗﺔ
⎤ ⎞ ⎡⎛ ∞ j −1 ⎥ E ⎡(1 − φ1 B ) at ⎤ = E ⎢⎜ ∑φ1 B j ⎟ at ⎣ ⎦ ⎥⎦ ⎠ ⎣⎢⎝ j =0 ⎞ ⎡⎛ ∞ j j ⎤ ⎥ ) = ⎢⎜ ∑φ1 B ⎟ E ( at ⎢⎣⎝ j =0 ⎦⎥ ⎠ =0, ∀t وﻳﻜﻮن
٢٤
δ
) (1 − φ1 او
= ) E ( zt
δ
=µ
) (1 − φ1 ) ∴δ = µ (1 − φ1
وﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻓﻲ ﺻﻴﻐﺔ اﻟﻨﻤﻮذج ﻧﺠﺪ
zt = δ + φ1 zt −1 + at
= µ (1 − φ1 ) + φ1 zt −1 + at = µ + φ1 ( zt −1 − µ ) + at ﻧﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ أي
( zt − µ ) − φ1 ( zt −1 − µ ) = at
E ⎡⎣( zt −k − µ )( zt − µ ) ⎤⎦ − φ1E ⎡⎣( zt −k − µ )( zt −1 − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L أي
γ k − φ1γ k −1 = E ⎡⎣( zt −k − µ ) at ⎤⎦ , k = 0, ±1, ±2,L
وذﻟﻚ ﻣﻦ ﺗﻌﺮﻳﻒ ٨و ﺗﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ آﻤﺎ ﻳﻠﻲ: ⎦⎤ k = 0 : γ 0 − φ1γ 1 = E ⎡⎣( zt − µ ) at ﻹﻳﺠﺎد اﻟﻄﺮف اﻷﻳﻤﻦ ﻧﻘﻮم ﺑﺎﻟﺘﺎﻟﻲ: ) E ⎡⎣ at ( zt − µ ) ⎤⎦ − φ1 E ⎡⎣ at ( zt −1 − µ ) ⎤⎦ = E ( at at E ⎡⎣ at ( zt − µ ) ⎤⎦ − φ1 × ( 0 ) = σ 2 ∴ E ⎡⎣ at ( zt − µ ) ⎤⎦ = σ 2 إذا
γ 0 − φ1γ 1 = σ k = 1: γ 1 − φ1γ 0 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ = 0 2
ﻓﻲ اﻟﺤﻘﻴﻘﺔ
ﺑﻘﺴﻤﺔ اﻟﻤﻌﺎدﻟﺔ اﻷﺧﻴﺮة ﻋﻠﻲ γ 0
γ k − φ1γ k −1 = 0, k = 1, 2,L ﻧﺠﺪ
ρ k − φ1 ρ k −1 = 0, k = 1, 2,L
أو
وﺑﻤﺎ ان ρ 0 = 1
ρ k = φ1 ρ k −1 , k = 1, 2,L ﻓﺈن:
ρ1 = φ1 ρ 0 = φ1 ρ 2 = φ1 ρ1 = φ12
٢٥
M
ρ k = φ1k أو ﺑﺸﻜﻞ داﻟﺔ
ρ k = φ , k = 0, ±1, ±2,L وذﻟﻚ ﻷن ρ − k = ρ k , ∀kﺳﻮف ﻧﻨﻈﺮ ﻣﻦ اﻵن وﺻﺎﻋﺪا ﻟﻠﺸﻖ اﻟﻤﻮﺟﺐ ﻣﻦ ρ kأي k ρ k = φ1 , k = 0,1, 2,L هﺬﻩ اﻟﺪاﻟﺔ ﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 > 0 k 1
Autocorrelation function of AR(1) Model 0.5 0.4
ACF
0.3 0.2 0.1 0.0 10
9
8
7
6
4
5
3
2
1
0
Lag
-٢ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 < 0 Autocorrelation function of AR(1) Model 0.3 0.2 0.1 0.0
-0.2 -0.3 -0.4 -0.5 10
9
8
7
6
4
5
Lag
ﻧﺸﺘﻖ اﻵن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﻦ ﺗﻌﺮﻳﻒ ١١ﻧﺠﺪ
٢٦
3
2
1
0
ACF
-0.1
φ00 = 1, by definition φ11 = ρ1 = φ1 , by definition 1
φ22 =
ρ1 1
ρ1
1 φ1 ρ1 ρ 2 φ1 φ12 0 = = =0 1 φ1 1 − φ12 ρ1 1 φ1 1
M 1
ρ1
ρ1
1 M
M
ρ ρ k −2 φkk = k −1 1 ρ1 1 ρ1 M
M
ρ k −1 ρ k −2
L L L L L L L L
ρ1 ρ2 M
1
φ1
φ1
1 M
M
ρk φ = ρ k −1 1 ρ k −2 φ1
φ1k −2 φ1 L
k −1 1
M 1
φ
L φ1 L φ12 L M L φ1k
M
φ
k −1 1
φ1k L φ1k −1
1 M
L L
k −2 1
=
0 >0
M 1
وﻧﻜﺘﺐφ1 ﻣﺤﺪدة اﻟﺒﺴﻂ ﺗﺴﺎوي ﺻﻔﺮا ﻷن اﻟﻌﺎﻣﻮد اﻷﺧﻴﺮ ﻳﺴﺎوي اﻟﻌﺎﻣﻮد اﻷول ﻣﻀﺮوﺑﺎ ﻓﻲ :داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻋﻠﻲ اﻟﺸﻜﻞ اﻟﺪاﻟﻲ ⎧ 1, k = 0 ⎪ φkk = ⎨φ1 , k = 1 ⎪ 0, k ≥ 2 ⎩ :وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ φ1 > 0 ﻋﻨﺪﻣﺎ ﺗﻜﻮن-١ Partial Autocorrelation function of AR(1) Model 0.5
PACF
0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
Lag
φ1 < 0 ﻋﻨﺪﻣﺎ ﺗﻜﻮن-٢
٢٧
Partial Autocorrelation function of AR(1) Model 0.0 -0.1
-0.3
PACF
-0.2
-0.4 -0.5
10
9
8
7
6
4
5
3
2
1
0
Lag
ﻣﻼﺣﻈﺔ :داﺋﻤﺎ ﻻﺗﺮﺳﻢ أي ﻣﻦ ρ 0 = 1او φ00 = 1ﻓﻲ اﻷﺷﻜﺎل اﻟﺒﻴﺎﻧﻴﺔ.
ﻣﻨﺎﻗﺸﺔ اﻟﻨﻤﻮذج: -١ -٢ -٣ -٤
ﻋﻨﺪﻣﺎ ﺗﻜﻮن ) φ1 < 1ﺷﺮط اﻹﺳﺘﻘﺮار( ﻓﺈن ) E ( zt ) = δ (1 − φ1وهﻮﺛﺎﺑﺖ ﻟﺠﻤﻴﻊ ﻗﻴﻢ t داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ داﻟﺔ ﻟﻠﺘﺨﻠﻒ kﻓﻘﻂ وﻻﺗﻌﺘﻤﺪ ﻋﻠﻲ اﻟﺰﻣﻦ t داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ إﺑﺘﺪاءا ﻣﻦ ρ1ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 > 0وﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻣﺘﺮددة ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺗﻜﻮن φ1 < 0 داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ ﻗﻴﻤﺔ واﺣﺪة ﻏﻴﺮ ﺻﻔﺮﻳﺔ ) ﻣﻊ ﻋﺪم اﻟﻨﻈﺮ اﻟﻲ ( φ00وﻳﻜﻮن إﺗﺠﺎهﻬﺎ ﺣﺴﺐ إﺷﺎرة φ1وﻣﻘﺪارهﺎ ﻳﺴﺎوي φ1
ﺛﺎﻟﺜﺎ :ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ): ARMA(٢،٠) = AR(٢ وﻳﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ 2 ( B ) zt = δ + θ 0 ( B ) a t = δ + at ) WN ( 0,σ 2
(1 − φ B + φ B ) z 2
t
2
1
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at
آﺎﻟﺴﺎﺑﻖ ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: (1 − φ1B − φ2 B ) zt = δ + at 2
−1 δ + (1 − φ1 B − φ2 B 2 ) at ) (1 − φ1 − φ2 −1 δ ⎤ + E ⎡(1 − φ1 B − φ2 B 2 ) at ⎦⎥ ⎣⎢ ) (1 − φ1 − φ2
٢٨
= zt
= ) E ( zt
∞ ⎛ ⎞ اﻟﺤﺪ اﻟﺜﺎﻧﻲ ﻓﻲ اﻟﻄﺮف اﻷﻳﻤﻦ ﻣﺠﻤﻮع ﻻﻧﻬﺎﺋﻲ ﻋﻠﻰ اﻟﺸﻜﻞ ⎟ E ⎜ ∑ψ j at − jوﻟﻜﻲ ﻧﺪﺧﻞ اﻟﺘﻮﻗﻊ ⎝ j =0 ⎠ ∞
داﺧﻞ اﻟﺘﺠﻤﻴﻊ اﻟﻼﻧﻬﺎﺋﻲ ﻻﺑﺪ ان ﺗﻜﻮن
∑ψ a
j t− j
ﻣﺘﻘﺎرﺑﺔ ﻓﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺮﺑﻊ وهﺬا ﻳﺘﺤﻘﻖ إذا
j =0
وﻓﻘﻂ إذا آﺎن ∞ <
2 j
∞
∑ψ
وهﺬا ﻳﺘﺤﻘﻖ إذا ﺣﻘﻘﺖ ﻣﻌﺎﻟﻢ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺸﺮوط اﻟﺘﺎﻟﻴﺔ:
j =0
φ2 − φ1 < 1 φ2 + φ1 < 1 −1 < φ2 < 1 واﻟﺘﻲ ﺗﺴﻤﻲ ﺑﺸﺮوط اﻹﺳﺘﻘﺮار ) هﺬﻩ اﻟﺸﺮوط ﺗﻨﺘﺞ اﻳﻀﺎ ﻣﻦ آﻮن ﺟﺰور او أﺻﻔﺎر (1 − φ1 B − φ2 B 2 ) = 0ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ( .إذا ﺗﺤﻘﻘﺖ ﺷﺮوط اﻹﺳﺘﻘﺮار ﻓﺈن −1 −1 E ⎡(1 − φ1 B − φ2 B 2 ) at ⎤ = ⎡(1 − φ1 B − φ2 B 2 ) E ( at ) ⎤ = 0, ∀t ⎢⎣ ⎦⎥ ⎦⎥ ⎢⎣
وﻳﻜﻮن
δ ) (1 − φ1 − φ2
و ﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻓﻲ ﺻﻴﻐﺔ اﻟﻨﻤﻮذج ﻧﺠﺪ
= ) µ = E ( zt
δ = (1 − φ1 − φ2 ) µ zt = (1 − φ1 − φ2 ) µ + φ1 zt −1 + φ2 zt −2 + at = µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at
( zt − µ ) − φ1 ( zt −1 − µ ) − φ2 ( zt −2 − µ ) = at ﻧﻀﺮب اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻓﻲ zt −k − µوﻧﺄﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ:
⎦⎤ ) E ⎡⎣( zt − µ )( zt −k − µ ) − φ1 ( zt −1 − µ )( zt −k − µ ) − φ2 ( zt −2 − µ )( zt −k − µ = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
أي ⎦⎤ ) E ⎡⎣( zt − µ )( zt −k − µ ) ⎦⎤ − φ1 E ⎡⎣ ( zt −1 − µ )( zt −k − µ ) ⎤⎦ − φ2 E ⎡⎣( zt −2 − µ )( zt −k − µ = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
أو
γ k − φ1γ k −1 − φ2γ k −2 = E ⎡⎣ at ( zt −k − µ ) ⎤⎦ , k = 0, ±1, ±2,...
وذﻟﻚ ﻣﻦ ﺗﻌﺮﻳﻒ ٨اﻵن ﻧﺤﻞ هﺬﻩ اﻟﻌﻼﻗﺔ ﺗﻜﺮارﻳﺎ آﻤﺎ ﻳﻠﻲ:
٢٩
k = 0 : γ 0 − φ1γ −1 − φ2γ −2 = E ⎡⎣ at ( zt − µ ) ⎤⎦ = σ 2 ⇒ γ 0 = φ1γ 1 − φ2γ 2 + σ 2 وذﻟﻚ ﻣﻦ ﻗﺎﻋﺪة ١ k = 1: γ 1 − φ1γ 0 − φ2γ 1 = 0 ⇒ γ 1 = φ1γ 0 − φ2γ 1 k = 2 : γ 2 − φ1γ 1 − φ2γ 0 = 0 ⇒ γ 2 = φ1γ 1 − φ2γ 0 وﺑﺸﻜﻞ ﻋﺎم k ≥ 1: γ k = φ1γ k −1 + φ2γ k −2 ﺑﻘﺴﻤﺔ اﻟﻄﺮﻓﻴﻦ ﻋﻠﻲ γ 0ﻧﺠﺪ ρ k = φ1 ρ k −1 + φ2 ρ k −2 , k = 1, 2, ... ) ﻣﻼﺣﻈﺔ :ﺑﻮﺿﻊ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻋﻠﻲ اﻟﺸﻜﻞ ρ k − φ1 ρ k −1 − φ2 ρ k −2 = 0, k = 1, 2,... ﻧﺠﺪ اﻧﻬﺎ ﻣﻌﺎدﻟﺔ ﻓﺮوﻗﻴﺔ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ واﻟﺘﻲ ﻳﻤﻜﻦ ﺣﻠﻬﺎ ﺑﺸﻜﻞ ﻣﻐﻠﻖ ﺑﺈﺳﺘﺨﺪام ﻃﺮق ﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻔﺮوﻗﻴﺔ وﻟﻜﻦ هﺬا ﺧﺎرج ﻧﻄﺎق اﻟﻤﻘﺮر اﻟﺤﺎﻟﻲ( ﺳﻮف ﻧﺤﻞ اﻟﻌﻼﻗﺔ اﻟﺴﺎﺑﻘﺔ ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺘﻜﺮارﻳﺔ واﻟﺘﻲ ﺗﺤﺘﺎج اﻟﻲ ﻗﻴﻤﺘﻴﻦ اوﻟﻴﺘﻴﻦ: 1 − ρ0 = 1 φ 2 − ρ1 = φ1 ρ 0 + φ2 ρ −1 ⇒ ρ1 = 1 1 − φ2 وﻣﻨﻬﺎ ﻧﺠﺪ 2 φ ρ 2 = φ1 ρ1 + φ2 ρ0 ⇒ ρ 2 = 1 + φ2 1 − φ2 وهﻜﺬا اﻟﺦ… اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ ) AR ( 2 -١اﻟﺸﻜﻞ )φ1 = 0.4, φ2 = 0.4 (١ -٢اﻟﺸﻜﻞ )φ1 = 1.5, φ2 = −0.8 (٢ -٣اﻟﺸﻜﻞ )φ1 = 0.5, φ2 = −0.6 (٣ ﺷﻜﻞ )(١ ACF 0.7 0.6 0.5
0.3 0.2 0.1 0.0 20
10
Lag
٣٠
0
ACF
0.4
(٢) ﺷﻜﻞ ACF 1.0
ACF
0.5
0.0
-0.5
0
10
20
Lag
(٣) ﺷﻜﻞ ACF
ACF
0.5
0.0
-0.5
0
10
20
Lag
: آﺎﻟﺘﺎﻟﻲAR ( 2 ) اﻵن ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ
φ00 = 1, by definition φ11 = ρ1 , by definition 1 ρ1 ρ1 ρ 2 ρ 2 − ρ12 = ≠0 φ22 = 1 ρ1 1 − ρ12 ρ1 1
٣١
=0
ρ1 = φ1 + φ2 ρ1 ρ 2 = φ1 ρ1 + φ2 ρ3 = φ1 ρ 2 + φ2 ρ1
ρ1 1
ρ1
>0
ρ1 1 ρ2 ρ1 ρ3 ρ 2 = ρ2 ρ1
ρ1
1
1
ρ1 ρ2
ρ1 ρ1 1
ρ1
1
= φ33
ρ1 ρ2
1 وذﻟﻚ ﻷن اﻟﻌﻤﻮد اﻷﺧﻴﺮ ﻓﻲ ﻣﺤﺪدة اﻟﺒﺴﻂ هﻮ ﺗﺮآﻴﺐ ﺧﻄﻲ ﻣﻦ اﻟﻌﻤﻮدﻳﻦ اﻷول واﻟﺜﺎﻧﻲ ،آﺬﻟﻚ 1 ρ1 L ρ1 1 ρ1 L ρ1 = φ1 ρ 0 + φ2 ρ1 ρ1 1 L ρ2 ρ1 1 L ρ 2 = φ1 ρ1 + φ2 ρ 0 M M L M M M L M ρ ρ k −2 L ρ k ρ ρ k −2 L ρ k = φ1 ρ k −1 + φ2 ρ k −2 φkk = k −1 = k −1 = 0, k = 3, 4,... 1 ρ1 L ρ k −1 >0 1 L ρ k −2 ρ1 M M L M ρ k −1 ρ k −2 L 1 وذﻟﻚ اﻳﻀﺎ ﻟﻨﻔﺲ اﻟﺴﺒﺐ اﻟﺴﺎﺑﻖ .إذا
k =0 ⎧ 1, ⎪ ρ, k =1 1 ⎪⎪ φkk = ⎨ ρ 2 − ρ12 ⎪ 1− ρ2 , k = 2 1 ⎪ k ≥3 ⎪⎩ 0,
اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻌﻤﻠﻴﺔ ) AR ( 2 -٤اﻟﺸﻜﻞ )φ1 = 0.4, φ2 = 0.4 (٤ -٥اﻟﺸﻜﻞ )φ1 = 1.5, φ2 = −0.8 (٥ -٦اﻟﺸﻜﻞ )φ1 = 0.5, φ2 = −0.6 (٦ ﺷﻜﻞ )(٤
PACF 0.7 0.6 0.5
0.3 0.2 0.1 0.0 20
10
Lag
٣٢
0
PACF
0.4
ﺷﻜﻞ )(٥ PACF 1
PACF
0
-1
20
0
10
Lag
ﺷﻜﻞ )(٦ PACF 0.3 0.2 0.1
-0.2
PACF
0.0 -0.1
-0.3 -0.4 -0.5 -0.6 20
0
10
Lag
راﺑﻌﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ ): ARMA(٠،١) = MA(١ وﺗﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ0 ( B ) zt = δ + θ1 ( B ) at
) WN ( 0,σ 2
zt = δ + (1 − θ1 B ) at zt = δ + at − θ1at −1 , at
اﻵن ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: E ( zt ) = E (δ + at − θ1at −1 ) = δ ∴µ = δ
وﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﺑﻀﺮب هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻓﻲ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
zt − µ = at − θ1at −1
E ⎡⎣( zt − µ )( zt −k − µ ) ⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
٣٣
او
γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ
⎦⎤ k = 0 : γ 0 = E ⎡⎣( zt − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt − µ ) at −1
ﻧﻮﺟﺪ آﻞ ﻣﻦ ⎦⎤ E ⎡⎣( zt − µ ) atو ⎦⎤ E ⎡⎣( zt − µ ) at −1آﺎﻵﺗﻲ: E ⎡⎣( zt − µ ) at ⎤⎦ = E ( at at ) − θ1 E ( at −1at ) = σ 2 E ⎡⎣( zt − µ ) at −1 ⎤⎦ = E ( at at −1 ) − θ1E ( at −1at −1 ) = −θ1σ 2
) ∴γ 0 = σ 2 − θ1 ( −θ1σ 2 ) = σ 2 (1 + θ12
⎦⎤ k = 1: γ 1 = E ⎡⎣( zt −1 − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −1 − µ ) at −1 −θ1 γ1 = γ 0 1 + θ12 وذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻟﻘﺎﻋﺪة ١
= ∴ γ 1 = −θ1σ 2 ⇒ ρ1
⎦⎤ k = 2 : γ 2 = E ⎡⎣( zt −2 − µ ) at ⎤⎦ − θ1 E ⎡⎣( zt −2 − µ ) at −1 ∴ γ 2 = 0 ⇒ ρ2 = 0
أﻳﻀﺎ ﻣﻦ ﻗﺎﻋﺪة ١وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن
k ≥ 2 : γ k = 0 ⇒ ρk = 0 وهﻜﺬا ﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ ) MA (1هﻲ ﻋﻠﻲ اﻟﺸﻜﻞ: ⎧ 1, k =0 ⎪ ⎪ −θ ρk = ⎨ 1 2 , k = 1 ⎪ 1 + θ1 ⎪⎩ 0 k ≥2 وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ θ1 = 0.8 ACF 0.0 -0.1
-0.3 -0.4 -0.5 20
0
10
Lag
٣٤
ACF
-0.2
-٢ﻋﻨﺪﻣﺎ θ1 = −0.8 ACF 0.5 0.4
0.2
ACF
0.3
0.1 0.0
20
10
0
Lag
اﻵن ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻷوﻟﻲ )MA (1 φ00 = 1, by definition
φ11 = ρ1 , by definition 1 ρ1 1 ρ1 ) −θ12 (1 − θ12 ρ1 ρ 2 ρ1 0 − ρ12 −θ12 = φ22 = = = = 1 ρ1 1 − ρ12 1 − ρ12 1 + θ12 + θ14 1 − θ16 ρ1 1 ρ1 1 ρ1 ρ1 ρ2 ρ1 1 0 ) −θ13 (1 − θ12 ρ3 0 ρ1 0 ρ13 = = = ρ2 1 ρ1 0 1 − 2 ρ12 1 − θ18 ρ1 ρ1 1 ρ1 1 0 ρ1 1 وﺑﺸﻜﻞ ﻋﺎم k >0
وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: -١ﻋﻨﺪﻣﺎ ﻋﻨﺪﻣﺎ θ1 = −0.8
٣٥
ρ1
1
1
ρ1 ρ2
ρ1 ρ1 1
ρ1
(1 − θ ) , 2 1
)2 k +1
1
= φ33
ρ1 ρ2 k 1
−θ
( 1 − θ1
= φkk
PACF 0.5 0.4 0.3
0.1 0.0
PACF
0.2
-0.1 -0.2 -0.3 20
0
10
Lag
-٢ﻋﻨﺪﻣﺎ θ1 = 0.8 PACF 0.0 -0.1
-0.3
PACF
-0.2
-0.4 -0.5 20
0
10
Lag
ﺧﺎﻣﺴﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ): ARMA(٠،٢) = MA(٢ وﺗﻜﺘﺐ ﻋﻠﻲ اﻟﺸﻜﻞ: φ 0 ( B ) zt = δ + θ 2 ( B ) a t zt = δ + (1 − θ1 B − θ 2 B 2 ) at
) WN ( 0,σ 2
zt = δ + at − θ1at −1 − θ 2 at −2 , at
اﻵن ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ وداﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ: E ( zt ) = E (δ + at − θ1at −1 − θ 2 at −2 ) = δ ∴µ = δ
٣٦
وﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﺑﻀﺮب هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻓﻲ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ
zt − µ = at − θ1at −1 − θ 2 at −2
⎦⎤ E ⎡⎣( zt − µ )( zt −k − µ ) ⎤⎦ = E ⎡⎣ ( zt −k − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −k − µ ) at −1 − θ 2 E ⎡⎣( zt −k − µ ) at −2 ⎤⎦ , k = 0, ±1, ±2,...
او − µ ) at −2 ⎤⎦ , k = 0, ±1, ±2,...
γ k = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ − θ 2 E ⎡⎣( zt −k
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ ﻧﺠﺪ 2
γ 0 = (1 + θ + θ ) σ 2 2
2 1
γ 1 = ( −θ1 + θ1θ 2 ) σ 2 γ 2 = −θ 2σ 2 γ k = 0, k > 2
وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻲ γ 0ﻧﺠﺪ
−θ1 + θ1θ 2 1 + θ12 + θ 22
= ρ1
−θ 2 1 + θ12 + θ 22
= ρ2
ρ k = 0, k > 2 وﺗﻜﺘﺐ ﻋﻠﻲ ﺷﻜﻞ داﻟﺔ 1, k =0 ⎧ ⎪ −θ + θ θ ⎪ 1 2 1 22 , k = 1 ⎪ 1 + θ1 + θ 2 ⎨ = ρk ⎪ −θ 2 , k =2 ⎪1 + θ12 + θ 22 ⎪ 0, k>2 ⎩
اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ هﻲ ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻌﻤﻠﻴﺔ ) MA ( 2 -٧اﻟﺸﻜﻞ )θ1 = 0.4, θ 2 = 0.4 (٧ -٨اﻟﺸﻜﻞ )θ1 = 1.5, θ 2 = −0.8 (٨ -٩اﻟﺸﻜﻞ )θ1 = 0.5, θ 2 = −0.6 (٩
٣٧
(٧) ﺷﻜﻞ
ACF 0.0
ACF
-0.1
-0.2
-0.3
0
10
20
Lag
(٨) ﺷﻜﻞ
ACF 0.2 0.1 0.0
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0
10
20
Lag
(٩) ﺷﻜﻞ ACF 0.4 0.3
0.2 0.1
ACF
ACF
-0.1
0.0 -0.1 -0.2 -0.3 -0.4 -0.5 0
10
Lag
٣٨
20
ﻣﻦ اﻟﺼﻌﺐ ﺟﺪا إﻳﺠﺎد ﺷﻜﻞ ﻣﻐﻠﻖ ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ) MA ( 2وﻟﻬﺬا ﺳﻮف ﻧﺴﺘﺨﺪم ﺗﻌﺮﻳﻒ ١١ب ﻟﺤﺴﺎﺑﻬﺎ ورﺳﻤﻬﺎ ﺗﻜﺮارﻳﺎ ﻟﻘﻴﻢ اﻟﻤﻌﺎﻟﻢ اﻟﺘﺎﻟﻴﺔ: -١٠اﻟﺸﻜﻞ )θ1 = 0.4, θ 2 = 0.4 (١٠ -١١اﻟﺸﻜﻞ )θ1 = 1.5, θ 2 = −0.8 (١١ -١٢اﻟﺸﻜﻞ )θ1 = 0.5, θ 2 = −0.6 (١٢
ﺷﻜﻞ )(١٠
PACF 0.0
-0.1
PACF
-0.2
-0.3
20
0
10
Lag
ﺷﻜﻞ )(١١ PACF 0.2 0.1 0.0 -0.1
-0.3 -0.4 -0.5 -0.6 -0.7
20
10
Lag
٣٩
0
PACF
-0.2
ﺷﻜﻞ )(١٢ PACF 0.4 0.3 0.2 0.1
PACF
0.0 -0.1 -0.2 -0.3 -0.4 -0.5 20
0
10
Lag
ﺳﺎدﺳﺎ :ﻧﻤﻮذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك-اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ ): ARMA(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ:
φ1 ( B ) zt = δ + θ1 ( B ) at
(1 − φ1B ) zt = δ + (1 − θ1B ) at WN ( 0, σ 2 ) , φ1 ≠ θ1
zt − φ1 zt −1 = δ + at − θ1at −1 zt = δ + φ1 zt −1 + at − θ1at −1 , at
ﺷﺮط اﻹﺳﺘﻘﺮار φ1 < 1وﺷﺮط اﻹﻧﻘﻼب θ1 < 1وهﻨﺎك ﺷﺮط ﺁﺧﺮ ﻳﺴﻤﻰ ﺷﺮط اﻹﻣﺘﺴﺎخ Degeneracy Conditionوهﻮ φ1 ≠ θ1وهﺬا اﻟﺸﺮط ﻳﻀﻤﻦ ﻋﺪم إﻣﺘﺴﺎخ اﻟﻨﻤﻮذج إﻟﻰ ﻧﻤﻮذج أﻗﻞ درﺟﺔ ﻓﻔﻲ ﺣﺎﻟﺔ آﻮن φ1 = θ1ﻓﻤﻦ اﻟﻌﻼﻗﺔ (1 − φ1B ) zt = δ + (1 − θ1B ) atوﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ) (1 − φ1Bﻧﺠﺪ أن اﻟﻨﻤﻮذج ﻳﺼﺒﺢ zt = δ ′ + atﺣﻴﺚ ﻧﻮﺟﺪ اﻟﻤﺘﻮﺳﻂ آﺎﻟﺘﺎﻟﻲ:
δ 1 − φ1
= δ ′وهﻮ ) ARMA ( 0,0
(1 − φ1B ) zt = δ + (1 − θ1B ) at (1 − θ1B ) a δ = zt + t ) 1 − φ1 (1 − φ1B (1 − θ1B ) E a δ = ) E ( zt + )( t ) 1 − φ1 (1 − φ1B
وذﻟﻚ ﻷن φ1 < 1وهﻜﺬا δ
أي
δ 1 − φ1
1 − φ1
= ) E ( zt
= E ( zt ) = µأو ) δ = µ (1 − φ1وﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ δﻧﺠﺪ zt = µ (1 − φ1 ) + φ1 zt −1 + at − θ1at −1
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1 وﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ ﺑﺎﻟﺤﺪ ( zt −k − µ ) , k = 0, ±1, ±2,...وأﺧﺬ اﻟﺘﻮﻗﻊ ﻟﻠﻄﺮﻓﻴﻦ ﻧﺠﺪ E ⎡⎣( zt −k − µ )( zt − µ )⎤⎦ − φ1E ⎡⎣ ( zt −k − µ )( zt −1 − µ )⎤⎦ = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﻣﻨﻬﺎ ٤٠
γ k − φ1γ k −1 = E ⎡⎣( zt −k − µ ) at ⎤⎦ − θ1E ⎡⎣ ( zt −k − µ ) at −1 ⎤⎦ , k = 0, ±1, ±2,...
وﺑﺤﻠﻬﺎ ﺗﻜﺮارﻳﺎ ﻧﺠﺪ
k = 0 γ 0 − φ1γ 1 = E ⎡⎣ ( zt − µ ) at ⎤⎦ − θ1E ⎡⎣ ( zt − µ ) at−1 ⎤⎦
ﺑﻀﺮب اﻟﻌﻼﻗﺔE ⎡⎣( zt − µ ) at −1 ⎤⎦ وE ⎡⎣( zt − µ ) at ⎤⎦ ﻧﻮﺟﺪ اﻵن آﻞ ﻣﻦ
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1
وأﺧﺬ اﻟﺘﻮﻗﻊat −1 وat ﻓﻲ آﻞ ﻣﻦ E ⎡⎣ ( zt − µ ) at ⎤⎦ − φ1E ⎡⎣ ( zt −1 − µ ) at ⎤⎦ = E [at at ] − θ1 E [at −1at ]
ﻧﺠﺪ١ وﻣﻦ اﻟﻘﺎﻋﺪة
E ⎡⎣( zt − µ ) at ⎤⎦ − φ1 ( 0 ) = σ − θ1 ( 0 ) 2
E ⎡⎣( zt − µ ) at ⎤⎦ = σ 2 E ⎡⎣( zt − µ ) at −1 ⎤⎦ − φ1E ⎡⎣( zt −1 − µ ) at −1 ⎤⎦ = E [at at −1 ] − θ1E [at −1at −1 ]
و
E ⎡⎣( zt − µ ) at −1 ⎤⎦ − φ1σ 2 = 0 − θ1σ 2 ∴ E ⎡⎣( zt − µ ) at −1 ⎤⎦ = σ 2 (φ1 − θ1 )
وﺑﺎﻟﺘﻌﻮﻳﺾ ﻓﻲ اﻟﺼﻴﻐﺔ اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ k = 0 γ 0 − φ1γ 1 = σ 2 − θ1σ 2 (φ1 − θ1 ) ∴γ 0 − φ1γ 1 = σ 2 ⎡⎣1 − θ1 (φ1 − θ1 )⎤⎦ k = 1 γ 1 − φ1γ 0 = E ⎡⎣ ( zt −1 − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −1 − µ ) at −1 ⎤⎦
و
∴ γ 1 − φ1γ 0 = −θ1σ 2 k = 2 γ 2 − φ1γ 1 = E ⎡⎣ ( zt −2 − µ ) at ⎤⎦ − θ1 E ⎡⎣ ( zt −2 − µ ) at −1 ⎤⎦ = 0
و
∴ k ≥ 2 γ k − φ1γ k −1 = 0
وﻣﻦ اﻟﻤﻌﺎدﻻت
γ 0 − φ1γ 1 = σ ⎡⎣1 − θ1 (φ1 − θ1 )⎤⎦ 2
و γ 1 − φ1γ 0 = −θ1σ
2
ﻧﺠﺪ γ0 = γ1 =
ρ1 =
1 + θ − 2φ1θ1 2 σ 1 − φ12 2 1
(1 − φ1θ1 )(φ1 − θ1 ) σ 2 1 − φ12
وﻣﻦ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ
γ 1 (1 − φ1θ1 )(φ1 − θ1 ) = 1 + θ12 − 2φ1θ1 γ0
٤١
وﻣﻦ اﻟﻌﻼﻗﺔ
γ k − φ1γ k −1 = 0, k ≥ 2
وﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ γ 0ﻧﺠﺪ ρ k − φ1 ρ k −1 = 0, k ≥ 2 وﻳﻤﻜﻦ ﺣﻞ هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﺗﻜﺮارﻳﺎ ﻟﺠﻤﻴﻊ ﻗﻴﻢ k ≥ 2ﺑﺈﺳﺘﺨﺪام اﻟﻘﻴﻢ اﻷوﻟﻴﺔ ρ0 = 1و ) (1 − φ1θ1 )(φ1 − θ1 = ρ1ﻓﻤﺜﻼ 1 + θ12 − 2φ1θ1 ρ2 = φ1 ρ1 ) (1 − φ1θ1 )(φ1 − θ1 ρ 2 = φ1 1 + θ12 − 2φ1θ1 ρ3 = φ1 ρ 2 ) (1 − φ1θ1 )(φ1 − θ1 ρ3 = φ12 1 + θ12 − 2φ1θ1
وهﻜﺬا. ﻧﻜﺘﺐ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﻋﻠﻰ اﻟﺸﻜﻞ 1, k =0 ⎧ ⎪ ) ⎪ (1 − φ1θ1 )(φ1 − θ1 , k =1 ⎨ = ρk 2 ⎪ 1 + θ1 − 2φ1θ1 ⎩⎪ k≥2 φ1 ρ k −1
ﺷﻜﻞ ١٣ﻳﻌﻄﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻘﻴﻢ φ1 = 0.9,θ1 = −0.5
ﺷﻜﻞ)(١٣
) A C F o f A R M A (1 ,1 1 .0 0 .9 0 .8 0 .7 0 .6 0 .4 0 .3 0 .2 0 .1 0 .0
15
5
10
Lag
ﺷﻜﻞ ١٤ﻳﻌﻄﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻘﻴﻢ φ1 = −0.9,θ1 = −0.5
٤٢
0
C1
0 .5
ﺷﻜﻞ)(١٤ ) A C F o f A R M A (1 ,1 0 .5
C1
0 .0
-0 .5
10
15
0
5
Lag
ﻧﻼﺣﻆ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ أو ﻣﺘﺮدد ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ وهﻲ ﻓﻲ هﺬا ﺗﺸﺒﻪ ﺗﻤﺎﻣﺎ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻨﻤﻮذج ) AR (1ﻣﺎﻋﺪى ان اﻟﺘﺨﺎﻣﺪ ﻳﺒﺪأ ﻣﻦ ) ρ1ﺑﺮهﻦ أن ( ρ k = φ1k −1 ρ1 , k ≥ 2 داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ φkkﺗﺤﺴﺐ ﻣﻦ ﺗﻌﺮﻳﻒ ١١أو ١١ب آﺎﻟﺘﺎﻟﻲ: ﻣﻦ ﺗﻌﺮﻳﻒ ١١ب ﻧﻮﺟﺪ φkkﺗﻜﺮارﻳﺎ φ00 = 1, by definition
) (1 − φ1θ1 )(φ1 − θ1 1 + θ12 − 2φ1θ1
= φ11 = ρ1
ρ 2 − φ11 ρ1 1 − φ11 ρ1 ρ −φ ρ −φ ρ φ33 = 3 21 2 22 1 , φ21 = φ11 − φ22φ11 1 − φ21 ρ1 − φ22 ρ 2
= φ22
وهﻜﺬا ﺗﺤﺴﺐ ﺑﻘﻴﺔ اﻟﻘﻴﻢ ﺗﻜﺮارﻳﺎ. ﻓﻤﺜﻼ ﻟﻠﻘﻴﻢ φ1 = 0.9,θ1 = −0.5ﻧﺠﺪ
φ11 = 0.944186 φ22 = -0.384471 φ33 = 0.183710 φ44 = -0.908462 φ55 = 0.452979 φ66 = -0.226337 φ77 = 0.113154 φ88 = -0.565702 φ99 = 0.282834
وﻧﺮﺳﻢ هﺬﻩ اﻟﻘﻴﻢ ﻓﻲ ﺷﻜﻞ ١٥ ﺷﻜﻞ ١٥ ) P A C F o f A R M A (1 ,1 1 .0
0 .5
C2 0 .0
15
5
10
Lag
٤٣
0
ﺷﻜﻞ ١٦ﻳﺒﻴﻦ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻘﻴﻢ φ1 = −0.9,θ1 = −0.5
ﺷﻜﻞ ١٦ ) P A C F o f A R M A (1 ,1 0 .3 0 .2 0 .1 0 .0
-0 .2
C2
-0 .1
-0 .3 -0 .4 -0 .5 -0 .6
15
5
10
0
Lag
ﻧﻼﺣﻆ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج ) ARMA (1,1ﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻓﻲ إﺗﺠﺎﻩ واﺣﺪ أو ﻣﺘﺮدد ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﻮﺟﺒﺔ واﻟﺴﺎﻟﺒﺔ وهﻲ ﻓﻲ هﺬا ﺗﺸﺒﻪ ﺗﻤﺎﻣﺎ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج )MA (1 ﻣﺎﻋﺪى ان اﻟﺘﺨﺎﻣﺪ ﻳﺒﺪأ ﺑﻌﺪ اﻟﻘﻴﻤﺔ اﻷوﻟﻴﺔ . φ11 = ρ1
ﺧﻮاص ﻧﻤﺎذج ): ARMA(p,q أوﻻ :ﻧﻤﻮذج )AR(p وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: -١داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ. -٢داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ ﺗﺘﻜﻮن ﻣﻦ أﺻﻔﺎر ﻟﻘﻴﻢ اﻟﺘﺨﻠﻔﺎت k > pأي φ11 = φ22 = φ33 = L = φ pp ≠ 0 φ p +1, p +1 = φ p +2, p + 2 = L = 0
وﻳﺴﻤﻰ هﺬا ﻗﻄﻌﺎ ﻓﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ . k > p ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج ): MA(q وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: -١داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺗﺘﻜﻮن ﻣﻦ أﺻﻔﺎر ﻟﻘﻴﻢ اﻟﺘﺨﻠﻔﺎت k > qأي ρ1 = ρ 2 = ρ3 = L = ρ q ≠ 0 ρ q+1,q+1 = ρ q+2,q +2 = L = 0
وﻳﺴﻤﻰ هﺬا ﻗﻄﻌﺎ ﻓﻲ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ . k > q -٢داﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ. ﻻﺣﻆ اﻹزدواﺟﻴﺔ Dualityﺑﻴﻦ ﻧﻤﻮذﺟﻲ ARو .MA ﺛﺎﻟﺜﺎ :اﻟﻨﻤﻮذج اﻟﻤﺨﺘﻠﻂ ):ARMA(p,q وﻳﺘﻤﻴﺰ ﺑﺎﻟﺘﺎﻟﻲ: دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﺨﺘﻠﻂ ﺗﻤﺘﺪ ﻻﻧﻬﺎﺋﻴﺎ وﺗﺘﻜﻮن ﻣﻦ ﺧﻠﻴﻂ ﻣﻦ اﻟﺘﺨﺎﻣﺪات اﻻﺳﻴﺔ واﻟﺘﺨﺎﻣﺪات اﻟﺠﻴﺒﻴﺔ اﻟﺘﻲ ﺗﻨﺘﻬﻲ إﻟﻰ اﻟﺼﻔﺮ آﻠﻤﺎ زاد اﻟﺘﺨﻠﻒ . kﻋﻨﺪﻣﺎ ﺗﻜﻮن k > q − pﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﺘﺤﺪد ﻣﻦ ﺟﺰء اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻟﻠﻨﻤﻮذج و ﻋﻨﺪﻣﺎ ﺗﻜﻮن k > p − qﻓﺈن داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺤﺪد ﻣﻦ ﺟﺰء اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻟﻠﻨﻤﻮذج. ٤٤
اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة Nonstationar Time :Series Models اوﻻ :ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ: ﻣﻦ ﺗﻌﺮﻳﻒ ٦ﻹﺳﺘﻘﺮار ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻧﺮى ان اﻟﺸﺮط اﻷول ﻟﻺﺳﺘﻘﺮار E ( zt ) = µ = constant ∀tﻳﺘﻄﻠﺐ أن ﻳﻜﻮن ﻣﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ ﺛﺎﺑﺖ ﻋﻠﻰ ﻃﻮل اﻟﺰﻣﻦ ،ﻓﻤﺜﻼ ﻟﻨﻤﻮذج اﻹﻧﺠﺮاف اﻟﺨﻄﻲ ) ∞ WN ( 0, σ 2 ) , b0 , b1 ∈ ( −∞,
zt = b0 + b1t + at , at
ﻧﺠﺪ ان اﻟﻤﺘﻮﺳﻂ هﻮ
E ( z )t = b0 + b1t
وهﻮ ﻏﻴﺮ ﺛﺎﺑﺖ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﺰﻣﻦ ،اي ان ﺷﺮط اﻹﺳﺘﻘﺮار اﻷول ﻏﻴﺮ ﻣﺘﺤﻘﻖ ﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ. ﻟﻨﺤﺎول اﻟﺘﺤﻮﻳﻞ ∇ztوذﻟﻚ ﺑﺘﻄﺒﻴﻖ ﻋﺎﻣﻞ اﻟﺘﻔﺮﻳﻖ ﻋﻠﻰ اﻟﻨﻤﻮذج ﻧﺠﺪ
wt = ∇zt = zt − zt −1 = b0 + b1t + at − b0 − b1 ( t − 1) − at −1 =b1 + at − at −1 = b1 + ct
) WN ( 0,ν 2
) ﺗﻤﺮﻳﻦ :أوﺟﺪ اﻟﻌﻼﻗﺔ ﺑﻴﻦ ν 2و ( σ 2 اﻵن ﻧﺠﺪ ﻣﺘﻮﺳﻂ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺠﺪﻳﺪة wt
∴ wt = b1 + ct , ct
E ( wt ) = b1 = constant ∀t
أي ان ﺗﻄﺒﻴﻖ اﻟﺘﺤﻮﻳﻞ ) ∇ = (1 − Bﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ) أي أﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول ﻟﻠﻤﺘﺴﻠﺴﻠﺔ( ﺣﻮﻟﻬﺎ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. آﻤﺜﺎل ﺁﺧﺮ ﻧﻤﻮذج اﻹﻧﺠﺮاف اﻟﺘﺮﺑﻴﻌﻲ ) ∞ WN ( 0, σ 2 ) , b0 , b1 , b2 ∈ ( −∞,
zt = b0 + b1t + b2t 2 + at , at
ﺑﺈﻳﺠﺎد اﻟﻤﺘﻮﺳﻂ 2
E ( zt ) = b0 + b1t + b2t
وهﻮ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ ،أي ان اﻟﻨﻤﻮذج ﻏﻴﺮ ﻣﺴﺘﻘﺮ .ﺑﺄﺧﺬ اﻟﺘﺤﻮﻳﻞ ) ∇ 2 ztأﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻟﺜﺎﻧﻲ( ﻧﺠﺪ
) ∇2 zt = ∇ 2 ( b0 + b1t + b2t 2 + at
) + b1t + b2t 2 + at
(1 − 2 B + B ) z = (1 − 2 B + B )( b 2
2
0
t
wt = {b0 − 2b0 + b0 } + {b1t − 2b1 ( t − 1) + b1 ( t − 2 )} +
}
− 2b2 ( t − 1) + b2 ( t − 2 ) + 2
2
2
{b t 2
} {at − 2at −1 + at −2 } = 2b2 + {at − 2at −1 + at −2 ) WN ( 0,τ 2
وهﻜﺬا
٤٥
=b′ + ht , ht
) WN ( 0,τ 2
wt = ∇2 zt = b′ + ht , ht
E ( wt ) = b′ = constant ∀t
أي ان ﺗﻄﺒﻴﻖ اﻟﺘﺤﻮﻳﻞ ) ∇2أي اﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻟﺜﺎﻧﻲ( ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﺣﻮﻟﻬﺎ اﻟﻰ ﻣﺴﺘﻘﺮة. 2 2 ) ﺗﻤﺮﻳﻦ :أوﺟﺪ اﻟﻌﻼﻗﺔ ﺑﻴﻦ τو .( σ ﺑﺸﻜﻞ ﻋﺎم إذا آﺎن اﻟﻨﻤﻮذج ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮ ﻋﻠﻰ اﻟﺸﻜﻞ
) ∞ WN ( 0, σ 2 ) , b0 , b1 ,L , bd ∈ ( −∞,
zt = b0 + b1t + L + bd t + at , at d
ﻓﺈن اﻟﺘﺤﻮﻳﻞ ∇d ztﻳﺤﻮﻟﻪ إﻟﻰ ﻧﻤﻮذج ﻣﺴﺘﻘﺮ ،أي ان wt = ∇ d ztهﻮ ﻧﻤﻮذج ﻣﺴﺘﻘﺮ.
ﺗﻌﺮﻳﻒ :١٦ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ) ∞ zt = b0 + b1t + L + bd t + at , at WN ( 0, σ 2 ) , b0 , b1 ,L , bd ∈ ( −∞, اﻟﺘﺤﻮﻳﻞ ∇d ztوهﻮ اﻟﺘﻔﺮﻳﻖ ﻟﻠﺪرﺟﺔ dﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻳﺤﻮﻟﻬﺎ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. d
ﺛﺎﻧﻴ ًﺎ :ﻋﺪم اﻹﺳﺘﻘﺮار ﻓﻲ اﻟﺘﺒﺎﻳﻦ: ﻣﻦ ﺗﻌﺮﻳﻒ ٦ﻹﺳﺘﻘﺮار ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ،اﻟﺸﺮط اﻟﺜﺎﻧﻲ
ﻳﺘﻄﻠﺐ أن ﻳﻜﻮن اﻟﺘﺒﺎﻳﻦ ﺛﺎﺑﺖ ﻟﺠﻤﻴﻊ ﻗﻴﻢ . t ﻓﻤﺜﻼ ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ
V ( zt ) = γ 0 = constant ∀t
) WN ( 0, σ 2
zt = zt −1 + at , at
ﻧﺠﺪ ﻣﻦ اﻟﺘﻌﻮﻳﺾ اﻟﻤﺘﻜﺮر zt = a1 + a2 + L + at
وﺑﺈﺧﺬ اﻟﺘﻮﻗﻊ واﻟﺘﺒﺎﻳﻦ
E ( zt ) = 0 = constant ∀t V ( zt ) = tσ 2
وﻧﻼﺣﻆ أن اﻟﺘﺒﺎﻳﻦ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ . t ﺑﺄﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول wt = ∇zt = zt − zt −1 = at
وﺑﺈﺧﺬ اﻟﺘﻮﻗﻊ واﻟﺘﺒﺎﻳﻦ E ( wt ) = 0 = constant ∀t V ( wt ) = σ 2 = constant ∀t
إذًا اﻟﺘﻔﺮﻳﻖ اﻷول ﺣﻮل اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ إﻟﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة. ﺑﺸﻜﻞ ﻋﺎم ﻟﻮ آﺎن اﻟﺘﺒﺎﻳﻦ داﻟﺔ ﻟﻤﺴﺘﻮى )ﻣﺘﻮﺳﻂ( ﻣﺘﻐﻴﺮ ﻋﻠﻰ اﻟﺸﻜﻞ
) V ( zt ) = cf ( µt
ﺣﻴﺚ c > 0ﺛﺎﺑﺖ و )⋅( fداﻟﺔ ﻣﻌﺮوﻓﺔ ﺗﻌﻄﻰ ﻗﻴﻤﺔ ﻏﻴﺮ ﺳﺎﻟﺒﺔ و µtﻣﺴﺘﻮى أو ﻣﺘﻮﺳﻂ ﻳﺘﻐﻴﺮ ﻣﻊ اﻟﺰﻣﻦ و ﺑﺎﻟﺘﺎﻟﻲ ﻓﺈن اﻟﺘﺒﺎﻳﻦ ﻳﻌﺘﻤﺪ ﻋﻠﻰ اﻟﺰﻣﻦ وهﻨﺎ ﻧﺤﺎول إﻳﺠﺎد ﺗﺤﻮﻳﻞ ) T ( ztأي إﻳﺠﺎد داﻟﺔ )⋅ ( Tﻹﺳﺘﻘﺮار اﻟﺘﺒﺎﻳﻦ. ٤٦
اﻟﺘﺤﻮﻳﻞ λ
zt − 1
λ
= ) yt = T ( zt
ﻳﻌﻄﻲ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ ﺣﻴﺚ ) ∞ λ ∈ ( −∞,هﻮ ﻣﻌﻠﻢ اﻟﺘﺤﻮﻳﻞ .اﻟﺠﺪول اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﻘﻴﻢ اﻷآﺜﺮ إﺳﺘﺨﺪاﻣﺎ ﻟﻠﻤﻌﻠﻢ λﻣﻊ اﻟﺘﺤﻮﻳﻼت اﻟﻤﻘﺎﺑﻠﺔ ﻟﻬﺎ: 1.0 zt
0.5
0.0
-0.5
zt
ln zt
1 zt
-0.1
λ
1 zt
yt
ﻣﺜﺎل: اﻟﺸﻜﻞ)ا( ﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ واﻟﺘﺒﺎﻳﻦ zt O r ig in a l S e r ie s 400
300
)z(t 200
100 90
80
70
60
50
40
30
20
10
In d e x
اﻟﺸﻜﻞ)ب( اﻟﻤﺘﺴﻠﺴﻠﺔ ﺑﻌﺪ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ ﺑﺈﺟﺮاء اﻟﺘﺤﻮﻳﻞ yt = ln zt T r a n s f o r m e d S e r ie s 6 .0
5 .5
)ln z(t 5 .0
90
80
70
60
50
40
30
20
اﻟﺸﻜﻞ)ج( اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺤﻮﻟﺔ ytﺑﻌﺪ إﺟﺮاء اﻟﺘﻔﺮﻳﻖ اﻷول ∇yt = yt − yt −1
٤٧
10
In d e x
D if f e re n c e d a n d T ra n s f o rm e d S e rie s 0 .2
0 .1 )y(t)-y(t-1
0 .0
-0 .1
-0 .2
90
80
70
60
50
40
30
20
10
In d e x
ﻻﺣﻆ آﻴﻒ اﺻﺒﺤﺖ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﻤﺘﻮﺳﻂ واﻟﺘﺒﺎﻳﻦ.
ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )(p,d,q Autoregressive-Integrated-Moving Average Models )ARIMA(p,d,q ﻳﻤﻜﻦ ﻧﻤﺬﺟﺔ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺴﺘﻘﺮة wt = ∇ d ztﻋﻠﻰ ﺷﻜﻞ ﻧﻤﻮذج أﻧﺤﺪار ذاﺗﻲ-ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, qآﺎﻟﺘﺎﻟﻲ: ) WN ( 0, σ 2
أو
)
2
φ p ( B ) wt = φ p ( B ) ∇ d zt = δ + θ q ( B ) at , at
φ p ( B )(1 − B ) zt = δ + θ q ( B ) at , at WN ( 0,σ d
وهﺬا اﻟﻨﻤﻮذج ﻳﺴﻤﻰ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ( p, d , q ﺣﻴﺚ ) ∞ δ ∈ ( −∞,ﻣﻌﻠﻢ اﻹﻧﺠﺮاف. أﻣﺜﻠﺔ ﻋﻠﻰ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) : ( p, d , q اوﻻ :ﻧﻤﻮذج اﻹﻧﺤﺪاراﻟﺬاﺗﻲ-اﻟﺘﻜﺎﻣﻠﻲ ﻣﻦ اﻟﺪرﺟﺔ ) (١،١أو )ARIMA(١،١،٠ ): =ARI(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ1 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at WN ( 0, σ 2
(1 − φ1B )(1 − B ) zt = δ + at {1 − (φ1 + 1) B + φ1B 2 } zt = δ + at أي φ1 < 1
),
2
WN ( 0, σ
zt = δ + (φ1 + 1) zt −1 − φ1 zt −2 + at , at
ﺛﺎﻧﻴﺎ :ﻧﻤﻮذج اﻟﺘﻜﺎﻣﻠﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ )(١،١ أو ): ARIMA(٠،١،١) = IMA(١،١ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ0 ( B )(1 − B ) zt = δ + θ1 ( B ) at , at WN ( 0, σ 2
WN ( 0,σ 2 ) , θ1 < 1
at
(1 − B ) zt = δ + (1 − θ1B ) at ,
WN ( 0, σ 2 ) , θ1 < 1
zt − zt −1 = δ + at − θ1at , at
أي ٤٨
WN ( 0, σ 2 ) ,
θ1 < 1
zt = δ + zt −1 + at − θ1at , at
ﺛﺎﻟﺜﺎ :ﻧﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ﺑﺈﻧﺠﺮاف Random Walk with Trend Modelأو ): ARIMA(٠،١،٠ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
φ0 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at
) WN ( 0, σ 2
أي
)
2
WN ( 0, σ
at
(1 − B ) zt = δ + at ,
zt = δ + zt −1 + at , at
داﻟﺔ اﻷوزان ) ψ ( Bوﺗﻤﺜﻴﻞ ﻧﻤﺎذج ):ARMA(p,q ﺳﺒﻖ أن آﺘﺒﻨﺎ ﻧﻤﺎذج ) ARMA ( p, qﻋﻠﻰ اﻟﺸﻜﻞ
) WN ( 0, σ 2
أو ﺑﺸﻜﻞ اﻹﻧﺤﺮاف ﻋﻦ اﻟﻤﺘﻮﺳﻂ
)
2
WN ( 0, σ
φ p ( B ) zt = δ + θ q ( B ) a t , a t φ p ( B )( zt − µ ) = θ q ( B ) at , at
ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﺑﺎﻟﻘﺴﻤﺔ ﻋﻠﻰ ﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ) φ p ( Bﻧﺠﺪ ) WN ( 0, σ 2
) θq ( B a , at φp (B) t
) WN ( 0, σ 2
+
δ
)φ p (1
) θq ( B a , at φp (B) t
= zt
= zt − µ
) θq ( B ﻻﺣﻆ ان ﻟﻠﻨﻤﺎذج اﻟﻤﺴﺘﻘﺮة اﻟﻨﺴﺒﺔ )φp (B δ ﺗﻘﻊ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة اﻳﻀﺎ = µ وﻟﻬﺬا ﺳﻮف ﻧﻜﺘﻔﻲ ﺑﺸﻜﻞ اﻹﻧﺤﺮاف ﻋﻦ اﻟﻤﺘﻮﺳﻂ ﻓﻲ )φ p (1
ﺗﺸﻜﻞ ﻣﺴﻠﺴﻠﺔ ﻣﺘﻘﺎرﺑﺔ وذﻟﻚ ﻷن ﺟﺬور φ p ( B ) = 0
ﻣﻨﺎﻗﺸﺘﻨﺎ اﻟﺘﺎﻟﻴﺔ ) WN ( 0, σ 2
اﻟﻤﺴﻠﺴﻠﺔ اﻟﻤﺘﻘﺎرﺑﺔ
واﻟﺘﻲ ﺗﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
)θ (B zt − µ = q a , at φp (B) t ) θq ( B )φp (B
)θq ( B = ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + L , ψ 0 = 1 )φp (B
ﺗﺴﻤﻰ داﻟﺔ اﻷوزان.
٤٩
= )ψ (B
= )ψ (B
ﺗﻌﺮﻳﻒ :١٧داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ) ARMA ( p, qاﻟﻤﺴﺘﻘﺮ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ )θq ( B = ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + L , ψ 0 = 1 )φp (B
= )ψ (B
∞ )θq ( B = ∑ψ B j , ψ 0 = 1 φ p ( B ) j =0 j
= )ψ (B
ﺣﻴﺚ اﻷوزان هﻲ ψ 0 = 1,ψ 1 ,ψ 2 ,ψ 3 ,L ) θq ( B a , at φp (B) t
ﻣﻼﺣﻈﺔ :اﻟﻨﻤﻮذج اﻟﺬي ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
= zt − µﻳﺴﻤﻰ ﺗﻤﺜﻴﻞ
اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻼﻧﻬﺎﺋﻲ ﻟﻨﻤﺎذج ) . ARMA ( p, q
أﻣﺜﻠﺔ ﻟﺪاﻟﺔ اﻷوزان ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ): AR(١ ﻧﻤﻮذج ) AR(١ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ1 ( B )( zt − µ ) = θ 0 ( B ) at , at WN ( 0, σ 2
(1 − φ1B )( zt − µ ) = at 1 a (1 − φ1B ) t
= zt − µ
zt − µ = ψ ( B ) at
ﺣﻴﺚ 1 ) (1 − φ1B
= )ψ (B
ﺳﻮف ﻧﻮﺟﺪ اﻷوزان ψ 1 ,ψ 2 ,ψ 3 ,Lﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺘﺎﻟﻴﺔ 1 ) (1 − φ1B
= )ψ (B
ψ ( B )(1 − φ1B ) ≡ 1
B 2 + ψ 3 B 3 + L) (1 − φ1B ) ≡ 1
2
(1 + ψ B + ψ 1
ﻻﺣﻆ أن اﻟﻌﻼﻗﺔ اﻷﺧﻴﺮة هﻲ ﻋﻼﻗﺔ ﺗﻜﺎﻓﺆ أي ان ﻣﻌﺎﻣﻼت B jﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻣﺘﺴﺎوﻳﺔ. وﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت B jﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪ φ1 < 1
+ ψ 3 B 3 + L) (1 − φ1B ) ≡ 1,
2
(1 + ψ B + ψ B 2
1
B 0 : (1)(1) ≡ 1 B1 : ψ 1 − φ1 ≡ 0 ⇒ ψ 1 = φ1 B 2 : ψ 2 − ψ 1φ1 ≡ 0 ⇒ ψ 2 = ψ 1φ1 = φ12 B 3 : ψ 3 − ψ 2φ1 ≡ 0 ⇒ ψ 3 = ψ 2φ1 = φ13 M B j : ψ j − ψ j −1φ1 ≡ 0 ⇒ ψ j = ψ j −1φ1 = φ1j
أي ان اﻷوزان ﻟﻨﻤﻮذج ) AR(١هﻲ ٥٠
ψ j = φ1j , φ1 < 1
: MA(١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞMA(١) ﻧﻤﻮذج
φ0 ( B )( zt − µ ) = θ1 ( B ) at , at WN ( 0, σ 2 )
( zt − µ ) = (1 − θ1B ) at zt − µ = ψ ( B ) at ﺣﻴﺚ
ψ ( B ) = (1 − θ1B )
ﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪB j ﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت
ψ 1 = −θ1 , ψ 2 = ψ 3 = L = 0
أي j=0 j =1
⎧ 1, ⎪ ψ j = ⎨ −θ1 , ⎪ 0, ⎩
j≥2
φ2 ( B )( zt − µ ) = θ 0 ( B ) at , at WN ( 0,σ 2 )
(1 − φ B − φ B ) ( z − µ ) = a
: AR(٢) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞAR(٢) ﻧﻤﻮذج
2
1
zt − µ =
2
t
t
1 a (1 − φ1B − φ2 B 2 ) t
zt − µ = ψ ( B ) at
ﺣﻴﺚ ψ (B) =
1 (1 − φ1B − φ2 B 2 )
ψ ( B ) (1 − φ1B − φ2 B ) ≡ 1
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
2
(1 + ψ B + ψ B 1
2
2
+ ψ 3 B 3 + L)(1 − φ1B − φ2 B 2 ) ≡ 1
B1 : ψ 1 − φ1 = 0 ⇒ ψ 1 = φ1 2 B 2 : ψ 2 − φψ 1 1 − φ2 = 0 ⇒ ψ 2 = φψ 1 1 + φ2 = φ1 + φ 2
B 3 : ψ 3 − φψ 1 2 − φ2ψ 1 = 0 ⇒ ψ 3 = φψ 1 2 + φ2ψ 1 M B j : ψ j − φψ 1 j −1 − φ2ψ j − 2 = 0 ⇒ ψ j = φψ 1 j −1 + φ 2ψ j − 2
هﻲAR(٢) أي ان اﻷوزان ﻟﻨﻤﻮذج
٥١
j=0
⎧ 1, ⎪φ , ⎪ 1 ψj =⎨ 2 ⎪ φ1 + φ2 , ⎪⎩φψ 1 j −1 + φ2ψ j − 2 ,
j =1 j=2 j≥3
: MA(٢) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞMA(٢) ﻧﻤﻮذج
φ0 ( B )( zt − µ ) = θ 2 ( B ) at , at WN ( 0,σ 2 )
( zt − µ ) = (1 − θ1B − θ 2 B 2 ) at zt − µ = ψ ( B ) at ψ ( B ) = (1 − θ1B − θ 2 B
2
ﺣﻴﺚ
)
ﻋﻠﻰ ﻃﺮﻓﻲ اﻟﻌﻼﻗﺔ ﻧﺠﺪB j ﺑﻤﺴﺎواة ﻣﻌﺎﻣﻼت
ψ 1 = −θ1 , ψ 2 = −θ 2 , ψ 3 = ψ 4 = ψ 5 L = 0
أي j=0 j =1 j=2 j≥2
⎧ 1, ⎪ −θ , ⎪ ψj =⎨ 1 ⎪ −θ 2 , ⎪⎩ 0,
φ1 ( B )( zt − µ ) = θ1 ( B ) at , at
: ARMA(١،١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞARMA(١،١) ﻧﻤﻮذج
WN ( 0, σ 2 )
(1 − φ1B )( zt − µ ) = (1 − θ1B ) at (1 − θ1B ) a zt − µ = (1 − φ1B ) t zt − µ = ψ ( B ) a t ψ (B) =
(1 − θ1B ) (1 − φ1B )
ψ ( B )(1 − φ1B ) ≡ (1 − θ1B )
(1 + ψ B + ψ 1
ﺣﻴﺚ
2
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
B 2 + ψ 3 B 3 + L) (1 − φ1B ) ≡ (1 − θ1B )
B1 : ψ 1 − φ1 = −θ1 ⇒ ψ 1 = φ1 − θ1 B 2 : ψ 2 − φψ 1 1 = 0 ⇒ ψ 2 = φψ 1 1 = φ1 (φ1 − θ1 ) 2 B 3 : ψ 3 − φψ 1 2 = 0 ⇒ ψ 3 = φψ 1 2 = φ1 (φ1 − θ1 )
M
j −1 B j : ψ j − φψ (φ1 − θ1 ) 1 j −1 = 0 ⇒ ψ j = φψ 1 j −1 = φ1
هﻲARMA(١،١) أي ان اﻷوزان ﻟﻨﻤﻮذج ٥٢
j −1 ψ j = φψ (φ1 − θ1 ) , 1 j −1 = φ1
j ≥ 1,
φ1 < 1, φ1 ≠ θ1
: ARI(١) داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞARI(١) ﻧﻤﻮذج
φ1 ( B )(1 − B )( zt − µ ) = at , at WN ( 0,σ 2 ) zt − µ =
1 a (1 − φ1B )(1 − B ) t
zt − µ = ψ ( B ) at
ﺣﻴﺚ ψ ( B) =
1 (1 − φ1B )(1 − B )
ﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔψ 1 ,ψ 2 ,ψ 3 ,L و ﻧﻮﺟﺪ اﻷوزان
ψ ( B )(1 − φ1B )(1 − B ) ≡ 1
(1 +ψ B +ψ (1 +ψ B +ψ 1
2
1
2
B 2 + ψ 3 B 3 + L) (1 − φ1B )(1 − B ) ≡ 1
B 2 + ψ 3 B 3 + L) (1 − (φ1 + 1) B + φ1B 2 ) ≡ 1
B1 : ψ 1 − (φ1 + 1) = 0 ⇒ ψ 1 = φ1 + 1 B 2 : ψ 2 − (φ1 + 1)ψ 1 + φ1 = 0 ⇒ ψ 2 = (φ1 + 1)ψ 1 + φ1 = (φ1 + 1) + φ1 2
B 3 : ψ 3 − (φ1 + 1)ψ 2 + φψ 1 1 = 0 ⇒ ψ 3 = (φ1 + 1)ψ 2 − φψ 1 1 M
B j : ψ j − (φ1 + 1)ψ j −1 + φψ 1 j −2 1 j − 2 = 0 ⇒ ψ j = (φ1 + 1)ψ j −1 − φψ
هﻲARI(١) أي ان اﻷوزان ﻟﻨﻤﻮذج j=0 j =1
⎧ 1, ⎪ φ + 1, ⎪ 1 ψj =⎨ 2 ⎪ (φ1 + 1) + φ1 , ⎪(φ1 + 1)ψ j −1 − φψ 1 j −2 , ⎩
j=2 j≥3
وأﺧﻴﺮا ﻧﻮﺟﺪ ARIMA(١،٠،١) أوRandom Walk Mdel داﻟﺔ اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ وﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ zt = zt −1 + at , at
WN ( 0, σ 2 )
zt − zt −1 = at , at
WN ( 0, σ
2
أي
)
(1 − B ) zt = at zt =
1 a (1 − B ) t
ﺣﻴﺚ 1 ψ ( B) = (1 − B )
٥٣
و ﻧﻮﺟﺪ اﻷوزان ψ 1 ,ψ 2 ,ψ 3 ,Lﺑﺎﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔ
ψ ( B )(1 − B ) ≡ 1 + ψ 3 B 3 + L) (1 − B ) ≡ 1
2
(1 + ψ B + ψ B 1
2
B1 : ψ 1 − 1 = 0 ⇒ ψ 1 = 1 B 2 :ψ 2 −ψ 1 = 0 ⇒ ψ 2 = ψ 1 = 1 B 3 :ψ 3 −ψ 2 = 0 ⇒ ψ 3 = ψ 2 = 1 M B j : ψ j − ψ j −1 = 0 ⇒ ψ j = ψ j −1 = 1
أي ان اﻷوزان ﻟﻨﻤﻮذج اﻟﻤﺸﻲ اﻟﻌﺸﻮاﺋﻲ ) ARIMA ( 0,1,0هﻲ ψ j = 1,
j ≥1
ﺑﻌﺾ ﺧﻮاص داﻟﺔ اﻷوزان ) : ψ ( B ﺳﺒﻖ أن آﺘﺒﻨﺎ ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ARMA(p,qﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
وﺑﻜﺘﺎﺑﺔ هﺬﻩ اﻟﻌﻼﻗﻪ ﻋﻠﻰ اﻟﺸﻜﻞ
zt − µ = ψ ( B ) a t , a t
zt − µ = at + ψ 1at −1 + ψ 2 at −2 + ψ 3at −3 + L ∞
= ∑ψ j at − j , ψ 0 = 1 j =0
وإذا اﻓﺘﺮﺿﻨﺎ ان اﻷوزان ﺗﺘﻘﺎرب اي ∞ < ψ 2j
∞
j =0
∑
ﻓﺈﻧﻪ ﻳﻤﻜﻦ إﺛﺒﺎت اﻟﻨﻈﺮﻳﺔ اﻟﺘﺎﻟﻴﺔ:
ﻧﻈﺮﻳﺔ :١ ﻟﻨﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ-اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ) ARMA(p,qاﻟﻤﺴﺘﻘﺮ واﻟﺬي ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ∞ < WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j ∞
∞
zt − µ =∑ψ j at − j , at j =0
-١اﻟﻤﺘﻮﺳﻂ هﻮ
E ( zt ) = µ , ∀t
-٢داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ∞
j +k
, k = 0,1, 2,L
∑ψ ψ j
j =0
2 j
∞
∑ψ
= ρk
j=0
ﻣﺜﺎل :ﻟﻨﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ ) AR(١وﺟﺪﻧﺎ ﺳﺎﺑﻘﺎ داﻟﺔ اﻷوزان ψ j = φ , φ1 < 1 j 1
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
٥٤
= φ1k , k = 0,1, 2,L
φ1k 1 − φ12 1 1 − φ12
∞
=
∑φ1jφ1j+k j =0 ∞
∑φ
2j 1
j =0
∞
=
∑ψ jψ j+k j =0
2 j
∑ψ
وهﻲ ﻧﻔﺲ اﻟﻨﺘﻴﺠﺔ اﻟﺴﺎﺑﻘﺔ ﺗﻤﺮﻳﻦ :ﺑﺄﺳﺘﺨﺪام ﻧﻈﺮﻳﺔ (٢) ٢أوﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻨﻤﺎذج اﻟﺘﺎﻟﻴﺔ )AR(٢), MA(١), MA(٢), ARMA(١،١), ARMA(٢،١), ARMA(١،٢
٥٥
∞
j=0
= ρk
اﻟﻔﺼﻞ اﻟﺮاﺑﻊ اﻟﺘﻨﺒﺆات ذات ﻣﺘﻮﺳﻂ ﻣﺮﺑﻊ اﻟﺨﻄﺄ اﻷدﻧﻰ ﻟﻨﻤﺎذج )ARMA(p,q Minimum Mean Square Error Forecasts for ARMA(p,q) Models ﻓﻲ ﺍﻟﻔﻘﺭﺓ ﺍﻟﺴﺎﺒﻘﺔ ﻜﺘﺒﻨﺎ ﻨﻤﻭﺫﺝ ﺍﻹﻨﺤﺩﺍﺭ ﺍﻟﺫﺍﺘﻲ-ﺍﻟﻤﺘﻭﺴﻁ ﺍﻟﻤﺘﺤﺭﻙ ﻤﻥ ﺍﻟﺩﺭﺠﺔ )ARMA(p,q ﺍﻟﻤﺴﺘﻘﺭ ﻋﻠﻰ ﺍﻟﺸﻜل ∞ < WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j ∞
∞
zt − µ =∑ψ j at − j , at j =0
ﺃﻭ
zt − µ = at + ψ 1at −1 + ψ 2at −2 + ψ 3at −3 +L ∞
=∑ψ j at − j , ψ 0 = 1 j =0
ﻤﻼﺤﻅﺔ :ﻫﺫﺍ ﻴﻨﻁﺒﻕ ﺃﻴﻀﺎ ﻋﻠﻰ ﻨﻤﺎﺫﺝ ) ARIMA(p,d,qﺒﺸﻜل ﻋﺎﻡ. ﻟﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﺓ } {z1 , z2 ,L, zn −1 , znﺍﻟﺘﻨﺒﺅﺍﺕ zn ( l ) , l ≥ 1ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﻴﻤﻜﻥ ﺍﻥ ﺘﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل zn ( l ) = ξ 0an + ξ1an −1 + ξ 2an −2 + L, l ≥ 1
ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﺘﻜﺘﺏ ﺒﺩﻻﻟﺔ ﺍﻟﻨﻤﻭﺫﺝ ﻜﺎﻟﺘﺎﻟﻲ
zn + l − µ = an +l + ψ 1an + l−1 + L + ψ l−1an +1 + ψ l an + ψ l+1an −1 + L, l ≥ 1
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﻴﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ )ﺃﻨﻅﺭ ﺘﻌﺭﻴﻑ ( ٥
2
⎦⎤E ⎡⎣ zn + l − zn ( l )⎤⎦ = E ⎡⎣ an + l + ψ 1an + l−1 + L + ψ l−1an +1 + (ψ l − ξ 0 ) an + (ψ l+1 − ξ1 ) an −1 + L 2
∞
= (1 + ψ 12 + L + ψ l2−1 ) σ 2 + ∑ (ψ l+ j − ξ j ) σ 2 2
j =0
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ ﻴﻨﺘﺞ ﻤﻥ ﺘﺼﻐﻴﺭ ﺍﻟﻌﻼﻗﺔ ﺍﻟﺴﺎﺒﻘﺔ ﺒﺎﻟﻨﺴﺒﺔ ﻟﻸﻭﺯﺍﻥ ξ jﻟﺠﻤﻴﻊ ﻗﻴﻡ j ﻭﻫﺫﺍ ﻴﻤﻜﻥ ﺇﺫﺍ ﻭﻓﻘﻁ ﺇﺫﺍ ﺤﻘﻘﺕ ﺍﻷﻭﺯﺍﻥ ξ jﺍﻟﻌﻼﻗﺔ ﺍﻟﺘﺎﻟﻴﺔ j = 0,1, 2,L , l ≥ 1
ξ j = ψ l+ j ,
ﻭﻋﻠﻴﻪ ﻓﺈﻥ ﺍﻟﺘﻨﺒﺅﺍﺕ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ zn ( l ) = ψ l an + ψ l+1an −1 + ψ l+2an −2 + L, l ≥ 1
ﻨﻅﺭﻴﺔ :٢ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ:
en ( l ) = zn +l − zn ( l ) = an +l + ψ 1an +l−1 + ψ 2an +l−2 + L + ψ l−1an+1 , l ≥ 1
ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ: V ⎡⎣ en ( l )⎤⎦ = σ 2 (1 + ψ 12 + ψ 22 + L + ψ l2−1 ) , l ≥ 1
٥٦
ﺍﻟﺼﻴﻐﺔ zn ( l ) = ψ lan + ψ l+1an−1 + ψ l+2an −2 + L, l ≥ 1ﻏﻴﺭ ﻋﻤﻠﻴﺔ ﻹﻴﺠﺎﺩ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn+l , l ≥ 1ﻭﺫﻟﻙ ﻷﻨﻨﺎ ﻨﺤﺘﺎﺝ ﺇﻟﻰ ﻤﻌﺭﻓﺔ ﺍﻟﻘﻴﻡ } . {a1 , a2 ,L, an−1 , an
ﺘﻌﺭﻴﻑ : ١٨ﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ )} I ({z1 , z2 ,L , zn −1 , znﺘﻜﺎﻓﺊ ﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ )} I ({a1 , a2 ,L , an −1 , anﻭﺫﻟﻙ ﺒﺎﻟﻤﻌﻨﻰ ﺃﻥ ﺍﻟﻤﺠﻤﻭﻋﺔ } {a1 , a2 ,L, an−1 , anﺘﺤﺘﻭﻯ ﻋﻠﻰ ﻨﻔﺱ ﺍﻟﻤﻌﻠﻭﻤﺎﺕ ﻋﻥ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } . {z1 , z2 ,L, zn −1 , zn ﻤﻼﺤﻅﺔ :ﺍﻟﻤﺘﺴﻠﺴﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﻭﻗﻴﺎﺴﻬﺎ ﻭﻟﻜﻥ ﺍﻟﻤﺘﻠﺴﻠﺔ } {a1 , a2 ,L, an−1 , anﻻﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﺃﻭ ﻗﻴﺎﺴﻬﺎ.
ﻨﻅﺭﻴﺔ : ٣ﺍﻟﻤﺘﻨﺒﺊ ﺫﺍ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﻴﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1 ﺃﻱ ﻫﻭ ﺍﻟﺘﻭﻗﻊ ﺍﻟﺸﺭﻁﻲ ﻟﻠﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l ≥ 1ﻤﻌﻁﻰ } . {z1 , z2 ,L, zn −1 , zn ﺘﺴﺘﺨﺩﻡ ﻨﻅﺭﻴﺔ ٢ﻋﻤﻠﻴﺎ ﻹﻴﺠﺎﺩ ﻗﻴﻡ ﺍﻟﺘﻨﺒﺅﺍﺕ ﺒﺩﻻ ﻤﻥ ﺍﻟﺼﻴﻐﺔ zn ( l ) = ψ l an + ψ l+1an −1 + ψ l+2an −2 + L, l ≥ 1
ﻭﺫﻟﻙ ﺘﺒﻌﺎ ﻟﻠﻤﻼﺤﻅﺔ ﺍﻟﺴﺎﺒﻘﺔ.
ﻗﺎﻋﺩﺓ :٢ ⎧a , j ≤ 0 1 − E ( an + j zn , zn −1 ,L) = ⎨ n + j j>0 ⎩ 0, j≤0 ⎧ zn + j , ⎨ = )2 − E ( zn + j zn , zn −1 ,L ⎩ zn ( j ) , j > 0
ﻨﻅﺭﻴﺔ ٣ﻤﻊ ﺍﻟﻘﺎﻋﺩﺓ ٢ﺘﻌﻁﻲ ﻁﺭﻴﻘﺔ ﻋﻤﻠﻴﺔ ﻭﺴﻬﻠﺔ ﻹﻴﺠﺎﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn + l , l ≥ 1
ﺍﻟﺩﺍﻟﺔ zn ( l ) , l ≥ 1ﻜﺩﺍﻟﺔ ﻟﺯﻤﻥ ﺍﻟﺘﻘﺩﻡ
l ≥1
ﺘﻌﺭﻴﻑ :١٩ ﻋﻨﺩ ﻨﻘﻁﺔ ﺍﻻﺼل ﻟﻠﺯﻤﻥ nﺘﺴﻤﻰ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ.
دوال اﻟﺘﻨﺒﺆ ﻟﻨﻤﺎذج ): ARIMA(p,d,q ﺍﻭﻻ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): AR(١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل
٥٧
) ∞ WN ( 0, σ 2 ) , φ1 < 1, µ ∈ ( −∞,
zt − µ = φ1 ( zt −1 − µ ) + at , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ +E ⎡⎣ ⎡⎣φ1 ( zn + l−1 − µ ) + an +l ⎤⎦ zn , zn −1 ,L⎤⎦ , l ≥ 1
=µ +E ⎡⎣φ1 ( zn + l−1 − µ ) zn , zn −1 ,L + an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
= µ +φ1E ⎡⎣( zn + l−1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﺃﻱ zn ( l ) = µ +φ1E ⎣⎡ ( zn + l−1 zn , zn −1 ,L) − µ ⎦⎤ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
⎦⎤l = 1: zn (1) = µ +φ1E ⎣⎡ ( zn zn , zn −1 ,L) − µ ⎦⎤ + E ⎡⎣ an +1 zn , zn −1 ,L ) = µ +φ1 ( zn − µ
⎦⎤l = 2 : zn ( 2 ) = µ +φ1E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + 2 zn , zn −1 ,L ⎦⎤ = µ +φ1 ⎡⎣ zn (1) − µ
⎦⎤l = 3 : zn ( 3) = µ +φ1E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +3 zn , zn −1 ,L ⎦⎤ = µ +φ1 ⎡⎣ zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 1
ﻭﻫﻲ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ )AR(١
ﺘﻌﺭﻴﻑ :٢٠ ﺸﺭﻁ ﺍﻹﺴﺘﻤﺭﺍﺭ Continuity Conditionﻴﺘﻁﻠﺏ ﺃﻨﻪ ﻋﻨﺩﻤﺎ ﺘﻜﻭﻥ l = 1ﻓﺈﻥ zn ( l − 1) = zn ( 0 ) = zn
ﻤﻥ ﻨﻅﺭﻴﺔ ٢ﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﺘﻌﻁﻰ ﻤﻥ ﺍﻟﻌﻼﻗﺔ V ⎡⎣ en ( l )⎤⎦ = σ 2 (1 + ψ 12 + ψ 22 + L + ψ l2−1 ) , l ≥ 1
ﺴﺒﻕ ﺃﻥ ﺍﺸﺘﻘﻘﻨﺎ ﺩﺍﻟﺔ ﺍﻷﻭﺯﺍﻥ ﻟﻨﻤﻭﺫﺝ ) AR(١ﻭﻫﻲ ψ j = φ1j , φ1 < 1
), l ≥1
ﻭﺒﺎﻟﺘﻌﻭﻴﺽ ﻓﻲ ﺼﻴﻐﺔ ﺍﻟﺘﺒﺎﻴﻥ ﻨﺠﺩ
(
( V ⎡⎣ en ( l )⎤⎦ = σ 2 1 + φ12 + φ14 + L + φ1
)2 l −1
1 − φ12 l , l ≥1 1 − φ12
=σ2
ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻟﻨﻤﻭﺫﺝ ٥٨
) WN ( 0,0.024
zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻬﺎ. ﺍﻟﺤل :ﻤﻥ ﺍﻟﺼﻴﻐﺔ zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 1ﻨﺠﺩ ) z156 (1) = 0.97+0.85 ( z156 − 0.97
= 0.97+0.85 ( 0.49 − 0.97 ) = 0.56 ) z156 ( 2 ) = 0.97+0.85 ( z156 (1) − 0.97 = 0.97+0.85 ( 0.56 − 0.97 ) = 0.62 ) z156 ( 3) = 0.97+0.85 ( z156 ( 2 ) − 0.97 = 0.97+0.85 ( 0.62 − 0.97 ) = 0.68
ﻭﺍﻟﺘﺒﺎﻴﻨﺎﺕ 1 − φ12 l , l ≥1 1 − φ12
V ⎡⎣ en ( l )⎤⎦ = σ 2
V ⎡⎣ e156 (1)⎤⎦ = 0.024 = 0.041 = 0.054
4
)1 − ( 0.85
2
)1 − ( 0.85
6
)1 − ( 0.85
2
)1 − ( 0.85
V ⎡⎣ e156 ( 2 )⎤⎦ = 0.024 V ⎡⎣ e156 ( 2 )⎤⎦ = 0.024
ﺜﺎﻨﻴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): AR(٢
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(٢ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل WN ( 0, σ 2 ) , µ ∈ ( −∞, ∞ ) ,
zt = µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at , at
φ2 − φ1 < 1, φ2 + φ1 < 1, φ2 < 1
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ +E ⎡⎣ ⎡⎣φ1 ( zn + l−1 − µ ) + φ2 ( zn + l−2 − µ ) + an + l ⎤⎦ zn , zn −1 ,L⎤⎦ , l ≥ 1
=µ +E ⎡⎣φ1 ( zn + l−1 − µ ) zn , zn −1 ,L + φ2 ( zn + l−2 − µ ) zn , zn −1 ,L + an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
= µ +φ1E ⎡⎣( zn +l−1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn + l−2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
ﺃﻱ
٥٩
zn ( l ) = µ +φ1E ⎡⎣ ( zn + l −1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣ ( zn + l −2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + l zn , zn −1 ,L⎤⎦ , l ≥ 1
٢ ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ
l = 1: zn (1) = µ +φ1E ⎡⎣( zn zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn −1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +1 zn , zn −1 ,L⎤⎦ = µ +φ1 ( zn − µ ) + φ2 ( zn −1 − µ )
l = 2 : zn ( 2 ) = µ +φ1E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣ ( zn zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +2 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn (1) − µ ⎤⎦ + φ2 ( zn − µ )
l = 3 : zn ( 3) = µ +φ1E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn +1 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an +3 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn ( 2 ) − µ ⎤⎦ +φ2 ⎡⎣ zn (1) − µ ⎤⎦
l = 4 : zn ( 4 ) = µ +φ1E ⎡⎣( zn +3 zn , zn −1 ,L) − µ ⎤⎦ + φ2 E ⎡⎣( zn + 2 zn , zn −1 ,L) − µ ⎤⎦ + E ⎡⎣ an + 4 zn , zn −1 ,L⎤⎦ = µ +φ1 ⎡⎣ zn ( 3) − µ ⎤⎦ +φ2 ⎡⎣ zn ( 2 ) − µ ⎤⎦
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ +φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ + φ2 ⎡⎣ zn ( l − 2 ) − µ ⎤⎦ , l ≥ 1
AR(٢) ﻭﻫﻲ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ﻭﺩﺍﻟﺔ ﺍﻷﻭﺯﺍﻥ٢ ﻭﻴﻤﻜﻥ ﺤﺴﺎﺏ ﺘﺒﺎﻴﻨﺎﺕ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻤﻥ ﻨﻅﺭﻴﺔ ⎧ 1, ⎪φ , ⎪ 1 ψj =⎨ 2 ⎪ φ1 + φ2 , ⎪⎩φψ 1 j −1 + φ2ψ j − 2 ,
j=0 j =1 j=2 j≥3
: ARIMA(٠،١،١) ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ:ﺜﺎﻟﺜﺎ
ﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎn { ﺤﺘﻰ ﺍﻟﺯﻤﻥz1 , z2 ,L, zn −1 , zn } ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜلARIMA(٠،١،١) ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ zt = zt −1 + at − θ1at −1 , at
WN ( 0, σ 2 )
. zn+l , l ≥ 1 ﺃﻭ ﺒﺸﻜل ﻋﺎﻡzn +1 , zn +2 , zn+3 ,L ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻨﺠﺩ٣ ﻤﻥ ﻨﻅﺭﻴﺔ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=E ( zn +l−1 zn , zn −1 ,L) + E ( an +l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) , l ≥ 1
٢ ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ
٦٠
zn ( l ) =E ( zn + l−1 zn , zn −1 ,L) + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =E ( zn zn , zn −1 ,L) + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = zn − θ1an
)l = 2 : zn ( 2 ) =E ( zn +1 zn , zn −1 ,L) + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L )= zn (1 )l = 3 : zn ( 3) =E ( zn + 2 zn , zn −1 ,L) + E ( an +3 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L ) = zn ( 2
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = zn ( l − 1) , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) ARIMA(٠،١،١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ zn − θ1an , l = 1 ⎨ = ) zn ( l ⎩ zn ( l − 1) , l > 1
ﺭﺍﺒﻌﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): MA(١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) MA(١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل ) WN ( 0, σ 2
zt = µ + at − θ1at −1 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + E ( an + l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
zn ( l ) =µ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = µ − θ1an
)l = 2 : zn ( 2 ) =µ + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L =µ )l = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L =µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) MA(١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ µ − θ1an , l = 1 ⎨ = ) zn ( l l≥2 ⎩ µ,
ﺨﺎﻤﺴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): MA(٢
٦١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) MA(٢ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل ) WN ( 0, σ 2
zt = µ + at − θ1at −1 − θ 2 at − 2 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn+l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + E ( an + l zn , zn −1 ,L) − θ1E ( an +l−1 zn , zn −1 ,L) − θ 2 E ( an +l−2 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
zn ( l ) =µ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) − θ 2 E ( an + l−2 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L) − θ 2 E ( an −1 zn , zn −1 ,L = µ − θ1an − θ 2an −1
)l = 2 : zn ( 2 ) =µ + E ( an + 2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L) − θ 2 E ( an zn , zn −1 ,L = µ − θ 2 an
)l = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L) − θ 2 E ( an +1 zn , zn −1 ,L =µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ , l ≥ 3
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) MA(٢ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ ⎧ µ − θ1an − θ 2an −1 , l = 1 ⎪ zn ( l ) = ⎨ µ − θ 2 an , l=2 ⎪ µ, l≥3 ⎩
ﺴﺎﺩﺴﺎ :ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ): ARMA(١،١
ﻟﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ } {z1 , z2 ,L, zn −1 , znﺤﺘﻰ ﺍﻟﺯﻤﻥ nﻭﺍﻟﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) ARMA(١،١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل WN ( 0, σ 2 ) , φ1 ≠ θ1 , φ1 < 1
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +1 , zn +2 , zn+3 ,Lﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +l , l ≥ 1 ﻤﻥ ﻨﻅﺭﻴﺔ ٣ﻨﺠﺩ zn ( l ) = E ( zn + l zn , zn −1 ,L) , l ≥ 1
=µ + φ1E ⎡⎣( zn + l−1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an + l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﺎﻋﺩﺓ ٢
٦٢
zn ( l ) =µ + φ1 E ⎡⎣( zn + l−1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +l zn , zn −1 ,L) − θ1E ( an + l−1 zn , zn −1 ,L) , l ≥ 1
)l = 1: zn (1) =µ + φ1E ⎣⎡( zn − µ ) zn , zn −1 ,L⎦⎤ + E ( an +1 zn , zn −1 ,L) − θ1E ( an zn , zn −1 ,L = µ + φ1 ( zn − µ ) − θ1an
)l = 2 : zn ( 2 ) =µ + φ1 E ⎡⎣( zn +1 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +2 zn , zn −1 ,L) − θ1E ( an +1 zn , zn −1 ,L ⎦⎤ = µ + φ1 ⎡⎣ zn (1) − µ
)l = 3 : zn ( 3) =µ + φ1 E ⎡⎣( zn + 2 − µ ) zn , zn −1 ,L⎤⎦ + E ( an +3 zn , zn −1 ,L) − θ1E ( an + 2 zn , zn −1 ,L ⎦⎤ = µ + φ1 ⎡⎣ zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( l ) = µ + φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ , l ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻟﻨﻤﻭﺫﺝ ) ARMA(١،١ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ l =1 l≥2
⎧⎪ µ + φ1 ( zn − µ ) − θ1an , ⎨ = ) zn ( l ⎪⎩ µ + φ1 ⎡⎣ zn ( l − 1) − µ ⎤⎦ ,
ﺘﻤﺭﻴﻥ: ﻟﻨﻤﻭﺫﺝ ) ARMA(١،١ﻭﺍﻟﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻟﺸﻜل
WN ( 0, σ 2 ) , φ1 ≠ θ1 , φ1 < 1
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﺒﺭﻫﻥ ﺍﻥ ﻋﻨﺩﻤﺎ ﺘﺅﻭل φ1 → 1ﻓﺈﻥ ) ARMA(1,1) → IMA(1,1ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﻟﻨﻤﻭﺫﺝ ). IMA(١،١ ﺘﻤﺭﻴﻥ: ﺃﻭﺠﺩ ﺩﻭﺍل ﺍﻟﺘﻨﺒﺅ ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻜل ﻤﻥ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﺘﺎﻟﻴﺔ: ARIMA(١،١،١), ARIMA(٢،١،٠), ARIMA(٠،١،٢), ARIMA(١،٢،٠), ARIMA(٠،٢،١), ARIMA(٠،٢،٠). ﺤﺩﻭﺩ ﺍﻟﺘﻨﺒﺅ : Forecasting Limits ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ zn ( l ) , l ≥ 1ﻋﻨﺩ ﻗﻴﻤﺔ ﻤﻌﻴﻨﺔ ﺘﻌﻁﻲ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻟﻨﻘﻁﺔ Point Forecastﻭﺍﻟﺫﻱ ﻻﻴﻜﻔﻲ ﺍﻭ ﻴﻔﻴﺩ ﻓﻲ ﺇﺘﺨﺎﺫ ﻗﺭﺍﺭﺍﺕ ﺇﺤﺼﺎﺌﻴﺔ ﻋﻥ ﺍﻟﻅﺎﻫﺭﺓ ﺍﻟﻌﺸﻭﺍﺌﻴﺔ ﺍﻟﻤﺩﺭﻭﺴﺔ ﻷﻥ P ( Z n + m = zn ( m ) ) = 0, for some m > 0
ﺃﻱ ﺃﻥ ﻤﻘﺩﺍﺭ ﺘﺄﻜﺩﻨﺎ ) ﺃﻭ ﺇﺤﺘﻤﺎل( ﻤﻥ ﺃﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻟﻤﺭﺍﺩ ﺍﻟﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﺴﺎﻭﻱ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻁﺎﺓ ﻤﻥ ﺩﺍﻟﺔ ﺍﻟﺘﻨﺒﺅ ﺘﺴﺎﻭﻱ ﺍﻟﺼﻔﺭ ﺃﻱ ﺍﻨﻨﺎ ﻏﻴﺭ ﻤﺘﺄﻜﺩﻴﻥ ﺇﻁﻼﻗﺎ ﻭﺒﺎﻟﺘﺎﻟﻲ ﻻﻓﺎﺌﺩﺓ ﻤﻥ ﺍﻟﺘﻨﺒﺅ. ﻟﻠﺘﻐﻠﺏ ﻋﻠﻰ ﺫﻟﻙ ﻭﺃﻹﺴﺘﻔﺎﺩﺓ ﻤﻥ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻨﺴﺘﺨﺩﻡ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻟﻔﺘﺭﺓ Interval Forecast ﻭﻫﻲ ﻋﺒﺎﺭﺓ ﻋﻥ ﻓﺘﺭﺓ ﻤﺜل ] [a, bﻋﻠﻰ ﺨﻁ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ ﺒﺤﻴﺙ ﻴﻜﻭﻥ ) P ( a ≤ Z n +m ≤ b ) = (1 − α
ﻭﺒﻬﺫﺍ ﻨﺴﺘﻁﻴﻊ ﺃﻥ ﻨﺤﺩﺩ ﺩﺭﺠﺔ ﺘﺄﻜﺩﻨﺎ ﻤﻥ ﺃﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻟﻤﺭﺍﺩ ﺍﻟﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﻘﻊ ﺒﻴﻥ ﺍﻟﻘﻴﻡ a ﻭ bﺒﺩﺭﺠﺔ ﺘﺄﻜﺩ ﺃﻭ ﺇﺤﺘﻤﺎل ) 1 − αﺃﻭ ( 100 × (1 − α ) %ﻓﻤﺜﻼ ﻟﻭ ﻜﺎﻨﺕ α = 0.05ﻓﺈﻨﻨﺎ ﻨﻜﻭﻥ ﻤﺘﺄﻜﺩﻴﻥ ﻭﺒﺈﺤﺘﻤﺎل ٩٥٪ﺍﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺘﻘﻊ ﺒﻴﻥ ﺍﻟﻘﻴﻡ aﻭ . b
٦٣
ﺘﻌﺭﻴﻑ :٢١ﻋﻠﻰ ﺇﻓﺘﺭﺍﺽ ﺃﻥ ) at N ( 0,σ 2ﻓﺈﻥ ﺤﺩﻭﺩ 100 × (1 − α ) %ﻓﺘﺭﺓ ﺘﻨﺒﺅ ﻟﻠﻘﻴﻤﺔ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn+l , l ≥ 1ﺘﻌﻁﻰ ﺒﺎﻟﻌﻼﻗﺔ
{
}
12
⎦⎤) zn ( l ) ± uα 2 V ⎡⎣ en ( l α ⎟⎞ 100 ⎛⎜ 1 −ﻟﻠﺘﻭﺯﻴﻊ ). N ( 0,1 ⎠2 ⎝
ﺤﻴﺙ uα 2ﺍﻟﻤﺌﻴﻥ
ﻓﻤﺜﻼ ﻋﻨﺩﻤﺎ α = 0.05ﻓﺈﻥ . u0.025 = 1.96 ﻤﻼﺤﻅﺔ :ﻓﻲ ﺍﻟﺘﻌﺭﻴﻑ ﺇﻓﺘﺭﻀﻨﺎ ﺃﻥ ﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﻀﺠﺔ ﺍﻟﺒﻴﻀﺎﺀ ) at N ( 0, σ 2ﻭﻫﺫﺍ ﻤﻤﻜﻥ ﺇﻋﺘﻤﺎﺩﺍ ﻋﻠﻰ ﻨﻅﺭﻴﺔ ﻨﻬﺎﻴﺔ ﻤﺭﻜﺯﻴﺔ. ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻟﻨﻤﻭﺫﺝ ) N ( 0,0.024
zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻟﻬﺎ ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ ٩٥٪ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ. ﺍﻟﺤل :ﺴﺒﻕ ﺃﻥ ﺤﺴﺒﻨﺎ ﻓﻲ ﻤﺜﺎل ﺴﺎﺒﻕ ﺍﻟﺘﻨﺒﺅﺍﺕ ﻭ ﺃﺨﻁﺎﺀ ﺍﻟﺘﻨﺒﺅ ﻜﺎﻟﺘﺎﻟﻲ: ﺍﻟﺘﻨﺒﺅﺍﺕ z156 (1) = 0.56, z156 ( 2 ) = 0.62, z156 ( 3) = 0.68
ﻭﺍﻟﺘﺒﺎﻴﻨﺎﺕ V ⎡⎣ e156 (1)⎤⎦ = 0.024, V ⎡⎣ e156 ( 2 )⎤⎦ = 0.041, V ⎡⎣ e156 ( 2 )⎤⎦ = 0.054 ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ ٩٥٪ﻟﻠﻘﻴﻡ ﺍﻟﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159ﻨﻭﺠﺩﻫﺎ ﻤﻥ ﺼﻴﻐﺔ ﺘﻌﺭﻴﻑ ٢١
}
12
= 0.56 ± 1.96 0.024 = 0.56 ± 0.304
}
12
{
⎦⎤) zn ( l ) ± uα 2 V ⎡⎣ en ( l
{ {V ⎡⎣e {V ⎡⎣e
⎦⎤)1 − z156 (1) ± u0.025 V ⎡⎣ e156 (1
= 0.62 ± 1.96 0.041 = 0.62 ± 0.397
}⎦⎤) ( 2
156
2 − z156 ( 2 ) ± u0.025
= 0.68 ± 1.96 0.054 = 0.68 ± 0.455
}⎦⎤)( 3
156
3 − z156 ( 3) ± u0.025
12
12
ﺃﻱ ﺃﻥ ) z157 ∈ ( 0.256,0.864ﺒﺈﺤﺘﻤﺎل ٠,٩٥ﻭ ﻭﻜﺫﻟﻙ ) z158 ∈ ( 0.223,1.017ﻭ ﻜﺫﻟﻙ ﺃﻴﻀﺎ ). z159 ∈ ( 0.225,1.135
٦٤
٦٥
اﻟﻔﺼﻞ اﻟﺨﺎﻣﺲ ﺗﺼﻤﻴﻢ وﺑﻨﺎء ﻧﻈﺎم ﺗﻨﺒﺆ إﺣﺼﺎﺋﻲ Designing and Building : Statistical Forecasting System ﺳﺒﻖ أن ذآﺮﻧﺎ ان اﻟﺨﻄﻮة اﻷوﻟﻰ ﻟﺘﺼﻤﻴﻢ ﻧﻈﺎم ﺗﻨﺒﺆ هﻲ ﺑﻨﺎء ﻧﻤﻮذج .إن ﻋﻤﻠﻴﺔ ﺑﻨﺎء ﻧﻤﻮذج إﺣﺼﺎﺋﻲ هﻲ ﻋﻤﻠﻴﺔ ﺗﻜﺮارﻳﺔ Iterativeﺗﺘﻜﻮن ﻣﻦ ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ،ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج )وﻧﻘﺼﺪ ﺑﻬﺎ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج( و إﺧﺘﺒﺎر اﻟﻨﻤﻮذج. ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ): Model Identification (Specification ﻓﻲ ﻣﺮﺣﻠﺔ ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ﻧﺴﺘﺨﺪم اﻟﺒﻴﺎﻧﺎت أو اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ) اﻟﺘﺎرﻳﺦ( واي ﻣﻌﻠﻮﻣﺎت اﺧﺮى ﻋﻦ اﻟﻜﻴﻔﻴﺔ اﻟﺘﻲ ﺗﻮﻟﺪت ﺑﻬﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ وذﻟﻚ ﻹﻗﺘﺮاح ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻨﻤﺎذج اﻟﻤﻨﺎﺳﺒﺔ .وﻳﺘﻢ ﺗﻌﻴﻴﻦ أو ﺗﺤﺪﻳﺪ اﻟﻨﻤﻮذج ﺣﺴﺐ اﻟﺨﻄﻮات اﻟﻌﺮﻳﻀﺔ اﻟﺘﺎﻟﻴﺔ: اﻟﺨﻄﻮة اﻻوﻟﻰ :ﺗﺤﻮﻳﻞ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ : Variance-stabilizing Transformation ﺑﻌﺪ رﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotوإﺟﺮاء ﺑﻌﺾ اﻹﺧﺘﺒﺎرات اﻹﺣﺼﺎﺋﻴﺔ ﻟﻤﻌﺮﻓﺔ ﻓﻴﻤﺎ إذا آﺎن اﻟﺘﺒﺎﻳﻦ ﺛﺎﺑﺖ ،وﻓﻲ ﺣﺎﻟﺔ ﻋﺪم ﺛﺒﺎت اﻟﺘﺒﺎﻳﻦ او إذا آﺎن اﻟﺘﺒﺎﻳﻦ ﻳﺘﻐﻴﺮ ﻣﻊ ﻣﺴﺘﻮى اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﺈﻧﻨﺎ ﻧﻄﺒﻖ اﻟﺘﺤﻮﻳﻞ اﻟﻠﻮﻏﺎرﺗﻤﻲ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﻔﺤﺼﻬﺎ ﻣﻦ ﺟﺪﻳﺪ ﻓﺈذا ﺗﻢ ﺗﺜﺒﻴﺖ اﻟﺘﺒﺎﻳﻦ وإﻻ ﻧﻠﺠﺄ إﻟﻲ ﺗﻄﺒﻴﻖ أﺣﺪ اﻟﺘﺤﻮﻳﻼت اﻟﺘﻲ ذآﺮﻧﺎهﺎ ﻓﻲ ﺟﺪول ﺻﻔﺤﺔ .٤١ اﻟﺨﻄﻮة اﻟﺜﺎﻧﻴﺔ :إﺧﺘﻴﺎر درﺟﺔ اﻟﺘﻔﺮﻳﻖ : d إذا آﺎﻧﺖ اﻟﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ ﺗﺤﺪﻳﺪ درﺟﺔ اﻟﺘﻔﺮﻳﻖ dاﻟﺘﻲ ﺗﺠﻌﻞ اﻟﻤﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ وﻧﻘﻮم ﺑﺄﺧﺬ اﻟﺘﻔﺮﻳﻖ اﻷول ﺛﻢ ﻧﻔﺤﺺ اﻟﺘﺎﻟﻲ: -١اﻟﻤﺨﻄﻄﺎت اﻟﺰﻣﻨﻴﺔ ﻟﻠﻤﻤﺘﻠﺴﻠﺴﺔ أو ﺗﺤﻮﻳﻠﻬﺎ. -٢ﻣﺨﻄﻄﺎت داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ SACFو . SPACF -٣إﺟﺮاء ﺗﻔﺮﻳﻖ ﺁﺧﺮ إذا اﺣﺘﺎج اﻷﻣﺮ وإﻋﺎدة اﻟﺨﻄﻮات ١و ٢اﻟﺴﺎﺑﻘﺘﻴﻦ. اﻟﻤﺨﻄﻄﺎت اﻟﺰﻣﻨﻴﺔ ﻟﻠﻤﺘﺴﻠﺴﻼت ﻏﻴﺮ اﻟﻤﺴﺘﻘﺮة ﺗﺒﻴﻦ ﺗﻐﻴﺮ ﻓﻲ اﻟﻤﺴﺘﻮى وداﻟﺔ ﺗﺮاﺑﻂ ذاﺗﻲ ﻋﻴﻨﻲ ﻣﺘﺨﺎﻣﺪة ﺑﺒﻂء آﻤﺎ ان داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﺗﻌﻄﻲ ﻗﻴﻤﺔ واﺣﺪة ﻗﺮﻳﺒﺔ ﻣﻦ اﻟﻮاﺣﺪ اﻟﺼﺤﻴﺢ )ﺑﻐﺾ اﻟﻨﻈﺮ ﻋﻦ اﻹﺷﺎرة( وﺑﻘﻴﺔ اﻟﻘﻴﻢ ﻗﺮﻳﺒﺔ ﺟﺪا ﻣﻦ اﻟﺼﻔﺮ. ﻣﻼﺣﻈﺔ :درﺟﺔ اﻟﺘﻔﺮﻳﻖ dﻏﺎﻟﺒﺎ ﻣﺎ ﺗﻜﻮن ٠او ١او . ٢ اﻟﺨﻄﻮة اﻟﺜﺎﻟﺜﺔ :ﺗﺤﺪﻳﺪ pو : q ﺑﻌﺪ ان ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ ﻧﻘﻮم ﺑﺘﺤﺪﻳﺪ درﺟﺔ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ pودرﺟﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك qوذﻟﻚ ﺑﻤﻘﺎرﻧﺔ أﻧﻤﺎط داﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻣﻊ اﻷﻧﻤﺎط اﻟﻨﻈﺮﻳﺔ ﻟﺪاﻟﺘﻲ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻣﺴﺘﺮﺷﺪﻳﻦ ﺑﺨﻮاص ﻧﻤﺎذج ) ARMA(p,qاﻟﻤﺬآﻮرة ﻓﻲ ﺻﻔﺤﺔ ٣٨واﻟﺠﺪول اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ هﺬﻩ اﻟﺨﻮاص ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﺸﺎﺋﻌﺔ:
اﻟﻨﻤﻮذج ) (١,d,٠و)AR(١ ) (٢,d,٠و)AR(٢ ) (p,d,٠و)AR(p ) (٠,d,١و )MA(١ ) (٠,d,٢و )MA(٢
ACF ﺗﺨﺎﻣﺪ اﺳﻲ أواﺳﻲ ﻣﺘﺮدد ﺗﺨﺎﻣﺪ اﺳﻲ او ﺗﺨﺎﻣﺪ ﺟﻴﺒﻲ ﺗﺨﺎﻣﺪ اﺳﻲ و /او ﺗﺨﺎﻣﺪ ﺟﻴﺒﻲ ρ k = 0, k > 1
PACF φkk = 0, k > 1 φkk = 0, k > 2 φkk = 0, k > p
ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ او ﺟﻴﺒﻲ
ρ k = 0, k > 2
٦٦
) (٠,d,qو )MA(q ) (١,d,١و )ARMA(١،١ ) (p,d,qو )ARMA(p,q
ﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ و /او ﺟﻴﺒﻲ ρ k = 0, k > q ﺗﺘﻨﺎﻗﺺ وﻳﺴﻴﻄﺮ ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ ﺗﺘﻨﺎﻗﺺ وﺗﺘﺨﺎﻣﺪ اﺳﻴﺎ ﻣﻦ اﻟﺘﺨﻠﻒ ١ ﻣﻦ اﻟﺘﺨﻠﻒ ١ ﺗﺘﻨﺎﻗﺺ ﺑﻌﺪ اﻟﺘﺨﻠﻒ ﺗﺘﻨﺎﻗﺺ ﺑﻌﺪ اﻟﺘﺨﻠﻒ q - pوﺗﺘﺨﺎﻣﺪ p – qوﻳﺴﻴﻄﺮ اﺳﻴﺎ و /او ﺟﻴﺒﻴﺎ ﺑﻌﺪ اﻟﺘﺨﻠﻒ q – p ﻋﻠﻴﻬﺎ ﺗﺨﺎﻣﺪ اﺳﻲ و/او ﺟﻴﺒﻲ ﺑﻌﺪ اﻟﺘﺨﻠﻒ p – q
اﻟﺨﻄﻮة اﻟﺮاﺑﻌﺔ :إﺿﺎﻓﺔ ﻣﻌﻠﻢ إﻧﺠﺮاف: إذا آﺎﻧﺖ اﻟﻤﺘﺴﻠﺴﺔ ﺗﺤﺘﺎج إﻟﻰ ﺗﻔﺮﻳﻖ ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ اﻟﺘﺄآﺪ ﻓﻴﻤﺎ إذا آﺎن ﻋﻠﻴﻨﺎ إﺿﺎﻓﺔ إﻧﺠﺮاف ﻣﻌﻠﻮم δإﻟﻰ اﻟﻨﻤﻮذج وهﺬا ﻳﺘﻢ ﺑﻤﻘﺎرﻧﺔ ﻣﺘﻮﺳﻂ اﻟﻌﻴﻨﺔ wﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ اﻟﻤﺴﺘﻘﺮة ﻣﻊ اﻟﺨﻄﺄ اﻟﻤﻌﻴﺎري ﻟﻬﺬا اﻟﻤﺘﻮﺳﻂ 12
⎡c ⎤ ⎥) s.e ( w ) ≅ ⎢ 0 (1 + 2r1 + 2r2 + L + 2rK ⎣n ⎦
ﺣﻴﺚ c0 = γˆ0و r1 ,L, rKهﻲ اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﻌﻴﻨﻴﺔ اﻟﻤﻌﻨﻮﻳﺔ ﻟﻠﺪرﺟﺔ . Kوﻳﻜﻮن اﻹﺧﺘﺒﺎر
H0 : δ = 0 H1 : δ ± 0
w وﻧﺮﻓﺾ H 0ﻋﻨﺪ α = 0.05إذاآﺎﻧﺖ > 1.96 ) s.e ( w
.
