Problem 06 Rev 1 - Gmdh

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‫ﺷﮕﺮﺩﻫﺎﻱ ﻋﺪﺩﻱ ﺩﺭ ﭘﺎﺳﺨﻴﺎﺑﻲ ﭘﺮﺳﺸﻬﺎﻱ ﻣﻬﻨﺪﺳﻲ‬

‫ﻣﺴﺄﻟﻪ ‪ 4.New.2‬ـ ﮔﻤﺎﻥ ﮐﻨﻴﺪ ﺩﻭ ﻭﺭﻭﺩﻱ ‪ xi‬ﻭ ‪ x j‬ﺑﻪ ﻳﮏ ﻧﺮﻭﻥ ﻻﻳﻪ ﺩﻭﻡ ﺍﺯ ﻳﮏ ﺷﺒﮑﻪ ﻋﺼﺒﻲ ﻣﺼﻨﻮﻋﻲ ﻭﺍﺭﺩ ﺷﻮﻧﺪ ﮐﻪ‬ ‫ﺧﺮﻭﺟﻲ ﺯﻳﺮ ﺭﺍ ﺗﻮﻟﻴﺪ ﮐﻨﺪ‪.‬‬ ‫‪yˆ = G ( xi , x j ) = a1 + a 2 xi + a 3 x j + a 4 xi2 + a 5 xi x j + a 6 x 2j‬‬

‫ﺩﺭ ﺷﮑﻞ ﺯﻳﺮ ﭼﻨﻴﻦ ﻧﺮﻭﻧﻲ ﺭﺍ ﻧﻤﺎﻳﺶ ﺩﺍﺩﻩﺍﻳﻢ‪.‬‬

‫ﺭﻭﺷﻦ ﺍﺳﺖ ﮐﻪ ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﻣﺠﻤﻮﻉ ﻧﻤﺎﻫﺎﻱ ﺑﺎﻻﺗﺮ ‪ xi‬ﻭ ‪ x j‬ﻫﻢ ﺍﺳﺘﻔﺎﺩﻩ ﮐﺮﺩ‪ .‬ﺍﮔﺮ ﭼﻨﻴﻦ ﮐﻨﻴﻢ ﻭ ﺗﻌﺪﺍﺩ ﻭﺭﻭﺩﻳﻬﺎ ﺭﺍ ﻫﻢ ﺑﻴﺸﺘﺮ‬ ‫ﺍﺯ ﺍﻳﻦ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﻧﻬﺎﻳﺘﺎ ﺑﻪ ﻋﺒﺎﺭﺗﻲ ﺑﻪ ﺷﮑﻞ‬ ‫⋅ ⋅ ⋅ ‪yˆ = c 0 + ∑ ci xi + ∑∑ cij xi x j + ∑∑∑ cijk xi x j x k‬‬ ‫‪k‬‬

‫‪j‬‬

‫‪j‬‬

‫‪i‬‬

‫‪i‬‬

‫‪i‬‬

‫ﻣﻲﺭﺳﻴﻢ‪ ،‬ﮐﻪ ﺑﻪ ﺁﻥ ﭼﻨﺪﺟﻤﻠﻪﺍﻱ ﻛﻮﻟﻤﺎﮔﺎﺭﻭﻑ ـ ﮔﺎﺑﻮﺭ‪ ١‬ﻣﻲﮔﻮﻳﻨﺪ‪ .‬ﺑﺮﺍﻱ ﻣﺤﺎﺳﺒﻪ ﭼﻨﻴﻦ ﻋﺒﺎﺭﺗﻲ ﻣﻲﺗﻮﺍﻥ ﻫﻤﻪ ﺗﺮﮐﻴﺒﺎﺕ‬ ‫ﻳﮏﺗﺎﻳﻲ ﺗﺎ ‪ p‬ﺗﺎﻳﻲ ﻭﺭﻭﺩﻳﻬﺎ ﺭﺍ ﺑﺎ ﺍﺣﺘﺴﺎﺏ ﻣﺠﺎﺯ ﺑﻮﺩﻥ ﺗﮑﺮﺍﺭ ﻫﺮ ﻋﺪﺩ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩﻩ‪ ،‬ﺩﺭ ﻫﻢ ﺿﺮﺏ ﮐﺮﺩﻩ ﻭ ﺑﺮﺩﺍﺭﻱ ﺳﻄﺮﻱ‬ ‫ﺍﺯ ﺁﻧﻬﺎ ﺳﺎﺧﺖ‪ .‬ﺳﭙﺲ ﺍﻳﻦ ﺑﺮﺩﺍﺭ ﺳﻄﺮﻱ ﺭﺍ ﺩﺭ ﺑﺮﺩﺍﺭ ﺳﺘﻮﻧﻲ ﺿﺮﺍﻳﺐ ﭼﻨﺪﺟﻤﻠﻪﺍﻱ ﺿﺮﺏ ﮐﺮﺩ ﺗﺎ ˆ‪ y‬ﺑﻪ ﺩﺳﺖ ﺁﻳﺪ‪ .‬ﺑﺮﺍﻱ‬ ‫ﻧﻤﻮﻧﻪ‪ ،‬ﻫﻤﻪ ﺗﺮﮐﻴﺒﺎﺕ ﻳﮏﺗﺎﻳﻲ ﻭ ﺩﻭﺗﺎﻳﻲ ‪ xi‬ﻭ ‪ x j‬ﺑﺎ ﺍﺣﺘﺴﺎﺏ ﻣﺠﺎﺯ ﺑﻮﺩﻥ ﺗﮑﺮﺍﺭ ‪ xi‬ﻭ ‪ x j‬ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﻧﻮﺷﺖ‪.‬‬ ‫‪xi‬‬ ‫‪xj‬‬ ‫‪xi2‬‬ ‫‪xi x j‬‬ ‫‪x 2j‬‬