ﺗﻘﺪﻳﺮ اﻟﻨﻤﻮذج : Model Estimation ﺑﻌﺪ ﺗﺤﺪﻳﺪ ﺷﻜﻞ اﻟﻨﻤﻮذج ﻻﺑﺪ ﻣﻦ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج δو φ1 ,K ,φ pو θ1 ,K ,θ qو σوذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎرﻳﺨﻴﺔ اﻟﻤﺘﻮﻓﺮة ﻟﺪﻳﻨﺎ. ﻟﻨﻔﺘﺮض ان ﻟﺪﻳﻨﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة z1 , z2 ,K, zn −1 , znواﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح 2
أو
) N ( 0, σ 2
φ p ( B ) wt = δ + θ q ( B ) at , at
N ( 0, σ
φ p ( B ) zt = δ + θ q ( B ) at , at
)
2
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻻﻳﻮﺟﺪ ﺑﻴﻨﻬﺎ ﺟﺬور ﻣﺸﺘﺮآﺔ وﺟﺬور اﻟﻤﻌﺎدﻟﺔ φ p ( B ) = 0ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ) ﺷﺮط اﻹﺳﺘﻘﺮار(. هﻨﺎك ﻃﺮق آﺜﻴﺮة ﻟﺘﻘﺪﻳﺮ اﻟﻤﻌﺎﻟﻢ ﺳﻨﺬآﺮ ﻣﻨﻬﺎ هﻨﺎ ﻓﻘﻂ ﻃﺮﻳﻘﺘﻴﻦ ﺗﺪﺧﻞ ﺿﻤﻦ ﻧﻄﺎق هﺬا اﻟﻤﻘﺮر وهﻤﺎ ﻃﺮﻳﻘﺔ اﻟﻌﺰوم وﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ. أوﻻ :ﻃﺮﻳﻘﺔ اﻟﻌﺰوم : The Method of Moments وﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻣﺴﺎوات ﻋﺰوم اﻟﻌﻴﻨﺔ ﻣﺜﻞ ﻣﺘﻮﺳﻂ اﻟﻌﻴﻨﺔ zواﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ ﻟﻠﻌﻴﻨﺔ rkﺑﺎﻟﻌﺰوم اﻟﻨﻈﺮﻳﺔ ﻣﺜﻞ اﻟﻤﺘﻮﺳﻂ µوداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ρ kوﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻨﺎﺗﺠﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﺎﻟﻢ اﻟﻤﺮاد ﺗﻘﺪﻳﺮهﺎ. ﺳﻮف ﻧﺴﺘﻌﺮض اﻟﻄﺮﻳﻘﺔ ﻟﻠﻨﻤﻮذج ) AR(pآﺎﻟﺘﺎﻟﻲ: n -١ﻳﻘﺪر اﻟﻤﺘﻮﺳﻂ µﺑﺎﻟﻤﻘﺪر zاي µˆ = z = ∑i =1 zi n -٢ﻟﺘﻘﺪﻳﺮ φ1 ,K ,φ pﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺔ: ٦٧
ρ k = φ1 ρ k −1 + φ2 ρ k − 2 + L + φ p ρ k − p , k > 1
واﻟﺘﻲ ﺗﻨﺘﺞ ﻣﻦ ﺿﺮب اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻨﻤﻮذج ) AR(pﺑﺎﻟﺤﺪ zt −k − µوأﺧﺬ اﻟﺘﻮﻗﻊ .ﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﺑﻮﺿﻊ k = 1, 2,K , pﻧﺤﺼﻞ ﻋﻠﻰ ﻧﻈﺎم اﻟﻤﻌﺎدﻻت اﻟﻤﺴﻤﻰ ﻣﻌﺎدﻻت ﻳﻮل و ووآﺮ Yule-Walkerاﻟﺘﺎﻟﻲ:
ρ1 = φ1 + φ2 ρ1 + L + φ p ρ p −1
ρ 2 = φ1 ρ1 + φ2 + L + φ p ρ p −2 M
ρ p = φ1 ρ p −1 + φ2 ρ p −2 + L + φ p
و ﺑﺎﻟﺘﻌﻮﻳﺾ ﻋﻦ ρ kﺑﺎﻟﻤﻘﺪر rkﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ φˆ1 ,K,φˆpآﺎﻟﺘﺎﻟﻲ: ﺑﻮﺿﻊ ﻣﻌﺎدﻻت ﻳﻮل و ووآﺮ ﻋﻠﻰ اﻟﺸﻜﻞ اﻟﻤﺼﻔﻮﻓﻲ: ⎞ rp −1 ⎞ ⎛ φˆ1 ⎟ ⎜ ⎟ rp −2 ⎟⎟ ⎜ φˆ2 ⎟ M ⎟⎜ M ⎟ ⎜⎟ ⎟ ˆ1 ⎟⎠ ⎜ φ ⎠⎝ p
r2 L rp −2 r1 L rp −3 M M M rp −3 L r1
r1 ⎛ r1 ⎞ ⎛ 1 ⎜r ⎟ ⎜ r 1 ⎜ 2⎟=⎜ 1 ⎜M⎟ ⎜ M M ⎜⎜ r ⎟⎟ ⎜⎜ r ⎝ p ⎠ ⎝ p −1 rp −2
وﺑﺤﻞ هﺬﻩ اﻟﻤﻌﺎدﻟﺔ ﻟﻠﻤﻌﺎﻟﻢ ⎞ ⎛ r1 ⎟ ⎜r ⎟⎜ 2 ⎟⎜M ⎟⎟ ⎜⎜ ⎠ ⎝ rp
−1
ﺗﻘﺪر σ 2آﺎﻟﺘﺎﻟﻲ
⎞ rp −1 ⎟⎟ rp −2 ⎟ M ⎟ ⎠⎟ 1
r2 L rp −2 r1 L rp −3 M M M rp −3 L r1
)
r1 1 M rp −2
⎛ φˆ1 ⎞ ⎛ 1 ⎜ ⎟ ⎜ ⎜ φˆ2 ⎟ ⎜ r1 ⎜ M ⎟=⎜ M ⎜ ⎟ ⎜ ⎜ φˆ ⎟ ⎜⎝ rp −1 ⎠⎝ p
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2 −Lφˆp rp
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
هﻮ ﺗﺒﺎﻳﻦ اﻟﻌﻴﻨﺔ. ﺗﻘﺪﻳﺮ اﻟﻌﺰوم ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: -١ﻧﻤﻮذج )AR(١
) N ( 0, σ 2
= γˆ0
zt − µ = φ1 ( zt −1 − µ ) + at , at
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ φ1هﻮ φˆ1 = r1
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ µهﻮ µˆ = z
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ σ 2هﻮ
)
(
σˆ 2 = γˆ0 1 − φˆ1r1
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
٦٨
= γˆ0
-٢ﻧﻤﻮذج )MA(١
)
2
N ( 0, σ
zt − µ = at − θ1at −1 , at
ﻹﻳﺠﺎد ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺔ
−θ1 1 + θ12
= ρ1
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻌﺎﻟﻢ ﺑﻤﻘﺪراﺗﻬﺎ −θˆ1 1 + θˆ12
= r1
وﺑﺤﻞ اﻟﻤﻌﺎدﻟﺔ ﻟﻠﻤﻘﺪر θˆ1ﻧﺠﺪ −1 ± 1 − 4r1 2r1
= θˆ1
هﺬا اﻟﺤﻞ ﻳﻌﻄﻲ ﻗﻴﻤﺘﻴﻦ ﻟﻠﻤﻘﺪر θˆ1ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ اﻟﺘﻲ ﺗﺤﻘﻖ . θˆ1 < 1ﻓﻤﺜﻼ ﻟﻮ آﺎﻧﺖ r1 = −0.4ﻓﺈن (θˆ1 ) = −0.77و (θˆ1 ) = 3.27وﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1هﻮ . θˆ1 = −0.77 1
2
-٣ﻧﻤﻮذج )AR(٢
)
2
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + φ2 ( zt − 2 − µ ) + at , at
ﺑﺈﺳﺘﺨﺪام ﻣﻌﺎدﻻت ﻳﻮل ووآﺮ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ φ1و φ2هﻲ ⎞ ⎛ φˆ1 ⎞ ⎛ 1 r1 ⎞ −1 ⎛ r1 ⎜=⎟ ⎜ ⎠⎟ ⎜ φˆ ⎟ ⎝ r1 1 ⎟⎠ ⎜⎝ r2 ⎠⎝ 2
وﻣﻨﻬﺎ ﻧﺠﺪ r1 − r1r2 r −r , φˆ2 = 2 2 1 − r1 1− r
2 1 2 1
= φˆ1
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ µهﻮ µˆ = z
ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ σ 2هﻮ
)
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2
ﺣﻴﺚ 1 2 ) ( zt − z ∑ n t =1 n
-٤ﻧﻤﻮذج )MA(٢
)
2
N ( 0, σ
= γˆ0
zt − µ = at − θ1at −1 − θ 2 at − 2 , at
ﻹﻳﺠﺎد ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و θ2ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺎت
) −θ1 (1 − θ 2 −θ 2 = , ρ2 2 2 1 + θ1 + θ 2 1 + θ12 + θ 22
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻘﺪرات r1و r2ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و θ2
)
(
= ρ1
−θˆ1 1 − θˆ2 −θˆ2 , r = 2 1 + θˆ12 + θˆ22 1 + θˆ12 + θˆ22
٦٩
= r1
وﻧﺤﻞ ﻟﻜﻞ ﻣﻦ θˆ1و θˆ2وﻧﺄﺧﺬ اﻟﺤﻠﻮل اﻟﺘﻲ ﺗﺤﻘﻖ . θ 2 − θ1 < 1, θ 2 + θ1 < 1, θ 2 < 1 -٥ﻧﻤﻮذج )ARMA(١،١
)
2
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + at − θ1at −1 , at
ﻹﻳﺠﺎد ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و φ1ﻧﺴﺘﺨﺪم اﻟﻌﻼﻗﺎت
(1 − φ1θ1 )(φ1 − θ1 ) , ρ = (1 − φ1θ1 )(φ1 − θ1 ) φ 2 1 2 2 1 + θ1 − 2φ1θ1
1 + θ1 − 2φ1θ1
= ρ1
وﺑﺘﻌﻮﻳﺾ اﻟﻤﻘﺪرات r1و r2ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻠﻤﻌﺎﻟﻢ θ1و φ1 ) ˆ1 − φˆ θˆ )(φˆ − θ ( ˆφ = r 1
1 1
1
ˆ1 + θˆ 2 − 2φˆ θ
1
1 1
2
1
) ˆ1 − φˆ θˆ )(φˆ − θ ( = r , 1
1
1 1
ˆ1 + θˆ 2 − 2φˆ θ
1 1
1
1
وﺑﻘﺴﻤﺔ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r2ﻋﻠﻰ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r1ﻧﺠﺪ r2 r1
= φˆ1
وهﻮ ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ . φ1ﻹﻳﺠﺎد ﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ θ1ﻧﻌﻮض ﻋﻦ φˆ1ﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻟﻠﻤﻘﺪر r1ﻧﺠﺪ ⎞ ˆ ⎛ r2 ˆ ⎞ ⎛ r2 ⎟ ⎜ 1 − r θ1 ⎟ ⎜ r − θ1 1 ⎠⎝ 1 ⎠ ⎝ = r1 r 1 + θˆ12 − 2 2 θˆ1 r1 وﻧﺤﻞ اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ اﻟﻨﺎﺗﺠﺔ ﻟﻠﻤﻘﺪر θˆ1و ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ اﻟﺘﻲ ﺗﺤﻘﻖ . θˆ1 < 1
ﺗﻤﺎرﻳﻦ :أوﺟﺪ ﻣﻘﺪرات اﻟﻌﺰوم ﻟﻤﻌﺎﻟﻢ اﻟﻨﻤﺎذج اﻟﺘﺎﻟﻴﺔ ARIMA(١،١،١), ARIMA(٢،١،٠), ARIMA(٠،١،٢), ARIMA(١،٢،٠), ARIMA(٠،٢،١), ARIMA(٠،٢،٠). ﻣﻼﺣﻈﺔ :ﻣﻘﺪرات اﻟﻌﺰوم ﺗﺴﺘﺨﺪم آﻘﻴﻢ أوﻟﻴﺔ ﻹﻳﺠﺎد ﻣﻘﺪرات أآﺜﺮ دﻗﺔ.
ﺛﺎﻧﻴﺎ :ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ Conditional Least Square : Method ﻟﻨﻤﺎذج ) ARMA(p,qواﻟﺘﻲ ﺗﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
)
2
N ( 0, σ
φ p ( B )( zt − µ ) = θ q ( B ) at , at
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻻﻳﻮﺟﺪ ﺑﻴﻨﻬﺎ ﺟﺬور ﻣﺸﺘﺮآﺔ وﺟﺬور اﻟﻤﻌﺎدﻟﺔ θ q ( B ) = 0ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﺧﺎرج داﺋﺮة اﻟﻮﺣﺪة ) ﺷﺮط اﻹﻧﻘﻼب( .ﺑﺈﻋﺎدة آﺘﺎﺑﺔ اﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﻟﻸﺧﻄﺎء atآﺎﻟﺘﺎﻟﻲ: )φp (B )(z − µ θq ( B ) t
= at
اﻟﻄﺮف اﻷﻳﻤﻦ ﻳﻤﻜﻦ إﻋﺘﺒﺎرة آﺪاﻟﺔ ﻓﻲ اﻟﻤﻌﺎﻟﻢ } φ = {φ1 , φ2 ,K , φ pو } θ = {θ1 ,θ 2 ,K ,θ qو µ
و ﻳﻜﺘﺐ
)) (z − µ ) t
٧٠
p
−L − φ p B
2
p
−L −θ pB
2
(1 − φ B − φ B (1 − θ B − θ B 2
2
1
1
= ) at ( φ, θ, µ
ﺗﻌﺘﻤﺪ ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ و ﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة } z = {z1 , z2 ,K, znﻋﻠﻰ ﺗﺼﻐﻴﺮ اﻟﺪاﻟﺔ n
) ∑ a ( φ, θ, µ z 2 t
= ) min Sc ( φ, θ, µ φ,θ , µ
t = p +1
وﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻄﺒﻴﻌﻴﺔ Normal Equationsاﻟﻨﺎﺗﺠﺔ اﻟﺘﺎﻟﻴﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻘﺪرات. ∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑ at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ ∂φ ∂ φ t = p +1 ˆ θ =θ ˆθ =θ ˆµ = µ
ˆµ = µ
∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑p+1 at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ θ ∂θ ∂ t = ˆθ = θ ˆθ =θ ˆµ = µ
ˆµ = µ
∂ ∂ n 2 ) Sc ( φ, θ, µ = ∑ at ( φ, θ, µ z ) φ=φˆ = 0 ˆφ= φ ∂µ ∂µ t = p +1 ˆθ = θ ˆθ =θ ˆµ = µ
ˆµ = µ
هﺬﻩ اﻟﻤﻘﺪرات ﺗﺴﻤﻰ ﺷﺮﻃﻴﺔ ﻷﻧﻨﺎ هﻨﺎ ﻧﺸﺘﺮط ان اﻟﻘﻴﻢ a p = a p −1 = L = a p +1− q = 0أي ﻣﺴﺎوﻳﺔ ﻟﺘﻮﻗﻌﻬﺎ ) .ﻻﺣﻆ أن اﻟﺘﺠﻤﻴﻊ ﻓﻲ اﻟﻤﻌﺎدﻻت اﻟﺴﺎﺑﻘﺔ ﻳﺒﺪأ ﻣﻦ اﻟﻘﻴﻤﺔ .( t = p + 1 ﻳﻘﺪر اﻟﺘﺒﺎﻳﻦ σ 2ﻣﻦ
)
(
Sc φˆ , θˆ , µ
)n − ( p + q + 1
= σˆ 2
ﺗﻘﺪﻳﺮات اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج: -١ﻧﻤﻮذج )AR(١
)
ﻟﺘﺒﺴﻴﻂ اﻹﺷﺘﻘﺎﻗﺎت ﺳﻮف ﻧﺴﺘﺒﺪل µﺑﻤﻘﺪرهﺎ z
2
)
N ( 0, σ
zt − µ = φ1 ( zt −1 − µ ) + at , at
N ( 0, σ
zt − z = φ1 ( zt −1 − z ) + at , at
2
ﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة } z = {z1 , z2 ,K, znﻧﻜﺘﺐ اﻷﺧﻄﺎء
at (φ1 ) = ( zt − z ) − φ1 ( zt −1 − z ) , t = 2,3,L, n
وﺗﺮﺑﻴﻊ اﻟﻄﺮﻓﻴﻦ واﻟﺠﻤﻊ ﻋﻠﻰ آﻞ اﻟﻤﺸﺎهﺪات a (φ1 ) = ⎡⎣( zt − z ) − φ1 ( zt −1 − z )⎤⎦ , t = 2,3,L , n 2
2
2 t
n
n
t =2
t =2
⎦⎤) Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ⎡⎣( zt − z ) − φ1 ( zt −1 − z
وهﺬﻩ داﻟﺔ ﻟﻠﻤﻌﻠﻢ φ1ﻓﻘﻂ ،ﻧﺸﺘﻖ اﻟﻤﻌﺎدﻟﺔ اﻟﺴﺎﺑﻘﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻌﻠﻢ φ1وﺗﻜﻮن اﻟﻨﺘﻴﺠﺔ ﻣﺴﺎوﻳﺔ ﻟﻠﺼﻔﺮ ﻋﻨﺪﻣﺎ φ1 = φˆ1أي
٧١
2
n
n
⎦⎤) Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ⎡⎣( zt − z ) − φ1 ( zt −1 − z t =2
t =2
2 ∂ ∂ = ) Sc (φ1 ⎦⎤) ⎡⎣( zt − z ) − φ1 ( zt −1 − z ∑ ∂φ1 ∂φ1 t =2 n
n
⎦⎤) = ∑ −2 ( zt −1 − z ) ⎡⎣ ( zt − z ) − φ1 ( zt −1 − z t= 2
n ∂ Sc (φ1 ) ˆ = ∑ −2 ( zt −1 − z ) ⎡⎣( zt − z ) − φˆ1 ( zt −1 − z )⎦⎤ = 0 φ1 =φ1 ∂φ1 t =2 n
∴ ∑ ( zt −1 − z ) ⎡⎣( zt − z ) − φˆ1 ( zt −1 − z )⎤⎦ = 0 t =2
n
n
t =2
t= 2
2 ∑ ( zt −1 − z )( zt − z ) − φˆ1 ∑ ( zt −1 − z ) = 0
أي ) − z )( zt − z
n
t −1
∑( z t =2
2
)−z
n
t −1
∑( z
= φˆ1
t =2
وهﻮ ﻣﻘﺪر اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ اﻟﺸﺮﻃﻴﺔ ﻟﻠﻤﻌﻠﻢ . φ1 ﺗﻤﺮﻳﻦ :ﻗﺎرن ﺑﻴﻦ هﺬا اﻟﻤﻘﺪر وﻣﻘﺪر اﻟﻌﺰوم ﻟﻠﻤﻌﻠﻢ . φ1 -٢ﻧﻤﻮذج )MA(١
) N ( 0,σ 2
zt − µ = at − θ1at −1 , at
ﻟﺘﺒﺴﻴﻂ اﻹﺷﺘﻘﺎﻗﺎت ﺳﻮف ﻧﺴﺘﺒﺪل µﺑﻤﻘﺪرهﺎ zوﻧﻌﻤﻞ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻌﺪﻟﺔ ﻟﻠﻤﺘﻮﺳﻂ xt = zt − zﻓﻴﺼﺒﺢ اﻟﻨﻤﻮذج ) N ( 0,σ 2
وﺑﻜﺘﺎﺑﺔ اﻟﻤﻌﺎدﻟﺔ اﻷﺧﻴﺮة ﻋﻠﻰ اﻟﺸﻜﻞ
xt = at − θ1at −1 , at at = xt − θ1at −1
وﻟﻤﺸﺎهﺪات ﻣﻌﻄﺎة x1 , x2 ,K , xnو ﺑﻮﺿﻊ a0 = 0ﺷﺮﻃﻴﺎ ﻧﻜﺘﺐ اﻷﺧﻄﺎء
a1 = x1 a2 = x2 − θ1a1 a3 = x3 − θ1a2 M an = xn − θ1an −1
وﺑﺎﻟﺘﺎﻟﻲ n
Sc (θ1 ) = ∑ at2 t =1
اﻟﺪاﻟﺔ اﻟﺴﺎﺑﻘﺔ ﻏﻴﺮ ﺧﻄﻴﺔ ﻓﻲ اﻟﻤﻌﻠﻢ θ1و ﻳﻤﻜﻦ إﻳﺠﺎد ﻗﻴﻤﺔ θ1واﻟﺘﻲ ﺗﺼﻐﺮ ) Sc (θ1ﺑﻄﺮق اﻟﺒﺤﺚ اﻟﻌﺪدﻳﺔ ﻣﺜﻞ اﻟﺒﺤﺚ اﻟﺸﺒﻜﻲ ﻓﻲ اﻟﻤﺠﺎل ) (-١،١أو إﺳﺘﺨﺪام ﻃﺮﻳﻘﺔ ﺟﺎوس-ﻧﻴﻮﺗﻦ واﻟﺘﻲ ﺗﺘﻠﺨﺺ ﻓﻲ ﺗﻘﺮﻳﺐ ) at = at (θ1ﺑﺪاﻟﺔ ﺧﻄﻴﺔ ﻟﻠﻤﻌﻠﻢ θ1ﺣﻮل ﻗﻴﻤﺔ أوﻟﻴﺔ * θﻣﺜﻼ أي
٧٢
) * dat (θ
اﻟﻤﺸﺘﻘﺔ
) * dat (θ d θ1
dθ1
) * at (θ1 ) ≈ at (θ * ) + (θ1 − θ
ﻳﻤﻜﻦ ﺣﺴﺎﺑﻬﺎ ﺗﻜﺮارﻳﺎ وذﻟﻚ ﺑﺈﺷﺘﻘﺎق ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ at = xt − θ1at −1ﺑﺎﻟﻨﺴﺒﺔ
ﻟﻠﻤﻌﻠﻢ θ1ﻟﻨﺤﺼﻞ ﻋﻠﻰ
) da0 (θ1 و ﺑﻘﻴﻤﺔ أوﻟﻴﺔ = 0 dθ1
) dat (θ1 ) θ1dat −1 (θ1 = ) + at −1 (θ1 dθ1 dθ1
.اﻟﻤﻌﺎدﻟﺔ ) * dat (θ dθ1
)
*
−θ
1
) + (θ
*
at (θ1 ) ≈ at (θ
ﺧﻄﻴﺔ ﻓﻲ اﻟﻤﻌﻠﻢ θ1وﺑﺎﻟﺘﺎﻟﻲ ﺑﺎﻻﻣﻜﺎن ﺗﺼﻐﻴﺮ ﻣﺠﻤﻮع اﻟﻤﺮﺑﻌﺎت n
Sc (θ1 ) = ∑ at2 t=1
ﺗﺤﻠﻴﻠﻴﺎ ﻟﻨﺤﺼﻞ ﻋﻠﻲ ﻣﻘﺪر ﺟﺪﻳﺪ وأﻓﻀﻞ ﻟﻠﻤﻌﻠﻢ θ1وﻧﻜﺮر هﺬﻩ اﻟﻌﻤﻠﻴﺔ ﺑﺈﺳﺘﺒﺪال * θﺑﺎﻟﻤﻘﺪر اﻟﺠﺪﻳﺪ وﻧﺴﺘﻤﺮ ﺣﺘﻰ ﻳﺼﺒﺢ اﻟﻔﺮق ﺑﻴﻦ ﻣﻘﺪرﻳﻦ ﺗﺎﻟﻴﻴﻦ ﺻﻐﻴﺮ ﺟﺪا أو اﻟﻨﻘﺺ ﻓﻲ ﻣﺠﻤﻮع اﻟﻤﺮﺑﻌﺎت ﺻﻐﻴﺮ ﺟﺪا .ﻣﻤﻜﻦ إﺳﺘﺨﺪام ﻃﺮﻳﻘﺔ اﻟﻌﺰوم ﻹﻳﺠﺎد اﻟﻘﻴﻤﺔ أﻷوﻟﻴﺔ * θﻟﻜﻲ ﻧﺤﺼﻞ ﻋﻠﻰ ﺗﻘﺎرب ﺳﺮﻳﻊ .ﻃﺒﻌﺎ اﻟﻄﺮﻳﻘﺔ اﻟﺴﺎﺑﻘﺔ ﻻﺗﺘﻢ ﻳﺪوﻳﺎ ﺑﻞ ﺗﺤﺘﺎج إﻟﻰ ﺣﺎﺳﺐ ﻟﺬﻟﻚ. ﻳﻼﺣﻆ أن ﺗﻘﺪﻳﺮ اﻟﻤﻌﺎﻟﻢ ﻟﻠﻨﻤﻮذج ﻓﻲ ﺣﺎﻟﺔ ﻧﻤﺎذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك او اﻟﻨﻤﺎذج اﻟﻤﺨﺘﻠﻄﺔ اﻟﺘﻲ ﺗﺤﻮي ﻋﻠﻰ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﺗﺸﻜﻞ ﺗﻌﻘﻴﺪا ﻷﻧﻬﺎ ﺗﺤﻮى ﻣﻌﺎﻟﻢ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺑﺸﻜﻞ ﻏﻴﺮ ﺧﻄﻲ وﻟﻬﺬا ﺗﺤﺘﺎج إﻟﻰ ﻃﺮق ﻋﺪدﻳﺔ ﻟﺤﻠﻬﺎ آﻤﺎ ﺷﺎهﺪﻧﺎ ﻓﻲ ﺣﺎﻟﺔ اﻟﻨﻤﻮذج ) MA(١وهﻮ أﺑﺴﻄﻬﺎ ﺟﻤﻴﻌﺎ. ﺳﻮف ﻧﻜﺘﻔﻲ ﻓﻲ ﻣﻘﺮرﻧﺎ هﺬا ﻋﻠﻰ اﻟﻄﺮﻳﻘﺘﻴﻦ اﻟﺴﺎﺑﻘﺔ وﻟﻜﻦ ﻧﺬآﺮ ﺑﻌﺾ اﻟﻄﺮق اﻻﺧﺮى اﻟﻤﺴﺘﺨﺪﻣﺔ ﻓﻲ ﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج ﻣﺜﻞ: -١ﻃﺮﻳﻘﺔ اﻷرﺟﺤﻴﺔ اﻟﻌﻈﻤﻰ Maximum Likelihood Method -٢ﻃﺮﻳﻘﺔ اﻟﻤﺮﺑﻌﺎت اﻟﺪﻧﻴﺎ ﻏﻴﺮ اﻟﺸﺮﻃﻴﺔ Unconditional Least Squares Method -٣ﻃﺮق اﻟﺘﻘﺪﻳﺮﻏﻴﺮ اﻟﺨﻄﻴﺔ Nonlinear Estimation Methods ﺗﺸﺨﻴﺺ وإﺧﺘﺒﺎر اﻟﻨﻤﻮذج : Model Checking and Diagnostics ﺑﻌﺪ اﻟﺘﻌﺮف ﻋﻠﻰ ﻧﻤﻮذج ﻣﺒﺪﺋﻲ وﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ هﺬا اﻟﻨﻤﻮذج ﻧﺠﺮي ﺑﻌﺾ اﻟﺘﺸﺨﻴﺼﺎت ﻋﻠﻰ اﻟﺒﻮاﻗﻲ أو أﺧﻄﺎء اﻟﺘﻄﺒﻴﻖ )اﻧﻈﺮ ﺗﻌﺮﻳﻒ (٤ﻟﻨﺮى ﻣﺪى ﻣﻄﺎﺑﻘﺔ اﻟﻨﻤﻮذج ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ،وﻳﻔﺘﺮض أن اﻟﺒﻮاﻗﻲ هﻲ ﻣﻘﺪرات ﻟﻤﺘﺴﻠﺴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء atواﻟﺘﻲ ﻧﻔﺘﺮض اﻧﻬﺎ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ . σ 2اﻟﺒﻮاﻗﻲ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ et = zt − zˆt = aˆt , t = 1, 2,..., n
أي ان اﻟﺒﻮاﻗﻲ هﻲ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻧﺎﻗﺺ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ. ﻳﻘﻮم اﻟﺘﺸﺨﻴﺺ واﻹﺧﺘﺒﺎرات ﻋﻠﻰ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ واﻟﻨﻈﺮ ﻓﻲ ﻣﺪى ﺗﺤﻘﻴﻘﻬﺎ ﻟﻔﺮﺿﻴﺎت اﻟﻨﻤﻮذج واﻟﺘﻲ هﻲ: -١ﻣﺘﻮﺳﻂ ﺻﻔﺮي -٢اﻟﻌﺸﻮاﺋﻴﺔ ٧٣
-٣ﻋﺪم اﻟﺘﺮاﺑﻂ -٤ﻣﻮزﻋﺔ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ )ﻣﺴﺘﻘﻞ وﻣﺘﻄﺎﺑﻖ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ σ أي ) ( at IIDN ( 0, σ 2 2
ﻟﻬﺬا ﻓﺈﻧﻨﺎ ﻧﺠﺮي ﺗﺸﺨﻴﺺ وهﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ ﻟﻨﺮي ﻓﻴﻤﺎ إذا آﺎﻧﺖ ﺗﺤﻘﻖ هﺬﻩ اﻟﺸﺮوط وﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ ﻧﻌﺘﺒﺮ اﻟﻨﻤﻮذج اﻟﻤﻄﺒﻖ ﻣﻘﺒﻮﻻ أﻣﺎ إذا ﻓﺸﻞ اﺣﺪ هﺬﻩ اﻹﺧﺘﺒﺎرات ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ إﻋﺎدة اﻟﻨﻈﺮ وإﻗﺘﺮاح ﻧﻤﻮذج ﺁﺧﺮ أوﻻ :إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ: H 0 : E ( at ) = 0 H 1 : E ( at ) ≠ 0
وهﻮ إﺧﺘﺒﺎر ﺑﺬﻳﻠﻴﻦ وﻧﺴﺘﺨﺪم ﻓﻴﺔ اﻹﺣﺼﺎﺋﺔ
e
) se ( e
= uواﻟﺘﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ
ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ ان E ( at ) = 0إذا آﺎﻧﺖ ) u < 1.96هﺬا ﻋﻠﻲ إﻋﺘﺒﺎر ان ﺣﺠﻢ اﻟﻌﻴﻨﺔ اآﺒﺮ ﻣﻦ ٣٠وﺣﺪة وهﺬا داﺋﻤﺎ ﻣﺘﺤﻘﻖ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﻧﺪرﺳﻬﺎ ( ﺛﺎﻧﻴﺎ :إﺧﺘﺒﺎر اﻟﻌﺸﻮاﺋﻴﺔ: ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺠﺮي Runs testﺣﻮل اﻟﻤﺘﻮﺳﻂ وﺣﻮل اﻟﺼﻔﺮ وهﻮ اﺣﺪ اﻹﺧﺘﺒﺎرات اﻟﻼﻣﻌﻠﻤﻴﺔ ) ﻳﻮﺟﺪ آﺜﻴﺮ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻟﻠﻌﺸﻮاﺋﻴﺔ ﻳﺪرﺳﻬﺎ اﻟﻄﺎﻟﺐ ﻓﻲ اﻟﻤﻘﺮر ٢٤١ﺑﺤﺚ وﻟﻜﻦ ﻧﻜﺘﻔﻲ هﻨﺎ ﺑﻬﺬا اﻹﺧﺘﺒﺎر(. ﺛﺎﻟﺜﺎ :إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ أو اﻹﺳﺘﻘﻼل: ﻳﺨﺘﺒﺮ ﺗﺮاﺑﻂ أو إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation testوذﻟﻚ ﺑﺤﺴﺎب ورﺳﻢ اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﻌﻴﻨﻴﺔ SACFﻟﻠﺒﻮاﻗﻲ وﻣﻘﺎرﻧﺘﻬﺎ ﻣﻊ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻤﺘﺴﻠﺴﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء. اﻹﺧﺘﺒﺎر H 0 : ρ1 = 0
H1 : ρ1 ≠ 0 r1 ﺣﻴﺚ اﻹﺣﺼﺎﺋﺔ ) se ( r1
= uﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ
ان ρ1 = 0إذا آﺎﻧﺖ . u < 1.96 راﺑﻌﺎ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ: ﻧﺨﺘﺒﺮ ﻓﻲ ﻣﺎ إذا آﺎﻧﺖ اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ وذﻟﻚ ﺑﻌﺪة ﻃﺮق ﻣﺜﻞ: -١إﺧﺘﺒﺎر ﺣﺴﻦ اﻟﺘﻄﺎﺑﻖ Goodness of Fit Testوﻧﺴﺘﺨﺪم اﻹﺧﺘﺒﺎر اﻟﻼﻣﻌﻠﻤﻲ آﻮﻟﻤﻮﺟﻮروف -ﺳﻤﻴﺮﻧﻮف . Kolmogorov-Smirnov Test -٢ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ . Normal Probability Plot -٣ﻣﺨﻄﻂ اﻟﺮﺑﻴﻌﺎت-اﻟﺮﺑﻴﻌﺎت . Q-Q Plot ﺑﻌﺾ اﻟﻤﻌﺎﻳﻴﺮ اﻻﺧﺮى ﻹﺧﺘﻴﺎر ﻧﻤﻮذج اﻟﻤﻨﺎﺳﺐ: (١إﺣﺼﺎﺋﻴﺔ آﻴﻮ ﻟـ ﻟﺠﻨﻖ-ﺑﻮآﺲ Ljung-Box Q statistcوﺗﺨﺘﺼﺮ LBQوﺗﺴﺘﺨﺪم ﻹﺧﺘﺒﺎر اﻟﻔﺮﺿﻴﺔ: H 0 : ρ1 = ρ 2 = L = ρ K = 0 وﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ:
٧٤
)χ 2 ( K − m
rk2 k =1 n − k K
∑ )Q = n (n + 2
ﺣﻴﺚ mﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج. (٢ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ Automatic Information Criteriaوﺗﺨﺘﺼﺮ AICوﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ: 2 AIC ( m ) = n ln σ a + 2m ﺣﻴﺚ mﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲ ) min AIC ( m m
أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ : Examples and Case Studies -١اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة )z(t
59.9315 61.0344 60.6643 60.5777 59.9086 62.2658 61.9013 54.5045 55.8451 59.3572 61.8956 59.7252 58.4651 61.2797 58.0143 56.6085 56.8697 57.2961 59.6266 58.1105 61.9152 59.0778 58.2759 59.5015 63.4124
60.7196 62.8556 59.3191 59.5353 59.6004 59.1455 60.9492 58.1690 56.7437 63.8189 59.3402 59.9246 57.9624 57.8803 54.2185 57.0642 57.3940 59.3236 61.1107 58.7336 62.1957 56.9972 61.8685 56.6666 60.7356
57.2318 64.6886 60.5820 58.3755 60.1325 58.0151 59.5333 55.7339 58.9585 61.1520 59.0087 60.5289 59.1567 60.3373 55.4219 62.3728 60.1458 57.5307 61.5614 60.0377 60.8256 59.0780 63.1777 56.0309 59.2298
56.1346 63.5049 62.4654 59.3054 58.4174 59.0903 58.8802 63.7261 61.8370 61.6337 58.7564 60.8942 56.5413 61.4310 59.2086 60.8605 61.4451 53.8560 59.4119 59.3488 59.3839 59.8597 58.3583 56.1494 61.7218
56.4828 63.9622 59.1721 60.9225 58.1483 59.4554 61.7122 61.6708 58.5870 60.2990 58.2273 63.6776 54.6083 62.3827 57.8763 60.3843 63.5907 58.1711 59.9346 58.0423 55.4010 59.0997 59.5097 59.2927 61.1168
58.9275 63.1547 57.9813 61.1856 61.8108 59.1609 65.1325 62.6899 57.9363 59.9443 54.7163 60.0538 54.5550 63.3933 59.9569 60.0855 60.6919 61.1852 60.6201 60.7227 58.6501 59.0970 60.3563 60.2513 61.2179
59.5257 61.4230 58.7108 61.4761 62.1789 61.7008 60.5918 59.4444 57.3334 62.1017 57.3292 63.1070 58.1895 61.9205 56.2599 62.1362 58.4221 60.8962 60.3030 60.7021 58.5790 57.1459 60.9815 60.2052 60.9013
60.1815 61.0640 58.0059 61.2223 61.9753 60.4833 63.4411 59.3478 56.7241 58.1281 61.7840 60.9021 60.7001 61.9462 61.9448 60.9805 57.5151 59.2145 58.5278 60.3550 59.4242 60.3319 61.5555 59.7755 59.4755
اوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ MINITAB آﺎﻟﺘﺎﻟﻲ: ;')MTB > TSPlot 'z(t >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC ;Connect >SUBC Title "An obseved Time Series".
٧٥
A n o b s e v e d T im e S e r ie s
z(t)
65
60
55
50
In d e x
100
150
200
ﺛﺎﻧﻴﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮ MTB > %ACF 'z(t)'; SUBC> MAXLAG 20; SUBC> TITLE"SACF of observed Time Series". Executing from file: H:\MTBWIN\MACROS\ACF.MAC
Autocorrelation
S A C F o f o b s e rv e d T im e S e rie s 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
L a g C o rr
T
LBQ
0 .5 1 0 .2 0 -0 .0 0 -0 .0 5 -0 .0 8 -0 .1 8 -0 .1 9
7 .1 9 2 .3 2 - 0 .0 1 - 0 .5 9 - 0 .9 5 - 2 .0 5 - 2 .0 9
5 2 .4 8 6 0 .7 8 6 0 .7 8 6 1 .3 4 6 2 .8 2 6 9 .9 2 7 7 .5 8
1 2 3 4 5 6 7
10
15
L a g C o rr
T
LBQ
- 0 .1 4 - 0 .1 4 - 0 .0 9 - 0 .0 7 - 0 .0 8 - 0 .0 2 0 .0 3
-1 .5 0 -1 .5 2 -0 .9 0 -0 .7 1 -0 .7 9 -0 .2 1 0 .3 2
8 1 .7 6 8 6 .1 4 8 7 .7 3 8 8 .7 3 8 9 .9 7 9 0 .0 5 9 0 .2 7
8 9 10 11 12 13 14
20
L a g C o rr
T
LBQ
0 .0 7 0 .1 3 0 .1 7 0 .2 0 0 .1 2 0 .0 6
0 .6 8 1 .3 3 1 .7 5 2 .0 6 1 .2 1 0 .6 1
9 1 .2 3 9 4 .8 6 1 0 1 .3 3 1 1 0 .6 3 1 1 3 .9 8 1 1 4 .8 6
15 16 17 18 19 20
ﺛﺎﻟﺜﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮ MTB > %PACF 'z(t)'; SUBC> MAXLAG 20; SUBC> TITLE"SPACF of obseved Time Series". Executing from file: H:\MTBWIN\MACROS\PACF.MAC
٧٦
Partial Autocorrelation
S P A C F o f o b se ve d T im e S e rie s 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
10
L a g P AC
T
0 .5 1 -0 .0 8 -0 .1 0 0 .0 0 -0 .0 5 -0 .1 6 -0 .0 4
7 .1 9 -1 .0 7 -1 .3 8 0 .0 4 -0 .7 3 -2 .3 4 -0 .5 0
1 2 3 4 5 6 7
15
L a g P AC
T
-0 .0 1 -0 .1 2 0 .0 1 -0 .0 4 -0 .0 9 0 .0 3 0 .0 2
-0 .1 2 -1 .6 3 0 .1 6 -0 .6 0 -1 .3 4 0 .3 9 0 .3 2
8 9 10 11 12 13 14
20
L a g P AC
T
-0 .0 2 0 .0 9 0 .0 8 0 .0 6 -0 .0 3 0 .0 4
-0 .2 3 1 .2 8 1 .2 0 0 .8 6 -0 .4 5 0 .5 2
15 16 17 18 19 20
ﻣﻦ أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ و اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﺗﺘﺒﻊ ﻧﻤﻮذج وﻟﻬﺬا ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻋﻠﻲ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻷﻣﺮAR(١) MTB > Name c7 = 'RESI1' MTB > ARIMA 1 0 0 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 839.667 0.100 53.870 1 746.819 0.250 44.876 2 695.840 0.400 35.883 3 685.086 0.502 29.769 4 685.054 0.507 29.458 5 685.054 0.507 29.443 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.5073 0.0611 Constant 29.4429 0.1309 Mean 59.7571 0.2656
0.0010
T 8.30 224.98
Number of observations: 201 Residuals: SS = 685.020 (backforecasts excluded) MS = 3.442 DF = 199 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 10.8(DF=11) 27.6(DF=23) 35.9(DF=35)
48 45.0(DF=47)
Forecasts from period 201 Period 202 203
Forecast 59.7079 59.7322
95 Percent Limits Lower Upper 56.0707 63.3451 55.6537 63.8106
٧٧
Actual
204 205 206
59.7445 59.7507 59.7539
55.5600 55.5394 55.5357
zt = 59.76 + 0.51( zt −1 − 59.76) + at , at
63.9290 63.9620 63.9721
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
WN ( 0,3.44 )
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
( )
φˆ1 = 0.51, s.e. φˆ1 = 0.061, t = 8.3 µˆ = 59.76, s.e. ( µˆ ) = 0.66
( )
δˆ = 29.44, s.e. δˆ = 0.131, t = 224.98 σˆ 2 = 3.44, with d . f . = 199
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 1.855 'RESI1'; SUBC> Alternative 0; SUBC> GHistogram; SUBC> GDotplot; SUBC> GBoxplot.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 1.85 Variable RESI1
N 201
Mean -0.002
StDev 1.851
SE Mean 0.131
Z -0.01
P 0.99
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 94 The expected number of runs = 101.0796 107 Observations above K 94 below The test is significant at 0.3149 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣
٧٨
ACF of Residuals for z(t) (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
50
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ PACF of Residuals for z(t) (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
50
Lag
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء : إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ-٥ ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ-ا Histogram of the Residuals (response is z(t))
Frequency
30
20
10
0 -5
0
Residual
٧٩
5
: وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ.ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ Normal Probability Plot رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ-ب Normal Probability Plot for RESI1
99
Mean:
-1.6E-03
StDev:
1.85070
95 90
Percent
80 70 60 50 40 30 20 10 5
1
-5.0
-2.5
0.0
2.5
5.0
Data
:واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم ﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲK-S Test ﺑﺈﺧﺘﺒﺎر-ج MTB > %NormPlot 'RESI1'; SUBC> Kstest; SUBC> Title "Normal Test for Residuals". Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
Normal Test for Residuals
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -5
0
5
RESI1 Average: -0.0016272 StDev: 1.85070 N: 201
H 0 : Residuals
Kolmogorov-Smirnov Normality Test D+: 0.045 D-: 0.060 D : 0.060 Approximate P-Value: 0.074
:وﻧﻼﺣﻆ اﻟﺘﺎﻟﻲ اﻹﺧﺘﺒﺎر هﻮ
N ( 0,3.44 )
H1 : Residuals§ N ( 0,3.44 )
ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف +
−
D = 0.045, D = 0.06, D = 0.06 أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔα = 0.05 وهﻲ أآﺒﺮ ﻣﻦ٠٫٠٧٤ ﻟﻺﺧﺘﺒﺎر هﻲP-Value اﻟـ
.اﻟﺼﻔﺮﻳﺔ :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ٨٠
ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ٥ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 201 Period 202 203 204 205 206
Forecast 59.7079 59.7322 59.7445 59.7507 59.7539
95 Percent Limits Lower Upper 56.0707 63.3451 55.6537 63.8106 55.5600 63.9290 55.5394 63.9620 55.5357 63.9721
Actual
وﻧﺮﺳﻤﻬﺎ ﺑﺎﻷﻣﺮ اﻟﺘﺎﻟﻲ Plot C4*C8 C5*C8 C6*C8; SUBC> Connect; SUBC> Type 1; SUBC> Color 1; SUBC> Size 1; SUBC> Title "Forecast of 5 future value with 95% limits"; SUBC> Overlay.
Forecast of 5 future value with 95% limits 64 63 62
C4
61 60 59 58 57 56 55 1
2
3
4
5
C8
.واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺠﺰء اﻷﺧﻴﺮ ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﻊ اﻟﺘﻨﺒﺆات وﻓﺘﺮات اﻟﺘﻨﺒﺆ Forecast of 5 future value with 95% limits 64 63 62
C9
61 60 59 58 57 56 55 180
190
200
C8
اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة-٢ ٨١
494.948 503.170 496.806 504.340 496.372 503.506 496.062 501.521 507.886 506.345 507.735 497.295 497.642 510.287 496.029 503.134 501.634 499.611 500.388 505.987 493.858 494.857 497.861 493.017 498.931 497.794 513.900 504.717 502.209 497.992 501.387
496.208 498.429 496.227 491.923 499.260 500.356 494.033 497.393 495.431 496.285 507.673 503.415 499.783 496.181 497.133 494.878 497.943 499.299 498.787 499.467 502.933 495.430 511.741 501.999 496.078 497.413 495.353 499.134 503.130 504.174 501.701
507.382 502.233 505.884 496.665 500.074 492.286 508.489 504.965 500.664 507.072 485.991 500.921 503.852 498.380 498.060 500.199 504.785 505.517 505.475 497.462 504.817 504.336 490.887 503.465 506.649 505.218 498.477 496.680 491.202 495.699 492.716
498.440 496.678 493.371 506.329 498.598 516.373 499.217 495.000 492.352 495.423 505.577 501.819 501.175 504.666 509.407 504.408 501.417 492.318 496.757 498.403 491.707 505.429 514.220 502.414 491.418 496.150 498.016 501.542 507.590 497.647 504.640
488.539 506.040 501.605 497.785 502.891 491.981 493.161 505.581 501.064 497.883 500.744 493.866 495.868 494.885 494.814 495.954 493.552 497.130 497.626 506.259 504.085 490.683 487.344 494.614 507.438 512.122 505.367 504.012 497.231 505.234 496.877
511.026 489.348 498.229 502.545 503.107 496.830 507.020 495.355 504.712 504.072 495.737 503.580 501.700 507.582 501.928 503.325 500.484 502.066 501.082 493.843 498.571 501.703 498.599 500.256 497.643 490.619 500.324 499.779 497.956 502.681 502.823
496.650 501.649 502.969 506.459 494.559 500.508 498.319 498.304 501.527 494.833 504.129 507.594 498.294 496.932 504.394 499.982 498.640 502.173 504.346 504.913 500.712 504.204 506.403 502.721 503.746 506.001 501.827 496.063 505.709 493.502 497.421 497.048
)z(t 499.148 503.975 498.758 493.057 499.890 507.416 498.090 504.877 494.918 499.173 496.765 482.567 500.989 501.331 489.314 502.720 502.489 495.691 496.252 499.279 498.169 497.015 503.195 500.252 498.158 500.409 496.225 491.726 489.032 500.024 505.194 499.574
ﺍﻭﻻ :ﺴﻭﻑ ﻨﺭﺴﻡ ﻓﻘﻁ ٥٠ﻤﺸﺎﻫﺩﺓ ﻤﻥ ﻫﺫﻩ ﺍﻟﻤﺘﺴﻠﺴﻠﺔ
510
)z(t
500
490
50
40
30
ﺛﺎﻧﻴﺎ :ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ:
٨٢
20
10
In d e x
Autocorrelation
Autocorrelation Function for z(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag Corr
10
T
LBQ
1 -0.53 -8.38 2 -0.05 -0.64 3 0.12 1.52 4 0.04 0.52 5 -0.13 -1.61 6 0.11 1.39 7 -0.09 -1.13
71.03 71.68 75.39 75.83 80.08 83.33 85.52
Lag Corr
T
15
LBQ
8 0.10 1.19 87.98 9 -0.11 -1.31 91.01 10 0.15 1.77 96.64 11 -0.18 -2.16 105.32 12 0.11 1.25 108.35 13 0.05 0.56 108.97 14 -0.12 -1.36 112.61
Lag Corr
20
T
LBQ
15 0.05 0.60 113.33 16 -0.02 -0.26 113.47 17 0.07 0.82 114.85 18 -0.10 -1.16 117.65 19 0.07 0.75 118.81 20 -0.06 -0.73 119.93
: ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ:ﺛﺎﻟﺜﺎ Partial Autocorrelation
Partial Autocorrelation Function for z(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
10
15
Lag PAC
T
Lag PAC
T
1 -0.53 2 -0.46 3 -0.29 4 -0.07 5 -0.11 6 0.02 7 -0.09
-8.38 -7.28 -4.52 -1.08 -1.71 0.30 -1.36
8 0.04 9 -0.07 10 0.12 11 -0.07 12 -0.04 13 0.08 14 -0.04
0.65 -1.09 1.94 -1.15 -0.71 1.19 -0.59
Lag PAC
20
T
15 0.03 0.49 16 -0.13 -2.09 17 0.06 1.01 18 -0.08 -1.32 19 0.03 0.44 20 -0.13 -2.07
وﺑﺘﻄﺒﻴﻖ هﺬا اﻟﻨﻤﻮذج ﻧﺠﺪMA(١) ﻣﻦ اﻷﻧﻤﺎط اﻟﻤﺸﺎهﺪة ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﺔ ﻗﺪ ﺗﺘﺒﻊ ﻧﻤﻮذج MTB > Name c7 = 'RESI1' MTB > ARIMA 0 0 1 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 6081.19 0.100 500.046 1 5265.34 0.250 500.004 2 4615.22 0.400 499.980 3 4109.70 0.550 499.967 4 3766.60 0.700 499.960 5 3727.32 0.841 499.959 6 3687.70 0.797 499.963
٨٣
7 3687.08 0.790 499.962 8 3687.07 0.791 499.962 9 3687.07 0.790 499.962 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev MA 1 0.7905 0.0386 Constant 499.962 0.051 Mean 499.962 0.051
0.0010
T 20.50 9708.40
Number of observations: 250 Residuals: SS = 3684.13 (backforecasts excluded) MS = 14.86 DF = 248 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 26.7(DF=11) 35.9(DF=23) 63.1(DF=35)
48 82.8(DF=47)
Forecasts from period 250 Period 251 252 253 254 255
Forecast 502.256 499.962 499.962 499.962 499.962
95 Percent Limits Lower Upper 494.700 509.812 490.330 509.593 490.330 509.593 490.330 509.593 490.330 509.593
zt = 499.962 + at − 0.7905at −1 , at
WN ( 0,14.86 )
Actual
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
( )
θˆ1 = 0.7905, s.e. θˆ1 = 0.0386, t = 20.50
( )
µˆ = δˆ = 499.962, s.e. δˆ = 0.051, t = 9708.40 σˆ 2 = 14.86, with d . f . = 248
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 3.847 'RESI1'; SUBC> Alternative 0.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 3.85 Variable RESI1
N 250
Mean -0.007
StDev 3.847
SE Mean 0.243
Z -0.03
P 0.98
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1
٨٤
K =
0.0000
The observed number of runs = 134 The expected number of runs = 125.9920 126 Observations above K 124 below The test is significant at 0.3103 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣ A C F o f R e s id u a ls fo r z (t) ( w ith 9 5 % c o n f id e n c e l im it s f o r th e a u to c o r r e l a tio n s ) 1 .0
0 .8
Autocorrelation
0 .6 0 .4
0 .2 0 .0 -0 .2 -0 .4
-0 .6 -0 .8
-1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ P A C F o f R e s id u a ls f o r z (t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء : إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ-٥ ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ-ا
٨٥
H istogram of the R esiduals (resp on se is z(t))
Frequency
30
20
10
0 -10
0
10
R esidual
: وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ.ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ Normal Probability Plot رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ-ب Normal Probability Plot for RESI1
99
Mean:
-6.9E-03
StDev:
3.84651
95 90
Percent
80 70 60 50 40 30 20 10 5
1
-10
-5
0
5
10
Data
:واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم ﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲK-S Test ﺑﺈﺧﺘﺒﺎر-ج MTB > %Qqplot 'RESI1'; SUBC> Conf 95; SUBC> Ci. Executing from file: H:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis Normal Dist. Parameter Estimates Data Mean: StDev:
: RESI1 -6.9E-03 3.84651
MTB > %NormPlot 'RESI1'; SUBC> Kstest. Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
٨٦
Normal Probability Plot
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -10
0
10
RESI1 Average: -0.0069004 StDev: 3.84651 N: 250
Kolmogorov-Smirnov Normality Test D+: 0.034 D-: 0.051 D : 0.051 Approximate P-Value: 0.105
ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف +
−
D = 0.034, D = 0.051, D = 0.051 أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ انα = 0.05 وهﻲ أآﺒﺮ ﻣﻦ٠٫١٠٥ ﻟﻺﺧﺘﺒﺎر هﻲP-Value اﻟـ
.اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ٥ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 250 Period 251 252 253 254 255
95 Percent Limits Lower Upper 494.700 509.812 490.330 509.593 490.330 509.593 490.330 509.593 490.330 509.593
Forecast 502.256 499.962 499.962 499.962 499.962
Actual
ﺗﻨﺒﺆ٩٥% واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات ﻣﻊ ﻓﺘﺮات Forecast of 5 future values with 95% limits
C4
510
500
490 1
2
3
4
5
C8
اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ ﻣﺸﺎهﺪة-٣ ٨٧
224.678 227.500 221.632 234.178 233.005 232.432 228.189 231.718 222.468 231.163 226.479 234.207 230.455 233.640 233.725 233.089 228.655 220.958 231.531 231.363 228.285 229.320 235.587 228.729 232.700 240.884 236.017 229.321 228.141 228.958 226.114
226.760 224.063 225.663 238.463 232.209 230.102 231.121 229.848 223.115 228.418 224.483 232.738 228.163 232.232 236.562 233.427 225.484 220.782 232.030 230.938 229.757 230.636 230.713 230.360 234.625 241.821 230.865 226.222 225.985 229.352 228.837
226.641 222.515 229.136 239.577 227.621 229.564 231.633 227.982 227.859 228.618 228.989 229.792 225.447 234.825 235.224 233.044 224.207 222.819 232.315 233.127 229.733 227.948 227.046 230.662 234.107 238.762 230.499 225.771 225.358 229.369 232.881
226.778 221.562 232.308 232.653 222.156 229.331 231.319 225.734 230.122 228.225 235.122 228.198 224.928 234.707 231.665 229.575 224.747 226.865 232.027 233.852 232.160 224.258 228.606 234.243 231.265 233.628 226.994 225.616 225.291 230.296 236.029
229.903 222.482 236.488 223.408 218.067 229.359 234.668 225.721 230.888 229.851 235.024 226.724 227.812 233.841 228.922 227.089 228.954 232.152 232.135 232.684 235.038 223.994 228.203 239.883 229.486 230.342 223.795 226.271 223.250 228.920 235.339
230.260 223.390 236.033 217.433 217.123 235.744 235.767 224.927 228.472 228.227 236.659 225.196 231.266 233.891 230.327 227.032 232.820 232.461 228.727 231.411 233.799 225.273 231.898 243.963 225.070 227.643 223.381 230.088 225.447 227.934 235.210
227.346 225.772 234.323 213.619 221.484 236.419 233.918 226.765 228.200 225.799 236.399 222.523 232.976 232.067 230.148 227.077 233.256 231.062 224.050 232.032 233.622 223.599 229.100 242.860 225.123 230.082 228.468 235.623 226.745 229.727 232.891 224.096
)z(t 229.574 224.077 230.713 215.405 228.758 234.678 234.155 227.075 230.421 224.663 233.335 223.571 234.561 232.473 231.653 230.146 233.444 230.076 221.171 230.582 232.344 224.880 227.449 239.660 225.860 229.792 235.112 238.292 227.805 230.794 231.092 225.020
اوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ Time Plotﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻹﺣﺼﺎﺋﻴﺔ MINITAB آﺎﻟﺘﺎﻟﻲ ٥٠) :ﻣﺸﺎهﺪة ﻓﻘﻂ(
2 4 2
)z(t
2 3 2
2 2 2 5 0
4 0
3 0
ﺛﺎﻧﻴﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ
٨٨
2 0
1 0
In d e x
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
Lag 1 2 3 4 5 6 7
C o rr 0 0 0 -0 -0 -0 -0
.8 .5 .1 .1 .3 .3 .2
4 1 5 6 3 3 0
T 13 5 1 -1 -2 -2 -1
.2 .2 .3 .4 .9 .8 .7
7 0 5 6 8 7 3
10
LBQ 1 2 2 2 2 3 3
7 4 4 5 8 1 2
8 4 9 6 3 1 1
.2 .2 .7 .2 .7 .2 .9
1 9 1 1 5 3 5
Lag
1 1 1 1 1
C o rr
8 -0 .0 2 -0 9 0 .1 4 1 0 0 .2 3 1 1 0 .2 3 1 2 0 .1 5 1 3 0 .0 2 0 4 -0 .1 2 -0
T .1 .1 .9 .8 .2 .1 .9
7 8 5 8 2 5 6
15
LBQ 3 3 3 3 3 3 3
2 2 4 5 6 6 6
2 7 1 5 1 1 5
.0 .2 .4 .1 .1 .2 .0
Lag
5 4 8 6 7 5 4
1 1 1 1 1 2
5 6 7 8 9 0
C o rr -0 -0 -0 -0 0 0
.2 .2 .1 .1 .0 .0
1 2 8 0 0 8
20
T -1 -1 -1 -0 0 0
.6 .7 .4 .7 .0 .6
5 9 2 6 2 1
LBQ 3 3 3 4 4 4
7 8 9 0 0 0
6 9 8 1 1 3
.3 .9 .8 .3 .3 .0
2 3 1 7 7 5
ﺛﺎﻟﺜﺎ ﻧﺤﺴﺐ وﻧﺮﺳﻢ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
Lag 1 2 3 4 5 6 7
PAC
10
T
0 .8 4 1 3 .2 7 -0 .6 6 -1 0 .3 9 -1 .1 3 -0 .0 7 -1 .0 0 -0 .0 6 1 .7 8 0 .1 1 2 .1 8 0 .1 4 -0 .2 2 -0 .0 1
Lag
15
PAC
T
8 0 .0 7 9 -0 .0 7 1 0 0 .0 5 1 1 -0 .0 7 1 2 0 .0 6 1 3 -0 .1 5 1 4 0 .0 1
1 .0 6 -1 .1 2 0 .7 3 -1 .1 8 0 .9 0 -2 .4 1 0 .1 8
Lag
20
PAC
T
1 5 0 .0 4 1 6 -0 .0 6 1 7 0 .0 4 1 8 -0 .0 7 1 9 0 .0 9 2 0 -0 .0 9
0 .6 7 -0 .9 3 0 .6 0 -1 .1 4 1 .4 4 -1 .3 9
ﻣﻦ أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ و اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻲ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﺗﺘﺒﻊ ﻧﻤﻮذج وﻟﻬﺬا ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻋﻠﻲ اﻟﻤﺸﺎهﺪاتAR(٢) MTB > Name c7 = 'RESI1' MTB > ARIMA 2 0 0 'z(t)' 'RESI1'; SUBC> Constant; SUBC> Forecast 10 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 4257.23 0.100 0.100 1 3528.31 0.250 0.012 2 2889.23 0.400 -0.076 3 2338.97 0.550 -0.165 4 1877.39 0.700 -0.253 5 1504.46 0.850 -0.342 6 1220.13 1.000 -0.430
٨٩
183.784 169.535 155.360 141.201 127.051 112.913 98.789
7 1024.34 1.150 8 916.97 1.300 9 894.38 1.402 10 894.31 1.408 11 894.31 1.408 Relative change in each estimate
-0.519 84.690 -0.608 70.623 -0.668 61.154 -0.672 60.670 -0.672 60.646 less than 0.0010
Final Estimates of Parameters Type Coef StDev AR 1 1.4079 0.0473 AR 2 -0.6720 0.0474 Constant 60.6458 0.1203 Mean 229.638 0.456
T 29.78 -14.19 504.11
Number of observations: 250 Residuals: SS = 893.567 (backforecasts excluded) MS = 3.618 DF = 247 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 17.5(DF=10) 27.2(DF=22) 49.7(DF=34)
48 67.7(DF=46)
Forecasts from period 250 Period 251 252 253 254 255 256 257 258 259 260
Forecast 224.939 226.747 228.725 230.296 231.177 231.363 231.033 230.442 229.833 229.372
95 Percent Limits Lower Upper 221.211 228.668 220.308 233.186 220.642 236.808 221.546 239.045 222.311 240.044 222.494 240.233 222.070 239.996 221.327 239.558 220.600 239.067 220.090 238.655
zt = 60.6458 + 1.4079 zt −1 − 0.672 zt −2 + at , at
( ) s.e. (φˆ ) = 0.0474,
Actual
:وﻧﺴﺘﻨﺘﺞ اﻟﺘﺎﻟﻲ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ-١
WN ( 0,3.618)
: ﻟﻬﺎ هﻲt اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة وإﻧﺤﺮاﻓﻬﺎ اﻟﻤﻌﻴﺎري و ﻗﻴﻤﺔ-٢
φˆ1 = 1.4079, s.e. φˆ1 = 0.0473, t = 29.78 φˆ2 = −0.672,
2
t = −14.19
µˆ = 229.638, s.e. ( µˆ ) = 0.456
( )
δˆ = 60.6458, s.e. δˆ = 0.1203, t = 504.11 σˆ 2 = 3.618, with d . f . = 247
:راﺑﻌﺎ ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ-١ MTB > ZTest 0.0 3.618 'RESI1'; SUBC> Alternative 0.
Z-Test Test of mu = 0.000 vs mu not = 0.000 The assumed sigma = 3.62 Variable RESI1
N 250
Mean -0.005
StDev 1.894
SE Mean 0.229
٩٠
Z -0.02
P 0.98
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﻤﺘﻮﺳﻂ ﻳﺴﺎوي اﻟﺼﻔﺮ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ-٢ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 125 The expected number of runs = 125.8720 129 Observations above K 121 below The test is significant at 0.9119 Cannot reject at alpha = 0.05
ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺑﺄن اﻟﺒﻮاﻗﻲ ﻋﺸﻮاﺋﻴﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﺒﻮاﻗﻲ-٣ A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0 5
10
15
20
25
30
35
40
45
50
55
60
Lag
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ-٤ P A C F o f R e s id u a ls f o r z (t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0 5
10
15
20
25
30
Lag
٩١
35
40
45
50
55
60
ﻧﻼﺣﻆ ان أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﺗﺘﺒﻊ أﻧﻤﺎط ﻣﺘﺴﻠﺴﻠﺔ اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء -٥إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ : ا -ﻧﺮﺳﻢ اﻟﻤﻀﻠﻊ اﻟﺘﻜﺮاري ﻟﻠﺒﻮاﻗﻲ H istogram of the R esiduals ))(res p on s e is z(t 30
10
0 5
-5
0
R es idual
ﻧﻼﺣﻆ أﻧﻪ ﻣﺘﻨﺎﻇﺮ وﻟﺔ ﺷﻜﻞ اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﺗﻘﺮﻳﺒﺎ .وهﺬا ﻻﻳﻜﻔﻲ ﺑﻞ ﻳﺠﺐ ان ﻧﻨﻈﺮ اﻟﻰ: ب -رﺳﻢ اﻻﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ Normal Probability Plot N orm al P rob ab ility P lot for R E S I1
-4 .6 E -0 3
M ean:
1 .8 9 4 3 6
S tD e v:
99
95 90 80
20 10 5
1
4
2
0
D a ta
واﺿﺢ ﻣﻦ اﻟﺮﺳﻢ أن اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ وﻟﻠﺘﺄآﺪ ﻧﻘﻮم: ج -ﺑﺈﺧﺘﺒﺎر K-S Testﻟﻄﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ
٩٢
-2
-4
-6
Percent
70 60 50 40 30
Frequency
20
Normal Probability Plot
.999 .99
Probability
.95 .80 .50 .20 .05 .01 .001 -5
0
5
RESI1 Average: -0.0046305 StDev: 1.89436 N: 250
Kolmogorov-Smirnov Normality Test D+: 0.020 D-: 0.029 D : 0.029 Approximate P-Value > 0.15
وهﻲ أآﺒﺮ ﻣﻦ٠٫١٥ ﻟﻺﺧﺘﺒﺎر هﻲP-Value ﺳﻤﻴﺮﻧﻮف اﻋﻄﻰ اﻟـ-إﺧﺘﺒﺎر آﻮﻟﻤﻮﺟﻮروف . أي اﻧﻨﺎ ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲα = 0.05 :ﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ١٠ اﺳﺘﺨﺪﻣﻨﺎ اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ Forecasts from period 250 Period 251 252 253 254 255 256 257 258 259 260
95 Percent Limits Lower Upper 221.211 228.668 220.308 233.186 220.642 236.808 221.546 239.045 222.311 240.044 222.494 240.233 222.070 239.996 221.327 239.558 220.600 239.067 220.090 238.655
Forecast 224.939 226.747 228.725 230.296 231.177 231.363 231.033 230.442 229.833 229.372
Actual
ﻓﺘﺮات ﺗﻨﺒﺆ٩٥% واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات و Forecast of 10 future values with 95% limits
C4
240
230
220 0
1
2
3
4
5
6
7
8
9
C8
٩٣
10
Forecast of 10 future values with 95% limits 245
235
C9 225
215 200
0
100
C8
Forecast of 10 future values with 95% limits
240
C9
230
220 50
60
30
40
20
10
0
C8
اﻟﺸﻜﻞ اﻷول ﻳﺒﻴﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ ﺑﻜﺎﻣﻠﻬﺎ ﻣﻊ اﻟﺘﻨﺒﺆات واﻟﺸﻜﻞ اﻟﺜﺎﻧﻲ ﻟﻠﺨﻤﺴﻴﻦ ﻗﻴﻤﺔ اﻷﺧﻴﺮة ﻣﻊ اﻟﺘﻨﺒﺆات ﻟﺘﻮﺿﻴﺢ ﺷﻜﻞ داﻟﺔ اﻟﺘﻨﺒﺆ. ﺣﺎﻟﺔ دراﺳﺔ: اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺘﺎﻟﻴﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻴﻮﻣﻲ ﻟﻌﺪد اﻟﺘﻠﻔﺰﻳﻮﻧﺎت اﻟﻤﻌﻴﺒﺔ ﻓﻲ ﺧﻂ إﻧﺘﺎج ﻣﺼﻨﻊ ﻣﺎ )إﻗﺮأ ﺳﻄﺮا ﺑﺴﻄﺮ( 2.09 1.57 2.07 1.78
2.00 1.42 1.82 1.68
1.76 1.54 1.85 1.79
2.83 1.46 2.08 1.37
3.44 2.05 1.42 1.15
2.40 2.50 1.39 1.25
اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
٩٤
1.95 2.25 1.18 1.61
2.70 1.58 1.27 1.77
1.54 1.25 1.08 2.91
Defects 1.20 1.50 1.89 1.80 1.40 1.51 2.32 1.23 1.84
3 .5
3 .0
Defects
2 .5 2 .0 1 .5 1 .0
In d e x
10
20
30
40
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Autocorrelation
Autocorrelation Function for Defects 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
Lag
Corr
4
5
6
7
8
T
LBQ
Lag
Corr
T
LBQ
1 0.43 2.88 2 0.26 1.49 3 0.14 0.77 4 0.08 0.43 5 -0.09 -0.46 6 -0.07 -0.39 7 -0.21 -1.10
8.84 12.18 13.18 13.50 13.89 14.18 16.57
8 9 10 11
-0.11 -0.05 -0.01 -0.04
-0.57 -0.27 -0.04 -0.19
17.25 17.41 17.41 17.50
9
10
11
Partial Autocorrelation
Partial Autocorrelation Function for Defects 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
Lag PAC
4
5
T
1 0.43 2.88 2 0.09 0.63 3 -0.00 -0.01 4 0.00 0.00 5 -0.16 -1.07 6 0.00 0.02 7 -0.18 -1.19
6
Lag PAC
7
8
T
8 0.07 0.44 9 0.05 0.35 10 0.01 0.09 11 -0.03 -0.23
٩٥
9
10
11
: واﻟﺬي ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔAIC ﻹﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ ﺳﻮف ﻧﺴﺘﺨﺪم ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ AIC ( m ) = n ln σ a2 + 2m ﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲm ﺣﻴﺚ min AIC ( m ) m
:ﺳﻮف ﻧﻄﺒﻖ اﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺘﻮاﻟﻲ MTB > ARIMA 1 0 0 'Defects' 'RESI1'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.4421 0.1365 Constant 0.99280 0.06999 Mean 1.7795 0.1254
T 3.24 14.19
Number of observations: 45 Residuals: SS = 9.47811 (backforecasts excluded) MS = 0.22042 DF = 43 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.9(DF=11) 8.9(DF=23) 30.9(DF=35)
48 * (DF= *)
MTB > ARIMA 2 0 0 'Defects' 'RESI2'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.3999 0.1533 AR 2 0.0989 0.1531 Constant 0.89019 0.07047 Mean 1.7762 0.1406
T 2.61 0.65 12.63
Number of observations: 45 Residuals: SS = 9.38567 (backforecasts excluded) MS = 0.22347 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.0(DF=10) 8.1(DF=22) 28.8(DF=34) MTB > ARIMA 1 0 1 'Defects' 'RESI3'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.5983 0.2691 MA 1 0.1926 0.3294 Constant 0.71334 0.05693
T 2.22 0.58 12.53
٩٦
48 * (DF= *)
Mean
1.7759
0.1417
Number of observations: 45 Residuals: SS = 9.39423 (backforecasts excluded) MS = 0.22367 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.1(DF=10) 8.3(DF=22) 29.1(DF=34)
48 * (DF= *)
MTB > ARIMA 0 0 1 'Defects' 'RESI4'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev MA 1 -0.3409 0.1431 Constant 1.78480 0.09651 Mean 1.78480 0.09651
T -2.38 18.49
Number of observations: 45 Residuals: SS = 10.0362 (backforecasts excluded) MS = 0.2334 DF = 43 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 8.0(DF=11) 13.2(DF=23) 35.7(DF=35)
48 * (DF= *)
MTB > ARIMA 0 0 2 'Defects' 'RESI5'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev MA 1 -0.3869 0.1516 MA 2 -0.1816 0.1516 Constant 1.7839 0.1118 Mean 1.7839 0.1118
T -2.55 -1.20 15.96
Number of observations: 45 Residuals: SS = 9.61059 (backforecasts excluded) MS = 0.22882 DF = 42 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.6(DF=10) 9.2(DF=22) 31.0(DF=34) MTB > ARIMA 2 0 1 'Defects' 'RESI6'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 0.4134 1.5680
T 0.26
٩٧
48 * (DF= *)
AR 2 MA 1 Constant Mean
0.0929 0.0136 0.87675 1.7761
0.7113 1.5749 0.07036 0.1425
0.13 0.01 12.46
Number of observations: 45 Residuals: SS = 9.38561 (backforecasts excluded) MS = 0.22892 DF = 41 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.0(DF= 9) 8.1(DF=21) 28.8(DF=33)
48 * (DF= *)
MTB > ARIMA 1 0 2 'Defects' 'RESI7'; SUBC> Constant;
ARIMA Model ARIMA model for Defects * ERROR * Model cannot be estimated with these data MTB > ARIMA 2 0 2 'Defects' 'RESI8'; SUBC> Constant;
ARIMA Model ARIMA model for Defects Final Estimates of Parameters Type Coef StDev AR 1 1.6720 0.1165 AR 2 -0.7263 0.1251 MA 1 1.3199 0.0184 MA 2 -0.3196 0.0731 Constant 0.096224 0.003323 Mean 1.77238 0.06121
T 14.35 -5.80 71.63 -4.37 28.95
Number of observations: 45 Residuals: SS = 8.33225 (backforecasts excluded) MS = 0.20831 DF = 40 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 4.7(DF= 8) 8.9(DF=20) 29.7(DF=32)
48 * (DF= *)
:وﻧﻠﺨﺺ ذﻟﻚ ﺑﺎﻟﺠﺪول اﻟﺘﺎﻟﻲ Model __________ AR (1) AR ( 2 ) MA (1) MA ( 2 ) ARMA (1,1) ARMA ( 2,1) ARMA (1, 2 ) ARMA ( 2, 2 )
σˆ
2
________ 0.22042 0.22347 0.23340 0.22882 0.22367 0.22892 − 0.20831
m ___ 3 4 3 4 4 5 − 6
AIC _________ −62.0499 −59.4315 −59.4751 −58.3669 −59.3913 −56.3472 − −58.5928
٩٨
min AIC ( m ) = −62.0499 m
أي ان أﻓﻀﻞ ﻧﻤﻮذج هﻮ ). AR(١ ﻳﺘﺮك ﻟﻠﻄﺎﻟﺐ آﺘﻤﺮﻳﻦ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ وﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات.
ﺣﺎﻟﺔ دراﺳﺔ: اﻟﻤﺘﺴﻠﺴﺔ اﻟﺘﺎﻟﻴﺔ هﻲ دﺧﻞ اﻟﻤﺒﻴﻌﺎت اﻟﺴﻨﻮﻳﺔ ﺑﻤﻼﻳﻴﻦ اﻟﺮﻳﺎﻻت ﻟﺸﺮآﺔ ﻣﺎ 5.43 3.88 3.57 2.75 6.06 5.80 5.16 6.64 6.43 7.86
3.80 4.30 3.45 4.80 6.12 6.08 5.71 7.49 7.53 7.50
4.14 5.42 1.22 3.08 5.65 4.78 5.61 6.09 5.62 8.27
4.60 3.91 3.98 5.43 5.52 5.67 5.63 6.64 7.59 8.75
4.77 5.07 2.65 4.40 4.79 4.89 5.70 4.72 7.27 8.50
3.45 3.78 3.28 3.84 6.11 4.99 5.75 6.08 9.01 7.23
3.99 6.16 4.05 5.14 6.46 6.12 6.36 6.57 7.49 7.53
5.51 4.05 4.08 4.00 4.31 6.23 8.02 7.56 6.69 7.67
5.74 2.54 4.61 1.58 4.99 5.05 7.07 6.87 7.22 8.22
ﻣﺨﻄﻂ زﻣﻨﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
9 8 7 6
Sales
5 4 3 2 1 100
90
80
70
60
50
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ اﻟﻌﻴﻨﻴﺔ
٩٩
40
30
20
10
In d e x
Sales 3.49 3.96 2.89 2.52 5.77 3.20 5.13 7.20 7.26 6.42
Autocorrelation
Autocorrelation Function for Sales 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag Corr 1 2 3 4 5 6 7
0.71 0.60 0.65 0.64 0.59 0.59 0.51
T
LBQ
7.10 51.97 4.26 89.85 3.94 134.64 3.36 177.75 2.83 215.72 2.63 253.97 2.12 282.61
15
Lag Corr 8 9 10 11 12 13 14
0.56 0.49 0.49 0.51 0.42 0.38 0.45
T
LBQ
2.22 317.32 1.87 344.49 1.79 371.67 1.82 401.71 1.46 422.60 1.29 439.91 1.50 464.06
Lag Corr 15 16 17 18 19 20 21
0.41 0.35 0.31 0.30 0.36 0.31 0.26
T
25
LBQ
1.32 483.85 1.13 499.04 0.97 510.68 0.92 521.52 1.11 537.81 0.95 550.11 0.77 558.52
Lag Corr 22 23 24 25
0.22 0.17 0.21 0.25
T
LBQ
0.67 565.04 0.50 568.77 0.64 574.90 0.75 583.43
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5
Lag PAC
T
1 0.71 7.10 2 0.20 1.99 3 0.35 3.53 4 0.15 1.49 5 0.09 0.92 6 0.10 1.03 7 -0.13 -1.27
15
Lag PAC
25
T
Lag PAC
T
8 0.19 1.93 9 -0.17 -1.70 10 0.14 1.44 11 0.03 0.34 12 -0.15 -1.52 13 0.02 0.25 14 0.04 0.39
15 0.03 16 -0.10 17 -0.08 18 -0.01 19 0.14 20 -0.04 21 0.00
0.32 -0.96 -0.84 -0.10 1.43 -0.44 0.04
Lag PAC
T
22 -0.17 -1.68 23 -0.10 -1.00 24 0.15 1.55 25 0.03 0.34
.داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﻌﻴﻨﻲ ﺗﺪل ﻋﻠﻰ ﺗﺨﺎﻣﺪ ﺑﻄﻴﺊ ﻣﻤﺎ ﻗﺪ ﻳﺪل ﻋﻠﻰ ﻋﺪم إﺳﺘﻘﺮار ﻓﻲ اﻟﻤﺘﻮﺳﻂ وﻧﺮﺳﻤﻬﺎwt = ∇zt ﻟﻨﺠﺮب اﻟﺘﻔﺮﻳﻖ اﻷول ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
3 2 1
w(t)
Partial Autocorrelation
Partial Autocorrelation Function for Sales
0 -1 -2 -3
In d e x
10
20
30
40
50
60
70
80
90
100
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺴﺘﻘﺮة.ﺗﺒﺪو اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺴﺘﻘﺮة اﻵن
١٠٠
Autocorrelation
Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
Lag Corr
12
T
LBQ
1 -0.30 -3.00 2 -0.31 -2.86 3 0.11 0.90 4 0.04 0.31 5 -0.04 -0.33 6 0.14 1.15 7 -0.24 -1.98
9.26 19.32 20.49 20.63 20.80 22.80 29.02
Lag Corr
T
LBQ
8 0.19 1.53 9 -0.07 -0.53 10 -0.07 -0.52 11 0.16 1.27 12 -0.07 -0.53 13 -0.14 -1.04 14 0.15 1.10
33.07 33.58 34.08 37.12 37.67 39.85 42.37
Lag Corr
22
T
LBQ
15 0.00 0.00 16 0.02 0.13 17 -0.07 -0.52 18 -0.15 -1.09 19 0.20 1.48 20 0.01 0.05 21 -0.04 -0.25
42.37 42.40 42.99 45.65 50.73 50.74 50.90
Lag Corr
T
LBQ
22 0.05 0.35 51.22 23 -0.18 -1.28 55.44 24 0.03 0.20 55.55
Partial Autocorrelation
Partial Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
22
Lag PAC
T
Lag PAC
T
Lag PAC
T
1 -0.30 2 -0.44 3 -0.22 4 -0.21 5 -0.17 6 0.06 7 -0.26
-3.00 -4.41 -2.23 -2.05 -1.72 0.59 -2.55
8 0.12 9 -0.16 10 -0.06 11 0.09 12 -0.06 13 -0.02 14 -0.07
1.17 -1.59 -0.60 0.93 -0.59 -0.17 -0.71
15 0.06 16 0.07 17 -0.03 18 -0.17 19 -0.03 20 -0.10 21 0.09
0.59 0.70 -0.32 -1.65 -0.31 -0.95 0.85
Lag PAC
T
22 0.08 0.80 23 -0.15 -1.53 24 -0.05 -0.55
: واﻟﺬي ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔAIC ﻹﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ ﺳﻮف ﻧﺴﺘﺨﺪم ﻣﻌﻴﺎر اﻹﻋﻼم اﻟﺬاﺗﻲ AIC ( m ) = n ln σ a2 + 2m ﻋﺪد اﻟﻤﻌﺎﻟﻢ اﻟﻤﻘﺪرة ﻓﻲ اﻟﻨﻤﻮذج وﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲm ﺣﻴﺚ min AIC ( m ) m
:ﺳﻮف ﻧﻄﺒﻖ اﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺘﻮاﻟﻲ MTB > ARIMA 1 1 0 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.3114 0.0959
T -3.25
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 133.134 (backforecasts excluded)
١٠١
MS =
1.359
DF = 98
Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 31.9(DF=11) 51.2(DF=23) 62.8(DF=35)
48 81.0(DF=47)
MTB > ARIMA 2 1 0 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.4532 0.0897 AR 2 -0.4656 0.0901
T -5.05 -5.17
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 104.715 (backforecasts excluded) MS = 1.080 DF = 97 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 21.8(DF=10) 40.9(DF=22) 49.4(DF=34)
48 59.9(DF=46)
MTB > ARIMA 0 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev MA 1 0.7636 0.0648
T 11.78
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 101.411 (backforecasts excluded) MS = 1.035 DF = 98 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 12.6(DF=11) 27.8(DF=23) 35.9(DF=35)
48 48.5(DF=47)
MTB > ARIMA 0 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev MA 1 0.5756 0.0990 MA 2 0.2029 0.0998
T 5.81 2.03
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 99.2463 (backforecasts excluded) MS = 1.0232 DF = 97
١٠٢
Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 14.3(DF=10) 28.3(DF=22) 36.5(DF=34)
48 47.0(DF=46)
MTB > ARIMA 1 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 0.1283 0.1334 MA 1 0.8027 0.0799
T 0.96 10.04
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 100.421 (backforecasts excluded) MS = 1.035 DF = 97 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 13.3(DF=10) 27.9(DF=22) 36.1(DF=34)
48 48.2(DF=46)
MTB > ARIMA 2 1 1 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales * WARNING * Back forecasts not dying out rapidly Final Estimates of Parameters Type Coef StDev AR 1 -1.1389 0.0987 AR 2 -0.1440 0.0983 MA 1 -0.9889 0.0002
T -11.53 -1.47 -3987.49
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 134.250 (backforecasts excluded) MS = 1.398 DF = 96 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 35.1(DF= 9) 53.5(DF=21) 66.6(DF=33) MTB > ARIMA 1 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.3476 0.4077 MA 1 0.2422 0.3771 MA 2 0.4506 0.2656
T -0.85 0.64 1.70
Differencing: 1 regular difference
١٠٣
48 83.2(DF=45)
Number of observations: Original series 100, after differencing 99 Residuals: SS = 97.2357 (backforecasts excluded) MS = 1.0129 DF = 96 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 11.9(DF= 9) 25.0(DF=21) 32.1(DF=33)
48 41.8(DF=45)
MTB > ARIMA 2 1 2 'Sales'; SUBC> NoConstant.
ARIMA Model ARIMA model for Sales Final Estimates of Parameters Type Coef StDev AR 1 -0.0691 0.3618 AR 2 -0.2941 0.1450 MA 1 0.5637 0.3737 MA 2 0.0840 0.3266
T -0.19 -2.03 1.51 0.26
Differencing: 1 regular difference Number of observations: Original series 100, after differencing 99 Residuals: SS = 93.6368 (backforecasts excluded) MS = 0.9857 DF = 95 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 11.0(DF= 8) 23.4(DF=20) 30.1(DF=32)
48 36.5(DF=44)
:وﻧﻠﺨﺺ ذﻟﻚ ﺑﺎﻟﺠﺪول اﻟﺘﺎﻟﻲ Model σˆ __________ ________ ARI (1,1) 1.359 ARI ( 2,1) 1.080 IMA (1,1) 1.035 IMA (1, 2 ) 1.023 ARIMA (1,1,1) 1.035 ARIMA ( 2,1,1) 1.398 ARIMA (1,1, 2 ) 1.013 ARIMA ( 2,1, 2 ) 0.986 2
m ___ 2 3 2 3 3 4 4 5
AIC _________ 34.368 13.619 7.4057 8.2706 9.4057 41.169 9.2689 8.5741
min AIC ( m ) = 7.406 m
. ﻳﺘﺮك ﻟﻠﻄﺎﻟﺐ آﺘﻤﺮﻳﻦ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ وﺗﻮﻟﻴﺪ ﺗﻨﺒﺆات. IMA(١،١) أي ان أﻓﻀﻞ ﻧﻤﻮذج هﻮ
١٠٤
١٠٥
اﻟﻔﺼﻞ اﻟﺴﺎدس ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻴﺔ
Seasonal Autoregressive Integrated Moving Average Models اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ﺗﻌﻄﻲ اﻧﻤﺎط ﻣﺘﺸﺎﺑﻬﺔ ﺗﺘﻜﺮر ﻋﻠﻰ ﻓﺘﺮات زﻣﻨﻴﺔ ﻣﺘﺴﺎوﻳﺔ اﻟﺒﻌﺪ ﻣﺜﻞ ان ﻳﺘﻜﺮر اﻟﻨﻤﻂ آﻞ ارﺑﻌﺔ وﻋﺸﺮون ﺳﺎﻋﺔ او آﻞ ﺳﺒﻌﺔ اﻳﺎم او آﻞ ﺷﻬﺮ او ﺛﻼﺛﺔ اﺷﻬﺮ او ﺳﻨﺔ. اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﺗﺒﻴﻦ ﻣﺜﻞ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻼت S e a s o n a l T im e S e r ie s
70
)z(t 60
50
150
50
100
In d e x
S e a s o n a l T im e S e r ie s 1000
900
)z(t
800
700
600
150
100
In d e x
50
ﻓﻲ هﺬا اﻟﻔﺼﻞ ﺳﻮف ﻧﺴﺘﻌﺮض ﺧﻮاص اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ وﻃﺮق ﻧﻤﺬﺟﺘﻬﺎ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك SARIMA(p,d,q)(P,D,Q)sﻓﻤﺜﻼ اﻟﻨﻤﻮذج SARIMA(٠،١،١)(١،١،٠)١٢ﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ ) WN ( 0, σ 2
١٠٦
at
(1 − Φ B ) (1 − B ) z = (1 − θ B ) a , s
t
1
t
1
وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن ﻧﻤﻮذج اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﺘﻜﺎﻣﻠﻲ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺑﺎﻟﺪرﺟﺔ (p,d,q)(P,D,Q)s SARIMA(p,d,q)(P,D,Q)sﻳﻜﺘﺐ ﻋﻠﻰ اﻟﺸﻜﻞ
) φ p ( B ) Φ P ( B s ) (1 − B ) (1 − B s ) zt = δ + θ q ( B ) ΘQ ( B s ) at , at WN ( 0, σ 2 D
d
ﺣﻴﺚ ) φ p ( Bو ) θ q ( Bﻋﻤﺎل اﻹﻧﺤﺪار اﻟﺬاﺗﻲ واﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ واﻟﺘﻲ ﻣﺮت ﻋﻠﻴﻨﺎ ﺳﺎﺑﻘﺎ و Φ P ( B s ) = 1 + Φ1B s + Φ 2 B 2 s + L + Φ P B Psﻋﺎﻣﻞ اﻹﻧﺤﺪار اﻟﺬاﺗﻲ اﻟﻤﻮﺳﻤﻲ و ΘQ ( B s ) = 1 + Θ1B s + Θ2 B 2 s + L + ΘQ B Qsﻋﺎﻣﻞ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻤﻮﺳﻤﻲ وﻳﺴﻤﻰ هﺬا ﺑﺎﻟﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ .Multiplicative Seasonal Models ﻣﻼﺣﻈﺔ :ﻓﻲ ﺟﻤﻴﻊ اﻟﻨﻤﺎذج اﻟﻘﺎدﻣﺔ ﺳﻴﻜﻮن ﻣﻔﻬﻮﻣﺎ ﺿﻤﻨﻴﺎ أن ) at WN ( 0, σ 2
دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ: D d ﻓﻲ اﻹﺷﺘﻘﺎﻗﺎت اﻟﺘﺎﻟﻴﺔ ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﺴﺘﻘﺮة wt = (1 − B ) (1 − B s ) zt واﻟﺘﻲ ﺗﺘﺒﻊ اﻟﻨﻤﻮذج SARMA(p,q)(P,Q)s ) WN ( 0, σ 2
φ p ( B ) Φ P ( B ) wt = δ + θ q ( B ) ΘQ ( B ) at , at s
s
ﺳﻮف ﻧﺸﺘﻖ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ SARMA(٠،١)(١،١)١٢ ) WN ( 0, σ 2
wt = Φ wt −12 + at − θ at −1 − Θat −12 + θ Θat −13 , at
ﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻓﻲ wtوأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ γ 0 = Φ γ 12 + σ + θ σ − Θ ( Φ − Θ ) σ 2 + θ Θ ( −Φ θ + θ Θ ) σ 2 2
2
2
⎦⎤) =Φ γ 12 + σ 2 ⎡⎣(1 + θ 2 ) + Θ ( Φ − Θ ) (1 + θ 2 ⎦⎤) =Φ γ 12 + σ 2 (1 + θ 2 ) ⎡⎣1 + Θ ( Φ − Θ
وﺑﻀﺮب ﻃﺮﻓﻲ اﻟﻤﻌﺎدﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻓﻲ wt −12وأﺧﺬ اﻟﺘﻮﻗﻊ ﻧﺠﺪ 2
γ 12 = Φ γ 0 − Θσ + θ Θ ( −θ ) σ 2
) =Φ γ 0 − Θσ 2 (1 + θ 2
وﺑﺤﻞ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ 1 + Θ − 2ΦΘ 1 − Φ2 2 ⎡ ⎤ ) Φ (Θ − Φ 2 2 = σ (1 + θ ) ⎢Φ − Θ + ⎥ ⎦⎥ 1 − Φ 2 ⎣⎢ 2
أﻳﻀﺎ
) γ 0 = σ 2 (1 + θ 2 γ 12
) γ 1 = E ( wt wt −1 ) =Φ γ 11 − θσ 2 − ΘE ( at −12 wt −1 ) + θ ΘE ( at −13wt −1 =Φ γ 11 − θσ 2 + θ Θ ( Φ − Θ ) σ 2
و 2
وﺑﺤﻞ اﻟﻌﻼﻗﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ ﻧﺠﺪ
١٠٧
γ 11 = E ( wt wt −11 ) = Φγ 1 + Θθσ
⎡ ( Θ − Φ )2 ⎤ γ 1 = −θσ ⎢1 + ⎥ 1 − Φ 2 ⎦⎥ ⎣⎢ 2
⎡
γ 11 = θσ 2 ⎢ Θ − Φ − ⎣⎢
2 Φ (Θ − Φ ) ⎤ ⎥ 1 − Φ 2 ⎦⎥
وﺑﻨﻔﺲ اﻟﻄﺮﻳﻘﺔ ﻳﻤﻜﻦ إﺛﺒﺎت أن
γ 2 = γ 3 = L = γ 10 = 0 γ 13 = γ 11 γ k = Φγ k −12 , k > 13
وﻣﻦ ﺟﻤﻴﻊ اﻟﻌﻼﻗﺎت اﻟﺴﺎﺑﻘﺔ ﻧﻮﺟﺪ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ k =0 ⎧ 1, ⎪ θ ⎪ − , k =1 2 ⎪ 1+θ ⎪ 0, k = 2,...,10 ⎪ γ ⎪ θ ( Θ − Φ )(1 − ΦΘ ) ρk = k = ⎨ , k = 11 γ 0 ⎪1 + θ 2 1 + Θ2 − 2ΦΘ ⎪ ( Θ − Φ )(1 − ΦΘ ) k = 12 , ⎪− 2 1 + Θ − 2 ΦΘ ⎪ k = 13 ⎪ ρ11 , ⎪ Φρ , k > 13 k −12 ⎩
:دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ
wt = (1 − ΘB s ) at SARIMA(٠,d,٠)(٠,D,١)s ﻧﻤﻮذج-١
k =0 ⎧ 1, ⎪⎪ Θ , k=s ρk = ⎨− 2 1 + Θ ⎪ otherwise ⎪⎩ 0,
(1 − Φ B ) w s
t
= at
SARIMA(٠,d,٠)(١,D,١)s ﻧﻤﻮذج-٢
k =0 ⎧1, ⎪ ks ρk = ⎨Φ , k = s, 2 s,... ⎪ 0, otherwise ⎩
wt = (1 − θ B ) (1 − ΘB s ) at
١٠٨
SARIMA(٠,d,١)(٠,D,١)s ﻧﻤﻮذج-٣
k =0 ⎧ 1, ⎪ θ ⎪− , k =1 2 ⎪ 1+θ ⎪ θΘ , k = s −1 ⎪⎪ 2 2 ρ k = ⎨ (1 + θ )(1 + Θ ) ⎪ Θ ⎪− , k=s 2 ⎪ 1+ Θ ⎪ ρ s−1 , k = s +1 ⎪ otherwise ⎪⎩ 0,
(1 − ΦB ) w = (1 − ΘB ) a s
s
t
t
k =0 ⎧ 1, ⎪ ⎪ ( Θ − Φ )(1 − ΦΘ ) k s−1 ρk = ⎨− Φ , k = s, 2 s,... 2 ⎪ 1 + Θ − 2ΦΘ otherwise ⎪⎩ 0, (1 − ΦB s ) wt = (1 − θ B ) at
SARIMA(٠,d,٠)(١,D,١)s ﻧﻤﻮذج-٤
SARIMA(٠,d,١)(١,D,٠)s ﻧﻤﻮذج-٥
k =0 ⎧ 1, ⎪ θ ⎪− , k =1 2 ⎪ 1+θ ⎪ 0, k = 2,..., s − 2 ⎪ ρk = ⎨ θ Φ − , k = s −1 ⎪ 1+θ 2 ⎪ k=s ⎪ Φ, ⎪ ρ s−1 , k = s +1 ⎪ k > s +1 ⎪⎩ Φ ρ k − s ,
wt = (1 − θ1B − θ 2 B 2 )(1 − ΘB12 ) at
١٠٩
SARIMA(٠,d,٢)(٠,D,١)s ﻧﻤﻮذج-٦
k =0 k =1 k =2 k = s−2 k = s −1 k=s k = s +1 k = s+2 otherwise
⎧ 1, ⎪ ⎪ − θ1 (1 − θ 2 ) , ⎪ 1 + θ12 + θ 22 ⎪ θ2 ⎪− , ⎪ 1 + θ12 + θ 22 ⎪ θ 2Θ ⎪ , 2 ) ⎪⎪ (1 + θ1 + θ 22 )(1 + Θ2 ⎨ = ρk ) θ1Θ (1 − θ 2 ⎪ , ) ⎪ (1 + θ 2 + θ 2 )(1 + Θ2 1 2 ⎪ ⎪ Θ , ⎪− 2 + Θ 1 ⎪ ⎪ ρ s −1 , ⎪ ρ , ⎪ s −2 ⎪⎩ 0,
ﺳﻮف ﻧﺴﺘﻌﺮض ﺑﻌﺾ اﻟﺮﺳﻮﻣﺎت ﻟﺪاﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻷﻋﻄﺎء ﻓﻜﺮة ﻋﻦ أﺷﻜﺎﻟﻬﺎ. اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﻟﻨﻤﻮذج : SARIMA(٠,d,١)(١,D,٠)١٢ ﺷﻜﻞ )(١
Φ = 0.6, θ = 0.5 A ( 0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
o f S A R I M
A C F 0 .5
C1
0 .0
-0 .5
5 0
4 0
2 0
3 0
0
1 0
L a g
ﺷﻜﻞ )( ٢
Φ = 0.6, θ = −0.5 A ( 0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
o f S A R I M
A C F 0 .7 0 .6 0 .5
0 .3 0 .2 0 .1 0 .0
5 0
4 0
2 0
3 0
L a g
١١٠
1 0
0
C1
0 .4
ﺷﻜﻞ )(٣
Φ = −0.6, θ = 0.5 A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2 0 .5
C1
0 .0
-0 .5
50
40
20
30
10
0
Lag
ﺷﻜﻞ )(٤
Φ = −0.6, θ = −0.5 A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2
0 .5
C1
0 .0
-0 .5
5 0
4 0
2 0
3 0
1 0
0
L ag
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﻮذج اﻟﻤﻮﺳﻤﻲ اﻟﺘﻀﺎﻋﻔﻲ: ﻣﻦ اﻟﺼﻌﻮﺑﺔ إﺷﺘﻘﺎق وﺗﻔﺴﻴﺮ أﻧﻤﺎط داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻨﻤﺎذج اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ وﻟﻜﻨﻬﺎ وﺑﺸﻜﻞ ﻋﺎم ﻓﺈن أﺟﺰاء اﻟﻨﻤﻮذج اﻟﻤﻮﺳﻤﻴﺔ وﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ واﻟﺘﻲ ﺗﻨﻤﺬج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﺗﻌﻄﻲ ﺗﺨﺎﻣﺪات اﺳﻴﺔ وﺗﺨﺎﻣﺪات ﺟﻴﺒﻴﺔ ﻋﻨﺪ اﻟﺘﺨﻠﻔﺎت اﻟﻤﻮﺳﻤﻴﺔ وﻏﻴﺮاﻟﻤﻮﺳﻤﻴﺔ وﻓﻲ اﻟﻨﻤﺎذج اﻟﺘﻲ ﺗﺤﻮي إﻧﺤﺪار ذاﺗﻲ ﻓﺈن اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﺗﻌﻄﻲ ﻗﻄﻌﺎ . cut off اﻷﺷﻜﺎل اﻟﺘﺎﻟﻴﺔ ﻹﻋﻄﺎء ﻓﻜﺮة ﻋﻦ ﺑﻌﺾ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج : -١ﺷﻜﻞ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج wt = (1 − ΘB12 ) at ا( Θ = 0.6
١١١
A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2 0 .0 -0 .1
C1
-0 .2 -0 .3 -0 .4 -0 .5 -0 .6
0
1 0
2 0
3 0
4 0
5 0
L ag
Θ = −0.6 (ب A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2 0 .6 0 .5 0 .4
C1
0 .3 0 .2 0 .1 0 .0 -0 .1 -0 .2 0
1 0
2 0
3 0
4 0
5 0
L ag
(1 − Φ B ) w 12
t
= at ﺷﻜﻞ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻨﻤﻮذج-٢
Φ = 0.6 (ا A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2 0 .6 0 .5
C1
0 .4 0 .3 0 .2 0 .1 0 .0
0
1 0
2 0
3 0
4 0
5 0
L a g
Φ = −0.6 (ب
١١٢
A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2 0 .0 -0 .1 -0 .2
C1
-0 .3 -0 .4 -0 .5 -0 .6
5 0
4 0
2 0
3 0
0
1 0
L ag
ﺃﻤﺜﻠﺔ :ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﺍﻟﺯﻤﻨﻴﺔ ﺍﻟﻤﻭﺴﻤﻴﺔ )ﻓﻲ ﺠﻤﻴﻊ ﺍﻷﻤﺜﻠﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺇﻗﺭﺃ ﺴﻁﺭﺍ ﺒﺴﻁﺭ( 54.9 58.2 68.2 54.9 53.2 57.3 70.4 53.4 51.7 57.1 75.5
54.9 54.3 70.1 57.5 52.8 52.7 69.9 55.3 51.5 53.9 73.3
55.3 53.4 67.5 61.2 52.8 51.6 61.0 58.2 52.3 53.5 68.1 62.2
56.9 53.0 58.0 69.3 54.5 52.4 52.7 66.9 53.6 53.5 58.1 74.8
57.4 52.8 54.2 69.8 56.4 52.1 53.9 70.7 55.3 53.1 54.8 76.4
61.5 53.3 53.2 66.1 59.3 52.6 53.3 65.3 58.5 53.3 54.3 70.8
72.7 54.4 53.0 56.1 68.7 53.9 53.5 56.5 69.3 53.9 54.6 60.6
72.2 56.0 53.0 53.6 70.0 56.4 53.4 53.4 69.6 55.6 54.2 56.4
71.5 60.0 53.4 53.0 67.9 61.7 53.6 52.5 64.2 60.1 54.8 55.6
59.1 71.0 54.6 52.8 58.7 68.3 55.1 53.2 55.5 68.9 55.8 55.0
57.2 70.6 55.6 52.6 55.4 67.9 57.3 53.0 53.3 68.8 57.9 54.7
56.3 68.2 59.4 52.9 52.9 65.3 61.9 53.5 52.4 63.6 62.6 55.8
55.8 57.7 69.8 54.0 53.0 58.2 69.9 54.3 51.5 57.1 70.3 57.7
55.7 54.6 71.0 54.9 52.7 55.8 71.5 56.3 51.5 52.2 69.4 60.5
ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
7 0
)z(t
6 0
5 0
1 5 0
1 0 0
داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
١١٣
5 0
In d e x
)z(t 56.3 54.9 67.4 56.6 53.4 55.3 65.1 59.4 52.1 51.5 64.7 66.4
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
2 Lag 1 2 3 4 5 6 7 8 9 10 11 12
C o rr
12
T
LBQ
Lag
C o rr
0 .7 7 1 0 .3 1 2 .9 3 0 .3 3 -0 .1 1 -0 .9 9 -0 .3 8 -3 .2 8 -0 .5 2 -4 .2 2 -0 .5 6 -4 .1 6 -0 .5 2 -3 .5 0 -0 .3 8 -2 .3 9 -0 .1 2 -0 .7 1 1 .7 6 0 .2 9 4 .1 5 0 .6 9 4 .8 6 0 .8 8
1 0 8 .1 0 1 2 7 .3 4 1 2 9 .7 4 1 5 6 .8 3 2 0 7 .2 4 2 6 6 .5 5 3 1 6 .8 0 3 4 3 .7 1 3 4 6 .2 5 3 6 2 .0 8 4 5 3 .5 8 6 0 3 .8 7
13 14 15 16 17 18 19 20 21 22 23 24
0 .6 8 0 .2 8 -0 .1 2 -0 .3 6 -0 .4 9 -0 .5 3 -0 .4 9 -0 .3 7 -0 .1 4 0 .2 3 0 .6 1 0 .7 9
22 T
32
42
LBQ
Lag
C o rr
T
LBQ
Lag
C o rr
T
LBQ
3 .3 4 6 9 4 .2 8 1 .3 0 7 0 9 .8 2 -0 .5 3 7 1 2 .4 3 -1 .6 5 7 3 8 .3 0 -2 .2 1 7 8 6 .2 8 -2 .3 3 8 4 3 .0 4 -2 .1 0 8 9 2 .3 6 -1 .5 4 9 2 0 .3 9 -0 .5 6 9 2 4 .2 4 0 .9 5 9 3 5 .3 7 2 .4 6 1 0 1 1 .3 1 3 .1 0 1 1 4 0 .9 0
25 26 27 28 29 30 31 32 33 34 35 36
0 .6 1 0 .2 5 -0 .1 1 -0 .3 4 -0 .4 7 -0 .5 1 -0 .4 8 -0 .3 6 -0 .1 4 0 .1 9 0 .5 4 0 .7 2
2 .3 0 0 .9 1 -0 .4 2 -1 .2 4 -1 .6 7 -1 .8 0 -1 .6 5 -1 .2 3 -0 .4 9 0 .6 6 1 .8 2 2 .3 7
1 2 2 0 .0 7 1 2 3 3 .1 8 1 2 3 5 .9 8 1 2 6 1 .2 9 1 3 0 8 .3 2 1 3 6 4 .7 2 1 4 1 4 .0 8 1 4 4 2 .7 1 1 4 4 7 .3 4 1 4 5 5 .7 9 1 5 2 0 .8 9 1 6 3 6 .4 9
37 38 39 40 41 42 43 44
0 .5 7 0 .2 3 -0 .1 1 -0 .3 3 -0 .4 4 -0 .4 8 -0 .4 5 -0 .3 4
1 .8 2 0 .7 3 -0 .3 4 -1 .0 2 -1 .3 8 -1 .4 8 -1 .3 7 -1 .0 4
1 7 0 9 .3 6 1 7 2 1 .7 0 1 7 2 4 .4 3 1 7 4 9 .0 6 1 7 9 5 .0 0 1 8 4 9 .4 1 1 8 9 7 .0 5 1 9 2 5 .3 0
واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
2
12
22
32
42
Lag
P AC
T
Lag
P AC
T
Lag
P AC
T
Lag
P AC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .7 7 -0 .6 8 -0 .0 6 0 .0 1 -0 .4 2 -0 .1 8 -0 .1 1 -0 .2 2 0 .1 8 0 .5 7 0 .2 6 0 .1 6
1 0 .3 1 - 9 .0 1 - 0 .8 2 0 .0 7 - 5 .5 5 - 2 .4 5 - 1 .4 8 - 2 .9 6 2 .3 9 7 .6 5 3 .5 0 2 .1 6
13 14 15 16 17 18 19 20 21 22 23 24
- 0 .4 2 0 .2 9 0 .0 1 - 0 .0 7 0 .0 7 - 0 .0 1 - 0 .0 9 - 0 .0 5 - 0 .0 8 0 .1 0 - 0 .0 3 0 .0 3
- 5 .6 2 3 .8 9 0 .1 6 - 0 .9 2 0 .9 5 - 0 .1 1 - 1 .1 5 - 0 .7 2 - 1 .1 1 1 .3 8 - 0 .3 5 0 .3 7
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 3 - 0 .0 0 0 .0 4 - 0 .1 0 - 0 .0 3 0 .0 1 0 .0 1 - 0 .0 4 - 0 .0 4 0 .0 1 - 0 .0 2 0 .0 7
-1 .7 5 -0 .0 5 0 .5 0 -1 .3 0 -0 .3 4 0 .1 9 0 .0 7 -0 .5 5 -0 .5 2 0 .1 5 -0 .3 2 0 .9 0
37 38 39 40 41 42 43 44
-0 .0 6 -0 .0 5 0 .0 1 0 .0 0 -0 .0 5 0 .0 5 -0 .0 5 -0 .0 3
-0 .7 4 -0 .6 2 0 .1 7 0 .0 0 -0 .6 3 0 .6 6 -0 .6 4 -0 .4 2
.ﻧﻼﺣﻆ اﻷﻧﻤﺎط اﻟﻤﻮﺳﻤﻴﺔ واﺿﺤﺔ ﻓﻲ اﻷﺷﻜﺎل اﻟﺴﺎﺑﻘﺔ ﻤﺜﺎل ﺁﺨﺭ z(t) 589 673 678 621 713 796 801 747 826 898 908 827
561 742 639 602 667 858 764 711 799 957 867 797
640 716 604 635 762 826 725 751 890 924 815 843
656 660 611 677 784 783 723 804 900 881 812
727 617 594 635 837 740 690 756 961 837 773
697 583 634 736 817 701 734 860 935 784 813
640 587 658 755 767 706 750 878 894 791 834
599 565 622 811 722 677 707 942 855 760 782
568 598 709 798 681 711 807 913 809 802 892
577 628 722 735 687 734 824 869 810 828 903
553 618 782 697 660 690 886 834 766 778 966
582 688 756 661 698 785 859 790 805 889 937
600 705 702 667 717 805 819 800 821 902 896
566 770 653 645 696 871 783 763 773 969 858
653 736 615 688 775 845 740 800 883 947 817
ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ
١١٤
1000
z(t)
900
800
700
600
50
In d e x
100
150
اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
2 Lag 1 2 3 4 5 6 7 8 9 10 11 12
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.8 .7 .6 .4 .4 .3 .4 .4 .5 .6 .7 .8
T
12 LB Q
9 1 1 .5 6 1 3 5 .9 4 6 .2 7 2 4 0 .1 3 8 4 .1 2 3 0 6 .7 2 2 2 .9 5 3 4 7 .9 7 9 2 .4 7 3 8 0 .0 9 3 2 .1 0 4 0 5 .0 2 8 2 .2 5 4 3 5 .5 4 1 2 .4 0 4 7 2 .3 7 5 2 .8 7 5 2 9 .0 7 6 3 .3 4 6 1 4 .2 7 9 3 .5 2 7 2 1 .7 2 7 3 .6 1 8 5 2 .4 1 4
L ag 1 1 1 1 1 1 1 2 2 2 2 2
3 4 5 6 7 8 9 0 1 2 3 4
22
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.7 .6 .4 .3 .3 .2 .2 .3 .4 .5 .6 .6
4 4 9 6 1 5 9 2 2 3 0 7
T 2 2 1 1 1 0 1 1 1 1 2 2
.9 .4 .7 .3 .0 .9 .0 .1 .4 .8 .0 .2
6 11 91 11 91 01 11 21 41 11 31 11
LB Q 9 0 0 0 1 1 1 1 1 2 3 4
5 3 7 9 1 2 4 6 9 5 2 1
4 0 4 9 7 9 5 5 9 3 5 5
.6 .0 .8 .6 .3 .7 .5 .4 .1 .8 .5 .2
8 9 5 8 9 6 9 2 3 1 1 9
L ag 2 2 2 2 2 3 3 3 3 3 3 3
5 6 7 8 9 0 1 2 3 4 5 6
32
C o rr 0 0 0 0 0 0 0 0 0 0 0 0
.5 .4 .3 .2 .1 .1 .1 .2 .2 .3 .4 .5
8 9 5 4 9 4 7 0 8 8 5 2
T 1 1 1 0 0 0 0 0 0 1 1 1
.8 .5 .0 .7 .5 .4 .5 .6 .8 .1 .3 .5
61 21 91 31 71 31 11 01 51 51 51 31
LB Q 4 5 5 5 5 5 5 5 6 6 6 7
8 3 5 6 7 7 8 9 1 4 8 4
3 0 6 8 5 9 5 3 0 1 5 2
.1 .9 .4 .2 .4 .5 .5 .6 .3 .5 .2 .9
1 9 5 0 4 5 2 2 5 7 7 5
L ag 3 3 3 4 4 4
7 8 9 0 1 2
42 C o rr 0 0 0 0 0 0
.4 .3 .2 .1 .0 .0
3 5 2 2 6 2
T 1 1 0 0 0 0
.2 .0 .6 .3 .1 .0
71 01 41 31 81 51
LB Q 7 8 8 8 8 8
8 1 2 2 2 2
4 0 1 4 5 5
.1 .6 .6 .6 .5 .6
7 0 8 7 6 2
واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
2
12
22
32
42
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .8 9 -0 .0 8 -0 .2 8 0 .0 3 0 .3 5 -0 .0 8 0 .2 8 0 .0 9 0 .4 0 0 .3 0 0 .0 6 0 .2 2
1 1 .5 6 -1 .0 6 -3 .6 5 0 .4 2 4 .5 4 -1 .0 7 3 .6 7 1 .1 9 5 .1 7 3 .9 5 0 .8 1 2 .8 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .6 3 -0 .0 2 0 .0 7 -0 .0 4 -0 .0 9 -0 .0 4 -0 .0 5 0 .0 3 0 .0 4 0 .0 5 0 .0 5 0 .0 5
-8 .1 9 -0 .2 1 0 .9 5 -0 .5 2 -1 .1 2 -0 .4 9 -0 .6 0 0 .3 8 0 .4 6 0 .6 7 0 .6 0 0 .5 9
25 26 27 28 29 30 31 32 33 34 35 36
-0 .1 8 0 .0 8 0 .0 6 -0 .0 3 -0 .0 4 0 .0 0 -0 .0 6 -0 .0 1 -0 .0 1 0 .0 3 0 .0 0 0 .0 1
-2 .3 6 1 .0 6 0 .7 3 -0 .4 4 -0 .4 7 0 .0 2 -0 .7 2 -0 .1 1 -0 .1 8 0 .3 8 0 .0 3 0 .0 9
37 38 39 40 41 42
-0 .1 1 -0 .0 2 0 .0 4 -0 .0 3 -0 .0 8 0 .0 1
-1 .3 7 -0 .2 2 0 .5 1 -0 .4 2 -1 .0 6 0 .0 8
١١٥
ﻣﺜﺎل ﺁﺧﺮ z(t) 302 107 055 237 079 035 256
262 056 048 247 045 056 250
218 049 115 215 040 097 198
175 047 185 182 038 210 136
100 047 276 080 041 260 073
077 071 220 046 069 257 039
043 151 181 065 152 210 032
047 244 151 040 232 125 030
049 280 083 044 282 080 031
069 230 055 063 255 042 045
152 185 049 085 161 035
205 148 042 185 107 031
246 098 046 247 053 032
294 061 074 231 040 050
242 046 103 167 039 092
181 045 200 117 034 189
وﻟﻬﺎ اﻟﺸﻜﻞ
3 0 0
z(t)
2 0 0
1 0 0
0
1 0
In d e x
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
وداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t ) 1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
5
1 2 3 4 5 6 7
C o rr 0 0 -0 -0 -0 -0 -0
.8 .4 .0 .4 .6 .7 .6
1 3 3 3 9 8 8
T 8 2 -0 -2 -4 -3 -3
.3 .8 .1 .7 .0 .9 .0
8 9 7 0 3 9 9
L B Q
1 1 2 2
7 9 9 1 6 3 8
2 2 2 3 6 5 9
.2 .3 .4 .1 .3 .1 .3
L a g
2 2 0 3 7 7 3
1 1 1 1 1
C o rr
T
8 -0 .4 3 -1 .7 8 3 1 9 -0 .0 4 -0 .1 8 3 1 0 0 .3 8 1 .5 4 3 2 1 0 .7 1 2 .8 2 3 8 2 0 .8 4 3 .1 0 4 7 3 0 .7 1 2 .4 2 5 3 4 0 .3 7 1 .2 0 5 5
L B Q 0 0 8 9 6 8 6
.6 .8 .2 .7 .2 .8 .1
L a g
6 8 2 2 2 4 0
1 1 1 1 1 2 2
5 6 7 8 9 0 1
2 5
C o rr -0 -0 -0 -0 -0 -0 -0
.0 .3 .6 .6 .5 .3 .0
3 8 0 7 9 7 4
T -0 -1 -1 -2 -1 -1 -0
.0 .1 .8 .0 .7 .0 .1
8 9 8 3 2 5 0
L B Q 5 5 6 6 7 7 7
5 7 2 7 2 4 4
6 4 0 8 4 2 2
.1 .0 .2 .3 .1 .2 .3
8 7 2 6 7 1 8
L a g 2 2 2 2 2
2 3 4 5 6
C o rr 0 0 0 0 0
.3 .6 .7 .5 .3
3 1 1 9 1
T 0 1 1 1 0
.9 .7 .9 .5 .8
2 1 3 5 0
L B Q 7 8 8 9 9
5 0 7 2 4
6 8 9 8 2
.9 .9 .4 .4 .4
8 7 2 1 6
وداﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t ) Partial Autocorrelation
L a g
1 5
1 0 0 0 0 0 -0 -0 -0 -0 -1
.0 .8 .6 .4 .2 .0 .2 .4 .6 .8 .0
5
L a g 1 2 3 4 5 6 7
P A C 0 -0 -0 -0 -0 -0 -0
.8 .7 .2 .2 .1 .3 .1
1 0 9 2 6 2 6
T 8 -7 -2 -2 -1 -3 -1
.3 .1 .9 .2 .6 .2 .6
8 7 8 8 1 9 0
1 5
L a g
1 1 1 1 1
P A C
8 -0 .0 3 9 0 .1 9 0 0 .2 7 1 0 .1 8 2 0 .0 7 3 -0 .1 0 4 -0 .1 5
-0 1 2 1 0 -0 -1
.3 .9 .7 .8 .7 .9 .5
T
L a g
1 1 6 2 4 9 2
1 1 1 1 1 2 2
١١٦
2 5
P A C
0 .1 5 0 .0 6 0 .1 7 0 .0 8 9 -0 .0 0 -0 .0 0 .0 1
9 8 1 4 2 3 4
1 0 1 0 -0 -0 0
.9 .8 .1 .3 .2 .2 .4
T
L a g
3 4 0 9 2 9 6
2 2 2 2 2
P A C
2 0 .0 3 0 .0 4 -0 .0 5 -0 .0 6 0 .1
1 7 7 2 0
T 0 0 -0 -0 1
.1 .7 .7 .2 .0
5 0 2 3 1
وآﻞ هﺬﻩ اﻟﻤﺘﻠﺴﻼت ﺗﺒﺪي اﻧﻤﺎط ﻣﻮﺳﻤﻴﺔ واﺿﺤﺔ.