‫‪xi‬‬ ‫‪xj‬‬ ‫‪xi‬‬ ‫‪xj‬‬ ‫‪xj‬‬

‫‪ :‬ﺗﺮﮐﻴﺒﺎﺕ ﻳﮏﺗﺎﻳﻲ‬ ‫‪xi‬‬ ‫‪xi‬‬ ‫‪xj‬‬

‫‪ :‬ﺗﺮﮐﻴﺒﺎﺕ ﺩﻭﺗﺎﻳﻲ‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎ ﮔﻤﺎﻥ ‪ a = [a1 a 2 a3 a 4 a5 a6 ]T‬ﻣﻲﺗﻮﺍﻥ ﻧﻮﺷﺖ‪:‬‬ ‫‪x 2j ]a‬‬

‫‪xi x j‬‬

‫‪xi2‬‬

‫‪xj‬‬

‫‪yˆ = [1 xi‬‬

‫ﺑﺮﺍﻱ ﺁﻣﻮﺯﺵ ﭼﻨﻴﻦ ﺷﺒﮑﻪﺍﻱ ﻣﻲﺗﻮﺍﻥ ﺑﺮﺍﻱ ﻫﺮﮐﺪﺍﻡ ﺍﺯ ﻣﺸﺎﻫﺪﺍﺕ ﻣﺴﺄﻟﻪ ﻳﮏ ﻣﻌﺎﺩﻟﻪ ﻫﻤﺎﻧﻨﺪ ﻣﻌﺎﺩﻟﻪ ﺑﺎﻻ ﺗﺸﮑﻴﻞ ﺩﺍﺩ‪ .‬ﺍﮔﺮ ﺷﻤﺎﺭ‬ ‫ﻣﺸﺎﻫﺪﺍﺕ ‪ m‬ﺑﺎﺷﺪ‪ m ،‬ﻣﻌﺎﺩﻟﻪ ﺑﻪ ﺩﺳﺖ ﻣﻲﺁﻳﺪ‪ .‬ﺭﻭﺷﻦ ﺍﺳﺖ ﮐﻪ ﺍﮔﺮ ﺷﻤﺎﺭ ﻣﺸﺎﻫﺪﺍﺕ ﺍﺯ ﻃﻮﻝ ‪ a‬ﺑﻴﺸﺘﺮ ﺑﺎﺷﺪ‪ ،‬ﺩﺳﺘﮕﺎﻩ‬ ‫ﻣﻌﺎﺩﻻﺕ ‪ over-determined‬ﺧﻮﺍﻫﺪ ﺷﺪ‪ ،‬ﮐﻪ ﺑﻪ ﻳﺎﺭﻱ ‪ SVD‬ﻣﻲﺗﻮﺍﻥ ﺑﻬﺘﺮﻳﻦ ﭘﺎﺳﺦ ﺁﻥ ﺭﺍ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩ‪ .‬ﺍﮐﻨﻮﻥ ﺗﺎﺑﻌﻲ‬ ‫ﺑﻨﻮﻳﺴﻴﺪ ﮐﻪ ﺑﺎ ﮔﺮﻓﺘﻦ ﺑﺮﺩﺍﺭ ﻭﺭﻭﺩﻳﻬﺎ ﻭ ﺩﺭﺟﻪ ﭼﻨﺪﺟﻤﻠﻪﺍﻱ‪ ،‬ﺑﺮﺩﺍﺭ ﺿﺮﺍﻳﺐ ﻣﺘﻨﺎﻇﺮ ﺑﺎ ﭼﻨﺪﺟﻤﻠﻪﺍﻱ ﻛﻮﻟﻤﺎﮔﺎﺭﻭﻑ ـ ﮔﺎﺑﻮﺭ ﺭﺍ ﺑﻪ‬ ‫ﺩﺳﺖ ﺁﻭﺭﺩ‪ .‬ﺑﻪ ﻳﺎﺭﻱ ﺍﻳﻦ ﺗﺎﺑﻊ ﻭ ﺑﺎ ﺍﻧﺘﺨﺎﺏ ﻣﻨﺎﺳﺐ ﺩﺭﺟﻪ ﭼﻨﺪﺟﻤﻠﻪﺍﻱ ‪ XOR‬ﺭﺍ ﺷﺒﻴﻪﺳﺎﺯﻱ ﮐﻨﻴﺪ‪.‬‬ ‫‪Kolmogorov-Gabor‬‬

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