إﺷﺘﻘﺎق دوال ﺗﻨﺒﺆ ﻟﺒﻌﺾ ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ اﻟﺘﻀﺎﻋﻔﻴﺔ: ﺑﻤﺎ ان ﻧﻤﺎذج اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ هﻲ ﺣﺎﻟﺔ ﺧﺎﺻﺔ ﻣﻦ ﻧﻤﺎذج ARIMAﻓﺈن ﻃﺮق اﻟﺘﻌﺎﻣﻞ ﻣﻌﻬﺎ هﻲ ﻧﻔﺲ اﻟﻄﺮق اﻟﺴﺎﺑﻘﺔ ﻣﻦ ﺣﻴﺚ اﻟﺘﻌﺮف ﻋﻠﻲ ﺷﻜﻞ اﻟﻨﻤﻮذج وﺗﻘﺪﻳﺮ ﻣﻌﺎﻟﻢ اﻟﻨﻤﻮذج واﻹﺧﺘﺒﺎرات اﻟﺘﻔﺤﺼﻴﺔ وﻣﻦ ﺛﻢ اﻟﺘﻨﺒﺆ .ﺟﻤﻴﻊ اﻟﻄﺮق واﻟﻤﻌﺎدﻻت اﻟﺘﻲ درﺳﻨﺎهﺎ ﺳﺎﺑﻘﺎ ﻟﻠﻨﻤﺎذج ﻏﻴﺮ اﻟﻤﻮﺳﻤﻴﺔ ﺗﻨﻄﺒﻖ هﻨﺎ .ﺳﻮف ﻧﺸﺘﻖ دوال اﻟﺘﻨﺒﺆ ﻟﺒﻌﺾ اﻟﻨﻤﺎذج ﻟﻠﺘﻮﺿﻴﺢ ﻓﻘﻂ. -١داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج : SARIMA(٠،٠،٠)(٠،١،١)١٢ وﻳﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞ
(1 − B ) z = (1 − ΘB ) a 12
t
12
t
ﻣﻦ اﻟﻤﻌﺎدﻟﺔ اﻟﻔﺮوﻗﻴﺔ zn + l = zn +l−12 + an + l − Θan + l−12
ﻳﻤﻜﻦ اﻟﺤﺼﻮل ﻋﻠﻰ اﻟﺘﻨﺒﺆات آﺎﻟﺘﺎﻟﻲ
zn (1) = zn −11 − Θan −11 zn ( 2 ) = zn −10 − Θan −10
M
zn (12 ) = zn − Θan zn ( l ) = zn ( l − 12 ) , l ≥ 12
أو l = 1, 2,...,12
l > 12
واﺿﺢ أن
⎧ zn + l−12 − Θan + l−12 , ⎨ = ) zn ( l ⎩ zn ( l − 12 ) , zn (1) = zn (13) = zn ( 25) = L
zn ( 2 ) = zn (14 ) = zn ( 26 ) = L
M
zn (12 ) = zn ( 24 ) = zn ( 36 ) = L
ﺗﺒﺎﻳﻦ أﺧﻄﺎء اﻟﺘﻨﺒﺆ
)
2 l −1
+ L +ψ
2 1
(1 + ψ
2
V ⎣⎡ en ( l )⎦⎤ = σ
وداﻟﺔ اﻷوزان ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ )ﺑﺮهﻦ ذﻟﻚ( j = 12, 24,... otherwise
⎧1 − Θ, ⎩ 0,
⎨= ψj
وﺑﺘﻌﻮﻳﺾ اﻷوزان ﻓﻲ ﺻﻴﻐﺔ ﺗﺒﺎﻳﻦ أﺧﻄﺎء اﻟﺘﻨﺒﺆ ﻧﺠﺪ ⎥⎧ ⎢ l − 1 ⎫2 ⎢ V ⎡⎣ en ( l )⎤⎦ = σ 2 ⎨1 + ⎬ ) (1 − Θ ⎥ ⎦ ⎩ ⎣ 12 ⎭
ﺣﻴﺚ ⎦⎥ ⎢⎣ xﺗﻌﻨﻲ اﻟﺠﺰء اﻟﺼﺤﻴﺢ ﻣﻦ . x -٢داﻟﺔ اﻟﺘﻨﺒﺆ ﻟﻠﻨﻤﻮذج : SARIMA(٠،١،١)(٠،١،١)١٢ وﻳﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞ
(1 − B ) (1 − B12 ) zt = (1 − θ B ) (1 − ΘB12 ) at
١١٧
ﻣﻦ اﻟﻤﻌﺎدﻟﺔ اﻟﻔﺮوﻗﻴﺔ
zn + l = zn +l−1 + zn + l−12 − zn + l−13 + an + l − θ an +l−1 − Θan + l−12 + θ Θan+ l−13
ﻳﻤﻜﻦ اﻟﺤﺼﻮل ﻋﻠﻰ اﻟﺘﻨﺒﺆات آﺎﻟﺘﺎﻟﻲ
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12 zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11 M
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1 zn (13) = zn (12 ) + zn (1) − zn + θ Θan )zn ( l ) = zn ( l − 1) + zn ( l − 12 ) − zn ( l − 13
وهﻜﺬا ﺑﻘﻴﻢ أوﻟﻴﺔ
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12 zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11
M
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1 zn (13) = zn (12 ) + zn (1) − zn + θ Θan
وﻋﻼﻗﺔ ﺗﻜﺮارﻳﺔ
zn ( l ) = zn ( l − 1) + zn ( l − 12 ) − zn ( l − 13) , l > 13
ﻳﻤﻜﻦ ﺗﻮﻟﻴﺪ اﻟﻌﺪد اﻟﻤﻄﻠﻮب ﻣﻦ اﻟﺘﻨﺒﺆات. أﻣﺜﻠﺔ وﺣﺎﻻت دراﺳﺔ ﻟﺒﻌﺾ اﻟﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﻤﻮﺳﻤﻴﺔ: ﻣﺜﺎل ) : (١ﺳﻮف ﻧﺤﺎول إﻳﺠﺎد ﻧﻤﻮذج ﻣﻦ ﻋﺎﺋﻠﺔ SARIMAﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ: 653 736 615 688 775 845 740 800 883 947 817
566 770 653 645 696 871 783 763 773 969 858
600 705 702 667 717 805 819 800 821 902 896
582 688 756 661 698 785 859 790 805 889 937
553 618 782 697 660 690 886 834 766 778 966
577 628 722 735 687 734 824 869 810 828 903
568 598 709 798 681 711 807 913 809 802 892
599 565 622 811 722 677 707 942 855 760 782
واﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻠﻤﺸﺎهﺪات
١١٨
640 587 658 755 767 706 750 878 894 791 834
697 583 634 736 817 701 734 860 935 784 813
727 617 594 635 837 740 690 756 961 837 773
656 660 611 677 784 783 723 804 900 881 812
640 716 604 635 762 826 725 751 890 924 815 843
561 742 639 602 667 858 764 711 799 957 867 797
)z(t 589 673 678 621 713 796 801 747 826 898 908 827
1000
900
)z(t
800
700
600
150
50
100
In d e x
ﻳﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ ﻟﺬﻟﻚ ﻧﺜﺒﺖ اﻟﺘﺒﺎﻳﻦ أوﻻ ﺑﺘﺤﻮﻳﻞ ﻟﻮﻏﺎرﺛﻤﻲ أي ) yt = ln ( ztوﻧﺮﺳﻢ اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ ﻟﻬﺎ 6 .9 6 .8 6 .7
)y(t
6 .6 6 .5 6 .4 6 .3
1 5 0
5 0
1 0 0
In d e x
ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﺳﺘﻘﺮت ﻓﻲ اﻟﺘﺒﺎﻳﻦ وﻟﻜﻦ ﻻﺗﺰال ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂ ﻟﺬﻟﻚ ﻧﺄﺧﺬ اﻟﻔﺮق اﻷول ) xt = (1 − B ) yt = (1 − B ) ln ( ztوﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ 0 .1 5 0 .1 0
0 .0 0
)y(t)-y(t-1
0 .0 5
-0 .0 5 -0 .1 0 1 5 0
1 0 0
5 0
In d e x
اﻟﻤﺘﺴﻠﺴﻠﺔ اﻵن ﻣﺴﺘﻘﺮة ﻓﻲ آﻞ ﻣﻦ اﻟﺘﺒﺎﻳﻦ واﻟﻤﺘﻮﺳﻂ .ﻟﻨﻨﻈﺮ إﻟﻰ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ
١١٩
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
10
20
30
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
0 .0 1 0 .2 5 - 0 .0 8 - 0 .3 7 - 0 .0 6 - 0 .5 0 - 0 .0 4 - 0 .3 5 - 0 .0 5 0 .2 3 0 .0 1 0 .9 0
0 .1 2 3 .2 3 - 1 .0 1 - 4 .4 8 - 0 .6 8 - 5 .4 1 - 0 .3 3 - 3 .3 0 - 0 .4 3 2 .0 1 0 .1 1 7 .7 4
0 .0 2 1 0 .7 0 1 1 .8 8 3 5 .6 0 3 6 .2 8 7 9 .9 7 8 0 .1 9 1 0 2 .3 7 1 0 2 .8 0 1 1 2 .2 9 1 1 2 .3 2 2 6 1 .4 1
13 14 15 16 17 18 19 20 21 22 23 24
0 .0 2 0 .2 3 - 0 .0 7 - 0 .3 4 - 0 .0 6 - 0 .4 6 - 0 .0 3 - 0 .3 2 - 0 .0 5 0 .2 1 0 .0 1 0 .8 2
0 .1 2 1 .5 1 - 0 .4 7 - 2 .1 8 - 0 .3 9 - 2 .8 8 - 0 .2 0 - 1 .9 3 - 0 .2 6 1 .2 0 0 .0 8 4 .7 6
2 6 1 .4 7 2 7 1 .2 9 2 7 2 .3 0 2 9 3 .7 3 2 9 4 .4 5 3 3 4 .6 7 3 3 4 .8 8 3 5 4 .9 6 3 5 5 .3 6 3 6 3 .5 5 3 6 3 .5 8 4 9 7 .3 0
25 26 27 28 29 30 31 32 33 34 35 36
0 .0 2 0 .2 1 - 0 .0 6 - 0 .3 1 - 0 .0 6 - 0 .4 2 - 0 .0 2 - 0 .3 0 - 0 .0 5 0 .1 8 0 .0 0 0 .7 6
0 .0 8 1 .1 0 - 0 .2 9 - 1 .5 7 - 0 .2 9 - 2 .0 9 - 0 .1 2 - 1 .4 7 - 0 .2 4 0 .8 9 0 .0 2 3 .6 3
4 9 7 .3 5 5 0 6 .5 0 5 0 7 .1 6 5 2 6 .4 5 5 2 7 .1 4 5 6 3 .0 5 5 6 3 .1 6 5 8 2 .0 5 5 8 2 .5 8 5 8 9 .8 4 5 8 9 .8 5 7 1 3 .1 4
40
L ag
T
LBQ
3 7 0 .0 1 0 .0 5 3 8 0 .2 0 0 .9 0 3 9 - 0 .0 4 - 0 .1 9 4 0 - 0 .2 7 - 1 .2 1 4 1 - 0 .0 5 - 0 .2 4
C o rr
7 1 3 .1 6 7 2 2 .0 6 7 2 2 .4 5 7 3 8 .8 6 7 3 9 .5 1
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t i o n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
10
20
30
40
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .0 1 0 .2 5 - 0 .0 9 - 0 .4 6 - 0 .0 2 - 0 .3 5 - 0 .1 4 - 0 .4 8 - 0 .4 1 - 0 .2 3 - 0 .5 8 0 .6 3
0 .1 2 3 .2 3 - 1 .2 0 - 5 .9 7 - 0 .2 2 - 4 .5 2 - 1 .8 2 - 6 .2 6 - 5 .3 6 - 2 .9 9 - 7 .5 5 8 .0 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 4 -0 .3 3 0 .0 0 0 .1 8 0 .0 1 0 .0 8 -0 .0 8 0 .0 8 0 .0 2 -0 .0 4 0 .0 2 0 .0 8
- 0 .4 8 - 4 .2 6 0 .0 6 2 .2 9 0 .0 7 1 .0 6 - 1 .0 4 0 .9 8 0 .2 9 - 0 .5 3 0 .2 9 1 .0 4
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 3 - 0 .0 5 0 .0 6 0 .0 0 - 0 .0 7 0 .0 3 0 .0 7 - 0 .0 1 - 0 .1 0 0 .0 6 0 .0 3 0 .0 1
-1 .6 7 -0 .6 1 0 .8 4 0 .0 5 -0 .9 4 0 .3 4 0 .9 3 -0 .0 9 -1 .3 1 0 .8 0 0 .3 9 0 .1 3
37 38 39 40 41
- 0 .0 4 0 .0 2 0 .0 4 0 .0 6 - 0 .0 2
- 0 .5 1 0 .3 0 0 .4 8 0 .8 1 - 0 .2 0
و١٢ ﻳﻼﺣﻆ ﻣﻦ داﻟﺔ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ان اﻟﻤﺘﺴﻠﺴﻠﺔ ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻣﻮﺳﻤﻴﺎ ﻷن ﻗﻴﻤﻬﺎ ﻋﻨﺪ اﻟﺘﺨﻠﻔﺎت wt = (1 − B12 ) (1 − B ) ln ( zt ) ﺗﺘﺨﺎﻣﺪ ﺑﺒﻂء ﻟﺬﻟﻚ ﻧﺄﺧﺬ اﻟﻔﺮق اﻟﻤﻮﺳﻤﻲ اﻷول٣٦ و٢٤ وﻧﺮﺳﻤﻬﺎ ﺑﻌﺪ هﺬا اﻟﺘﻔﺮﻳﻖ
y(t)-y(t-1)12
0 .0 5
0 .0 0
-0 .0 5 In d e x
50
100
150
ﻧﻮﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻬﺎ
١٢٠
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
5
15
25
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .2 1 - 0 .0 1 0 .1 0 - 0 .1 3 - 0 .1 0 - 0 .0 2 0 .1 2 0 .0 5 - 0 .0 5 0 .1 3 - 0 .0 1 - 0 .4 4
- 2 .6 5 - 0 .1 3 1 .1 4 - 1 .4 9 - 1 .1 1 - 0 .2 8 1 .3 3 0 .5 4 - 0 .5 8 1 .4 5 - 0 .0 9 - 4 .9 0
7 .1 5 7 .1 7 8 .6 3 1 1 .2 0 1 2 .6 7 1 2 .7 7 1 4 .9 5 1 5 .3 1 1 5 .7 5 1 8 .4 9 1 8 .5 0 5 0 .8 8
13 14 15 16 17 18 19 20 21 22 23 24
0 .1 8 - 0 .0 7 - 0 .0 5 0 .0 3 0 .1 2 - 0 .0 0 - 0 .1 1 0 .0 3 - 0 .0 2 - 0 .0 9 0 .1 1 - 0 .0 4
1 .8 0 - 0 .7 0 - 0 .4 8 0 .2 6 1 .1 3 - 0 .0 2 - 1 .0 8 0 .2 5 - 0 .2 0 - 0 .8 0 0 .9 9 - 0 .3 8
5 6 .6 4 5 7 .5 4 5 7 .9 7 5 8 .1 0 6 0 .5 8 6 0 .5 8 6 2 .9 1 6 3 .0 4 6 3 .1 2 6 4 .4 5 6 6 .5 2 6 6 .8 2
25 26 27 28 29 30 31 32 33 34 35 36
0 .0 7 - 0 .0 0 - 0 .0 6 0 .0 3 - 0 .1 0 0 .0 1 0 .0 4 0 .0 0 0 .0 2 0 .0 0 0 .0 9 - 0 .0 6
0 .6 2 - 0 .0 2 - 0 .5 7 0 .3 0 - 0 .9 0 0 .0 8 0 .3 8 0 .0 4 0 .1 9 0 .0 3 0 .8 0 - 0 .5 1
6 7 .6 7 6 7 .6 7 6 8 .3 8 6 8 .5 8 7 0 .4 0 7 0 .4 1 7 0 .7 4 7 0 .7 4 7 0 .8 3 7 0 .8 3 7 2 .3 6 7 2 .9 9
35 L ag
T
LBQ
3 7 - 0 .0 6 - 0 .5 4 3 8 0 .0 3 0 .2 6
C o rr
7 3 .7 1 7 3 .8 8
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5
15
25
35
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .2 1 - 0 .0 6 0 .0 8 - 0 .0 9 - 0 .1 5 - 0 .1 0 0 .1 1 0 .1 1 - 0 .0 4 0 .0 6 0 .0 5 - 0 .4 2
- 2 .6 5 - 0 .7 3 1 .0 6 - 1 .1 6 - 1 .8 2 - 1 .1 9 1 .4 1 1 .4 2 - 0 .5 1 0 .8 0 0 .5 9 - 5 .2 3
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 0 -0 .0 2 0 .0 0 -0 .1 1 0 .0 4 0 .0 1 -0 .0 4 -0 .0 3 -0 .0 3 0 .0 3 0 .0 7 -0 .2 7
- 0 .0 3 - 0 .2 5 0 .0 0 - 1 .4 0 0 .4 7 0 .1 2 - 0 .5 2 - 0 .3 5 - 0 .3 1 0 .3 2 0 .9 1 - 3 .4 1
25 26 27 28 29 30 31 32 33 34 35 36
0 .1 2 - 0 .0 1 - 0 .1 2 - 0 .0 7 0 .0 4 - 0 .0 4 - 0 .0 5 0 .0 2 - 0 .0 1 - 0 .0 2 0 .1 8 - 0 .2 1
1 .5 1 -0 .1 5 -1 .4 4 -0 .9 1 0 .4 4 -0 .5 3 -0 .6 0 0 .1 9 -0 .1 2 -0 .1 9 2 .2 1 -2 .5 7
37 38
0 .1 0 - 0 .0 6
1 .2 3 - 0 .7 9
wt = (1 − B12 ) (1 − B ) ln ( zt ) ﻣﻦ اﻧﻤﺎط دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ
اي ﻧﻄﺒﻖ اﻟﻨﻤﻮذج٠ و٠ اﻟﻤﻤﻜﻨﺔ هﻲq وp ﻧﺠﺪ ان ﻗﻴﻢ
(1 − B ) (1 − B ) ln ( z ) = (1 − ΘB ) a 12
12
t
t
ﻳﻄﺒﻖ هﺬا اﻟﻨﻤﻮذجMINITAB اﻷﻣﺮ اﻟﺘﺎﻟﻲ ﻓﻲSARIMA(٠،١،٠)(٠،١،١)١٢ هﻮ ARIMA 0 1 ٠ 0 1 1 12 'y(t)' ; NoConstant. zt = e y وﻟﻠﺤﺼﻮل ﻋﻠﻰ اﻟﻨﺘﺎﺋﺞ اﻟﻨﻬﺎﺋﻴﺔ ﻧﺠﺮي اﻟﺘﺤﻮﻳﻞyt = ln ( zt ) ﻻﺣﻆ اﻧﻨﺎ اﺳﺘﺨﺪﻣﻨﺎ :اﻟﻨﺘﺎﺋﺞ t
MTB > Name c14 = 'RESI3' c15 = 'FITS3' MTB > ARIMA 0 1 0 0 1 1 12 'y(t)' 'RESI3' 'FITS3'; SUBC> NoConstant; SUBC> Forecast 24 c7 c8 c9; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model
١٢١
ARIMA model for y(t) Estimates at each iteration Iteration SSE Parameters 0 0.0228597 0.100 1 0.0204943 0.250 2 0.0187066 0.400 3 0.0174234 0.550 4 0.0169841 0.684 5 0.0169841 0.683 6 0.0169841 0.683 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev SMA 12 0.6831 0.0610
0.0010
T 11.20
Differencing: 1 regular, 1 seasonal of order 12 Number of observations: Original series 168, after differencing 155 Residuals: SS = 0.0165799 (backforecasts excluded) MS = 0.0001077 DF = 154 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 9.0(DF=11) 29.9(DF=23) 44.5(DF=35)
48 59.4(DF=47)
Forecasts from period 168 Period 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192
Forecast 6.76750 6.70901 6.83815 6.85381 6.92288 6.89349 6.84654 6.80008 6.74395 6.75028 6.70664 6.75999 6.79052 6.73203 6.86117 6.87684 6.94590 6.91651 6.86956 6.82310 6.76697 6.77330 6.72966 6.78301
95 Percent Limits Lower Upper 6.74716 6.78784 6.68024 6.73778 6.80292 6.87338 6.81313 6.89450 6.87739 6.96836 6.84366 6.94331 6.79272 6.90035 6.74255 6.85761 6.68293 6.80497 6.68596 6.81461 6.63918 6.77410 6.68952 6.83045 6.71514 6.86590 6.65203 6.81203 6.77680 6.94554 6.78832 6.96535 6.85342 7.03838 6.82023 7.01279 6.76962 6.96950 6.71963 6.92657 6.66009 6.87385 6.66312 6.88349 6.61627 6.84305 6.66649 6.89952
(1 − B ) (1 − B ) ln ( z ) = (1 − 0.683B ) a , 12
12
t
t
Actual
أي أن اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
at
N ( 0, 0.0001077 )
ﻻﺣﻆ ان
Θ = 0.683, s.e. ( Θ ) = 0.061, t = 11.2
.أي ان اﻟﻤﻌﻠﻢ ﻋﺎﻟﻲ اﻟﻤﻌﻨﻮﻳﺔ :ﻓﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ MTB > ZTest 0.0 0.0103778 'RESI3'; SUBC> Alternative 0.
١٢٢
Z-Test Test of mu = 0.000000 vs mu not = 0.000000 The assumed sigma = 0.0104 Variable RESI3
N Mean StDev 155 -0.000111 0.010375
SE Mean 0.000834
Z -0.13
P 0.89
اي ﻻﻧﺮﻓﺾ أن ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﺻﻔﺮا٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦP-value=٠٫٨٩ ﻻﺣﻆ ان اﻟـ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 0 'RESI3'.
Runs Test RESI3 K =
0.0000
The observed number of runs = 70 The expected number of runs = 78.1097 72 Observations above K 83 below The test is significant at 0.1893 Cannot reject at alpha = 0.05
اي اﻧﻨﺎ ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ٠٫١٨٩٣ اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ :إﺧﺘﺒﺎر إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ A C F o f R e s id u a ls f o r y ( t ) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 3
6
9
12
15
18
21
Lag
١٢٣
24
27
30
33
36
39
P A C F o f R e s id u a ls f o r y ( t ) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0 3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
.ﻧﻼﺣﻆ اﻧﻬﺎ ﺗﻌﻄﻲ اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء أي اﻧﻬﺎ ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ وإذا آﺎﻧﺖ ﻃﺒﻴﻌﻴﺔ ﻓﻬﻲ ﻣﺴﺘﻘﻠﺔ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Histogram of the Residuals (response is y(t))
Frequency
30
20
10
0 -0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
Residual
Normal Probability Plot of the Residuals (response is y(t)) 0.04 0.03 0.02
Residual
Partial Autocorrelation
0 .8
0.01 0.00 -0.01 -0.02 -0.03 -3
-2
-1
0
1
2
Normal Score
١٢٤
3
K-S test for Residuals
.999 .99 .95
.50 .20
Probability
.80
.05 .01 .001 0.03
0.02
0.01
-0.01
0.00
-0.02
-0.03
RESI3 Average: -0.0001115 StDev: 0.0103754 N: 155
Kolmogorov-Smirnov Normality Test D+: 0.074 D-: 0.045 D : 0.074 Approximate P-Value: 0.041
ﻻﺣﻆ ان اﻟـ P-valueﻹﺧﺘﺒﺎر K-Sﻳﻌﻄﻲ ٠٫٠٤١وهﻲ اﻗﻞ ﻣﻦ ٠٫٠٥اذا اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ α = 0.05اي ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ. اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻨﻤﻮذج: ﻓﻲ اﻟﻤﺨﺮﺟﺎت اﻟﺴﺎﺑﻘﺔ ﻗﻤﻨﺎ ﺑﺎﻟﺘﻨﺒﺆ ﻋﻦ ٢٤ﻗﻴﻤﺔ ﻣﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ ٩٥%ﻓﺘﺮات ﺗﻨﺒﺆ وﻧﺮﺳﻤﻬﺎ ﺑﺎﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ: 1150
1050
850
750
25
20
10
15
T im e
واﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻣﻊ اﻟﺘﻨﺒﺆات وﺣﺪود اﻟﺘﻨﺒﺆ
١٢٥
5
0
Forecast
950
1150
1050
850 750
Forecast
950
650 550 200
0
100
Time
ﺣﺎﻟﺔ دراﺳﺔ : ١ ﺳﻮف ﻧﺤﺎول إﻳﺠﺎد ﻧﻤﻮذج ﻣﻦ ﻋﺎﺋﻠﺔ SARIMAﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ: 57.4 60.0 69.8 70.1 66.1 58.7 55.8 51.6 53.5 53.0 52.1 53.6 55.6 62.6 75.5 76.4
61.5 71.0 71.0 67.5 56.1 55.4 55.3 52.4 53.4 53.5 53.4 55.3 60.1 70.3 73.3 70.8
72.7 70.6 67.4 58.0 53.6 52.9 53.2 52.1 53.6 54.3 55.3 58.5 68.9 69.4 68.1 60.6
72.2 68.2 58.2 54.2 53.0 53.0 52.8 52.6 55.1 56.3 58.2 69.3 68.8 64.7 58.1 56.4
71.5 57.7 54.3 53.2 52.8 52.7 52.8 53.9 57.3 59.4 66.9 69.6 63.6 57.1 54.8 55.6
59.1 54.6 53.4 53.0 52.6 53.4 54.5 56.4 61.9 70.4 70.7 64.2 57.1 53.9 54.3 55.0
57.2 54.9 53.0 53.0 52.9 54.9 56.4 61.7 69.9 69.9 65.3 55.5 52.2 53.5 54.6 54.7
56.3 54.9 52.8 53.4 54.0 57.5 59.3 68.3 71.5 61.0 56.5 53.3 51.5 53.5 54.2 55.8
55.7 55.3 54.4 55.6 56.6 69.3 70.0 65.3 57.3 53.9 52.5 51.5 51.5 53.3 55.8 60.5 62.2
55.8 54.9 53.3 54.6 54.9 61.2 68.7 67.9 65.1 52.7 53.4 52.4 51.7 53.1 54.8 57.7
اﻟﻤﺨﻄﻂ اﻟﺰﻣﻨﻲ
7 0
)z(t 6 0
5 0 1 5 0
5 0
1 0 0
١٢٦
In d e x
)z(t 56.3 56.9 56.0 59.4 68.2 69.8 67.9 58.2 52.7 53.3 53.2 51.5 52.3 53.9 57.9 66.4 74.8
ﻧﺠﺪwt = (1 − B ) zt ﺑﺄﺧﺬ اﻟﻔﺮق اﻷول ﻹﺳﺘﻘﺮار اﻟﻤﺘﻮﺳﻂ
y(t)
1 0
0
-1 0
5 0
In d e x
1 0 0
1 5 0
وﻟﻬﺎ دوال ﺗﺮاﺑﻂ ذاﺗﻲ وﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ Autocorrelation
Autocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
0.47 -0.02 -0.37 -0.28 -0.21 -0.19 -0.21 -0.27 -0.32 -0.00 0.46 0.86
6.32 -0.22 -4.10 -2.82 -2.00 -1.77 -1.92 -2.46 -2.85 -0.02 3.91 6.71
40.60 40.67 65.83 80.10 87.92 94.41 102.29 115.89 135.47 135.48 176.37 318.37
13 14 15 16 17 18 19 20 21 22 23 24
0.43 -0.01 -0.33 -0.25 -0.19 -0.17 -0.19 -0.25 -0.31 -0.01 0.42 0.79
2.76 -0.03 -2.00 -1.49 -1.12 -0.97 -1.09 -1.42 -1.76 -0.06 2.35 4.28
354.69 354.69 375.77 388.04 395.15 400.68 407.71 419.89 439.21 439.24 475.10 602.75
25 26 27 28 29 30 31 32 33 34 35 36
0.41 -0.00 -0.29 -0.23 -0.17 -0.16 -0.17 -0.23 -0.28 -0.01 0.37 0.72
2.06 -0.01 -1.43 -1.08 -0.82 -0.78 -0.83 -1.06 -1.31 -0.06 1.70 3.27
638.60 638.60 656.89 667.77 674.14 679.93 686.56 697.63 714.89 714.93 745.11 860.42
37 38 39 40 41 42 43 44
0.40 0.02 -0.27 -0.21 -0.17 -0.15 -0.16 -0.20
1.72 0.07 -1.14 -0.89 -0.71 -0.61 -0.65 -0.83
896.48 896.54 913.29 923.75 930.47 935.55 941.37 950.82
2
12
22
32
42
Partial Autocorrelation
Partial Autocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
Lag PAC
T
0.47 -0.32 -0.30 0.09 -0.23 -0.25 -0.16 -0.43 -0.64 -0.31 -0.16 0.41
6.32 -4.21 -4.00 1.22 -3.05 -3.33 -2.09 -5.67 -8.56 -4.11 -2.18 5.42
1 2 3 4 5 6 7 8 9 10 11 12
Lag PAC 13 14 15 16 17 18 19 20 21 22 23 24
-0.27 -0.01 0.01 -0.07 -0.01 0.02 0.03 0.07 -0.12 -0.02 -0.06 0.13
22 T -3.62 -0.17 0.13 -0.95 -0.16 0.24 0.44 0.89 -1.62 -0.26 -0.86 1.74
32
Lag PAC 25 26 27 28 29 30 31 32 33 34 35 36
-0.02 -0.06 0.07 -0.00 -0.02 -0.04 0.01 0.03 -0.02 0.00 -0.09 0.05
T -0.33 -0.74 0.99 -0.05 -0.26 -0.60 0.20 0.37 -0.25 0.06 -1.17 0.61
42
Lag PAC 37 38 39 40 41 42 43 44
0.03 -0.03 -0.01 0.02 -0.04 0.03 -0.00 0.03
T 0.38 -0.35 -0.17 0.23 -0.56 0.35 -0.00 0.43
وﻳﻨﺘﺞ اﻟﻤﺘﺴﻠﺴﻠﺔwt = (1 − B12 ) (1 − B ) zt أي١٢ ﻧﺮى اﻧﻬﺎ ﺗﺤﺘﺎج إﻟﻰ ﺗﻔﺮﻳﻖ ﻣﻮﺳﻤﻲ ﻣﻦ اﻟﺮﺗﺒﺔ اﻟﺘﺎﻟﻴﺔ
١٢٧
w(t)
5
0
-5 In d e x
50
100
150
وﻟﻬﺎ دوال ﺗﺮاﺑﻂ ذاﺗﻲ وﺗﺮاﺑﻂ ذاﺗﻲ ﺟﺰﺋﻲ Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r w ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 .0
10
20
30
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1 2 3 4 5 6 7 8 9 10 11 12
-0 .0 1 -0 .1 9 -0 .2 4 -0 .0 0 0 .0 0 -0 .0 4 -0 .0 4 -0 .0 3 0 .2 2 0 .1 0 0 .0 8 -0 .4 2
- 0 .1 9 - 2 .4 8 - 2 .9 7 - 0 .0 3 0 .0 3 - 0 .5 0 - 0 .4 9 - 0 .3 1 2 .5 6 1 .1 6 0 .9 3 - 4 .6 3
0 .0 4 6 .3 1 1 6 .1 0 1 6 .1 0 1 6 .1 0 1 6 .4 1 1 6 .7 1 1 6 .8 3 2 5 .2 6 2 7 .1 3 2 8 .3 7 5 9 .4 0
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 8 0 .1 0 0 .1 2 -0 .0 2 -0 .0 3 0 .1 0 0 .0 7 -0 .0 1 -0 .2 0 -0 .0 2 0 .1 0 0 .1 4
-0 .8 4 0 .9 8 1 .1 5 -0 .1 7 -0 .2 8 1 .0 1 0 .7 0 -0 .0 5 -1 .9 4 -0 .2 3 0 .9 5 1 .3 0
6 0 .6 9 6 2 .5 0 6 5 .0 1 6 5 .0 7 6 5 .2 2 6 7 .2 4 6 8 .2 3 6 8 .2 4 7 5 .9 4 7 6 .0 5 7 8 .0 2 8 1 .7 8
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .0 8 - 0 .1 3 - 0 .0 1 0 .0 6 0 .1 2 - 0 .1 5 - 0 .0 8 - 0 .0 3 0 .1 9 0 .1 7 - 0 .1 0 - 0 .2 2
-0 .7 6 -1 .1 8 -0 .1 3 0 .5 6 1 .0 9 -1 .3 6 -0 .7 2 -0 .2 8 1 .6 6 1 .4 9 -0 .8 6 -1 .9 2
8 3 .0 9 8 6 .2 8 8 6 .3 3 8 7 .0 7 8 9 .9 4 9 4 .4 7 9 5 .8 0 9 6 .0 1 1 0 3 .2 0 1 0 9 .1 9 1 1 1 .2 4 1 2 1 .7 7
40
L ag
T
LBQ
3 7 0 .0 5 0 .4 6 3 8 0 .1 0 0 .8 4 3 9 0 .0 2 0 .1 5 4 0 -0 .0 2 - 0 .1 6 4 1 -0 .1 2 - 1 .0 1
C o rr
1 2 2 .4 0 1 2 4 .5 2 1 2 4 .5 9 1 2 4 .6 7 1 2 7 .8 4
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r w ( t ) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
10
20
30
40
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
- 0 .0 1 - 0 .1 9 - 0 .2 6 - 0 .0 7 - 0 .1 1 - 0 .1 4 - 0 .1 1 - 0 .1 3 0 .1 4 0 .0 7 0 .1 7 - 0 .3 2
- 0 .1 9 - 2 .4 8 - 3 .2 8 - 0 .8 5 - 1 .4 3 - 1 .8 4 - 1 .4 6 - 1 .6 6 1 .7 6 0 .9 5 2 .1 9 - 4 .0 8
13 14 15 16 17 18 19 20 21 22 23 24
-0 .0 5 0 .0 1 -0 .0 4 -0 .0 3 -0 .0 2 0 .0 6 0 .0 5 -0 .0 3 -0 .0 5 0 .0 5 0 .1 9 0 .0 1
- 0 .6 9 0 .1 5 - 0 .5 0 - 0 .3 3 - 0 .2 3 0 .7 8 0 .6 2 - 0 .4 2 - 0 .6 8 0 .6 3 2 .5 0 0 .0 9
25 26 27 28 29 30 31 32 33 34 35 36
- 0 .1 2 - 0 .0 8 - 0 .0 3 - 0 .0 3 0 .0 7 - 0 .1 1 - 0 .0 2 - 0 .1 0 - 0 .0 1 0 .1 8 0 .0 7 - 0 .0 9
-1 .4 9 -0 .9 7 -0 .3 2 -0 .3 6 0 .8 7 -1 .3 9 -0 .2 9 -1 .2 9 -0 .0 9 2 .3 1 0 .9 5 -1 .1 8
37 38 39 40 41
0 .0 2 - 0 .0 8 0 .0 3 0 .0 5 - 0 .0 3
0 .2 4 - 1 .0 6 0 .4 4 0 .6 3 - 0 .3 6
ﻣﻦ اﻷﻧﻤﺎط اﻟﻤﺸﺎهﺪة ﻟﺪوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻗﺪ ﻳﻜﻮن اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ هﻮ أيSARIMA(١،١،١)(٠،١،١)١٢
١٢٨
(1 − φ B ) (1 − B12 ) (1 − B ) zt = (1 − θ B ) (1 − ΘB12 ) at :ﻧﻄﺒﻖ هﺬا اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎﻟﺘﺎﻟﻲ MTB > ARIMA 1 1 1 0 1 1 12 'z(t)' 'RESI2'; SUBC> NoConstant; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram; SUBC> GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 307.653 0.100 0.100 1 281.217 0.100 0.100 2 262.275 0.226 0.231 3 262.027 0.376 0.381 4 261.770 0.526 0.531 5 261.426 0.675 0.681 6 260.905 0.824 0.831 7 260.036 0.970 0.981 8 227.926 0.835 0.980 9 221.838 0.748 0.980 10 221.665 0.738 0.980 11 221.637 0.738 0.980 12 221.610 0.737 0.980 13 221.585 0.737 0.980 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.7374 0.0620 MA 1 0.9796 0.0017 SMA 12 0.5898 0.0736
0.100 0.250 0.400 0.401 0.401 0.402 0.403 0.405 0.536 0.576 0.586 0.589 0.589 0.590 0.0010
T 11.89 582.86 8.01
Differencing: 1 regular, 1 seasonal of order 12 Number of observations: Original series 178, after differencing 165 Residuals: SS = 214.393 (backforecasts excluded) MS = 1.323 DF = 162 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 15.7(DF= 9) 30.9(DF=21) 61.6(DF=33)
48 67.1(DF=45)
أي أن اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ هﻮ
(1 − 0.74 B ) (1 − B ) (1 − B ) zt = (1 − 0.98B ) (1 − 0.59 B12 ) at , 12
at
N ( 0,1.323)
ﻻﺣﻆ ان
φ = 0.74, s.e. (φ ) = 0.062, t = 11.89 θ = 0.96, s.e. (θ ) = 0.0017, t = 582.86 Θ = 0.59, s.e. ( Θ ) = 0.074, t = 8.01
.أي ان اﻟﻤﻌﺎﻟﻢ ﻋﺎﻟﻴﺔ اﻟﻤﻌﻨﻮﻳﺔ :ﻓﺤﺺ اﻟﺒﻮاﻗﻲ إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ MTB > ZTest 0.0 1.15 'RESI1'; SUBC> Alternative 0.
Z-Test
١٢٩
Test of mu = 0.0000 vs mu not = 0.0000 The assumed sigma = 1.15 Variable RESI1
N 165
Mean -0.0144
StDev 1.1433
SE Mean 0.0895
Z -0.16
P 0.87
اي ﻻﻧﺮﻓﺾ أن ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﺻﻔﺮا٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦP-value=٠٫٨٧ ﻻﺣﻆ ان اﻟـ إﺧﺘﺒﺎر ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 0 'RESI1'.
Runs Test RESI1 K =
0.0000
The observed number of runs = 67 The expected number of runs = 82.4061 73 Observations above K 92 below The test is significant at 0.0149
اي اﻧﻨﺎ ﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ وهﺬا ﻳﺤﺘﺎج إﻟﻰ إﺟﺮاء٠٫٠٥ اﻹﺧﺘﺒﺎر ﻏﻴﺮﻣﻌﻨﻮي ﻋﻨﺪ : ﻋﻠﻰ اﻟﻮﺳﻴﻂ اﻟﺘﺎﻟﻲSign Test إﺧﺘﺒﺎر ﺁﺧﺮ أآﺜﺮ ﻗﻮة ﻣﻦ إﺧﺘﺒﺎر اﻟﺠﺮي ﻣﺜﻞ إﺧﺘﺒﺎر اﻹﺷﺎرة MTB > STest 0.0 'RESI1'; SUBC> Alternative 0.
Sign Test for Median Sign test of median = 0.00000 versus RESI1
N 165
N* 13
Below 92
Equal 0
not = Above 73
0.00000 P 0.1611
Median -0.08139
وﻟﻠﺘﺄآﺪ ﻧﺠﺮي إﺧﺘﺒﺎر وﻟﻜﻮآﺴﻮن ﻹﺷﺎرات اﻟﺮﺗﺐ ﻋﻠﻰ اﻟﻮﺳﻴﻂ٠٫١٦١١ واﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ اﻟﺘﺎﻟﻲ MTB > WTest 0.0 'RESI1'; SUBC> Alternative 0.
Wilcoxon Signed Rank Test Test of median = 0.000000 versus median not = 0.000000
RESI1
N 165
Number Missing 13
N for Test 165
Wilcoxon Statistic 6321.0
P 0.392
Estimated Median -0.05940
٠٫٣٩٢ واﻹﺧﺘﺒﺎر اﻳﻀﺎ ﻣﻌﻨﻮي ﻋﻨﺪ :إﺧﺘﺒﺎر إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
١٣٠
A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s ) 1 .0 0 .8
Autocorrelation
0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
36
39
Lag
P A C F o f R e s id u a ls f o r z ( t) ( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s ) 1 .0
Partial Autocorrelation
0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
3
6
9
12
15
18
21
24
27
30
33
Lag
.ﻧﻼﺣﻆ اﻧﻬﺎ ﺗﻌﻄﻲ اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء أي اﻧﻬﺎ ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ وإذا آﺎﻧﺖ ﻃﺒﻴﻌﻴﺔ ﻓﻬﻲ ﻣﺴﺘﻘﻠﺔ :إﺧﺘﺒﺎر ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Histogram of the Residuals (response is z(t)) 50
Frequency
40
30
20
10
0 -6
-5
-4
-3
-2
-1
0
1
2
Residual
١٣١
3
4
Normal Probability Plot of the Residuals (response is z(t)) 4 3 2
Residual
1 0 -1 -2 -3 -4 -5 -3
-2
-1
0
1
2
3
Normal Score
K-S Test for Residuals
.999
Probability
.99 .95 .80 .50 .20 .05 .01 .001 -5
-4
-3
-2
-1
0
1
2
3
RESI1 Average: -0.0144171 StDev: 1.14327 N: 165
Kolmogorov-Smirnov Normality Test D+: 0.117 D-: 0.140 D : 0.140 Approximate P-Value < 0.01
ايα = 0.05 اذا اﻹﺧﺘﺒﺎر ﻣﻌﻨﻮي ﻋﻨﺪ٠٫٠١ أﻗﻞ ﻣﻦK-S ﻹﺧﺘﺒﺎرP-value ﻻﺣﻆ ان اﻟـ .ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ :اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻨﻤﻮذج ﻓﺘﺮات ﺗﻨﺒﺆ٩٥% ﻗﻴﻤﺔ ﻣﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ٣٦ ﺳﻨﻘﻮم ﺑﺎﻟﺘﻨﺒﺆ ﻋﻦ Forecasts from period 178 Period 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193
Forecast 57.7885 55.8516 54.7429 54.1820 54.6298 54.9152 55.6640 59.3778 68.6924 74.0698 73.9650 63.5866 58.9095 56.7768 55.5237
95 Percent Limits Lower Upper 55.5332 60.0437 53.0220 58.6812 51.6264 57.8594 50.9063 57.4578 51.2601 57.9994 51.4875 58.3430 52.1986 59.1294 55.8869 62.8688 65.1833 72.2015 70.5472 77.5925 70.4317 77.4983 60.0447 67.1286 55.1843 62.6347 52.9407 60.6128 51.6169 59.4305
١٣٢
Actual
50.9022 51.2380 51.4409 52.1278 55.7947 65.0729 70.4217 70.2940 59.8968 55.0418 52.7962 51.4676 50.7472 51.0774 51.2749 51.9568 55.6189 64.8927 70.2374 70.1057 59.7046
58.8105 59.2131 59.4653 60.1905 63.8885 73.1929 78.5647 78.4575 68.0793 63.5664 61.5364 60.3514 59.7315 60.1357 60.3903 61.1184 64.8194 74.1270 79.5021 79.3983 69.0235
194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214
54.8564 55.2256 55.4531 56.1592 59.8416 69.1329 74.4932 74.3758 63.9881 59.3041 57.1663 55.9095 55.2394 55.6066 55.8326 56.5376 60.2191 69.5099 74.8697 74.7520 64.3640
وﻧﺮﺳﻤﻬﺎ ﺑﺎﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ: 8 0
Forecast
7 0
6 0
5 0
3 0
4 0
2 0
0
1 0
T im e
اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﺑﻜﺎﻣﻠﻬﺎ ﻣﻊ اﻟﺘﻨﺒﺆات وﻓﺘﺮات اﻟﺘﻨﺒﺆ
80
60
50
200
100
T im e
١٣٣
0
Forecast
70
١٣٤
اﻟﻔﺼﻞ اﻟﺴﺎﺑﻊ ورﻗﺔ ﺗﺪرﻳﺐ ﻋﻤﻠﻲ ﻋﻠﻲ اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻧﻤﺎذج اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك-اﻹﻧﺤﺪار اﻟﺬاﺗﻲ Forecasting By ARMA Models اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺴﺠﻠﺔ ﻋﻠﻲ ﺷﻜﻞ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ 28.0 28.0 29.5
23.5 30.5 32.3
21.3 29.6 30.6
24.5 36.5 26.5
23.5 36.5 20.7 16.4
15.3 25.0 16.0 23.4
15.5 25.3 19.0 26.4
21.0 17.3 19.7 32.2
12.0 24.0 26.0 28.3
20.5 15.5 21.5 31.3
وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ: ;MTB > TSPlot C1 >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
30
C 1 20
10 30
10
20
اوﻻ ﻧﻮﺟﺪ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
١٣٥
Index
MTB > %ACF C1.
Autocorrelation
Autocorrelation Function for C1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag
Corr
T
LBQ
1 2 3 4 5 6 7
0.63 0.30 0.14 -0.05 -0.24 -0.30 -0.24
3.79 1.35 0.59 -0.20 -1.01 -1.22 -0.94
15.62 19.27 20.07 20.17 22.71 26.71 29.40
5
Lag
6
Corr
7
T
LBQ
8 -0.20 -0.78 9 -0.12 -0.44
31.42 32.10
8
9
MTB > %PACF C1. Executing from file: E:\MTBWIN\MACROS\PACF.MAC
Partial Autocorrelation
Partial Autocorrelation Function for C1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag PAC 1 2 3 4 5 6 7
0.63 -0.16 0.04 -0.20 -0.18 -0.04 0.02
5
6
T
Lag PAC
T
3.79 -0.98 0.22 -1.20 -1.09 -0.23 0.12
8 -0.07 9 0.05
-0.39 0.29
7
8
9
ARMA (1,1) ﻧﻼﺣﻆ ﻣﻦ اﻧﻤﺎط اﻟﺪاﻟﺘﻴﻦ ان اﻟﻤﺸﺎهﺪات ﻗﺪ ﺗﻜﻮن ﻣﻦ ﻧﻤﻮذج ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح MTB > Name c17 = 'RESI1' MTB > ARIMA 1 0 1 C1 'RESI1'; SUBC> Constant; SUBC> Forecast 5 c14 c15 c16; SUBC> GACF; SUBC> GPACF; SUBC> GNormalplot.
١٣٦
ARIMA Model ARIMA model for C1 Estimates at each iteration Iteration SSE Parameters 0 1337.71 0.100 0.100 21.918 1 936.95 0.250 -0.049 18.193 2 849.78 0.211 -0.199 19.106 3 751.53 0.215 -0.349 18.941 4 658.66 0.266 -0.499 17.594 5 592.30 0.372 -0.649 14.890 6 580.80 0.433 -0.699 13.314 7 579.30 0.455 -0.714 12.698 8 579.11 0.464 -0.719 12.470 9 579.08 0.467 -0.721 12.386 10 579.08 0.468 -0.722 12.356 11 579.08 0.468 -0.722 12.345 Relative change in each estimate less than 0.0010 Final Estimates of Parameters Type Coef StDev AR 1 0.4684 0.1755 MA 1 -0.7221 0.1380 Constant 12.345 1.154 Mean 23.221 2.170
T 2.67 -5.23 10.70
Number of observations: 36 Residuals: SS = 523.365 (backforecasts excluded) MS = 15.860 DF = 33 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 7.2(DF=10) 15.9(DF=22) * (DF= *) * (DF= *) Forecasts from period 36 Period 37 38 39 40 41
Forecast 14.7649 19.2606 21.3663 22.3524 22.8143
95 Percent Limits Lower Upper 6.9578 22.5720 7.1228 31.3985 8.4715 34.2610 9.2975 35.4074 9.7245 35.9041
zt = 12.345 + 0.4684 zt −1 + at − 0.7221at −1 , at ﺣﻴﺚ φˆ1 = 0.4684 se φˆ1 = 0.1755 t = 2.67
Actual
:اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ WN ( 0,15.86 ) ∀t
( ) θˆ = −0.7221 se (θˆ ) = 0.1380 t = −5.23 δˆ = 12.345 se (δˆ ) = 1.154 t = 10.70 1
1
σˆ 2 = 15.86 df = 33 ﻓﻤﺜﻼ اﻟﻔﺮﺿﻴﺔα = 0.05 وﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻘﺪرات ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪ
١٣٧
H 0 : φ1 = 0 H1 : φ1 ≠ 0 أيα = 0.05 وهﻲ ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪt =
0.4684 φˆ1 = = 2.6689 ﻧﺨﺘﺒﺮهﺎ ﺑﺎﻹﺣﺼﺎﺋﺔ 0.1755 se φˆ1
( )
وﺑﺎﻟﻤﺜﻞ ﻟﺠﻤﻴﻊ اﻟﻘﺪرات اﻻﺧﺮىφ1 = 0 اﻧﻨﺎ ﻧﺮﻓﺾ ان
ﺛﺎﻧﻴﺎ ﻓﺤﺺ اﻟﺒﻮاﻗﻲ :إﺧﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ اﻹﺧﺘﺒﺎر هﻮ
H 0 : µa = 0, H1 : µa ≠ 0
MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean Test of mu = 0.000 vs mu not = 0.000 Variable RESI1
N 36
Mean 0.344
StDev 3.851
SE Mean 0.642
T 0.54
P 0.60
أي ان اﻹﺧﺘﺒﺎرα = 0.05 وهﻲ اآﺒﺮ ﻣﻦ0.6 ﻟﻬﺎ هﻲP-Value واﻟـt = 0.54 ﻻﺣﻆ ان ﻏﻴﺮ ﻣﻌﻨﻮي أي ﻳﻤﻜﻦ إﻋﺘﺒﺎر ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ ﻳﺴﺎوي اﻟﺼﻔﺮ :إﺧﺘﺒﺎرﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ وﻧﺴﺘﺨﺪم ﻟﺬﻟﻚ إﺧﺘﺒﺎر اﻟﺠﺮيRuns Test MTB > Runs 'RESI1'.
Runs Test RESI1 K =
0.3443
The observed number of runs = 21 The expected number of runs = 19.0000 18 Observations above K 18 below The test is significant at 0.4989 Cannot reject at alpha = 0.05
α = 0.05 ﻻﻳﻤﻜﻨﻨﺎ رﻓﺾ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﻋﻨﺪ :إﺧﺘﺒﺎر ﺗﺮاﺑﻂ اﻟﺒﻮاﻗﻲ وﻧﺴﺘﺨﺪم ﻟﺬﻟﻚ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ
ACF of Residuals for C1 (with 95% confidence limits for the autocorrelations) 1.0 0.8
orrelation
0.6 0.4 0.2 0.0
١٣٨
PACF of Residuals for C1 (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1
2
3
4
5
6
7
8
9
Lag
ﻧﻼﺣﻆ اﻧﻪ ﻻﻳﻮﺟﺪ أي ﺗﺮاﺑﻂ ﻣﻦ أي درﺟﺔ ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﺨﺘﻠﻔﺔ ﻟﻠﺒﻮاﻗﻲ أي اﻧﻬﺎ ﺗﻈﻬﺮ اﻧﻤﺎط ﺗﺘﻤﺸﻲ ﻣﻊ آﻮﻧﻬﺎ ﻣﺘﺴﻠﺴﺔ ﺿﺠﺔ ﺑﻴﻀﺎء Normal Probability Plot واﺧﻴﺮا ﻧﺨﺘﺒﺮ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﺎﻟـ
Normal Probability Plot of the Residuals (response is C1)
Residual
10
0
-10
-2
-1
0
Normal Score
١٣٩
1
2
( وهﻮ ﻣﻘﺒﻮل ) ﻧﻮﻋﺎ إذا ﻳﻤﻜﻦ إﻋﺘﺒﺎر اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح ﻣﻨﺎﺳﺒﺎ ﺗﻨﺒﺆ ﻟﻬﺎ95% اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﺘﻨﺒﺆات ﻟﺨﻤﺴﺔ ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ وﻓﺘﺮات
MTB > TSPlot C14 C15 C16; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect; SUBC> overlay.
Time Series Plot for C1 (with forecasts and their 95% confidence limits)
36
C1
26
16
6
5
10
15
20
25
30
35
Time
:ﻣﺜﺎل ﺁﺧﺮ اﻟﻤﺸﺎهﺪات اﻟﺘﺎﻟﻴﺔ ﻟﻈﺎهﺮة ﻋﺸﻮاﺋﻴﺔ ﻣﺴﺠﻠﺔ ﻋﻠﻲ ﺷﻜﻞ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ 10.38 11.68 9.10 10.01 9.51 9.48 7.23 10.85 6.89
11.86 11.17 9.09 9.37 9.24 7.38 8.42 10.41 5.96
10.97 10.53 9.35 8.69 8.66 6.90 9.61 9.96 6.80
10.80 10.01 8.82 8.19 8.86 6.94 9.05 9.61 7.68
9.79 9.91 9.32 8.67 8.05 6.24 9.26 8.76 8.38
10.39 9.14 9.01 9.55 7.79 6.84 9.22 8.18 8.52
10.42 9.16 9.00 8.92 6.75 6.85 9.38 7.21 9.74
10.82 9.55 9.80 8.09 6.75 6.90 9.10 7.13 9.31
11.40 9.67 9.83 9.37 7.82 7.79 7.95 9.10 9.89
11.32 11.44 8.44 8.24 9.72 9.89 10.13 10.14 8.64 10.58 8.18 7.51 8.12 9.75 8.25 7.91 9.96
MTB > TSPlot C10; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
12 11
C 1 0
10 9 8
١٤٠
7 6 Index
10
20
30
40
50
60
70
80
90
ﻧﻔﺤﺺ دوال اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
Autocorrelation
Autocorrelation Function for C10 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
Lag Corr 1 2 3 4 5 6 7
0.83 0.61 0.46 0.37 0.33 0.28 0.26
T
12
LBQ
8.24 69.92 3.91 107.90 2.56 129.56 1.95 143.87 1.65 155.04 1.40 163.68 1.28 171.23
Lag Corr 8 9 10 11 12 13 14
0.26 0.26 0.18 0.09 0.04 0.03 0.04
T
LBQ
22
Lag Corr
1.26 178.83 1.21 186.14 0.84 189.86 0.43 190.87 0.20 191.09 0.13 191.19 0.19 191.39
T
LBQ
15 0.05 0.21 191.63 16 0.04 0.16 191.78 17 0.00 0.02 191.78 18 -0.03 -0.15 191.91 19 -0.05 -0.24 192.26 20 -0.05 -0.24 192.60 21 0.01 0.07 192.63
Lag Corr
T
LBQ
22 0.10 0.47 194.02 23 0.18 0.81 198.12 24 0.20 0.89 203.24
Partial Autocorrelation
Partial Autocorrelation Function for C10 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
12
Lag PAC
T
Lag PAC
T
1 0.83 2 -0.27 3 0.13 4 0.03 5 0.06 6 -0.02 7 0.09
8.24 -2.64 1.29 0.34 0.61 -0.21 0.91
8 0.05 9 0.00 10 -0.20 11 0.02 12 0.01 13 0.01 14 0.03
0.45 0.03 -1.98 0.19 0.09 0.12 0.34
22
Lag PAC 15 16 17 18 19 20 21
-0.01 -0.03 -0.07 -0.03 0.06 0.02 0.21
T
Lag PAC
T
-0.15 -0.25 -0.73 -0.26 0.60 0.20 2.03
22 0.05 23 0.06 24 -0.07
0.51 0.59 -0.65
AR ( 2 ) ﻧﻼﺣﻆ ان اﻷﻧﻤﺎط ﺗﻘﺘﺮح ﻧﻤﻮذج إﻧﺤﺪار ذاﺗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻧﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح MTB > Name c17 = 'RESI1' MTB > ARIMA 2 0 0 C10 'RESI1'; SUBC> Constant;
١٤١
SUBC> SUBC> SUBC> SUBC> SUBC>
Forecast 5 c14 c15 c16; GSeries; GACF; GPACF; GNormalplot.
ARIMA Model ARIMA model for C10 Estimates at each iteration Iteration SSE Parameters 0 126.398 0.100 0.100 1 103.515 0.250 0.043 2 84.535 0.400 -0.014 3 69.407 0.550 -0.071 4 58.132 0.700 -0.128 5 50.724 0.850 -0.184 6 47.212 1.000 -0.239 7 46.918 1.053 -0.256 8 46.916 1.054 -0.255 9 46.916 1.054 -0.255 10 46.916 1.054 -0.255 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 1.0542 0.0992 AR 2 -0.2547 0.0993 Constant 1.81360 0.07092 Mean 9.0480 0.3538
7.283 6.434 5.586 4.738 3.887 3.030 2.163 1.838 1.816 1.814 1.814 0.0010
T 10.63 -2.56 25.57
Number of observations: 98 Residuals: SS = 46.7518 (backforecasts excluded) MS = 0.4921 DF = 95 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 7.2(DF=10) 13.7(DF=22) 21.3(DF=34) 28.8(DF=46) Forecasts from period 98 Period 99 100 101 102 103
Forecast 9.7950 9.6033 9.4432 9.3232 9.2375
95 Percent Limits Lower Upper 8.4198 11.1703 7.6050 11.6016 7.1234 11.7629 6.8446 11.8018 6.6825 11.7925
Actual
اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ zt = 1.8136 + 1.0542 zt −1 − 0.2547 zt −2 + at , at WN ( 0,0.4921) ∀t ﻟﻬﺎ هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻢ φˆ1 = 1.0542 se φˆ1 = 0.0992 t = 10.63
( ) φˆ = −0.2547 se (φˆ ) = 0.0993 t = −2.56 δˆ = 1.8136 se (δˆ ) = 0.07092 t = 25.57 2
2
σˆ 2 = 0.4921 df = 95
α = 0.05 ﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻘﺪرات ﻣﻌﻨﻮﻳﺔ ﻋﻨﺪ ١٤٢
ﻟﻜﻲ ﻧﻘﺒﻞ ﺑﻬﺬا اﻟﻨﻤﻮذج ﻋﻠﻲ اﻧﻪ ﻣﻨﺎﺳﺐ ﻟﻠﺘﻨﺒﺆ ﻧﺠﺮي إﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean Test of mu = 0.0000 vs mu not = 0.0000 Variable RESI1
N 98
Mean StDev -0.0082 0.6942
SE Mean 0.0701
T -0.12
P 0.91
هﻲ إﺣﺘﻤﺎل ان اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔP-Value واﺿﺢ ﺟﺪا ان اﻷﺧﺘﺒﺎر ﻏﻴﺮ ﻣﻌﻨﻮي ) ﻣﻼﺣﻈﺔ اﻟـ ( ﺻﺤﻴﺤﺔ ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ MTB > Runs 'RESI1'.
Runs Test RESI1 K =
-0.0082
The observed number of runs = 47 The expected number of runs = 49.9184 47 Observations above K 51 below The test is significant at 0.5529 Cannot reject at alpha = 0.05
أي ﻻﻳﻤﻜﻨﻨﺎ رﻓﺾ ﻓﺮﺿﻴﺔ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﻧﻔﺤﺺ اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ
ACF of Residuals for C10 (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
4
6
8
10
12
14
Lag
١٤٣
16
18
20
22
24
PACF of Residuals for C10 (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
2
4
6
8
10
12
14
16
18
20
22
24
Lag
واﺿﺢ ﺟﺪا اﻧﻤﺎط اﻟﻀﺠﺔ اﻟﺒﻴﻀﺎء ﻳﺒﻘﻲ ﻓﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ Normal Probability Plot of the Residuals (response is C10) 2
Residual
1
0
-1
-2 -3
( at
-2
-1
0
1
Normal Score
2
3
IIDN ( 0,0.4921) وﻧﺴﺘﻄﻴﻊ ان ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ) أي ﺗﻨﺒﺆ ﻟﻬﺎ95% اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﺘﻨﺒﺆات ﻟﺨﻤﺴﺔ ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﺔ وﻓﺘﺮات
MTB > TSPlot C14 C15 C16; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect; SUBC> overlay.
Time Series Plot for C10 (with forecasts and their 95% confidence limits) 12 11
C10
10 9 8 7 6
١٤٤
ﺗﻤﺮﻳﻦ :ﻃﺒﻖ ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻧﻤﻮذج ) AR (1وﻗﺎرن ﺑﻴﻦ اﻟﻨﺘﺎﺋﺞ
١٤٥
١٤٦
اﻟﻔﺼﻞ اﻟﺜﺎﻣﻦ ﻣﺜﺎل ﻋﻠﻰ ﺗﺤﻠﻴﻞ اﻟﺒﻮاﻗﻲ وﻣﻌﺎﻳﻴﺮ إﺧﺘﻴﺎر اﻟﻨﻤﻮذج اﻟﻤﻨﺎﺳﺐ Example on Residual Analysis and Model Selection : Criteria ﻟﻘﺪ ﻋﺮﻓﻨﺎ ﺳﺎﺑﻘﺎ اﻟﺒﻮاﻗﻲ ﻋﻠﻲ اﻧﻬﺎ اﻟﻘﻴﻢ اﻟﻤﺸﺎهﺪة ﻧﺎﻗﺺ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻓﻤﻦ ﻣﺸﺎهﺪات ﻣﻌﻄﺎة z1 , z2 ,..., znوﻧﻤﻮذج ﻣﻄﺒﻖ ﻳﻨﺘﺞ ﻟﺪﻳﻨﺎ ﻗﻴﻢ ﻣﻄﺒﻘﺔ zˆ1 , zˆ2 ,..., zˆnوﺗﻜﺘﺐ اﻟﺒﻮاﻗﻲ: ei = zi − zˆi , i = 1, 2,..., n واﻟﺒﻮاﻗﻲ هﻲ ﻣﻘﺪرات اﻷﺧﻄﺎء ﻓﻲ اﻟﻨﻤﻮذج أي aˆi = ei , i = 1, 2,..., nوﻟﻬﺬا ﻳﺠﺐ ان ﺗﺤﻘﻖ اﻟﺸﺮوط اﻟﻤﻔﺮوﺿﺔ ﻋﻠﻲ اﻷﺧﻄﺎء ﻓﻲ هﺬا اﻟﻨﻤﻮذج واﻟﺘﻲ ﻣﻨﻬﺎ: -١ﻣﺘﻮﺳﻂ اﻷﺧﻄﺎء ﻳﺴﺎوي اﻟﺼﻔﺮ -٢اﻷﺧﻄﺎء ﻋﺸﻮاﺋﻴﺔ و ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ أو ﻣﺴﺘﻘﻠﺔ ) وﻓﻲ آﺜﻴﺮ ﻣﻦ اﻟﻨﻤﺎذج ﻧﻔﺘﺮض ان اﻷﺧﻄﺎء ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻣﺴﺘﻘﻞ وﻣﺘﻄﺎﺑﻖ ﺑﻤﺘﻮﺳﻂ ﺻﻔﺮي وﺗﺒﺎﻳﻦ σ 2أي ) ( at IIDN ( 0, σ 2 ﻟﻬﺬا ﻓﺈﻧﻨﺎ ﻧﺠﺮي ﺗﺤﻠﻴﻼ وهﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻹﺧﺘﺒﺎرات ﻋﻠﻲ اﻟﺒﻮاﻗﻲ ﻟﻨﺮي ﻓﻴﻤﺎ إذا آﺎﻧﺖ ﺗﺤﻘﻖ هﺬﻩ اﻟﺸﺮوط وﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ ﻧﻌﺘﺒﺮ اﻟﻨﻤﻮذج اﻟﻤﻄﺒﻖ ﻣﻘﺒﻮﻻ أﻣﺎ إذا ﻓﺸﻞ اﺣﺪ هﺬﻩ اﻹﺧﺘﺒﺎرات ﻓﻴﺠﺐ ﻋﻠﻴﻨﺎ إﻋﺎدة اﻟﻨﻈﺮ وإﻗﺘﺮاح ﻧﻤﻮذج ﺁﺧﺮ أوﻻ :إﺧﺘﺒﺎر اﻟﻤﺘﻮﺳﻂ H 0 : E ( at ) = 0 H 1 : E ( at ) ≠ 0
وهﻮ إﺧﺘﺒﺎر ﺑﺬﻳﻠﻴﻦ وﻧﺴﺘﺨﺪم ﻓﻴﺔ اﻹﺣﺼﺎﺋﺔ
e
) se ( e
= uواﻟﺘﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻗﻴﺎﺳﻲ ﻓﻌﻨﺪ
ﻣﺴﺘﻮى ﻣﻌﻨﻮﻳﺔ α = 0.05ﻧﻌﺘﺒﺮ ان E ( at ) = 0إذا آﺎﻧﺖ ) u < 1.96هﺬا ﻋﻠﻲ إﻋﺘﺒﺎر ان ﺣﺠﻢ اﻟﻌﻴﻨﺔ اآﺒﺮ ﻣﻦ ٣٠وﺣﺪة وهﺬا داﺋﻤﺎ ﻣﺘﺤﻘﻖ ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﺰﻣﻨﻴﺔ اﻟﺘﻲ ﻧﺪرﺳﻬﺎ (
ﻣﺜﺎل: ﺳﻮف ﻧﻌﻮد اﻟﻲ ﻣﺜﺎل ﺗﻄﺒﻴﻖ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ ﻋﻠﻲ اﻟﻤﺘﻐﻴﺮ Metals MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'. Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW Worksheet was saved on 6/ 5/1996 ;'MTB > TSPlot 'Metals >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
50
M e ta ls
45
40 60
50
40
30
20
10
Index
'MTB > Name c4 = 'AVER1' c5 = 'FITS1' c6 = 'RESI1
١٤٧
MTB > %MA 'Metals' 3; SUBC> Averages 'AVER1'; SUBC> Fits 'FITS1'; SUBC> Residuals 'RESI1'. Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average Data Length NMissing
Metals 60.0000 0
Moving Average Length: 3 Accuracy Measures MAPE: 1.55036 MAD: 0.70292 MSD: 0.76433
Moving Average
Actual
Predicted
M etals
50
Actual Predicted
45 Moving Average Length:
40 0
10
20
30
40
50
3
MAPE:
1.55036
MAD:
0.70292
MSD:
0.76433
60
Time
ﻻﺣﻆ اﻧﻨﺎ ﺧﺰﻧﺎ اﻟﺒﻮاﻗﻲ ﻓﻲ اﻟﻌﻤﻮد اﻟﺴﺎدس واﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻓﻲ اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ MTB > print c3 c6 c5
Data Display
Row
Metals
RESI1
FITS1
1 2 3 4 5 6 7 8 9
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4
* * * -0.90000 -1.23333 0.76667 0.90000 0.96667 -0.10000
* * * 44.3000 44.0333 43.5333 43.5000 43.8333 44.5000
١٤٨
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
-1.43333 -1.50000 -0.96667 -0.50000 -0.60000 -2.03333 0.63333 1.10000 1.60000 -0.10000 0.46667 0.33333 -0.10000 -0.03333 -0.20000 -0.36667 -0.16667 -0.33333 0.43333 1.00000 1.76667 0.76667 0.73333 0.06667 0.13333 0.30000 0.23333 -0.10000 0.36667 0.96667 1.23333 1.33333 1.46667 0.76667 1.06667 0.33333 0.63333 0.86667 0.56667 -0.20000 0.03333 -0.23333 1.00000 0.63333 0.56667 -0.26667 0.50000 -0.16667 -0.50000 -1.53333 -1.96667
44.5333 44.1000 43.3667 42.7000 42.4000 42.1333 41.3667 41.3000 41.5000 42.5000 42.6333 42.8667 42.9000 43.0333 43.0000 42.8667 42.7667 42.6333 42.4667 42.6000 42.9333 43.7333 44.2667 44.7333 44.7667 44.9000 44.9667 45.1000 45.1333 45.2333 45.5667 46.1667 46.8333 47.5333 48.0333 48.5667 48.7667 49.1333 49.4333 49.8000 49.8667 49.8333 49.7000 50.0667 50.3333 50.7667 50.7000 50.8667 50.8000 50.7333 50.0667
اﻵن ﻧﺨﺘﺒﺮ ﻣﺘﻮﺳﻂ اﻟﺒﻮاﻗﻲ MTB > TTest 0.0 'RESI1'; SUBC> Alternative 0.
T-Test of the Mean
١٤٩
Test of mu = 0.000 vs mu not = 0.000 Variable RESI1
N 57
Mean 0.158
StDev 0.868
SE Mean 0.115
T 1.37
P 0.17
ﻳﺴﺘﺨﺪم ﺑﺮﻧﺎﻣﺞ ﻋﺎم ﻋﻨﺪﻣﺎ ﻳﻜﻮن اﻹﻧﺤﺮاف اﻟﻤﻌﻴﺎري ) او اﻟﺘﺒﺎﻳﻦ ( ﻏﻴﺮMinitab ﻓﻲ:ﻣﻼﺣﻈﺔ أي١٫٩٦ وهﻲ اﻗﻞ ﻣﻦT=١٫٣٧ ﻻﺣﻆ ان ﻗﻴﻤﺔ اﻹﺣﺼﺎﺋﺔ هﻲ. Ttest ﻣﻌﺮوف وﻳﻄﻠﻖ ﻋﻠﻴﻪ ﻻﻧﺮﻓﺾ اﻟﻔﺮﺿﻴﺔ اﻟﺼﻔﺮﻳﺔ ﺣﻮل اﻟﻤﺘﻮﺳﻂ وﺣﻮل اﻟﺼﻔﺮRuns test ﻧﺨﺘﺒﺮ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺠﺮي:ﺛﺎﻧﻴﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > Runs 'RESI1'.
Runs Test
RESI1 K =
0.1579
The observed number of runs = 17 The expected number of runs = 29.4211 30 Observations above K 27 below The test is significant at 0.0009 MTB > Runs 0 'RESI1'.
Runs Test
RESI1 K =
0.0000
The observed number of runs = 17 The expected number of runs = 28.7895 33 Observations above K 24 below The test is significant at 0.0013
ﻧﻼﺣﻆ اﻧﻪ ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﻻﻧﺮﻓﺾ ﻋﺸﻮاﺋﻴﺔ اﻟﺒﻮاﻗﻲ Autocorrelation test ﻧﺨﺘﺒﺮ ﺗﺮاﺑﻂ أو إﺳﺘﻘﻼل اﻟﺒﻮاﻗﻲ ﺑﻮاﺳﻄﺔ إﺧﺘﺒﺎر اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ:ﺛﺎﻟﺜﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > %ACF 'RESI1'. Executing from file: E:\MTBWIN\MACROS\ACF.MAC
١٥٠
Autocorrelation
Autocorrelation Function for RESI1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
4
9
14
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1 2 3 4 5 6 7
0.56 0.24 -0.01 0.04 0.02 -0.07 -0.14
4.24 1.39 -0.06 0.21 0.11 -0.38 -0.81
18.93 22.31 22.32 22.40 22.42 22.72 24.07
8 9 10 11 12 13 14
-0.04 0.12 0.29 0.24 0.20 0.05 0.03
-0.22 0.68 1.64 1.30 1.05 0.25 0.17
24.17 25.21 31.36 35.67 38.71 38.90 38.99
أيρ1 = 0 أي اﻧﻨﺎ ﻧﺮﻓﺾ ان٤٫٢٤ ﻟﻠﺘﺮاﺑﻂ اﻟﺬاﺗﻲ ﻋﻨﺪ اﻟﺘﺨﻠﻒ اﻷول ﺗﺴﺎويT ﻻﺣﻆ ان اﻟـ ﻳﻮﺟﺪ ﺗﺮاﺑﻂ ﺑﻴﻦ اﻟﺒﻮاﻗﻲ ﻣﻦ اﻟﺪرﺟﺔ اﻻوﻟﻰ ﻓﻲ اﻹﺧﺘﺒﺎر H 0 : ρ1 = 0
H1 : ρ1 ± 0 r1 = 4.24 ﺣﻴﺚ اﻹﺣﺼﺎﺋﺔ هﻲ se ( r1 ) ﻧﺨﺘﺒﺮ ﻓﻲ ﻣﺎ إذا آﺎﻧﺖ اﻟﺒﻮاﻗﻲ ﻣﻮزﻋﺔ ﻃﺒﻴﻌﻴﺎ:راﺑﻌﺎ :ﺗﺎﺑﻊ اﻟﻤﺜﺎل MTB > %NormPlot 'RESI1'; SUBC> Kstest. Executing from file: E:\MTBWIN\MACROS\NormPlot.MAC
Normal Probability Plot
.999
P ro b a b ility
.99 .95 .80 .50 .20 .05 .01 .001 -2
-1
0
1
RESI1 Average: 0.157895 StDev: 0.867525 N: 57
Kolmogorov-Smirnov Normality Test D+: 0.054 D-: 0.084 D : 0.084 Approximate P-Value > 0.15
أي ﻻﻧﺮﻓﺾ ﻓﺮﺿﻴﺔ اﻟﺘﻮزﻳﻊ٠٫٠٥ وهﻲ اآﺒﺮ ﻣﻦ٠٫١٥ اﻟﻨﺎﺗﺠﺔ ﺗﺴﺎويP-Value ﻻﺣﻆ اﻟـ واﻟﺬي ﻳﺒﻴﻦ ﻣﺪيQ-Q Plot هﻨﺎك اﻳﻀﺎ إﺧﺘﺒﺎر ﺁﺧﺮ ﻟﻠﻄﺒﻴﻌﻴﺔ هﻮ اﻟـα = 0.05 اﻟﻄﺒﻴﻌﻲ ﻋﻨﺪ ﺗﻄﺎﺑﻖ ﻣﺸﺎهﺪات ﻣﺎ ﻣﻊ ﺗﻮزﻳﻊ ﻣﻌﻴﻦ MTB > %Qqplot 'RESI1'; SUBC> Table; SUBC> Conf 95; SUBC> Ci. Executing from file: E:\MTBWIN\MACROS\Qqplot.MAC
١٥١
Distribution Function Analysis
Normal Dist. Parameter Estimates Data
: RESI1
Mean: StDev:
0.157895 0.867525
Normal Probability Plot for RESI1
99
0.157895
Mean:
0.867525
StDev: 95 90 80
60 50 40
Percent
70
30 20 10 5
1
2
0
1
-1
-2
Data
ﻻﺣﻆ ان ﻓﻲ آﻠﺘﺎ اﻟﺤﺎﻟﺘﻴﻦ ﻓﺈﻧﻨﺎ ﻻﻧﺮﻓﺾ ان اﻟﺒﻮاﻗﻲ ﻟﻬﺎ ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻣﻼﺣﻈﺔ اﺧﻴﺮة :ﻳﺒﺪو ان اﻟﺒﻮاﻗﻲ ﺗﺤﻘﻖ ﻣﻌﻈﻢ اﻟﺸﺮوط ﻓﻴﻤﺎ ﻋﺪي اﻟﺘﺮاﺑﻂ اﻟﺬي ﻳﻮﺟﺪ ﺑﻴﻦ اﻟﻘﻴﻢ اﻟﻤﺘﺘﺎﻟﻴﺔ وهﺬا ﻳﺠﻌﻠﻨﺎ ﻧﺮﻓﺾ ﺟﻮدة اﻟﺘﻄﺒﻴﻖ ﻟﻄﺮﻳﻘﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ ﺣﻴﺚ ادي اﻟﻲ ﺑﻮاﻗﻲ ﻣﺘﺮاﺑﻄﺔ.
١٥٢
اﻟﻔﺼﻞ اﻟﺘﺎﺳﻊ ﺗﺤﻠﻴﻞ او ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻲ ﻣﺮآﺒﺎت :Decomposition Method ﻳﻨﻈﺮ إﻟﻰ اﻟﻤﺘﻠﺴﻠﺴﺔ اﻟﺰﻣﻨﻴﺔ ﻋﻠﻰ اﻧﻬﺎ ﻣﻜﻮﻧﺔ ﻣﻦ ﻋﺪة ﻣﺮآﺒﺎت أو اﺟﺰاء ﻣﺘﺤﺪة ﻣﻊ ﺑﻌﻀﻬﺎ ﻟﺘﻜﻮﻳﻦ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ، ﻟﻨﻔﺘﺮض ان ﻟﺪﻳﻨﺎ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪﻩ . z1 , z2 ,..., znﻟﻘﺪ وﺟﺪ ﺑﺎﻟﺘﺠﺮﺑﺔ أﻧﻪ ﻳﻤﻜﻦ ﻧﻤﺬﺟﺘﻬﺎ ﻋﻠﻰ اﻟﺸﻜﻞ zt = Tt + St + Ct + Et , t = 1, 2,..., n
ﺣﻴﺚ Ztاﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻤﺸﺎهﺪة و Ttﻣﺮآﺒﺔ اﻹﻧﺠﺮاف وهﻲ اﻟﺘﻲ ﺗﻨﻤﺬج اﻹﺗﺠﺎﻩ اﻟﻌﺎم اﻟﺬي ﺗﻨﺤﻲ أو ﺗﻨﺠﺮف اﻟﻴﻪ اﻟﻤﺘﺴﻠﺴﻠﺔ و Stﻣﺮآﺒﺔ ﻣﻮﺳﻤﻴﺔ وﺗﻨﻤﺬج اﻟﺘﺄﺛﻴﺮ اﻟﻤﻮﺳﻤﻲ )إذا وﺟﺪ( وهﻮ اﻟﺘﻐﻴﺮ اﻟﺬي ﻳﺤﺪث ﻟﻠﻤﺘﺴﻠﺴﻠﺔ ﻧﺘﻴﺠﺔ اﻟﺘﺄﺛﻴﺮات اﻟﻤﻮﺳﻤﻴﺔ ﻣﺜﻞ اﻟﺸﻬﺮﻳﺔ واﻟﺴﻨﻮﻳﺔ و Ctﻣﺮآﺒﺔ دورﻳﺔ )إذا وﺟﺪت( و ﺗﻨﻤﺬج ﻣﻨﺤﻰ أو إﺗﺠﺎة ﻳﺘﻜﺮر ﺑﻌﺪ ﻓﺘﺮات زﻣﻨﻴﺔ ﻃﻮﻳﻠﺔ ﻏﻴﺮ ﻣﻮﺳﻤﻴﺔ و Et ﻣﺮآﺒﺔ اﻟﺨﻄﺄ وﺗﺸﻤﻞ ﺟﻤﻴﻊ اﻟﻌﻮاﻣﻞ اﻻﺧﺮى اﻟﺘﻲ ﺗﺆﺛﺮ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ واﻟﺘﻲ ﻻﻳﻤﻜﻦ ﻧﻤﺬﺟﺘﻬﺎ ﺿﻤﻨﻴﺎ أو اﻟﺘﻲ ﻻﻳﻤﻜﻦ ﻣﺸﺎهﺪﺗﻬﺎ او ﻗﻴﺎﺳﻬﺎ .اﻟﻨﻤﻮذج اﻟﺴﺎﺑﻖ ﻳﺴﻤﻲ ﺑﺎﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ Additive Modelوذﻟﻚ ﻷن آﻞ اﻟﻤﺮآﺒﺎت ﺗﺪﺧﻞ ﺑﺸﻜﻞ إﺿﺎﻓﻲ ﻓﻲ اﻟﻨﻤﻮذج .هﻨﺎك أﺷﻜﺎل اﺧﺮى ﻣﺜﻞ zt = Tt St + Ct + Et , t = 1, 2,..., n zt = Tt St Ct + Et , t = 1, 2,..., n
واﻟﺘﻲ ﺗﺴﻤﻰ ﺑﺎﻟﻨﻤﺎذج اﻟﺘﻀﺎﻋﻔﻴﺔ . Multiplicative Models ﻓﻲ هﺬا اﻟﻤﺴﺘﻮى ﺳﻮف ﻧﻬﻤﻞ اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ Ctوذﻟﻚ ﻷن اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ ﻧﺎدرا ﻣﺎﺗﻜﻮن ﻣﻮﺟﻮدة ﻓﻲ اﻟﻤﺘﺴﻠﺴﻼت اﻟﻘﺼﻴﺮة أو اﻟﻄﻮﻳﻠﺔ ﻧﺴﺒﻴﺎ ﻷﻧﻬﺎ ﺗﺤﺘﺎج اﻟﻰ ﻣﺸﺎهﺪات ﻃﻮﻳﻠﺔ ﺟﺪا ﻋﻠﻰ ﻣﺪي ﻋﺪد آﺒﻴﺮ ﻣﻦ اﻟﻌﻘﻮد. وﻧﻜﺘﻔﻲ ﺑﺎﻟﻨﻤﺎذج ﻋﻠﻲ اﻟﺸﻜﻞ zt = Tt + St + Et , t = 1,2,..., n
zt = Tt St + Et , t = 1,2,..., n أﻧﻈﺮ آﺘﺎب FORECASTING: METHODS AND APPLICATIONSﻟﻠﻤﺆﻟﻔﻴﻦ MAKRIDAKIS/ WHEELWRIGHT/ McGEEص ١٤١-١٣١ ﺳﻮف ﻧﺴﺘﻌﺮض ﻓﻲ اﻟﻤﺜﺎل اﻟﺘﺎﻟﻲ ﻃﺮق اﻟﺘﺤﻠﻴﻞ اﻹﺿﺎﻓﻴﺔ واﻟﺘﻀﺎﻋﻔﻴﺔ ﺑﺪون اﻟﻤﺮآﺒﺔ اﻟﺪورﻳﺔ أي اﻟﻨﻤﺎذج zt = Tt + St + Et , t = 1,2,..., n
zt = Tt St + Et , t = 1,2,..., n اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ هﻲ اﻟﻄﻠﺐ ﻋﻠﻲ اﻟﺒﻨﺰﻳﻦ ﺑﻤﻼﻳﻴﻦ اﻟﻠﺘﺮات ﻓﻲ ﻣﺪﻳﻨﺔ اوﻧﺘﺎرﻳﻮ ﺑﻜﻨﺪة ﻣﻦ ﺳﻨﺔ ١٩٦٠ وﺣﺘﻰ ﺳﻨﺔ ١٩٧٥ GasDemand MONTHLY GASOLINE DEMAND ONTARIO GALLON MILLIONS 1960-1975 87695 86890 96442 98133 113615 123924 128924 134775 117357 114626 107677 108087 92188 88591 98683 99207 125485 124677 132543 140735 124008 121194 111634 111565 101007 94228 104255 106922 130621 125251 140318 146174 122318 128770 117518 115492 108497 100482 106140 118581 132371 132042 151938 150997 130931 137018 121271 123548 109894 106061 112539 125745 136251 140892 158390 148314 144148 140138 124075 136485 109895 109044 122499 124264 142296 150693 163331 165837 151731 142491 140229 140463 116963 118049 137869 127392 154166 160227 165869 173522 155828 153771 143963 143898 124046 121260 138870 129782 162312 167211 172897 189689 166496 160754 155582 145936 139625 137361 138963 155301 172026 165004 185861 190270 163903 174270 160272 165614 146182 137728 148932 156751 177998 174559 198079 189073 175702 180097 155202 174508
١٥٣
154277 174176 198688 218099 227443 193522 199024 217775
144998 184416 190474 229001 233038 212870 191813 227621
159644 158167 194502 203200 234119 248565 195997
168646 156261 190755 212557 255133 221532 208684
166273 176353 166286 197095 216478 252642 244113
190176 175720 170699 193693 232868 255007 243108
205541 193939 181468 188992 221616 206826 255918
193657 201269 174241 175347 209893 233231 244642
182617 218960 210802 196265 194784 212678 237579
189614 209861 212262 203526 189756 217173 237579
أوﻻ ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻓﻲ ﻣﺨﻄﻂ زﻣﻨﻲ MTB > TSPlot 'GasDemand'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
G asD em and
250
200
150
100
100 150 اﻟﻰ اﻻﻋﻠﻰ ﻣﻮﺳﻤﻴﺔ وﻣﻨﺠﺮﻓﺔ ﻧﻼﺣﻆ ان اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ zt = Tt + St + Et , t = 1, 2,..., n : ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ:اوﻻ 50
Index
SUBC> SUBC> SUBC> SUBC>
MTB > %Decomp 'GasDemand' 12; Additive ; Forecasts 24; Title "Forecast of Gasoline Demand"; Start 1.
Time Series Decomposition Data Length NMissing
GasDeman 192.000 0
Trend Line Equation Yt = 96.4074 + 0.680579*t Seasonal Indices Period 1 2 3 4 5 6 7 8 9 10 11 12
Index -20.5625 -26.8125 -14.8958 -11.0625 9.89583 11.8958 22.7708 25.1875 5.64583 7.27083 -4.81250 -4.52083
١٥٤
Accuracy of Model MAPE: MAD: MSD:
3.6952 5.6622 52.7851
Forecasts Row
Period
Forecast
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
207.197 201.627 214.225 218.738 240.377 243.058 254.614 257.711 238.850 241.155 229.753 230.725 215.364 209.794 222.391 226.905 248.544 251.225 262.780 265.878 247.017 249.322 237.919 238.892
Forecast of Gasoline Demand Seasonal Indices
Original Data, by Seasonal Period
30
250
20 10
200
0 150
-10 -20
100
-30 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period
Residuals, by Seasonal Period 30 20
10
10 0 5
-10 -20
0
-30 1 2 3 4 5 6 7 8 9 10 11 12
١٥٥
1 2 3 4 5 6 7 8 9 10 11 12
(١) ﺷﻜﻞ
ﺷﻜﻞ )(٢ Forecast of Gasoline Demand Original Data
Detrended Data 50 40 30 20 10 0 -10 -20 -30 -40 100
200
250 200 150 100 200
0
Seasonally Adj. and Detrended Data
0
100
Seasonally Adjusted Data 250
30 20 10
200
-10
150
0 -20 -30 100
200
ﺷﻜﻞ )(٣
100 200
0
0
100
Forecast of Gasoline Demand
Actual Predicted
250
Actual Predicted Forecast
200
Forecast
GasDeman
150
100 3.6952 5.6622 52.7851
MAPE: MAD: MSD:
200
100
0
Time
ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ: ﺷﻜﻞ ) (١ﻳﻮﺿﺢ اﻟﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ ، Seasonal Indicesﻓﺎﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﺴﺎر ﻳﺒﻴﻦ ﺗﺄﺛﺮاﻟﻄﻠﺐ ﻓﻲ اﻷﺷﻬﺮ اﻟﻤﺨﺘﻠﻔﺔ ﻣﻦ اﻟﺴﻨﺔ ﻓﻔﻲ اﻷﺷﻬﺮ ١١و ١٢و ١و ٢و ٣و ٤ﻳﺤﺪث ﻧﻘﺺ ﻓﻲ اﻟﻄﻠﺐ إذ ﻳﺘﻨﺎﻗﺺ ﺗﺪرﻳﺠﻴﺎ ﺣﺘﻰ ﻳﺼﻞ إﻟﻰ أﻗﻞ ﻣﻌﺪل ﻟﻪ ﻓﻲ اﻟﺸﻬﺮ ٢ﺛﻢ ﻳﺘﺰاﻳﺪ ﺣﺘﻰ ﻳﺼﺒﺢ ﻣﻮﺟﺒﺎ ﻓﻲ اﻟﺸﻬﺮ ٥وﻳﺘﺰاﻳﺪ ﺣﺘﻰ ﻳﺼﻞ اﻗﺼﻰ ﻗﻴﻤﺔ ﻣﻮﺟﺒﺔ ﻓﻲ اﻟﺸﻬﺮ ٨ﺛﻢ ﻳﻨﻘﺺ ﺑﺸﻜﻞ آﺒﻴﺮ ﺑﻌﺪﺋﺬ .اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﻤﻴﻦ ﻳﻌﻄﻲ رﺳﻢ اﻟﺼﻨﺪوق Box Plotﻟﻠﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ﻣﻮزﻋﺔ ﻋﻠﻰ اﻷﺷﻬﺮ وهﻮ ﻳﻮﺿﺢ ﺗﻮزﻳﻊ وإﻧﺘﺸﺎر اﻟﻤﺸﺎهﺪات ﻋﻠﻰ آﻞ ﺷﻬﺮ واﻟﻘﻴﻢ اﻟﺨﺎرﺟﺔ . Out Liers اﻟﺸﻜﻞ اﻷﺳﻔﻞ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﺘﻐﻴﺮ اﻟﻨﺴﺒﻲ اﻟﻤﺌﻮي ﻋﻠﻰ اﻟﻔﺘﺮات اﻟﻤﻮﺳﻤﻴﺔ )اﻷﺷﻬﺮ( .اﻟﺸﻜﻞ اﻷﺳﻔﻞ اﻷﻳﻤﻦ ﻳﻌﻄﻲ رﺳﻢ اﻟﺼﻨﺪوق ﻟﻠﺒﻮاﻗﻲ أو اﻷﺧﻄﺎء ﻣﻮزﻋﺔ ﻋﻠﻰ اﻷﺷﻬﺮ. ﺷﻜﻞ ) (٢اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﻤﻴﻦ ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ،اﻟﺸﻜﻞ اﻷﻋﻠﻰ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات ﺑﻌﺪ إزاﺣﺔ ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف أي wt = zt − Tt , t = 1, 2,..., n =St + Et , t = 1, 2,..., n
١٥٦
اﻟﺸﻜﻞ اﻷﺳﻔﻞ ﻣﻦ اﻟﻴﺴﺎر ﻳﻌﻄﻲ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ ﺑﻌﺪ إزاﺣﺔ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ أي yt = zt − St , t = 1, 2,..., n =Tt + Et , t = 1, 2,..., n
أو اﻟﺒﻮاﻗﻲ ﺑﻌﺪ إزاﺣﺔ ﻣﺮآﺒﺘﻲ اﻹﻧﺠﺮافEt اﻟﺸﻜﻞ اﻷﺳﻔﻞ اﻷﻳﻤﻦ ﻳﻌﻄﻲ ﻣﺮآﺒﺔ اﻟﺨﻄﺄ واﻟﻤﻮﺳﻤﻴﺔ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ أي et = zt − Tt − St , t = 1, 2,..., n =Et , t = 1, 2,..., n
. اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ ﻣﻘﺎﻳﻴﺲ دﻗﺔ اﻟﺘﻄﺒﻴﻖ٢٤ ( ﻳﻌﻄﻲ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ٣) ﺷﻜﻞ zt = Tt St + Et , t = 1, 2,..., n
: ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:ﺛﺎﻧﻴﺎ
MTB > %Decomp 'GasDemand' 12; SUBC> Forecasts 24; SUBC> Title "Forecast of Gasoline Demand"; SUBC> Start 1. Executing from file: D:\MTBWIN\MACROS\Decomp.MAC Macro is running ... please wait
Time Series Decomposition Data Length NMissing
GasDeman 192.000 0
Trend Line Equation Yt = 96.4074 + 0.680579*t Seasonal Indices Period
Index
1 2 3 4 5 6 7 8 9 10 11 12
0.860355 0.828555 0.892431 0.936273 1.06124 1.07274 1.15775 1.17075 1.03409 1.05059 0.966300 0.968923
Accuracy of Model MAPE: MAD: MSD:
3.6338 5.7720 56.8996
Forecasts Row
Period
Forecast
1 2
193 194
195.954 189.275
١٥٧
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
204.474 215.156 244.596 247.977 268.415 272.227 241.154 245.718 226.660 227.935 202.980 196.042 211.762 222.803 253.263 256.738 277.870 281.789 249.599 254.298 234.552 235.848
(٤) ﺷﻜﻞ Forecast of Gasoline Demand Seasonal Indices
Original Data, by Seasonal Period
1.2
250
1.1
200
1.0 150 0.9 100 0.8 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period 14 12 10 8 6 4 2 0
Residuals, by Seasonal Period 20 10 0 -10 -20 -30
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
(٥) ﺷﻜﻞ Forecast of Gasoline Demand Original Data
Detrended Data
250
1.3 1.2
200
1.1 1.0
150
0.9 100
0.8 0
100
200
0
Seasonally Adjusted Data 240 220 200 180 160 140
١٥٨
100
200
Seasonally Adj. and Detrended Data 20 10 0 -10
Forecast of Gasoline Demand
ﺷﻜﻞ )(٦ Actual
280
Predicted Forecast
180
3.6338 5.7720 56.8996
MAPE: MAD: MSD:
GasDeman
Actual Predicted Forecast
80
200
100
0
Time
ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ: اﻷﺷﻜﺎل ) (٤و ) (٥و ) (٦ﻟﻬﺎ ﻧﻔﺲ اﻟﺘﻔﺴﻴﺮ آﻤﺎ ﻓﻲ اﻷﺷﻜﺎل ) (١و ) (٢و ).(٣ ﺑﻤﺎ اﻧﻨﺎ ﻃﺒﻘﻨﺎ ﻧﻤﻮذﺟﻴﻦ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺸﺎهﺪة ﻓﻴﺠﺐ أن ﻧﺨﺘﺎر اﻓﻀﻞ ﻧﻤﻮذج ،وهﻨﺎ ﻳﺄﺗﻲ دور ﻣﻘﺎﻳﻴﺲ دﻗﺔ اﻟﺘﻄﺒﻴﻖ واﻟﺘﻲ ﺗﻨﺘﺞ ﻣﻦ اﻟﺒﺮﻧﺎﻣﺞ ،ﻟﺪﻳﻨﺎ ﺛﻼﺛﺔ ﻣﻘﺎﻳﻴﺲ دﻗﺔ: -١ﻣﺘﻮﺳﻂ اﻟﺨﻄﺄ اﻟﻨﺴﺒﻲ اﻟﻤﻄﻠﻖ Mean Absolute Percentage Errorأو MAPE وﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ zt − zˆt zt × 100, zt ≠ 0 n
n
∑ t =1
= MAPE
-٢ﻣﺘﻮﺳﻂ اﻹﻧﺤﺮاف اﻟﻤﻄﻠﻖ Mean Absolute Deviationأو MADوﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ n
− zˆt
t
∑z t =1
n
= MAD
-٣ﻣﺘﻮﺳﻂ أﻹﻧﺤﺮاف اﻟﻤﺮﺑﻊ )أو ﻣﺘﻮﺳﻂ اﻟﺨﻄﺄ اﻟﻤﺮﺑﻊ( ) MSDأو ( MSEوﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ 2
) − zˆt
n
t
∑( z
n
t =1
= MSD
ﺑﺈﺧﺘﻴﺎر أﺣﺪ هﺬة اﻟﻤﻘﺎﻳﻴﺲ ﻧﺨﺘﺎر اﻟﻨﻤﻮذج اﻟﺬي ﻳﻌﻄﻲ أﻗﻞ ﻗﻴﻤﺔ ﻟﻬﺬا اﻟﻤﻘﻴﺎس ،اﻟﻤﻘﻴﺎس اﻷآﺜﺮ إﺳﺘﺨﺪاﻣﺎ وﺷﻴﻮﻋﺎ هﻮ MSDأو MSEوهﻮ اﻟﺬي ﺳﻮف هﻨﺎ. ﻟﻠﻨﻤﻮذج اﻹﺿﺎﻓﻲ ﻣﻘﺎﻳﻴﺲ اﻟﺪﻗﺔ هﻲ: 3.6952 ١٥٩
MAPE:
MAD: MSD:
5.6622 52.7851 و ﻟﻠﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:
MAPE: 3.6338 MAD: 5.7720 MSD: 56.8996 ﻧﻼﺣﻆ أن اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ اﻋﻄﻰ اﻗﻞ ﻗﻴﻤﺔ ﻟﻠﻤﻘﻴﺎس MSDوﻟﺬﻟﻚ ﻧﻘﺮر إﺳﺘﺨﺪام هﺬا اﻟﻨﻤﻮذج ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻄﻠﺐ.
ﺗﻮﺿﻴﺢ ﻃﺮﻳﻘﺔ ﺗﺤﻠﻴﻞ او ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻲ ﻣﺮآﺒﺎت Decomposition :Method ﺳﻮف ﻧﻮﺿﺢ ﺑﺎﻟﻤﺜﺎل اﻟﺘﺎﻟﻲ ﻃﺮﻳﻘﺔ ﺗﺤﻠﻴﻞ أو ﺗﻔﻜﻴﻚ اﻟﻤﺘﺴﻠﺴﻠﺔ إﻟﻰ ﻣﺮآﺒﺎت .اﻟﺒﻴﺎﻧﺎت اﻟﺘﺎﻟﻴﺔ إﻧﺘﺎج ١٦٨ﻳﻮﻣﺎ ﻟﻠﺤﻠﻴﺐ ﻓﻲ أﺣﺪ اﻟﻤﺰارع ﺑﺎﻟﻜﻴﻠﻮ ﺟﺮام MTB > Read "E:\Mtbwin\milk.dat" c1. Entering data from file: E:\Mtbwin\milk.dat 168 rows read. 'MTB > name c1='MilkProd MTB > print c1
Data Display MilkProd 582 598 634 635 688 698 711 734 751 800 805 802 813 843
553 565 594 602 645 660 677 690 711 763 766 760 773 797
577 587 611 621 667 687 706 723 747 800 810 791 812 827
568 583 604 615 661 681 701 725 740 790 809 784 815 817
599 617 639 653 697 722 740 764 783 834 855 837 867 858
640 660 678 702 735 767 783 801 819 869 894 881 908 896
697 716 736 756 798 817 826 845 859 913 935 924 947 937
727 742 770 782 811 837 858 871 886 942 961 957 969 966
640 653 688 709 736 762 775 785 807 860 890 883 889 892
656 673 705 722 755 784 796 805 824 878 900 898 902 903
561 566 618 622 635 667 696 690 707 756 799 773 778 782
589 600 628 658 677 713 717 734 750 804 826 821 828 834
وﻧﺮﺳﻢ اﻟﺒﻴﺎﻧﺎت اﻟﺴﺎﺑﻘﺔ ﺑﺎﻷواﻣﺮ: ;'MTB > TSPlot 'MilkProd >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol >SUBC Connect.
1000
900
700
600
150
100
١٦٠
50
Index
MilkProd
800
: ﺗﻄﺒﻴﻖ اﻟﻨﻤﻮذج اﻹﺿﺎﻓﻲ:اوﻻ ﻟﻜﻲzt = Tt + St + Et , t = 1, 2,..., n ﻧﻜﺘﺐ اﻟﻨﻤﻮذج ﻋﻠﻰ اﻟﺸﻜﻞz1 , z2 ,..., zn ﻟﻤﺸﺎهﺪات :ﻧﻘﻮم ﺑﺘﻔﻜﻴﻚ هﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺰﻣﻨﻴﺔ اﻟﻲ اﻟﻤﺮآﺒﺎت اﻟﺴﺎﺑﻘﺔ ﺳﻮف ﻧﺴﺘﻌﺮض ﻃﺮﻳﻘﺘﻴﻦ ﻣﻤﻜﻨﺔ :اﻟﻄﺮﻳﻘﺔ اﻻوﻟﻲ : أيTt ﻧﻄﺒﻖ إﻧﺤﺪار ﺧﻄﻲ ﺑﺴﻴﻂ ﻟﻠﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ ﻟﺘﻘﺪﻳﺮ ﻣﺮآﺒﺔ اﻻﻧﺠﺮاف-١ Tˆt ≡ zˆt = a + bt , t = 1, 2,...,168
:أي MTB > set c2 DATA> 1:168 DATA> end MTB > name c1='MilkProd' c2='Time' c3='Trend' c5='Detrend' c6='Index' c8='Fitted' c9='Resid' MTB > regr c1 1 c2; SUBC> fits c3.
Regression Analysis The regression equation is MilkProd = 612 + 1.69 Time Predictor Constant Time S = 60.74
Coef StDev T P 611.682 9.414 64.97 0.000 1.69262 0.09663 17.52 0.000 R-Sq = 64.9% R-Sq(adj) = 64.7%
Analysis of Variance Source Regression Error Total
DF 1 166 167
SS 1132003 612439 1744443
MS 1132003 3689
F 306.83
P 0.000
وﺷﻜﻞ اﻹﻧﺠﺮاف هﻮ
900
Trend
800
700
600
Index
50
100
150
ﻧﻄﺮح ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف ﻣﻦ اﻟﻤﺸﺎهﺪات اﻻﺻﻠﻴﺔ ﻓﻨﺤﺼﻞ ﻋﻠﻲ ﻣﺎﻳﺴﻤﻲ ﺑﺎﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ-٢ zt − zˆt = zt − Tˆt , t = 1, 2,...,168 أيDetrended Series اﻹﻧﺠﺮاف MTB > let c5=c1-c3
:وﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
١٦١
Detrend
100
0
-100 Index
50
100
150
zt − Tˆt = St + Et , t = 1, 2,...,168 ﻧﻼﺣﻆ اﻧﻬﺎ اﺻﺒﺤﺖ ﻣﻮﺳﻤﻴﺔ ﻓﻘﻂ ﻷن : آﺎﻟﺘﺎﻟﻲSeasonal Indices ﻟﺘﻘﺪﻳﺮ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻧﻮﺟﺪ اﻟﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ-٣ اﻟﻤﺆﺷﺮ اﻟﻤﻮﺳﻤﻲ ﻟﻠﺸﻬﺮI1 ﺣﻴﺚI s , s = 1, 2,...,12 ﻟﻨﺮﻣﺰ ﻟﻠﻤﺆﺷﺮات اﻟﻤﻮﺳﻤﻴﺔ ﺑﺎﻟﺮﻣﺰ d t = zt − Tˆt , t = 1, 2,...,168 اﻟﻤﺆﺷﺮ اﻟﻤﻮﺳﻤﻲ ﻟﻠﺸﻬﺮ اﻟﺜﺎﻧﻲ وهﻜﺬا وﻟﻨﺮﻣﺰ ﺑـI 2 اﻷول و :ﺗﻘﺪر هﺬﻩ اﻟﻤﺆﺷﺮات آﺎﻟﺘﺎﻟﻲ
1 ( d1 + d13 + d 25 + L + d157 ) 14 1 I 2 = ( d 2 + d14 + d 26 + L + d158 ) 14 M I1 =
I12 =
1 ( d12 + d 24 + d 36 + L + d168 ) 14 :وﻳﺘﻢ ذﻟﻚ ﺑﺈﺳﺘﺨﺪام اﻷواﻣﺮ اﻟﺘﺎﻟﻴﺔ
MTB > DATA> DATA> MTB > SUBC> SUBC> MTB > & CONT> MTB > MTB > MTB > DATA> DATA> MTB >
set c4 14(1:12) end stat c5; by c4; mean c6. Stack 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' 'Index' c7. let c8=c3+c7 let c9=c1-c8 set c10 1:12 end print c10 c6
Data Display
١٦٢
-٤اﻟﺘﻨﺒﺆات ﺗﻮﻟﺪ آﺎﻟﺘﺎﻟﻲ:
Index
Season
Row
-18.328 -57.806 34.716 49.595 110.616 82.281 32.517 -9.747 -52.297 -48.775 -79.754 -43.018
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
z168 ( l ) = 612 + 1.69 ( l + 168) + I ( l mod 12 ) , l = 1,2,...
ﻓﻤﺜﻼ اﻟﺘﻨﺒﺆ ﻋﻨﺪ اﻟﻴﻮم ١٦٩هﻮ
z168 (1) = 612 + 1.69 (169 ) + I1 =897.61 + ( −18.328 ) = 879.282
اﻟﻄﺮﻳﻘﺔ اﻟﺜﺎﻧﻴﺔ: وهﻲ اﻟﺘﻲ ﻳﺴﺘﺨﺪﻣﻬﺎ ﺑﺮﻧﺎﻣﺞ : Minitab -١آﺎﻟﻄﺮﻳﻘﺔ اﻻوﻟﻰ ﻧﻄﺒﻖ إﻧﺤﺪار ﺧﻄﻲ ﺑﺴﻴﻂ ﻟﻠﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ ﻟﺘﻘﺪﻳﺮ ﻣﺮآﺒﺔ اﻻﻧﺠﺮاف Ttﻓﻨﺤﺼﻞ ﻋﻠﻰ ﻧﻔﺲ اﻟﻨﺘﻴﺠﺔ آﻤﺎ ﻓﻲ اﻟﻄﺮﻳﻘﺔ اﻻوﻟﻰ )(١ -٢اﻳﻀﺎ هﻨﺎ ﻧﻄﺮح ﻣﺮآﺒﺔ اﻹﻧﺠﺮاف ﻣﻦ اﻟﻤﺸﺎهﺪات اﻻﺻﻠﻴﺔ ﻓﻨﺤﺼﻞ ﻋﻠﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ اﻹﻧﺠﺮاف Detrended Series -٣ﻧﻄﺒﻖ اﻵن ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ درﺟﺔ اﻟﻤﻮﺳﻢ وﻧﻮﺳﻄﻪ اذا اﺣﺘﺎج اﻻﻣﺮ -٤ﻧﻄﺮح اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ ﻣﻦ ﻧﻈﻴﺮاﺗﻬﺎ ﻓﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ ﻣﺰاﻟﺔ اﻹﻧﺠﺮاف ﻓﻨﺤﺼﻞ ﻋﻠﻲ ﻣﺘﺴﻠﺴﻠﺔ ﺗﺤﻮي اﻟﻤﺮآﺒﺎت اﻟﻤﻮﺳﻤﻴﺔ -٥ﺗﻘﺪر اﻟﻤﺮآﺒﺎت اﻟﻤﻮﺳﻤﻴﺔ آﺎﻟﺘﺎﻟﻲ: ) I1 = Median ( d1 , d13 , d 25 ,L, d157
) I 2 = Median ( d 2 , d14 , d 26 ,L, d158
M
) I12 = Median ( d12 , d 24 , d 36 ,L, d168 -٦ﺗﻮﻟﺪ اﻟﺘﻨﺒﺆات آﺎﻟﺴﺎﺑﻖ وﺳﻮف ﻧﺴﺘﻌﺮض هﺬا آﺎﻟﺘﺎﻟﻲ:
MTB > Read "E:\Mtbwin\milk.dat" c1. Entering data from file: E:\Mtbwin\milk.dat 168 rows read. 'MTB > name c1='MilkProd MTB > set c2 DATA> 1:168 DATA> end 'MTB > name c2='Time ;MTB > regr c1 1 c2 SUBC> fits c3.
Regression Analysis
١٦٣
The regression equation is MilkProd = 612 + 1.69 Time Predictor Constant Time
Coef 611.682 1.69262
S = 60.74
StDev 9.414 0.09663
R-Sq = 64.9%
T 64.97 17.52
P 0.000 0.000
R-Sq(adj) = 64.7%
Analysis of Variance Source Regression Error Total
DF 1 166 167
SS 1132003 612439 1744443
Unusual Observations Obs Time MilkProd 113 113 942.00 125 125 961.00
MS 1132003 3689
Fit 802.95 823.26
F 306.83
StDev Fit 5.44 6.11
P 0.000
Residual 139.05 137.74
St Resid 2.30R 2.28R
R denotes an observation with a large standardized residual MTB MTB MTB MTB
> > > >
name c3='Trend' let c4=c1-c3 name c4='Detrend' Name c5 = 'AVER1'
: وﻧﻮﺳﻄﻪ١٢ ﻓﻲ اﻟﺨﻄﻮة اﻟﺘﺎﻟﻴﺔ ﻧﻄﺒﻖ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ MTB > %MA 'Detrend' 12; SUBC> Center; SUBC> Averages 'AVER1'. Executing from file: E:\MTBWIN\MACROS\MA.MAC Macro is running ... please wait
Moving average Data Length NMissing
Detrend 168.000 0
Moving Average Length: 12 Accuracy Measures MAPE: 111.68 MAD: 52.36 MSD: 3564.77
ﻧﻄﺮح اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ اﻟﻤﻮﺳﻄﺔ ﻣﻦ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﺰال إﻧﺠﺮاﻓﻬﺎ MTB > MTB > MTB > DATA> DATA> MTB > SUBC> SUBC> MTB >
let c6=c4-c5 name c6='DeSeason' set c2 14(1:12) end stat c6; by c2; median c7. name c7='SeasInx'
Data Display Row
Season
SeasInx
١٦٤
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
-20.750 -58.958 35.625 50.083 109.542 81.292 33.917 -10.000 -52.792 -50.250 -79.958 -44.375
:ﻧﻘﺎرن اﻟﺤﺴﺎﺑﺎت اﻟﺘﻲ اﺟﺮﻳﻨﺎهﺎ ﻣﻊ اﻟﺒﺮﻧﺎﻣﺞ اﻷﺻﻠﻲ MTB > %Decomp 'MilkProd' 12; SUBC> Additive ; SUBC> Start 1. Executing from file: E:\MTBWIN\MACROS\Decomp.MAC Macro is running ... please wait
Time Series Decomposition Data Length NMissing
MilkProd 168.000 0
Trend Line Equation Yt = 611.682 + 1.69262*t Seasonal Indices Period 1 2 3 4 5 6 7 8 9 10 11 12
Index -20.1979 -58.4062 36.1771 50.6354 110.094 81.8437 34.4687 -9.44792 -52.2396 -49.6979 -79.4063 -43.8229
Accuracy of Model MAPE: MAD: MSD:
1.583 12.088 244.406
.وﺑﻤﻘﺎرﻧﺔ اﻟﻨﺘﻴﺠﺘﻴﻦ ﻧﺠﺪ اﻧﻬﻤﺎ ﺗﻘﺮﻳﺒﺎ ﻣﺘﺴﺎوﻳﺘﺎن : آﺎﻟﺘﺎﻟﻲ%Decomp ﺳﻮف ﻧﻮﻟﺪ ﺗﻨﺒﺆات ﺑﺎﺳﺘﺨﺪام اﻟﺒﺮﻧﺎﻣﺞ MTB > %Decomp 'MilkProd' 12; SUBC> Additive ; SUBC> Forecasts 12; SUBC> Start 1.
:واﻟﺘﻲ ﺗﻌﻄﻲ اﻟﺘﻨﺒﺆات Forecasts
١٦٥
Forecast
Period
Row
877.54 841.02 937.30 953.45 1014.60 988.04 942.36 900.13 859.04 863.27 835.25 872.53
169 170 171 172 173 174 175 176 177 178 179 180
1 2 3 4 5 6 7 8 9 10 11 12
ﺗﻤﺮﻳﻦ :وﻟﺪ ﺗﻨﺒﺆات ﻹﻧﺘﺎج اﻟﺤﻠﻴﺐ اﻟﻴﻮﻣﻲ ﺑﺈﺳﺘﺨﺪام اﻟﻄﺮﻳﻘﺘﻴﻦ اﻟﻤﻌﻄﺎة وﻗﺎرﻧﻬﺎ ﺑﺎﻟﻘﻴﻢ اﻷﺧﻴﺮة اﻟﻨﺎﺗﺠﺔ ﻣﻦ اﻟﺒﺮﻧﺎﻣﺞ
١٦٦
اﻟﻔﺼﻞ اﻟﻌﺎﺷﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك Using Moving Average Smoothing for Forecasting ﻳﺴﺘﺨﺪم اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻟﺘﻤﻬﻴﺪ اﻟﻤﺸﺎهﺪات وذﻟﻚ ﺑﺘﻘﻠﻴﻞ ﺗﺒﺎﻳﻦ اﻷﺧﻄﺎء ﻓﻤﺜﻼ ﻟﻮ آﺎن ﻟﺪﻳﻨﺎ ﻣﺸﺎهﺪات ﻣﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻓﺎﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ m ﻟﻠﻤﺸﺎهﺪات ﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ 1 zˆt = ( zt + zt −1 + zt −2 + L + zt −m+1 ) , t = m, m + 1,..., n m أو 1 zˆt = zˆt −1 + ( zt − zt −m ) , t = m, m + 1,..., n m ﻻﺣﻆ ان ﻋﺪد اﻟﻤﺸﺎهﺪات اﺻﺒﺢ ﺑﻌﺪ اﻟﺘﻤﻬﻴﺪ . n − m + 1 ﻓﻤﺜﻼ ﻟﻮ آﺎﻧﺖ m=٣ﻓﺈن اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ اﻟﺜﺎﻟﺜﺔ هﻮ 1 ) ( z3 + z2 + z1 3 1 1 ) zˆ4 = ( z4 + z3 + z2 ) or zˆ4 = zˆ3 + ( z4 − z1 3 3 M 1 1 ) zˆn = ( zn + zn −1 + zn −2 ) or zˆn = zˆn −1 + ( zn − zn −3 3 3 = zˆ3
وﻟﻜﻲ ﻧﺮى آﻴﻒ ﻳﻌﻤﻞ اﻟﺘﻤﻬﻴﺪ ﻟﺘﻘﻠﻴﻞ ﺗﺒﺎﻳﻦ اﻷﺧﻄﺎء ﻟﻨﻔﺘﺮض ان اﻟﻤﺸﺎهﺪات ﺗﺘﺒﻊ اﻟﻨﻤﻮذج WN ( 0, σ 2 ) , t = 1, 2,..., n
ﻓﻴﻜﻮن
zt = µ + at , at
V ( zt ) = σ , ∀t 2
وﺑﺎﻟﺘﺎﻟﻲ 2
σ
= ) V ( zˆt
, t = m, m + 1,..., n m أي ان اﻟﻤﺸﺎهﺪات اﻟﻤﻤﻬﺪة اﺻﺒﺢ ﺗﺒﺎﻳﻨﻬﺎ أﺻﻐﺮ ﺑـ mﺿﻌﻒ ﻣﻦ اﻟﻤﺸﺎهﺪات اﻷﺻﻠﻴﺔ وهﺬا اﻟﺘﻤﻬﻴﺪ ﻟﻸﺧﻄﺎء ﻳﻈﻬﺮ أي ﻧﻤﻂ ﻓﻲ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎن ﻣﺪﻓﻮﻧﺎ او ﻣﻐﻄﻰ ﻣﻦ ﺗﺄﺛﻴﺮ اﻷﺧﻄﺎء. ﻣﻼﺣﻈﺔ :ﺗﺆﺧﺬ mداﺋﻤﺎ ﻓﺮدﻳﺔ وذﻟﻚ ﻟﻨﺘﺠﻨﺐ ﺗﻮﺳﻴﻂ اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة.
اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك: ﻳﺆﺧﺬ آﻤﺘﻨﺒﺊ ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك:
zn ( l ) = zˆn −1 , l > 0
ﻣﺜﺎل: ﺑﺈﺳﺘﺨﺪام اﻟﺤﺰﻣﺔ اﻷﺣﺼﺎﺋﻴﺔ MINITABﻧﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﻧﻨﻈﺮ ﻣﺎذا ﺗﺤﻮي ﻣﻦ ﻣﺘﻐﻴﺮات MTB > info
١٦٧
Information on the Worksheet Column C1 C2 C3
Count 60 60 60
Name Trade Food Metals
Metals ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻟﻠﻤﺘﻐﻴﺮ Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
44.3 42.6 43.1 43.6 45.5 50.0 51.2
44.4 42.4 43.2 44.7 46.2 50.0 50.7
43.4 42.2 42.8 44.5 46.8 49.6 50.3
42.8 41.8 43.0 45.0 47.5 49.9 49.2
44.3 40.1 42.8 44.8 48.3 49.6 48.1
44.4 42.0 42.5 44.9 48.3 50.7
44.8 42.4 42.6 45.2 49.1 50.7
44.4 43.1 42.3 45.2 48.9 50.9
:ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات MTB > TSPlot 'Metals'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
Metals
50
45
40 Index
10
20
30
40
50
60
وﻧﻮﺟﺪ ﺗﻨﺒﺆاتm=٣ ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ : ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﻪ٦ ﻟـ MTB > %MA 'Metals' 3; SUBC> Forecasts 6; SUBC> Title "Smoothing and Forecasting Metals". Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average Data Length NMissing
Metals 60.0000 0
Moving Average Length: 3 Accuracy Measures MAPE: 1.55036
١٦٨
0.70292 0.76433 Lower
Upper
50.9135 50.9135 50.9135 50.9135 50.9135 50.9135
Forecast
Period
49.2 49.2 49.2 49.2 49.2 49.2
61 62 63 64 65 66
47.4865 47.4865 47.4865 47.4865 47.4865 47.4865
MAD: MSD:
Row 1 2 3 4 5 6
Smoothing and Forecasting Metals
Actual Predicted
50
Forecast
45 Moving Average 3
Metals
Actual Predicted Forecast
Length:
MAPE: 1.55036 0.70292
MAD:
0.76433
MSD:
40 60
ﺛﺎﻧﻴﺎ :ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ ﻓﻲ اﻟﻤﺜﺎل اﻟﺤﺎﻟﻲ
50
40
30
20
10
0
Time
50.3 + 49.2 + 48.1 147.6 = = 49.2 3 3 ﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟـ ٦اﻟﻤﺴﺘﻘﺒﻠﻴﺔ أي ﻟﻠﻘﻴﻢ zn +1 , zn +2 ,..., zn+6أو ﻓﻲ هﺬا اﻟﻤﺜﺎل z61 , z62 ,..., z66آﺎﻟﺘﺎﻟﻲ: z60 (1) = z60 ( 2 ) = L = z60 ( 6 ) = 49.2
= zˆ59
ﻟﺤﺴﺎب ﻓﺘﺮات ﺗﻨﺒﺆ 95%ﻧﺤﺴﺐ اﻟﻜﻤﻴﺎت ⎡⎣ zn ( l ) ± 1.96σˆ ⎤⎦ , l > 0أي ] ˆ [ 49.2 ± 1.96σﻟﺠﻤﻴﻊ ﻗﻴﻢ اﻟﺘﻨﺒﺆات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ،ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ MSD = 0.76433آﻤﻘﺪر ﻟـ
σ 2أي σˆ 2 = 0.76433ﻓﻴﻜﻮن σˆ = 0.8743وﻋﻠﻴﻪ ﺗﻜﻮن ﻓﺘﺮة ﺗﻨﺒﺆ 95%ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ هﻲ: ]⎡⎣ 49.2 ± 1.96 ( 0.8743) ⎤⎦ = [ 49.2 ± 1.7135] = [ 47.4865,50.9135 أي: z60+l ∈ [ 47.4865,50.9135] , l > 0 with probability 0.95 ﻣﻼﺣﻈﺔ :ﺗﺤﺴﺐ MSDآﺎﻵﺗﻲ
١٦٩
) − zˆi
n −1
∑( z
i
i =2
n−2
= ˆMSD = σ 2
ﺗﻤﺮﻳﻦ: ﻃﺒﻖ ﻣﺘﻮﺳﻄﺎت ﻣﺘﺤﺮآﺔ ﻣﻦ اﻟﺪرﺟﺎت ٥و ٧ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ وﻗﺮر اﻳﻬﺎ اﻓﻀﻞ ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ؟. اﻟﻮﺳﻴﻂ اﻟﺠﺎري Running Median اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﺘﺄﺛﺮ آﺜﻴﺮا ﺑﺎﻟﻤﺸﺎهﺪات اﻟﺨﺎرﺟﺔ Outliersاو اﻟﻤﺘﻄﺮﻓﺔ ﻓﺎﻟﻘﻴﻤﺔ اﻟﻤﺘﻄﺮﻓﺔ اﻟﻮاﺣﺪة ﺗﺆﺛﺮ ﻋﻠﻲ mﻣﻦ اﻟﻤﺘﻮﺳﻄﺎت اﻟﻤﺘﺤﺮآﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ﻓﻤﺜﻼ ﻟﻮآﺎﻧﺖ ﻟﺪﻳﻦ اﻟﻤﺸﺎهﺪات 18
13
15
11
12
1500
10
6
8
9
)z(t 5 7 20
3
وﻟﻬﺎ اﻟﺸﻜﻞ 1500
1000
)z(t 500
0 10
In d e x
5
ﺑﺄﺧﺬ ﻣﺘﻮﺳﻂ ﻣﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ ٣ﻧﺠﺪ M o v in g A v e r a g e
A c tu a l
1500
P re d ic te d A c tu a l P re d ic te d
1000
)z(t
M o v in g A v e ra g e 3 1081
500
L e n g th : M APE :
273
M AD :
268811
MSD:
0
15
5
10
0
T im e
ﻻﺣﻆ ان اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك اﻟﻨﺎﺗﺞ ﺗﺄﺛﺮت ﻓﻴﺔ ﺛﻼﺛﺔ ﻗﻴﻢ ﺑﺎﻟﻘﻴﻤﺔ اﻟﻤﺘﻄﺮﻓﺔ. ﻟﻠﺘﻐﻠﺐ ﻋﻠﻰ ﻣﺜﻞ هﺬﻩ اﻟﺼﻌﻮﺑﺎت ﻳﺴﺘﺨﺪم اﻟﻮﺳﻴﻂ اﻟﺠﺎري ذا اﻟﻄﻮل اﻟﻔﺮدي آﻤﻤﻬﺪ ﻏﻴﺮ ﺧﻄﻲ واﻟﺬي ﻻﻳﺘﺄﺛﺮ ﺑﺎﻗﻴﻢ اﻟﻤﺘﻄﺮﻓﺔ. اﻟﻮﺳﻴﻂ اﻟﺠﺎري ذا اﻟﻄﻮل اﻟﻔﺮدي j = 2i + 1ﻟﻤﺸﺎهﺪات z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ
z%t = med ( zt −i ,..., zt ,..., zt +i ) , j = 2i + 1
١٧٠
ﻓﻤﺜﻼ ﻟﻘﻴﻤﺔ j = 3ﺗﺼﺒﺢ اﻟﻌﻼﻗﺔ ) z%t = med ( zt −1 , zt , zt +1وﺑﺄﺧﺬ وﺳﻴﻂ ﺟﺎري ذا اﻟﻄﻮل ٣ ﻟﻠﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ
15
)smoothz(t
10
5
12
8
10
4
6
2
In d e x
واذا آﺎﻧﺖ اﻟﻘﻴﻤﺔ اﻟﺤﻘﻴﻘﻴﺔ ﻟـ z9هﻲ ١٥وﻟﻴﺲ ١٥٠٠ﻓﺈن اﻟﻤﺸﺎهﺪات اﻟﺤﻘﻴﻘﻴﺔ ﻟﻬﺎ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻲ
20
15
)z(t 10
5
10
ﻗﺎرن ﺑﻴﻦ اﻟﻨﺘﻴﺠﺘﻴﻦ.
١٧١
5
Index
اﻟﻔﺼﻞ اﻟﺤﺎدي ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ Using Single : Exponential Smoothing for Forecasting اﻟﺘﻤﻬﻴﺪ ﺑﻮاﺳﻄﺔ اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﻌﻄﻲ ﺟﻤﻴﻊ اﻟﺒﻴﺎﻧﺎت ﻧﻔﺲ اﻷهﻤﻴﺔ وﺑﺎﻟﺘﺎﻟﻲ ﻓﺈن اﻟﻘﻴﻢ اﻟﻘﺪﻳﻤﺔ ﻧﻮﻋﺎ ﺗﺆﺛﺮ ﻧﻔﺲ اﻟﺘﺄﺛﻴﺮ آﺎﻟﻘﻴﻢ اﻟﺤﺪﻳﺜﺔ وهﺬا ﻗﺪ ﻻﻳﻜﻮن ﻣﻦ اﻟﻨﺎﺣﻴﺔ اﻟﻌﻤﻠﻴﺔ ﺻﺤﻴﺤﺎ ،اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ ﻋﻠﻰ اﻟﻌﻜﺲ ﻳﻌﻄﻲ اﻟﻘﻴﻢ اﻷآﺜﺮ ﺣﺪاﺛﺔ أهﻤﻴﺔ أآﺒﺮ واﻟﻘﻴﻢ اﻻﺧﺮي ﺗﻌﻄﻰ اهﻤﻴﺔ ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻣﻊ ﻗﺪﻣﻬﺎ .ﻓﻤﺜﻼ ﻟﻮ آﺎن ﻟﺪﻳﻨﺎ ﻣﺸﺎهﺪات ﻣﻦ ﻣﺘﺴﻠﺴﻠﺔ زﻣﻨﻴﺔ z1 , z2 , z3 ,K , zn −2 , zn −1 , znﻓﺎﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻣﻦ اﻟﺪرﺟﺔ m ﻟﻠﻤﺸﺎهﺪات ﻳﺤﺴﺐ ﻣﻦ اﻟﻌﻼﻗﺔ 1 zˆt = ( zt + zt −1 + zt −2 + L + zt −m+1 ) , t = m, m + 1,..., n m واﻟﺘﻲ ﻳﻤﻜﻦ آﺘﺎﺑﺘﻬﺎ 1 1 1 1 zˆt = zt + zt −1 + zt −2 + L + zt −m+1 , t = m, m + 1,..., n m m m m 1 = zˆt = β zt + β zt −1 + β zt −2 + L + β zt −m+1 , t = m, m + 1,..., n, β m أي ان اﻟﻤﺘﻮﺳﻂ اﻟﻤﺘﺤﺮك ﻳﻌﻄﻲ ﺟﻤﻴﻊ اﻟﺒﻴﺎﻧﺎت ﻧﻔﺲ اﻟﻮزن β اﻵن ﻟﻮ أﻋﻄﻴﻨﺎ اﻟﺒﻴﺎﻧﺎت اوزان ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻣﻊ ﺑُﻌﺪ اﻟﻤﺸﺎهﺪات ﻋﻦ اﻟﻘﻴﻤﺔ اﻟﺤﺎﺿﺮة znآﺎﻟﺘﺎﻟﻲ st = α zt + α (1 − α ) zt −1 + α (1 − α ) zt −2 + L , t = 1, 2,..., n, 0 < α < 1 2
اﻟﻘﻴﻤﺔ stهﻲ ﻣﺘﻮﺳﻂ ﻣﻮزون ﺑﺄوزان ﺗﺘﻨﺎﻗﺺ اﺳﻴﺎ ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﺴﺎﺑﻘﺔ وهﺬا ﻣﺎﻳﺴﻤﻰ ﺑﺎﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ وﻳﻜﺘﺐ ﺑﺸﻜﻞ ﺗﻜﺮاري
st = α zt + (1 − α ) st −1 , t = 1, 2,..., n, s0 = z
وﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات
zn ( l ) = sn , l ≥ 1
ﻣﺜﺎل:
ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ
EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Metals 44.4 43.1 42.3 45.2 48.9 50.9
44.8 42.4 42.6 45.2 49.1 50.7
44.4 42.0 42.5 44.9 48.3 50.7
44.3 40.1 42.8 44.8 48.3 49.6 48.1
42.8 41.8 43.0 45.0 47.5 49.9 49.2
43.4 42.2 42.8 44.5 46.8 49.6 50.3
44.4 42.4 43.2 44.7 46.2 50.0 50.7
44.3 42.6 43.1 43.6 45.5 50.0 51.2
Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات: ;'MTB > TSPlot 'Metals >SUBC ;Index >SUBC ;TDisplay 11 >SUBC ;Symbol
١٧٢
SUBC>
Connect.
M etals
50
45
40 Index
10
20
30
40
50
60
α = 0.2 ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ وﻧﺄﺧﺬ ﺛﺎﺑﺖ ﺗﻤﻬﻴﺪ : ﻗﻴﻢ ﻣﺴﺘﻘﺒﻠﻴﻪ٦ وﻧﻮﺟﺪ ﺗﻨﺒﺆات ﻟـ MTB > %SES 'Metals'; SUBC> Weight 0.2; SUBC> Forecasts 6; SUBC> Title "Smoothing and Forecasting Metals"; SUBC> Initial 6. Single Exponential Smoothing Data Metals Length 60.0000 NMissing 0 Smoothing Constant Alpha: 0.2 Accuracy Measures MAPE: 2.17304 MAD: 1.00189 MSD: 1.45392 Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
49.7216 49.7216 49.7216 49.7216 49.7216 49.7216
47.2670 47.2670 47.2670 47.2670 47.2670 47.2670
52.1763 52.1763 52.1763 52.1763 52.1763 52.1763
Smoothing and Forecasting Metals
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constant Alpha:
40 0
10
20
30
Time
١٧٣
40
50
60
0.200
MAPE:
2.17304
MAD:
1.00189
MSD:
1.45392
ﺛﺎﻧﻴﺎ :ﻣﻨﺎﻗﺸﺔ اﻟﻨﺘﺎﺋﺞ -١ﻳﺤﺴﺐ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ ﻟﻤﺸﺎهﺪات ﻣﻦ اﻟﻌﻼﻗﺔ اﻟﺘﻜﺮارﻳﺔ:
z1 , z2 ,K , zn −1 , zn −2ﺑﺜﺎﺑﺖ ﺗﻤﻬﻴﺪ α = 0.2
si = α zi + (1 − α ) si −1 , i = 1, 2,..., n ﻟﻜﻲ ﻧﺒﺪأ اﻟﻌﻼﻗﺔ اﻟﺘﻜﺮارﻳﺔ ﻟﺤﺴﺎب اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة اﺳﻴﺎ ﻧﺤﺘﺎج اﻟﻲ اﻟﻘﻴﻤﺔ اﻻوﻟﻴﺔ s0واﻟﺘﻲ ﺗﺤﺴﺐ ﺑﻌﺪة ﻃﺮق ،أﺣﺪ هﺬﻩ اﻟﻄﺮق واﻟﺘﻲ ﺳﻨﺴﺘﺨﺪﻣﻬﺎ هﻲ وﺿﻊ s0ﻣﺴﺎوﻳﺔ ﻟﻠﻤﺘﻮﺳﻂ m
) , m = 6 ( or n, if n<6
∑z
i
i =1
m
= s0ﻓﻔﻲ ﻣﺜﺎﻟﻨﺎ
44.2 + 44.3 + 44.4 + 43.4 + 42.8 + 44.3 = 43.9 6
= s0
وﺑﺎﻟﺘﺎﻟﻲ ﻳﻜﻮن s1 = α z1 + (1 − α ) s0 = 0.2 ( 44.2 ) + 0.8 ( 43.9 ) = 8.84 + 35.12 = 43.96 s2 = α z2 + (1 − α ) s1 = 0.2 ( 44.3) + 0.8 ( 43.96 ) = 8.86 + 35.168 = 44.028 وهﻜﺬا ﻧﺴﺘﻤﺮ ﺣﺘﻲ ﺁﺧﺮ ﻣﺸﺎهﺪة ﻓﻴﻨﺘﺞ اﻟﺘﺎﻟﻲ: RESI1
FITS1
SMOO1
Metals
Time
0.30000 0.34000 0.37200 -0.70240 -1.16192 0.57046 0.55637 0.84510 0.27608 -1.07914 -1.36331 -1.29065 -1.23252 -1.38601 -2.80881 -0.34705 0.12236 0.79789 -0.06169 0.65065 0.62052 0.09642 0.27713 0.02171 -0.28264 -0.12611 -0.40089 0.27929 0.92343 1.83875 1.27100 1.51680 1.01344 0.91075 1.02860
43.9000 43.9600 44.0280 44.1024 43.9619 43.7295 43.8436 43.9549 44.1239 44.1791 43.9633 43.6906 43.4325 43.1860 42.9088 42.3470 42.2776 42.3021 42.4617 42.4494 42.5795 42.7036 42.7229 42.7783 42.7826 42.7261 42.7009 42.6207 42.6766 42.8613 43.2290 43.4832 43.7866 43.9892 44.1714
43.9600 44.0280 44.1024 43.9619 43.7295 43.8436 43.9549 44.1239 44.1791 43.9633 43.6906 43.4325 43.1860 42.9088 42.3470 42.2776 42.3021 42.4617 42.4494 42.5795 42.7036 42.7229 42.7783 42.7826 42.7261 42.7009 42.6207 42.6766 42.8613 43.2290 43.4832 43.7866 43.9892 44.1714 44.3771
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
١٧٤
0.82288 0.45830 0.86664 1.39331 1.71465 2.07172 2.45738 1.96590 2.37272 1.69818 1.85854 2.08683 1.66947 0.93557 1.04846 0.53877 1.53101 1.22481 1.17985 0.54388 1.13510 0.40808 -0.07353 -1.15883 -2.02706
44.3771 44.5417 44.6334 44.8067 45.0853 45.4283 45.8426 46.3341 46.7273 47.2018 47.5415 47.9132 48.3305 48.6644 48.8515 49.0612 49.1690 49.4752 49.7202 49.9561 50.0649 50.2919 50.3735 50.3588 50.1271
44.5417 44.6334 44.8067 45.0853 45.4283 45.8426 46.3341 46.7273 47.2018 47.5415 47.9132 48.3305 48.6644 48.8515 49.0612 49.1690 49.4752 49.7202 49.9561 50.0649 50.2919 50.3735 50.3588 50.1271 49.7216
45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
اﻟﻌﻤﻮد اﻟﺮاﺑﻊ SMOO١ﻳﺤﻮى اﻟﻘﻴﻢ اﻟﻤﻤﻬﺪة أي si , i = 1, 2,...,60اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ FITS١ ﻳﺤﻮى اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ أي zˆi = si −1 , i = 1, 2,...,60اﻟﻌﻤﻮد اﻟﺨﺎﻣﺲ RESI١ﻳﺤﻮي اﻷﺧﻄﺎء )اﻟﺒﻮاﻗﻲ (Residualsأي ei = zi − zˆi , i = 1, 2,...,60 -٢ﻳﺆﺧﺬ آﻤﺘﻨﺒﺊ ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺁﺧﺮ ﻗﻴﻤﺔ ﻣﻤﻬﺪة أي: z n ( l ) = sn , l > 0 ﻓﻔﻲ اﻟﻤﺜﺎل اﻟﺤﺎﻟﻲ z60 ( l ) = 49.7216, l > 0 ﺗﺆﺧﺬ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟـ ٦اﻟﻤﺴﺘﻘﺒﻠﻴﺔ أي ﻟﻠﻘﻴﻢ zn +1 , zn +2 ,..., zn +6أو ﻓﻲ هﺬا اﻟﻤﺜﺎل z61 , z62 ,..., z66آﺎﻟﺘﺎﻟﻲ: z60 (1) = z60 ( 2 ) = L = z60 ( 6 ) = 49.7216
-٣ﻟﺤﺴﺎب ﻓﺘﺮات ﺗﻨﺒﺆ 95%ﻧﺤﺴﺐ اﻟﻜﻤﻴﺎت ⎡⎣ zn ( l ) ± 1.96σˆ ⎤⎦ , l > 0أي ] ˆ [ 49.7216 ± 1.96σﻟﺠﻤﻴﻊ ﻗﻴﻢ اﻟﺘﻨﺒﺆات اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ،ﻧﺄﺧﺬ اﻟﻘﻴﻤﺔ MSD = 1.45392آﻤﻘﺪر
ﻟـ σ 2أي σˆ 2 = 1.45392ﻓﻴﻜﻮن σˆ = 1.205786وﻋﻠﻴﻪ ﺗﻜﻮن ﻓﺘﺮة ﺗﻨﺒﺆ 95%ﻟﺠﻤﻴﻊ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ هﻲ: ]⎡⎣ 49.7216 ± 1.96 (1.205786 ) ⎤⎦ = [ 49.7216 ± 2.3633] = [ 47.35826,52.08494 أي: z60+l ∈ [ 47.3582,50.0849] , l > 0 with probability 0.95 ﻣﻼﺣﻈﺔ :ﺗﺤﺴﺐ MSDآﺎﻵﺗﻲ
١٧٥
) − zˆi
n
∑( z
i
n −1
i =1
= ˆMSD = σ 2
ﺗﻤﺮﻳﻦ: ﻃﺒﻖ ﺗﻤﻬﻴﺪ اﺳﻲ ﺑﺴﻴﻂ ﻋﻠﻲ اﻟﻤﺸﺎهﺪات اﻟﺴﺎﺑﻘﺔ ﻣﺴﺘﺨﺪﻣﺎ α = 0.3,0.4,0.5وﻗﺮر اﻳﻬﺎ اﻓﻀﻞ ﻟﻠﺘﻨﺒﺆ ﻋﻦ اﻟﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ؟.
١٧٦
اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﻤﺰدوج Using Double : Exponential Smoothing for Forecasting أوﻻ :ﻃﺮﻳﻘﺔ ﺑﺮاون :Brown’s Method
ﻟﻤﺸﺎهﺪات z1 , z2 ,K , zn −1 , zn −2وﻟﺜﺎﺑﺖ ﺗﻤﻬﻴﺪ 0 < α < 1ﻧﻮﺟﺪ اﻟﺘﺎﻟﻲ:
s = α zt + (1 − α ) st(−1) , t = 1,2,..., n )(1 t
1
ﺣﻴﺚ ) ( stﺗﻤﻬﻴﺪ اﺳﻲ ﺑﺴﻴﻂ ) st( ) = stاﻧﻈﺮ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺒﺴﻴﻂ( و ) (1ﺗﺮﻣﺰ اﻟﻲ درﺟﺔ هﺬا اﻟﺘﻤﻬﻴﺪ )(2 )(1 )(2 st = α st + (1 − α ) st −1 , t = 1,2,..., n 1
1
ﺣﻴﺚ ) ( stﺗﻤﻬﻴﺪ اﺳﻲ ﻣﺰدوج و ) ( 2ﺗﺮﻣﺰ اﻟﻲ درﺟﺔ هﺬا اﻟﺘﻤﻬﻴﺪ 2
at = 2 st( ) − st( ) , t = 1, 2,..., n 1
2
α 1 2 st( ) − st( ) , t = 1, 2,..., n 1−α
)
ﺗﺤﺴﺐ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻣﻦ اﻟﻤﻌﺎدﻟﺔ
t = 1, 2,..., n
وﺗﺤﺴﺐ اﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ zn +l , l > 0ﻣﻦ
zˆt = at + bt t ,
zn ( l ) = an + bn l, l > 0
ﺣﺴﺎب اﻟﻘﻴﻢ اﻻوﻟﻴﺔ ) ( s0و ) ( : s0ﻣﻦ اﻟﻌﻼﻗﺎت اﻟﺴﺎﺑﻘﺔ ﻧﺠﺪ 1
(
= bt
2
b0 b0 ﻧﻮﺟﺪ a0و b0ﺑﺈﻧﺤﺪار اﻟﻤﺸﺎهﺪات ﻋﻠﻲ اﻟﺰﻣﻦ t = 1, 2,..., n ˆ a0 = αو ˆb0 = β
1−α
α
1−α
α
s0( ) = a0 − 1
s0( ) = a0 − 2 2
zt = α + β t + et ,وﻳﻜﻮن
ﺛﺎﻧﻴﺎ :ﻃﺮﻳﻘﺔ هﻮﻟﺖ :Holt’s Method
ﻟﻤﺸﺎهﺪات z1 , z2 ,K , zn −1 , zn −2وﻟﺜﺎﺑﺘﻲ ﺗﻤﻬﻴﺪ 0 < α < 1و 0 < γ < 1ﻧﻮﺟﺪ اﻟﺘﺎﻟﻲ: st = α zt + (1 − α )( st −1 + bt −1 ) , t = 1, 2,..., n t = 1, 2,..., n ﻧﺤﺴﺐ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﻣﻦ
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 , t = 1, 2,..., n
واﻟﺘﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻦ
zˆt = st + bt t ,
zn ( l ) = sn + bn l, l > 0
ﻧﺤﺴﺐ اﻟﻘﻴﻢ اﻻوﻟﻴﺔ s0و b0ﻣﻦ ١٧٧
s0 = z1 b0 = z2 − z1
b0 =
or
( z2 − z1 ) + ( z3 − z2 ) = ( z3 − z1 )
or 2 2 ( z − z ) + ( z3 − z2 ) + ( z4 − z3 ) ( z4 − z1 ) b0 = 2 1 = 3 3 :ﻣﺜﺎل EMPLOY.MTW ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ MTB > Retrieve 'E:\Mtbwin\DATA\EMPLOY.MTW'.
Metals ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Metals 44.2 43.1 42.4 42.9 45.0 49.4 50.5
44.3 42.6 43.1 43.6 45.5 50.0 51.2
44.4 42.4 43.2 44.7 46.2 50.0 50.7
43.4 42.2 42.8 44.5 46.8 49.6 50.3
42.8 41.8 43.0 45.0 47.5 49.9 49.2
44.3 40.1 42.8 44.8 48.3 49.6 48.1
44.4 42.0 42.5 44.9 48.3 50.7
44.8 42.4 42.6 45.2 49.1 50.7
44.4 43.1 42.3 45.2 48.9 50.9
:ﻧﺮﺳﻢ هﺬﻩ اﻟﻤﺸﺎهﺪات MTB > TSPlot 'Metals'; SUBC> Index; SUBC> TDisplay 11; SUBC> Symbol; SUBC> Connect.
Metals
50
45
40 Index
10
20
30
١٧٨
40
50
60
ﻧﺴﺘﺨﺪم اﻵن،ﻧﻄﺒﻖ اﻵن ﺗﻤﻬﻴﺪا ﻟﻬﺬﻩ اﻟﻤﺸﺎهﺪات ﺑﺈﺳﺘﺨﺪام اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﻤﺰدوج ﺑﻄﺮﻳﻘﺔ ﺑﺮاون ﺑﺄوزان ﻣﺘﺴﺎوﻳﺔ ﺳﻮف ﻧﺄﺧﺬهﺎWEIGHT ﻣﻊ اﻻﻣﺮ اﻟﻔﺮﻋﻲ%DES (Macro) اﻟﺒﺮﻣﺞ 0.2 MTB > %DES 'Metals'; SUBC> Weight 0.2 0.2; SUBC> Forecasts 6; SUBC> Title "Brown's Double Exponential Smoothing"; SUBC> Table.
Double Exponential Smoothing Data Length NMissing
Metals 60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Accuracy Measures MAPE: 2.16187 MAD: 0.97032 MSD: 1.62936
Time Metals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7
Smooth
Predict
Error
41.7739 42.4976 43.1686 43.5546 43.7373 44.1459 44.4990 44.8575 45.0620 44.9391 44.6673 44.3271 43.9378 43.4769 42.7011 42.3565 42.1465 42.1286 42.0132 42.0763 42.1877 42.2374 42.3396 42.4078 42.4181 42.4495 42.4207 42.5129 42.7420 43.1797
41.1674 42.0470 42.8607 43.5933 43.9716 44.1074 44.5238 44.8719 45.2275 45.3989 45.1841 44.8088 44.3723 43.8962 43.3514 42.4456 42.0831 41.8857 41.9164 41.8204 41.9347 42.0967 42.1745 42.3098 42.3976 42.4119 42.4509 42.4161 42.5276 42.7996
3.03257 2.25303 1.53927 -0.19330 -1.17163 0.19257 -0.12377 -0.07189 -0.82751 -2.29891 -2.58407 -2.40884 -2.17229 -2.09617 -3.25142 -0.44557 0.31694 1.21426 0.48355 1.27964 1.26533 0.70327 0.82549 0.49024 0.10244 0.18809 -0.15090 0.48394 1.07245 1.90036
١٧٩
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0 49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
43.5507 43.9854 44.3338 44.6511 44.9749 45.2430 45.4157 45.6373 45.9491 46.3285 46.7909 47.3492 47.8339 48.4002 48.8413 49.2966 49.7849 50.1841 50.4162 50.6292 50.7104 50.9509 51.1334 51.3019 51.3407 51.4782 51.4770 51.3649 51.0127 50.4384
43.3133 43.7317 44.2172 44.5889 44.9187 45.2538 45.5197 45.6716 45.8863 46.2106 46.6136 47.1115 47.7173 48.2253 48.8267 49.2707 49.7311 50.2302 50.6202 50.8114 50.9880 51.0137 51.2417 51.4024 51.5509 51.5477 51.6712 51.6311 51.4659 51.0230
1.18668 1.26826 0.58280 0.31112 0.28133 -0.05376 -0.51967 -0.17162 0.31368 0.58938 0.88637 1.18850 0.58266 0.87469 0.07332 0.12930 0.26890 -0.23017 -1.02022 -0.91145 -1.38797 -0.31368 -0.54169 -0.50244 -1.05093 -0.34770 -0.97120 -1.33115 -2.26587 -2.92300
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
50.3318 50.2252 50.1186 50.0120 49.9054 49.7988
47.9545 47.7984 47.6384 47.4749 47.3080 47.1381
52.7091 52.6520 52.5987 52.5490 52.5027 52.4594
Brown's Double Exponential Smoothing
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 MAPE: MAD: MSD:
40 0
10
20
30
Time
١٨٠
40
50
60
2.16187 0.97032 1.62936
: b0 وa0 إﻳﺠﺎد MTB > DATA> DATA> MTB >
set c4 1:60 end regr c3 1 c4
Regression Analysis The regression equation is Metals = 41.0 + 0.152 C4
1−α
وﻣﻨﻬﺎ ﻧﺤﺴﺐb0 = 0.152 وa0 = 41.0 إذا
0.8 ( 0.152 ) = 41.608 α 0.2 1−α ⎛ 0.8 ⎞ 2 s0( ) = a0 − 2 b0 = 41.0 − 2 ⎜ ⎟ ( 0.152 ) = 42.216 α ⎝ 0.2 ⎠ 1 s1( ) = ( 0.2 )( 44.2 ) + ( 0.8 )( 41.608 ) = 42.1264 s0( ) = a0 − 1
b0 = 41.0 −
s1( ) = ( 0.2 )( 42.1264 ) + ( 0.8 )( 42.216 ) = 42.19808 2
a1 = ( 2 )( 42.1264 ) − 42.19808 = 42.05472
( 0.2 ) ( 42.19808 − 42.05472 ) = 0.03584 ( 0.8) zˆ1 = 42.05472 + ( 0.03584 )(1) = 42.09056
b1 =
… وهﻜﺬا اﻟﺦ . آﻤﺎ ﻓﻲ اﻷﻣﺜﻠﺔ اﻟﺴﺎﺑﻘﺔMSD ﺗﺤﺴﺐ ﻓﺘﺮات اﻟﺘﻨﺒﺆ ﺑﺈﺳﺘﺨﺪام :ﻣﺜﺎل WEIGHT ﻣﻊ اﻻﻣﺮ اﻟﻔﺮﻋﻲ%DES (Macro) ﻟﺘﻄﺒﻴﻖ ﻃﺮﻳﻘﺔ هﻮﻟﺖ ﻧﺴﺘﺨﺪم اﻵن اﻟﺒﺮﻣﺞ γ = 0.3 وα = 0.2 ﺑﺄوزان ﻣﺨﺘﻠﻔﺔ ﺳﻮف ﻧﺄﺧﺬ MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'. Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW Worksheet was saved on 6/ 5/1996 MTB > %DES 'Metals'; SUBC> Weight 0.2 0.3; SUBC> Forecasts 6; SUBC> Title "Holt's Double Exponential Smoothing"; SUBC> Table.
Double Exponential Smoothing Data Length NMissing
Metals 60.0000 0
١٨١
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.3 Accuracy Measures MAPE: 2.15656 MAD: 0.96328 MSD: 1.56274
Time
Metals
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
44.2 44.3 44.4 43.4 42.8 44.3 44.4 44.8 44.4 43.1 42.6 42.4 42.2 41.8 40.1 42.0 42.4 43.1 42.4 43.1 43.2 42.8 43.0 42.8 42.5 42.6 42.3 42.9 43.6 44.7 44.5 45.0 44.8 44.9 45.2 45.2 45.0 45.5 46.2 46.8 47.5 48.3 48.3 49.1 48.9 49.4 50.0 50.0
Smooth 41.7739 42.5461 43.2891 43.7501 43.9779 44.3895 44.7334 45.0685 45.2405 45.0676 44.7113 44.2595 43.7466 43.1634 42.2751 41.8139 41.5361 41.5057 41.4371 41.5799 41.8054 41.9895 42.2254 42.4205 42.5395 42.6522 42.6793 42.7982 43.0394 43.4861 43.8762 44.3257 44.6858 45.0007 45.3066 45.5449 45.6749 45.8384 46.0888 46.4159 46.8406 47.3799 47.8665 48.4419 48.9016 49.3693 49.8653 50.2702
Predict 41.1674 42.1076 43.0113 43.8376 44.2724 44.4118 44.8167 45.1356 45.4506 45.5595 45.2391 44.7244 44.1332 43.5043 42.8188 41.7674 41.3202 41.1072 41.1964 41.1999 41.4568 41.7869 42.0317 42.3257 42.5493 42.6653 42.7741 42.7728 42.8993 43.1826 43.7202 44.1571 44.6573 45.0259 45.3333 45.6312 45.8436 45.9230 46.0610 46.3199 46.6757 47.1499 47.7582 48.2774 48.9020 49.3617 49.8317 50.3378
Error 3.03257 2.19238 1.38868 -0.43760 -1.47237 -0.11184 -0.41671 -0.33560 -1.05057 -2.45952 -2.63911 -2.32443 -1.93322 -1.70426 -2.71884 0.23263 1.07985 1.99283 1.20365 1.90008 1.74322 1.01314 0.96829 0.47431 -0.04933 -0.06528 -0.47413 0.12724 0.70070 1.51743 0.77977 0.84285 0.14275 -0.12590 -0.13327 -0.43116 -0.84361 -0.42295 0.13895 0.48014 0.82428 1.15014 0.54181 0.82265 -0.00205 0.03832 0.16832 -0.33779
١٨٢
49 50 51 52 53 54 55 56 57 58 59 60
49.6 49.9 49.6 50.7 50.7 50.9 50.5 51.2 50.7 50.3 49.2 48.1
50.4979 50.6862 50.7296 50.9166 51.0532 51.1813 51.1869 51.2901 51.2673 51.1350 50.7591 50.1448
50.7224 50.8827 51.0121 50.9708 51.1415 51.2516 51.3586 51.3127 51.4092 51.3438 51.1489 50.6560
-1.12240 -0.98275 -1.41206 -0.27079 -0.44152 -0.35162 -0.85860 -0.11267 -0.70916 -1.04381 -1.94889 -2.55603
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6
61 62 63 64 65 66
49.8884 49.6319 49.3755 49.1190 48.8626 48.6061
47.5283 47.1597 46.7803 46.3915 45.9946 45.5908
52.2484 52.1041 51.9707 51.8466 51.7306 51.6215
Holt's Double Exponential Smoothing
Actual Predicted Forecast
Metals
50
Actual Predicted Forecast
45
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.300 MAPE: MAD: MSD:
40 0
10
20
30
40
50
2.15656 0.96328 1.56274
60
Time
ﻳﺘﻮﻗﻊ ﻋﺪم ﺗﻄﺎﺑﻖ: ﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ اﻟﺤﺴﺎﺑﺎت اﻟﺴﺎﺑﻘﺔ ﺑﺘﺘﺒﻊ ﺑﻌﺾ اﻟﻘﻴﻢ ﻳﺪوﻳﺎ ) ﻣﻼﺣﻈﺔ:ﺗﻤﺮﻳﻦ اﻟﺤﺴﺎﺑﺎت ﺗﻤﺎﻣﺎ وذﻟﻚ ﻹﺧﺘﻼف ﻃﺮﻳﻖ ﺗﻤﺜﻴﻞ اﻷﻋﺪاد ﺑﻴﻦ اﻟﺤﺎﺳﺐ واﻵﻟﺔ اﻟﺤﺎﺳﺒﺔ وآﺬﻟﻚ ﻓﻲ (ﺗﺨﺰﻳﻦ اﻷرﻗﺎم ﻓﻲ ذاآﺮات آﻞ ﻣﻨﻬﻤﺎ
١٨٣
اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻋﺸﺮ اﻟﺘﻤﻬﻴﺪ اﻻﺳﻲ اﻟﺜﻼﺛﻲ و اﻟﺘﻨﺒﺆ ﺑﻮاﺳﻄﺔ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ Triple Exponential Smoothing: Winters' ThreeParameter Trend and Seasonality Smoothing Method ﺟﻤﻴﻊ اﻟﻄﺮق اﻟﺴﺎﺑﻘﺔ اﻟﺘﻲ درﺳﻨﺎهﺎ ﻓﻲ هﺬا اﻟﻔﺼﻞ ﻻ ﺗﻨﻔﻊ ﻟﺘﺤﻠﻴﻞ اﻟﻈﻮاهﺮ اﻟﻤﻮﺳﻤﻴﺔ ﻣﺎﻋﺪى ﻃﺮﻳﻘﺔ اﻟﺘﻔﻜﻴﻚ Decomposition Methodوﻃﺮﻳﻘﺔ وﻧﺘﺮز Winters' trend and seasonal smoothingاﻟﺘﻲ ﺳﻮف ﻧﺴﺘﻌﺮﺿﻬﺎ هﻨﺎ
ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻟﻠﻤﺘﺴﻠﺴﻼت اﻟﻤﻮﺳﻤﻴﺔ اﻟﻤﻨﺠﺮﻓﺔ أوﻻ :ﺗﻤﻬﺪ اﻟﻤﺸﺎهﺪات ﺗﻤﻬﻴﺪا آﻠﻴﺎ ﺑﺎﻟﻌﻼﻗﺔ zt + (1 − α )( st −1 + bt −1 ) , t = 1, 2,..., n St − s ﺛﺎﻧﻴﺎ :ﺗﻤﻬﻴﺪ اﻹﻧﺠﺮاف
st = α
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 , t = 1, 2,..., n
ﺛﺎﻟﺜﺎ :ﺗﻤﻬﻴﺪ اﻟﻤﻮﺳﻤﻴﺔ zt + (1 − β ) St − s , t = 1, 2,..., n st ﺣﻴﺚ Siهﻲ اﻟﻤﺮآﺒﺔ اﻟﻤﻮﺳﻤﻴﺔ ﻋﻨﺪ اﻟﺰﻣﻦ iو sهﻲ دورة اﻟﻤﻮﺳﻤﻴﺔ اﻟﻘﻴﻢ اﻟﻤﻄﺒﻘﺔ ﺗﻌﻄﻲ ﺑﺎﻟﻌﻼﻗﺔ zˆt = ( st + bt t ) St − s , t = 1, 2,..., n واﻟﺘﻨﺒﺆات ﻣﻦ اﻟﻌﻼﻗﺔ zn ( l ) = ( sn + bn l ) Sn − s+l , l > 0
St = β
ﻣﻦ اﻟﺼﻌﺐ ﺟﺪا ﺗﺘﺒﻊ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﺑﺎﻟﺤﺴﺎﺑﺎت اﻟﻴﺪوﻳﺔ ﺣﻴﺚ ان اﻟﻘﻴﻢ اﻻوﻟﻴﺔ ﺗﺤﺴﺐ ﺑﺨﻮارزﻣﺎت ﻏﻴﺮ ﺧﻄﻴﺔ ﺑﺈﺳﺘﺨﺪام اﻟﺤﺎﺳﺐ وﻟﻬﺬا ﻟﻦ ﻧﺴﺘﻌﺮﺿﻬﺎ هﻨﺎ وﻧﻜﺘﻔﻲ ﺑﺎﻟﻨﺘﺎﺋﺞ اﻟﻤﺨﺮﺟﺔ ﻣﻦ اﻟﺤﺎﺳﺐ. ﻣﺜﺎل: ﺗﺤﻤﻞ اﻟﺒﻴﺎﻧﺎت ﻣﻦ ورﻗﺔ اﻟﻌﻤﻞ EMPLOY.MTW 'E:\Mtbwin\DATA\EMPLOY.MTW'.
MTB > Retrieve
ﺳﻮف ﻧﺴﺘﺨﺪم اﻟﻤﺸﺎهﺪات ﻓﻲ اﻟﺘﻐﻴﺮ Food Food
52.1 51.5 53.5 54.3
53.4 52.3 53.1 54.6
55.3 53.6 53.3 54.2
58.2 55.3 53.9 54.8
66.9 58.5 55.6 55.8
70.7 69.3 60.1 57.9 57.7
65.3 69.6 68.9 62.6 60.5
١٨٤
56.5 64.2 68.8 70.3 66.4
53.4 55.5 63.6 69.4 75.5
52.5 53.3 57.1 64.7 73.3
53.2 52.4 52.2 57.1 68.1
53.0 51.5 51.5 53.9 58.1
53.5 51.5 51.7 53.5 54.8
وﻧﺮﺳﻢ اﻟﻤﺸﺎهﺪات
75
Food
70 65 60 55 50 Index
10
20
30
40
50
60
: ﻧﻄﺒﻖ اﻵن ﻃﺮﻳﻘﺔ وﻧﺘﺮز آﺎﻟﺘﺎﻟﻲ١٢ ﻧﻼﺣﻆ ان اﻟﻈﺎهﺮة ﻣﻮﺳﻤﻴﺔ ﺑﺪورة Additive Model ﻟﻠﻨﻤﻮذج اﻹﺿﺎﻓﻲ:أوﻻ zt = bt + St + et , t = 1, 2,..., n MTB > %Wintadd 'Food' 12; SUBC> Weight 0.2 0.2 0.2; SUBC> Forecasts 12; SUBC> Title "Wintrs' Trend and Seasonal Smoothing"; SUBC> Table.
Winters' additive model Data Length NMissing
Food 60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Delta (seasonal): 0.2 Accuracy Measures MAPE: 1.94769 MAD: 1.15100 MSD: 2.66711
١٨٥
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Food 53.5 53.0 53.2 52.5 53.4 56.5 65.3 70.7 66.9 58.2 55.3 53.4 52.1 51.5 51.5 52.4 53.3 55.5 64.2 69.6 69.3 58.5 55.3 53.6 52.3 51.5 51.7 51.5 52.2 57.1 63.6 68.8 68.9 60.1 55.6 53.9 53.3 53.1 53.5 53.5 53.9 57.1 64.7 69.4 70.3 62.6 57.9 55.8 54.8 54.2 54.6 54.3 54.8 58.1 68.1 73.3 75.5 66.4 60.5
Smooth 48.7755 49.6020 51.0736 52.0733 53.5117 57.4851 66.2299 71.7852 71.8932 62.3206 57.5208 55.1544 55.0393 53.6493 53.1185 52.2661 52.6528 55.7616 63.8483 68.8787 68.0386 59.2825 55.0979 53.0432 53.0377 52.1394 51.9889 51.7214 52.1875 55.0750 63.7515 68.8427 68.0160 58.9061 55.3220 53.4419 53.3084 52.6717 52.9095 52.9745 53.7952 57.2065 65.2747 70.4039 69.6200 60.5655 57.0392 55.3721 55.2403 54.6681 54.8138 54.7366 55.3001 58.5310 66.4168 71.8806 71.8794 63.7240 60.3141
Predict 49.4303 50.4197 51.9944 53.0424 54.4591 58.3901 67.0593 72.5443 72.5785 62.7787 57.7958 55.3296 55.1373 53.6258 53.0100 52.0971 52.4960 55.6369 63.7181 68.7678 67.9610 59.2585 55.0435 52.9991 53.0177 52.0907 51.9165 51.6403 52.1009 54.9923 63.7531 68.8382 68.0099 58.9356 55.3981 53.5262 53.4076 52.7666 53.0177 53.1020 53.9386 57.3484 65.4066 70.5076 69.6794 60.6497 57.2014 55.5623 55.4399 54.8422 54.9622 54.8705 55.4112 58.6176 66.4827 72.0112 72.0616 64.0437 60.7281
Error 4.06965 2.58027 1.20556 -0.54244 -1.05914 -1.89013 -1.75932 -1.84430 -5.67851 -4.57874 -2.49577 -1.92957 -3.03734 -2.12584 -1.50996 0.30287 0.80401 -0.13695 0.48187 0.83218 1.33902 -0.75851 0.25650 0.60092 -0.71765 -0.59067 -0.21651 -0.14031 0.09913 2.10774 -0.15314 -0.03822 0.89007 1.16436 0.20190 0.37384 -0.10757 0.33345 0.48233 0.39803 -0.03858 -0.24838 -0.70661 -1.10758 0.62065 1.95031 0.69858 0.23773 -0.63993 -0.64220 -0.36218 -0.57048 -0.61119 -0.51765 1.61731 1.28878 3.43843 2.35629 -0.22810
١٨٦
60
57.7
58.6397
59.0446
-1.34455
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6 7 8 9 10 11 12
61 62 63 64 65 66 67 68 69 70 71 72
58.6167 58.3236 58.8195 58.9840 59.8723 63.4804 72.0757 77.4486 77.7540 68.9067 64.6434 62.7731
55.7968 55.4449 55.8775 55.9746 56.7913 60.3243 68.8410 74.1321 74.3528 65.4180 61.0647 59.1020
61.4366 61.2023 61.7614 61.9935 62.9532 66.6365 75.3104 80.7651 81.1552 72.3954 68.2221 66.4441
Wintrs' Trend and Seasonal Smoothing
Actual
80
Predicted Forecast Actual Predicted Forecast
Food
70
60
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 Delta (season):0.200
50
MAPE: MAD: MSD:
0
10
20
30
40
50
60
1.94769 1.15100 2.66711
70
Time
zt = bt St + et , t = 1, 2,..., n
Multiplicative Model ﻟﻠﻨﻤﻮذج اﻟﺘﻀﺎﻋﻔﻲ:ﺛﺎﻧﻴﺎ
MTB > %Wintmult 'Food' 12; SUBC> Weight 0.2 0.2 0.2; SUBC> Forecasts 12; SUBC> Title "Winters' Trend and Seasonal Smoothing"; SUBC> Table.
Winters' multiplicative model Data
Food
١٨٧
Length NMissing
60.0000 0
Smoothing Constants Alpha (level): 0.2 Gamma (trend): 0.2 Delta (seasonal): 0.2 Accuracy Measures MAPE: 1.88377 MAD: 1.12068 MSD: 2.86696
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Food 53.5 53.0 53.2 52.5 53.4 56.5 65.3 70.7 66.9 58.2 55.3 53.4 52.1 51.5 51.5 52.4 53.3 55.5 64.2 69.6 69.3 58.5 55.3 53.6 52.3 51.5 51.7 51.5 52.2 57.1 63.6 68.8 68.9 60.1 55.6 53.9 53.3 53.1 53.5 53.5 53.9 57.1 64.7 69.4
Smooth
48.7870 49.6755 51.1521 52.1675 53.6181 57.6509 66.6199 72.4105 72.5679 62.7837 57.9154 55.5108 54.4920 53.2117 52.8118 52.0929 52.5894 55.7388 63.7189 68.7087 67.9722 59.4594 55.4037 53.4103 52.6818 51.8659 51.8002 51.6271 52.1643 55.0424 63.6079 68.6702 67.9561 59.1021 55.6210 53.7881 53.0479 52.4502 52.7444 52.8747 53.7689 57.2790 65.4702 70.6713
Predict
49.3853 50.4303 52.0132 53.0746 54.5132 58.5541 67.5607 73.3280 73.3777 63.2634 58.1732 55.6485 54.5392 53.1621 52.6957 51.9302 52.4439 55.6209 63.5782 68.5838 67.8890 59.4361 55.3468 53.3536 52.6356 51.8071 51.7290 51.5549 52.0890 54.9676 63.6199 68.6823 67.9727 59.1487 55.7003 53.8609 53.1211 52.5294 52.8467 53.0029 53.9188 57.4374 65.6357 70.8095
Error
4.11470 2.56966 1.18677 -0.57458 -1.11323 -2.05414 -2.26072 -2.62800 -6.47768 -5.06337 -2.87320 -2.24849 -2.43920 -1.66212 -1.19573 0.46985 0.85611 -0.12087 0.62178 1.01617 1.41104 -0.93606 -0.04680 0.24639 -0.33562 -0.30705 -0.02902 -0.05492 0.11103 2.13244 -0.01988 0.11774 0.92727 0.95133 -0.10032 0.03912 0.17892 0.57055 0.65329 0.49714 -0.01879 -0.33743 -0.93567 -1.40954
١٨٨
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
70.3 62.6 57.9 55.8 54.8 54.2 54.6 54.3 54.8 58.1 68.1 73.3 75.5 66.4 60.5 57.7
69.8908 60.7552 57.1925 55.5181 54.8764 54.3244 54.5372 54.5298 55.1925 58.6054 66.7739 72.4056 72.3385 63.6729 60.0395 58.3023
69.9719 60.8370 57.3348 55.6775 55.0383 54.4749 54.6769 54.6661 55.3155 58.7141 66.8698 72.5622 72.5236 63.9378 60.3781 58.6338
0.32815 1.76302 0.56523 0.12253 -0.23826 -0.27486 -0.07694 -0.36612 -0.51551 -0.61410 1.23016 0.73784 2.97638 2.46217 0.12191 -0.93381
Row
Period
Forecast
Lower
Upper
1 2 3 4 5 6 7 8 9 10 11 12
61 62 63 64 65 66 67 68 69 70 71 72
57.8102 57.3892 57.8332 57.9307 58.8311 62.7415 72.1849 78.1507 78.5092 68.6689 63.9258 61.8189
55.0645 54.5864 54.9687 55.0005 55.8313 59.6686 69.0354 74.9215 75.1976 65.2721 60.4414 58.2446
60.5558 60.1921 60.6977 60.8609 61.8309 65.8145 75.3344 81.3798 81.8208 72.0657 67.4103 65.3933
Winters' Trend and Seasonal Smoothing
Actual
80
Predicted Forecast Actual Predicted Forecast
Food
70
60
Smoothing Constants Alpha (level): 0.200 Gamma (trend):0.200 Delta (season):0.200
50
MAPE: MAD: MSD:
0
10
20
30
40
50
60
1.88377 1.12068 2.86696
70
Time
:ﻣﻼﺣﻈﺎت ﺛﺎﺑﺖγ ﺛﺎﺑﺖ اﻟﺘﻤﻬﻴﺪ اﻟﻜﻠﻲ وα ﻟﺘﻄﺒﻴﻖ ﻃﺮﻳﻘﺔ وﻧﺘﺮز ﻧﺤﺘﺎج اﻟﻲ إﺧﺘﻴﺎر ﻗﻴﻢ ﺛﻼﺛﺔ ﻣﻌﺎﻟﻢ هﻲ ﻓﻲ ﺛﻼﺛﺔOptimization ﺛﺎﺑﺖ ﺗﻤﻬﻴﺪ اﻟﻤﻮﺳﻤﻴﺔ وهﺬﻩ ﻋﻤﻠﻴﺔ أﻓﻀﻠﻴﺔβ ﺗﻤﻬﻴﺪ اﻹﻧﺠﺮاف و ١٨٩
اﺑﻌﺎد ) ﻓﻀﺎء اﻟﻤﻌﺎﻟﻢ ( إﻣﺎ أن ﻧﺘﺮك ﻟﻠﺒﺮﻧﺎﻣﺞ اﻹﺣﺼﺎﺋﻲ اﻟﻤﺴﺘﺨﺪم ﺣﺴﺎﺑﻬﺎ ﺗﻠﻘﺎﺋﻴﺎ ﺑﺈﺳﺘﺨﺪام ﺧﻮارزﻣﺎت ﻏﻴﺮ ﺧﻄﻴﺔ ﻣﺒﻨﻴﺔ داﺧﻞ اﻟﺒﺮﻧﺎﻣﺞ أو ﻧﻘﻮم ﻧﺤﻦ ﺑﺈﻣﺪاد اﻟﺒﺮﻧﺎﻣﺞ ﺑﺘﻠﻚ اﻟﻘﻴﻢ. ﺗﻤﺮﻳﻦ :ﻓﻲ اﻷﻣﺜﻠﺔ اﻟﺴﺎﺑﻘﺔ اﺧﺬﻧﺎ . α = γ = β = 0.2ﺛﺒﺖ ﻓﻲ آﻞ ﻣﺮة ﻣﻌﻠﻤﻴﻦ وﻏﻴﺮ اﻟﺜﺎﻟﺚ ﺣﺘﻲ ﺗﺤﺼﻞ ﻋﻠﻲ أﻗﻞ MSD ﻣﺜﺎل ﻋﻤﻠﻲ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: ﺳﻮف ﻧﺴﺘﺨﺪم ورﻗﺔ اﻟﻌﻤﻞ CPI.MTWﻣﻦ ﻣﺠﻤﻮﻋﺔ اﻟﺒﻴﺎﻧﺎت ﻟﻠﺒﺮﻧﺎﻣﺞ MINITAB MTB > Retrieve
'C:\MTBWIN\STUDENT9\CPI.MTW'.
ﺳﻮف ﻧﺄﺧﺬ اﻟﻤﺘﻐﻴﺮ CPIChange 2.9 6.2 13.5 3.6
1.6 3.2 11.3 1.9
1.3 5.7 6.5 4.3 4.2
1.3 4.4 7.6 3.6 3.0
1.0 4.2 9.1 6.2 4.8
1.0 5.5 5.8 3.2 5.4
CPIChnge 1.7 3.1 11.0 10.3 4.1
ﻧﺮﺳﻢ اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﻮﺟﺪ اﻟﺘﺮﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻟﻌﻴﻨﻴﺔ MTB > TSPlot
14 12 10
6
CPIChnge
8
4 2 0 30
20
25
10
15
Index
5
MTB > %acf c2 Autocorrelation Function for CPIChnge
8
7
5
6
4
Corr
Lag
LBQ
T
41.23
8 -0.16 -0.51
١٩٠
3
2
Corr
1
Lag
LBQ
T
22.42 30.32 33.59 36.24 38.97 40.05 40.07
1 0.79 4.53 2 0.46 1.77 3 0.29 1.02 4 0.26 0.88 5 0.26 0.86 6 0.16 0.52 7 -0.02 -0.06
Autocorrelation
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
MTB > %pacf c2 Partial Autocorrelation
Partial Autocorrelation Function for CPIChnge 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
1
2
3
4
Lag PAC 1 2 3 4 5 6 7
0.79 -0.42 0.35 -0.04 0.08 -0.28 -0.05
5
6
T
Lag PAC
T
4.53 -2.44 2.01 -0.24 0.43 -1.62 -0.29
8 -0.09
-0.52
7
8
ARMA(١،١) ﻣﻦ اﻧﻤﺎط اﻟﺘﺮﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻟﻌﻴﻨﻴﺔ ﻗﺪ ﻳﻜﻮن اﻟﻨﻤﻮذج ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ٥ اﻷواﻣﺮ اﻟﺘﺎﻟﻴﺔ ﺗﻄﺒﻖ اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح وﺗﻮﻟﺪ،ﻳﻨﻄﺒﻖ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ MTB > SUBC> SUBC> SUBC> SUBC> SUBC> SUBC>
arima 1 0 1 c2; fore 5 c3 c4 c5; gser; gacf; gpacf; ghist; gnormal.
ARIMA Model ARIMA model for CPIChnge Estimates at each iteration Iteration SSE Parameters 0 323.251 0.100 0.100 4.522 1 200.616 0.250 -0.050 3.745 2 182.146 0.184 -0.200 4.067 3 163.067 0.135 -0.350 4.308 4 142.864 0.107 -0.500 4.434 5 121.402 0.111 -0.650 4.407 6 99.668 0.150 -0.800 4.197 7 77.036 0.268 -0.950 3.590 8 67.550 0.418 -0.956 2.828 9 62.802 0.568 -0.964 2.062 10 62.108 0.637 -0.973 1.687 11 62.030 0.644 -0.979 1.619 12 62.003 0.647 -0.982 1.584 13 61.996 0.651 -0.985 1.549 14 61.996 0.651 -0.986 1.539 Unable to reduce sum of squares any further Final Estimates of Parameters
١٩١
Type AR 1 MA 1 Constant Mean
Coef 0.6513 -0.9857 1.5385 4.412
StDev 0.1434 0.0516 0.4894 1.403
T 4.54 -19.11 3.14
Number of observations: 33 Residuals: SS = 61.8375 (backforecasts excluded) MS = 2.0613 DF = 30 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 9.6(DF=10) 17.0(DF=22) * (DF= *)
48 * (DF= *)
Forecasts from period 33 Period 34 35 36 37 38
95 Percent Limits Lower Upper 0.3216 5.9507 -1.8180 8.9801 -2.3061 10.0476 -2.4192 10.5380 -2.4201 10.7848
Forecast 3.1362 3.5810 3.8708 4.0594 4.1823
zt = 1.54 + 0.65zt −1 + at − 0.99at −1 , at
Actual
أي ان اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
N ( 0, 2.06 )
هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر ˆ ˆ φ1 = 0.6513, s.e. φ1 = 0.1434, t = 4.54
( ) θˆ = −0.9857, s.e. (θˆ ) = 0.0516, t = −19.11 δˆ = 1.5385, s.e. (δˆ ) = 0.4894, t = 3.14 1
1
σˆ 2 = 2.0613, with d . f . = 30
.ﻧﻼﺣﻆ ان ﺟﻤﻴﻊ اﻟﻤﻌﺎﻟﻢ ﻣﻌﻨﻮﻳﺔ
:اﻵن ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ
ACF of Residuals for CPIChnge (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
١٩٢
PACF of Residuals for CPIChnge (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1
2
3
4
5
6
7
8
Lag
أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ ﺗﺪل ﻋﻠﻰ أن اﻟﺒﻮاﻗﻲ ﺗﺘﺒﻊ ﺗﻮزﻳﻊ ﺿﺠﺔ : ﻟﻨﻔﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ،ﺑﻴﻀﺎء أي ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ رﺳﻢ اﻟﻤﺪرج اﻟﺘﻜﺮاري
Histogram of the Residuals (response is CPIChnge) 8 7
Frequency
6 5 4 3 2
١٩٣
.ﻳﺒﺪو ﻣﺘﻨﺎﻇﺮ ﺑﻌﺾ اﻟﺸﻴﺊ :ﻟﻨﻨﻈﺮ إﻟﻰ ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ ﻟﻠﺒﻮاﻗﻲ
Normal Probability Plot of the Residuals (response is CPIChnge) 4 3
Residual
2 1 0 -1 -2 -3 -2
-1
0
1
2
Normal Score
.ﻧﺴﺘﻄﻴﻊ أن ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ ﺗﻘﺮﻳﺒﺎ . ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ٥ اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺴﻠﺴﺔ ﻣﻊ
Time Series Plot for CPIChnge (with forecasts and their 95% confidence limits)
CPIChnge
10
5
0
١٩٤
: ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ آﺎﻟﺘﺎﻟﻲAR(٢) دﻋﻨﺎ ﻧﺤﺎول ﺗﻄﺒﻴﻖ ﻧﻤﻮذج MTB > arima 2 0 0 c2 Type AR 1 AR 2 Constant Mean
Coef 1.1872 -0.4657 1.3270 4.765
StDev 0.1625 0.1624 0.2996 1.076
T 7.31 -2.87 4.43
Number of observations: 33 Residuals: SS = 88.6206 (backforecasts excluded) MS = 2.9540 DF = 30 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 19.8(DF=10) 25.4(DF=22) * (DF= *)
zt = 1.33 + 1.187 zt −1 − 0.4657 zt −1 + at , at
N ( 0, 2.95)
48 * (DF= *)
أي ان اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
هﻲt ﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر
( ) φˆ = −0.4657, s.e. (θˆ ) = 0.1624, t = −2.87 δˆ = 1.327, s.e. (δˆ ) = 0.2996, t = 4.43
φˆ1 = 1.1872, s.e. φˆ1 = 0.1625, t = 7.31 2
1
σˆ 2 = 2.954, with d . f . = 30
ﻟﻨﻨﻈﺮ إﻟﻰ اﻹﺧﺘﺒﺎر
H 0 : φ2 = 0 H 1 : φ2 ± 0
اﻹﺣﺼﺎﺋﺔ t0 =
φˆ
2
( )
s.e. φˆ2
=
−0.4657 = −2.8676 0.1624
ﻟﻬﺎ ﺑﺎﻷﻣﺮP-value ﻧﻮﺟﺪ اﻟـ MTB > cdf -2.8676; SUBC> t 30.
Cumulative Distribution Function
١٩٥
Student's t distribution with 30 DF )P( X <= x 0.0037
x -2.8676
أي اﻟـ P-valueﻟﻬﺎ ﺗﺴﺎوي ٠٫٠٠٣٧وهﻲ أﻗﻞ ﻣﻦ ٠٫٠٥أي ﻻﻧﺮﻓﺾ ان φ2 = 0وﺑﺎﻟﺘﺎﻟﻲ ﻧﺮﻓﺾ اﻟﻨﻤﻮذج ). AR(٢ ﺗﻤﺮﻳﻦ: ﺣﺎول ﺗﻄﺒﻴﻖ ﻧﻤﺎذج اﺧﺮى ﻣﻨﺎﺳﺒﺔ ﻋﻠﻰ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﺴﺎﺑﻘﺔ وإﺧﺘﺎر أﻓﻀﻞ ﻧﻤﻮذج ،أﺟﺮي اﻹﺧﺘﺒﺎرات اﻟﻤﻨﺎﺳﺒﺔ واﺳﺘﺨﺪم اﻳﻀﺎ اﻟﻤﻌﻴﺎر . AIC ﻣﺜﺎل ﻋﻤﻠﻲ ﺁﺧﺮ ﻟﺒﻨﺎء ﻧﻤﻮذج ﺗﻨﺒﺆ: ﺳﻮف ﻧﺤﺎول ﺑﻨﺎء ﻧﻤﻮذج ﻟﻠﻤﺘﺴﻠﺴﻠﺔ )z(t -103.2 -391.4 -836.1 -1307.3 -1736.9 -2097.5 -2363.4 -2721.4 -3353.3 -4153.0 -5068.7 -5848.8 -6387.3 -6834.5 -7259.9 -7891.4 -8791.2 -9878.4 -11071.9 -12157.5 -13190.3 -14196.3 -15335.9 -16697.1 -18155.6 -19547.0 -20827.7 -21908.3
-76.7 -339.9 -766.7 -1242.9 -1679.0 -2055.4 -2328.5 -2651.9 -3253.4 -4028.2 -4937.2 -5760.6 -6317.8 -6773.3 -7193.3 -7784.7 -8649.8 -9713.3 -10903.2 -12011.7 -13039.8 -14051.7 -15158.3 -16492.7 -17951.5 -19356.3 -20655.7 -21762.1
-52.8 -291.3 -698.2 -1177.4 -1620.8 -2010.0 -2290.6 -2589.5 -3156.1 -3906.6 -4805.0 -5663.8 -6244.3 -6711.8 -7131.7 -7683.2 -8512.3 -9552.0 -10734.0 -11863.9 -12889.7 -13910.1 -14986.4 -16290.9 -17745.2 -19161.7 -20478.0 -21614.0
-33.1 -246.0 -631.6 -1111.3 -1561.5 -1960.6 -2250.5 -2533.0 -3060.8 -3788.7 -4673.3 -5558.0 -6167.9 -6649.0 -7073.1 -7586.4 -8379.1 -9394.6 -10564.0 -11713.8 -12740.8 -13769.9 -14819.8 -16091.5 -17535.7 -18965.1 -20294.5 -21463.7 -22474.4
وﻟﻬﺎ اﻟﺮﺳﻢ اﻟﺰﻣﻨﻲ اﻟﺘﺎﻟﻲ:
١٩٦
-19.2 -204.0 -566.7 -1044.2 -1500.0 -1906.9 -2210.8 -2482.0 -2968.6 -3675.0 -4542.3 -5444.4 -6090.4 -6584.6 -7015.6 -7495.3 -8249.2 -9239.6 -10392.1 -11560.2 -12593.4 -13629.3 -14657.5 -15895.8 -17324.4 -18766.2 -20107.3 -21310.9 -22335.1
-9.1 -165.9 -504.8 -975.1 -1436.5 -1851.1 -2173.3 -2437.6 -2880.4 -3564.3 -4412.0 -5323.6 -6011.8 -6518.9 -6957.1 -7411.2 -8124.0 -9086.3 -10219.0 -11402.0 -12447.5 -13486.3 -14499.7 -15705.0 -17113.5 -18564.2 -19919.5 -21154.4 -22195.6
-2.5 -132.4 -446.4 -905.3 -1371.8 -1794.5 -2136.4 -2398.7 -2797.6 -3456.9 -4281.7 -5197.8 -5931.6 -6453.5 -6896.0 -7332.9 -8004.6 -8936.4 -10047.2 -11238.9 -12302.5 -13339.6 -14345.9 -15518.4 -16904.2 -18360.1 -19733.5 -20993.7 -22053.2
O r ig in a l T im e S e r ie s
-1 0 0 0 0
-2 0 0 0 0
In d e x
50
100
150
200
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ هﻲ
Autocorrelation Function for z(t) Autocorrelation
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
10 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9813.93 196.91 0.97 8.00 388.73 0.95 6.15 575.44 0.94 5.15 757.06 0.92 4.50 933.59 0.91 4.041105.06 0.89 3.681271.47 0.88 3.391432.85 0.86 3.151589.23 0.84 2.951740.66 0.83 2.781887.17 0.81 2.622028.81
Lag Corr 13 14 15 16 17 18 19 20 21 22 23 24
20 T
LBQ
2.49 2165.64 2.37 2297.73 2.25 2425.14 2.15 2547.94 2.06 2666.22 1.97 2780.05 1.89 2889.51 1.82 2994.69 1.75 3095.67 1.68 3192.54 1.62 3285.39 1.56 3374.32
0.80 0.78 0.76 0.75 0.73 0.72 0.70 0.68 0.67 0.65 0.64 0.62
30
Lag Corr 25 26 27 28 29 30 31 32 33 34 35 36
0.61 0.59 0.58 0.56 0.55 0.53 0.52 0.50 0.49 0.47 0.46 0.45
T
LBQ
1.50 3459.42 1.45 3540.78 1.40 3618.50 1.35 3692.67 1.30 3763.38 1.26 3830.74 1.21 3894.84 1.17 3955.76 1.13 4013.61 1.09 4068.47 1.05 4120.44 1.01 4169.59
40
Lag Corr 37 38 39 40 41 42 43 44 45 46 47 48
0.43 0.42 0.41 0.39 0.38 0.36 0.35 0.34 0.32 0.31 0.30 0.28
T
LBQ
50
Lag Corr
T
LBQ
49 0.27 0.594589.24 50 0.26 0.564607.24
0.98 4216.02 0.94 4259.81 0.91 4301.04 0.87 4339.79 0.84 4376.14 0.81 4410.16 0.77 4441.93 0.74 4471.52 0.71 4498.99 0.68 4524.43 0.65 4547.90 0.62 4569.48
P artial A utocorrelation Function for z(t) Partial Autocorrelation
z(t)
0
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
10
20
30
40
Lag
PAC
T
Lag
PAC
T
Lag
PAC
T
Lag
PAC
T
1 2 3 4 5 6 7 8 9 10 11 12
0.98 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
13.93 -0.17 -0.17 -0.16 -0.16 -0.16 -0.15 -0.15 -0.15 -0.14 -0.14 -0.13
13 14 15 16 17 18 19 20 21 22 23 24
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.13 -0.12 -0.12 -0.11 -0.11 -0.11 -0.11 -0.10 -0.10 -0.09 -0.09 -0.09
25 26 27 28 29 30 31 32 33 34 35 36
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.09 -0.09 -0.09 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.09 -0.09
37 38 39 40 41 42 43 44 45 46 47 48
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.09 -0.09 -0.10 -0.10 -0.11 -0.11 -0.12 -0.12 -0.12 -0.13 -0.13 -0.13
50 Lag
PAC
T
49 -0.01 50 -0.01
-0.14 -0.14
. ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂzt واﺿﺢ ﺟﺪا ان اﻟﻤﺘﺴﻠﺴﻠﺔ وﻧﺮﺳﻤﻬﺎwt = zt − zt −1 ﻧﺄﺧﺬ اﻟﻔﺮوق اﻻوﻟﻰ ١٩٧
F irs t D if f e re n c e s w (t)= z (t)-z (t-1 )
w(t)
0
-1 0 0
-2 0 0 In d e x
50
100
150
200
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ هﻲ Autocorrelation
Autocorrelation Function for w(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9913.93 197.10 0.97 7.98 389.03 0.95 6.12 574.74 0.93 5.10 753.55 0.91 4.43 924.81 0.89 3.941087.93 0.86 3.561242.44 0.83 3.251388.10 0.81 2.981524.83 0.78 2.761652.58 0.75 2.561771.42 0.72 2.381881.49
15
Lag Corr 13 14 15 16 17 18 19 20 21 22 23 24
0.69 0.66 0.63 0.60 0.57 0.54 0.52 0.49 0.47 0.45 0.43 0.41
T
LBQ
2.21 1983.03 2.07 2076.38 1.93 2161.98 1.81 2240.33 1.70 2311.99 1.60 2377.42 1.50 2437.13 1.41 2491.57 1.33 2541.23 1.26 2586.61 1.19 2628.22 1.13 2666.48
25 Lag Corr 25 26 27 28 29 30 31 32 33 34 35 36
0.39 0.38 0.37 0.36 0.35 0.34 0.34 0.34 0.34 0.34 0.34 0.34
T
LBQ
1.08 2701.91 1.03 2734.99 1.00 2766.16 0.96 2795.84 0.94 2824.46 0.92 2852.35 0.91 2879.79 0.90 2907.03 0.89 2934.32 0.89 2961.85 0.89 2989.82 0.89 3018.34
35 Lag Corr 37 38 39 40 41 42 43 44 45 46 47 48
0.34 0.35 0.35 0.35 0.35 0.35 0.35 0.34 0.34 0.33 0.33 0.32
T
45 LBQ
0.90 3047.50 0.90 3077.32 0.90 3107.68 0.90 3138.42 0.90 3169.33 0.89 3200.21 0.88 3230.81 0.87 3260.99 0.85 3290.64 0.84 3319.59 0.82 3347.55 0.79 3374.17
Lag Corr
T
LBQ
49 0.31 0.763399.16
Partial Autocorrelation
P a rtia l A u to c o rre la tio n F u n c tio n fo r w (t) 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 -0 .2 -0 .4 -0 .6 -0 .8 -1 .0
5 Lag
P AC
T
1 2 3 4 5 6 7 8 9 10 11 12
0 .9 9 -0 .1 4 -0 .1 1 -0 .0 7 -0 .0 6 -0 .0 7 -0 .0 6 -0 .0 3 -0 .0 3 -0 .0 3 -0 .0 2 -0 .0 2
1 3 .9 3 -1 .9 7 -1 .5 1 -0 .9 6 -0 .9 0 -0 .9 7 -0 .7 8 -0 .4 8 -0 .3 7 -0 .3 7 -0 .3 4 -0 .2 2
15
Lag
P AC
T
1 3 -0 .0 1 1 4 -0 .0 0 1 5 0 .0 1 1 6 0 .0 2 1 7 0 .0 2 1 8 0 .0 1 1 9 0 .0 1 2 0 0 .0 1 2 1 0 .0 1 2 2 0 .0 3 2 3 0 .0 2 2 4 0 .0 2
-0 .1 6 -0 .0 2 0 .1 0 0 .2 3 0 .2 5 0 .1 0 0 .1 2 0 .1 1 0 .1 6 0 .3 5 0 .3 1 0 .2 7
25 Lag
35
45
P AC
T
Lag
P AC
T
2 5 0 .0 4 2 6 0 .0 4 2 7 0 .0 3 2 8 0 .0 4 2 9 0 .0 4 3 0 0 .0 2 3 1 0 .0 2 3 2 0 .0 1 3 3 0 .0 1 3 4 0 .0 1 3 5 0 .0 0 3 6 -0 .0 0
0 .5 2 0 .6 2 0 .4 7 0 .5 5 0 .5 6 0 .3 4 0 .2 2 0 .1 6 0 .1 8 0 .1 5 0 .0 4 -0 .0 7
37 38 39 40 41 42 43 44 45 46 47 48
-0 .0 1 -0 .0 2 -0 .0 4 -0 .0 4 -0 .0 4 -0 .0 4 -0 .0 3 -0 .0 0 0 .0 1 -0 .0 1 -0 .0 4 -0 .0 5
-0 .0 7 -0 .2 9 -0 .6 3 -0 .6 0 -0 .5 4 -0 .5 5 -0 .4 2 -0 .0 7 0 .0 8 -0 .1 8 -0 .5 6 -0 .7 6
Lag
P AC
T
4 9 -0 .0 3
-0 .4 8
. ﻻﺗﺰال ﻏﻴﺮ ﻣﺴﺘﻘﺮة ﻓﻲ اﻟﻤﺘﻮﺳﻂwt واﺿﺢ ﺟﺪا ان اﻟﻤﺘﺴﻠﺴﻠﺔ )ﻻﺣﻆ ان هﺬا اﻟﻔﺮق اﻟﺜﺎﻧﻲ ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻷﺻﻠﻴﺔ( وﻧﺮﺳﻤﻬﺎyt = wt − wt −1 ﻧﺄﺧﺬ اﻟﻔﺮوق اﻻوﻟﻰ
١٩٨
F ir s t D if f e r e n c e s
y (t)= w (t)-w (t-1 )
1 0
y(t)
5
0
-5
In d e x
5 0
1 0 0
1 5 0
2 0 0
:اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻟﻬﺬﻩ اﻟﻤﺘﺴﻠﺴﻠﺔ اﻟﻤﻔﺮﻗﺔ هﻲ Autocorrelation
A utocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag Corr 1 2 3 4 5 6 7 8 9 10 11 12
T
LBQ
0.9313.08 173.62 0.80 6.83 303.36 0.70 4.89 401.81 0.62 3.89 479.62 0.56 3.26 542.93 0.50 2.77 593.77 0.43 2.32 632.35 0.36 1.90 659.84 0.29 1.50 677.66 0.22 1.11 687.71 0.15 0.75 692.41 0.08 0.41 693.82
15
Lag Corr
T
LBQ
25 Lag Corr
13 0.01 0.06 693.85 14 -0.05 -0.23 694.31 15 -0.09 -0.46 696.12 16 -0.14 -0.70 700.34 17 -0.19 -0.97 708.51 18 -0.25 -1.24 722.16 19 -0.31 -1.53 743.30 20 -0.37 -1.83 774.40 21 -0.44 -2.13 818.29 22 -0.51 -2.40 876.73 23 -0.56 -2.57 948.00 1030.79 24 -0.60 -2.67
T
LBQ
1124.21 25 -0.64 -2.73 1224.05 26 -0.66 -2.72 1324.28 27 -0.66 -2.62 1418.06 28 -0.63 -2.44 1500.63 29 -0.59 -2.22 1572.45 30 -0.55 -2.01 1632.42 31 -0.50 -1.80 1677.44 32 -0.43 -1.53 1708.46 33 -0.36 -1.25 1728.63 34 -0.29 -1.00 1740.51 35 -0.22 -0.76 1746.42 36 -0.16 -0.53
35 Lag Corr
T
45 LBQ
1748.28 37 -0.09 -0.30 1748.29 38 -0.01 -0.03 1749.70 39 0.08 0.26 1755.14 40 0.15 0.50 1765.20 41 0.20 0.68 1778.55 42 0.23 0.78 1793.44 43 0.24 0.82 1810.08 44 0.25 0.86 1831.09 45 0.28 0.96 1860.92 46 0.34 1.13 1903.36 47 0.40 1.34 1956.81 48 0.45 1.49
Lag Corr
T
LBQ
49 0.48 1.562017.77
Partial Autocorrelation
P artial A utocorrelation Function for y(t) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
5 Lag PAC
T
0.93 -0.46 0.33 -0.14 0.12 -0.13 0.00 -0.06 -0.08 -0.04 -0.04 -0.09
13.08 -6.44 4.71 -1.97 1.72 -1.78 0.01 -0.81 -1.11 -0.55 -0.58 -1.21
1 2 3 4 5 6 7 8 9 10 11 12
15
Lag PAC 13 14 15 16 17 18 19 20 21 22 23 24
-0.05 0.05 -0.07 -0.09 -0.04 -0.08 -0.12 -0.11 -0.15 -0.07 -0.06 -0.15
25
T
Lag
PAC
T
-0.75 0.67 -0.95 -1.30 -0.55 -1.11 -1.71 -1.50 -2.12 -0.93 -0.87 -2.12
25 26 27 28 29 30 31 32 33 34 35 36
-0.03 -0.03 0.00 0.08 -0.01 -0.04 0.15 0.06 -0.00 0.04 0.03 0.04
-0.46 -0.44 0.04 1.13 -0.14 -0.56 2.17 0.88 -0.05 0.49 0.46 0.50
35 Lag PAC 37 38 39 40 41 42 43 44 45 46 47 48
0.08 0.11 0.04 0.04 -0.05 -0.05 -0.14 -0.01 0.07 0.06 -0.01 -0.09
45 T
1.16 1.56 0.51 0.52 -0.74 -0.66 -1.95 -0.10 0.92 0.79 -0.17 -1.24
Lag PAC 49
0.07
T 1.00
ﻧﻼﺣﻆ ﻣﻦ ﺷﻜﻞ اﻟﻤﺘﺴﻠﺴﻠﺔ و اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ اﻧﻬﺎ اﺻﺒﺤﺖ ﻣﺴﺘﻘﺮة . d=٢ ﻓﻲ اﻟﻤﺘﻮﺳﻂ اي ان ﻣﻦ اﻧﻤﺎط اﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ واﻟﺘﺮاﺑﻄﺎت اﻟﺬاﺗﻴﺔ اﻟﺠﺰﺋﻴﺔ ﻧﺮى اﻧﻬﺎ ﺗﺘﺨﺎﻣﺪ ﻣﻦ اﻟﺘﺨﻠﻒ اﻷول ﻣﻤﺎ وﺳﻮف ﻧﻄﺒﻖ هﺬا اﻟﻨﻤﻮذج ﺑﺎﻷﻣﺮzt ﻟﻠﻤﺘﺴﻠﺴﻠﺔ اﻷﺻﻠﻴﺔARIMA(١،٢،١) ﻳﺮﺷﺢ ﻧﻤﻮذج MTB > ARIMA 1 2 1 'z(t)' 'RESI2' 'FITS2'; SUBC> NoConstant; SUBC> Forecast 10 c4 c5 c6; SUBC> GACF; SUBC> GPACF; SUBC> GHistogram;
١٩٩
SUBC>
GNormalplot.
ARIMA Model ARIMA model for z(t) Estimates at each iteration Iteration SSE Parameters 0 2462.77 0.100 0.100 1 1345.58 0.250 -0.050 2 1170.63 0.203 -0.200 3 984.83 0.182 -0.350 4 782.47 0.200 -0.500 5 560.15 0.278 -0.650 6 363.93 0.428 -0.765 7 259.20 0.578 -0.814 8 202.76 0.728 -0.842 9 185.51 0.861 -0.859 10 185.36 0.873 -0.860 11 185.36 0.875 -0.860 12 185.36 0.875 -0.860 Relative change in each estimate less than Final Estimates of Parameters Type Coef StDev AR 1 0.8749 0.0353 MA 1 -0.8599 0.0357
0.0010
T 24.75 -24.12
Differencing: 2 regular differences Number of observations: Original series 200, after differencing 198 Residuals: SS = 183.717 (backforecasts excluded) MS = 0.937 DF = 196 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 Chi-Square 3.9(DF=10) 13.0(DF=22) 33.1(DF=34)
48 46.0(DF=46)
Forecasts from period 200 Period 201 202 203 204 205 206 207 208 209 210
Forecast -22615.2 -22757.2 -22900.4 -23044.5 -23189.5 -23335.2 -23481.5 -23628.4 -23775.8 -23923.7
95 Percent Limits Lower Upper -22617.1 -22613.3 -22764.6 -22749.9 -22917.2 -22883.5 -23075.3 -23013.7 -23238.8 -23140.1 -23407.8 -23262.6 -23582.1 -23380.9 -23761.8 -23495.0 -23946.7 -23605.0 -24136.6 -23710.7
zt = 0.875t −1 z + at − 0.859at −1 , at
N ( 0,0.937 )
Actual
اﻟﻨﻤﻮذج اﻟﻤﻘﺘﺮح هﻮ
هﻲt وﻣﻘﺪرات اﻟﻤﻌﺎﻟﻢ وإﻧﺤﺮاﻓﺎﺗﻬﺎ اﻟﻤﻌﻴﺎرﻳﺔ وﻗﻴﻤﺔ إﺧﺘﺒﺎر
٢٠٠
( ) s.e. (θˆ ) = 0.0357,
φˆ1 = 0.8749, s.e. φˆ1 = 0.0353, t = 24.75 θˆ1 = −0.8599,
1
t = −24.12
σˆ 2 = 0.937, with d . f . = 196
.ﻧﻼﺣﻆ ان اﻟﻤﻌﺎﻟﻢ ﻣﻌﻨﻮﻳﺔ :اﻵن ﻧﻔﺤﺺ اﻟﺒﻮاﻗﻲ ACF of Residuals for z(t) (with 95% confidence limits for the autocorrelations) 1.0 0.8
Autocorrelation
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
Lag
PACF of Residuals for z(t) (with 95% confidence limits for the partial autocorrelations) 1.0
Partial Autocorrelation
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 5
10
15
20
25
30
35
40
45
Lag
أﻧﻤﺎط اﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ واﻟﺘﺮاﺑﻂ اﻟﺬاﺗﻲ اﻟﺠﺰﺋﻲ ﻟﻠﺒﻮاﻗﻲ ﺗﺪل ﻋﻠﻰ أن اﻟﺒﻮاﻗﻲ ﺗﺘﺒﻊ ﺗﻮزﻳﻊ ﺿﺠﺔ : ﻟﻨﻔﺤﺺ ﻃﺒﻴﻌﻴﺔ اﻟﺒﻮاﻗﻲ،ﺑﻴﻀﺎء أي ﻏﻴﺮ ﻣﺘﺮاﺑﻄﺔ رﺳﻢ اﻟﻤﺪرج اﻟﺘﻜﺮاري
٢٠١
Histogram of the Residuals (response is z(t))
Frequency
20
10
0 -3
-2
-1
0
1
2
Residual
.ﻳﺒﺪو ﻣﺘﻨﺎﻇﺮ ﺑﻌﺾ اﻟﺸﻴﺊ :ﻟﻨﻨﻈﺮ إﻟﻰ ﻣﺨﻄﻂ اﻹﺣﺘﻤﺎل اﻟﻄﺒﻴﻌﻲ ﻟﻠﺒﻮاﻗﻲ Normal Probability Plot of the Residuals (response is z(t))
2
Residual
1
0
-1
-2
-3 -3
-2
-1
0
1
2
3
Normal Score
.ﻧﺴﺘﻄﻴﻊ أن ﻧﻘﻮل ان اﻟﺒﻮاﻗﻲ ﻃﺒﻴﻌﻴﺔ ﺗﻘﺮﻳﺒﺎ . ﻓﺘﺮات ﺗﻨﺒﺆ٩٥٪ ﺗﻨﺒﺆات ﻟﻠﻘﻴﻢ اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﻣﻊ١٠ اﻟﺮﺳﻢ اﻟﺘﺎﻟﻲ ﻟـ Forecast of 20 Future values with 95% limits -22500
z(t)
-23000
-23500
-24000
0
1
2
3
4
5
6
7
8
9
Time
٢٠٢
10
٢٠٣
:اﻟﻤﺮاﺟﻊ ١-
Abraham, B. and Ledoter, J. (1983). Statistical Methods for Forecasting, John Wiley, New York. ٢Anderson, T. W. (1971). The Statistical Analysis of Time Series, John Wiley, New York. ٣Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis
Forecasting and Control, 2nd ed., Holden-Day, San Francisco. Brillinger, D. R. (1975). Time Series: Data Analysis and Theory, Holt, Rinehart and Winston, New York. Fuller, W. A. (1976). Introduction to Statistical Time Series, John Wiley, New York. Granger, C. W. J. and Newbold, P. (1977). Forecasting Economic Time Series, Academic Press, New York. Hannan, E. J. (1970). Multiple Time Series, John Wiley, New York. Harvey, A. C. (1980). Time Series Models, Halsted Press, New York. P
٤٥٦٧٨-
P
٩-
Montgomery, D. C., Johnson, L. A. and Gardiner, J. S. (1990). Forecasting and Time Series Analysis, 2nd ed., McGraw-Hill International Edition. ١٠Makridakis, S., Wheelwright, S. C. and McGee, V. E. (1983). Forecasting Methods and Applications, 2nd ed., P
P
P
P
John Wiley, New York. Shumway, R. H. (1988). Applied Statistical Time Series ١١Analysis, Prentice-Hall, New York. Wei, W. W. S. (1990). Time Series Analysis Univariate and ١٢Multivariate Methods, Addison Wesley. ١٣Minitab Reference Manual, Release 11 for Windows. (1998).
T
ﻟﻸﺳﻒ اﻟﺸﺪﻳﺪ ﻻﺗﻮﺟﺪ ﺣﺴﺐ ﻋﻠﻤﻲ ﻣﺮاﺟﻊ ﻋﺮﺑﻴﺔ ﺗﻐﻄﻲ آﻞ أو ﺟﺰء ﻣﻦ ﻣﺤﺘﻮى اﻟﻤﺎدة اﻟﻤﻐﻄﺎة ﻓﻲ هﺬا اﻟﻜﺘﺎب وأرﺟﻮا ﻣﻦ أي ﻃﺎﻟﺐ أو ﺑﺎﺣﺚ أو ﻣﺪرس ﻳﻌﻠﻢ ﺑﻤﺜﻞ :هﺬا اﻟﻤﺮﺟﻊ او اﻟﻜﺘﺎب أن ﻳﺮﺳﻞ ﻟﻲ ﻣﻼﺣﻈﺔ ﻋﻠﻰ اﻟﺒﺮﻳﺪ اﻹﻟﻜﺘﺮوﻧﻲ [email protected] أو[email protected] .وﺷﻜﺮا T
T
T
٢٠٤
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