[presentation] Nonlinear Dynamics Of The Great Salt Lake: Short Term Forecasting And Probabilistic Cost Estimation

Introduction

Background

Methodology

Results

Conclusion

Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation Cameron Bracken Humboldt State University

E441 December, 2008

Introduction

Background

Methodology

Results

Conclusion

Great Salt Lake

I

Idaho

Terminal Lake I I

Nevada

Great Salt Lake

I

Wyo Utah

I I

Drainage Area 90,000 km2 Lake acts as a low pass filter. I

Outline of Great Salt Lake Basin

N

No Surface Outflow Only “outflows” are subsurface and evaporation High Salt Content

Reflects long term atmospheric trends.

Introduction

Background

Methodology

Results

Conclusion

Typical Terminal Lake Behavior Surface Elevation (ft.) 4195 4200 4205 4210

Erratic and drastic fluctuations in volume and stage.

1850 1900 1950 2000 GSL 1983-1987: Lake gained 1.5 MAF and rose 10 ft. Peak of 4211.60 ft.

Introduction

Background

Methodology

Results

Conclusion

Past efforts

I

Water balance model which require a TON of data (1979).

I

Correlate with other atmospheric data (2008). ARMA model directly on the time series?

I

I I I

Data is not stationary. ARMA models with huge (∼ 70) lags have failed. Really need a nonlinear autoregressive model.

Introduction

Background

Methodology

Results

Conclusion

Past efforts

I

Water balance model which require a TON of data (1979).

I

Correlate with other atmospheric data (2008). ARMA model directly on the time series?

I

I I I

I

Data is not stationary. ARMA models with huge (∼ 70) lags have failed. Really need a nonlinear autoregressive model.

Use the filtering effect to justify modeling as a low dimensional chaotic system.

Introduction

Background

Methodology

Results

Chaotic Systems Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994]

Conclusion

Introduction

Background

Methodology

Results

Chaotic Systems Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994] Lorenz System: x˙ = σ(y − x ) y˙ = rx − y − xz z˙ = xy − bz

Conclusion

Introduction

Background

Methodology

Results

Chaotic Systems Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994] Lorenz System: x˙ = σ(y − x ) y˙ = rx − y − xz z˙ = xy − bz →Nonlinear

Conclusion

Introduction

Background

Methodology

Results

Conclusion

Chaotic Systems Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994] Lorenz System:

Prediction fails out here

x˙ = σ(y − x ) y˙ = rx − y − xz z˙ = xy − bz →Nonlinear

thorizon t =0 Two indistinguishable initial conditions

Introduction

Background

Methodology

Results

Conclusion

Chaotic Systems Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994] Lorenz System:

y˙ = rx − y − xz z˙ = xy − bz

z

x˙ = σ(y − x )

→Nonlinear x

y

Introduction

Background

Methodology

Results

Conclusion

Chaotic Systems

Chaos is: “(1) Aperiodic long-term behavior in a (2) deterministic system that exhibits (3) sensitive dependence on initial conditions”. [Strogatz, 1994]

Lorenz System:

y˙ = rx − y − xz z˙ = xy − bz

z

x˙ = σ(y − x )

→Nonlinear x

Statistically, high dimensional chaos = randomness.

y

Introduction

Background

Methodology

Results

Conclusion

Attractor Reconstruction Technique of geometrically reconstructing an attractor from sample of a single coordinate of a dynamical system (just a time series)!

Introduction

Background

Methodology

Results

Conclusion

Attractor Reconstruction Technique of geometrically reconstructing an attractor from sample of a single coordinate of a dynamical system (just a time series)!

First define the time series st in terms of indexes: sn+T = st0 +(n+T )τs Then construct a vector of lagged or “embedded” time series: yn = [sn , sn+T , sn+2T , ..., sn+(dE −1)T ]

Introduction

Background

Methodology

Results

Conclusion

Attractor Reconstruction

zn+6

en+6

Technique of geometrically reconstructing an attractor from sample of a single coordinate of a dynamical system (just a time series)!

zn Lorenz reconstructed

zn+3

en

en+3

GSL Reconstructed

Introduction

Background

Methodology

Results

Conclusion

Forecast Model For a forecast starting at index I , the water surface elevation K steps in the future is a function of the current state of the system: sI +K = f (yI ) + εI where yI = [sI −(dE −1)T −1 , ..., sI −(dE −2)T −1 , sI −1 ] Any regression model can be used to construct f . If f is linear and T = 1 then this is a linear AR model.

Introduction

Background

Methodology

Results

Conclusion

Generating Ensembles

1. Construct f with locally weighted polynomial model with α and p. 2. Fit models to all combinations of parameters: dE , T , α, and p. 3. Evaluate goodness of fit. 4. Forecast with each model within acceptable range (10%).

Introduction

Background

Methodology

Forecast Results Time horizon for GSL ≈ 1 year. 1985 event.

Results

Conclusion

Introduction

Background

Methodology

Results

Conclusion

Forecast Results

Stage (ft. above MSL) 4206 4208 4210 4212

4206 4208 4210 4212

Time horizon for GSL ≈ 1 year. 1985 event.

1985 1986 1987 1988 1989 Blind Forecast10-step Forecast.

1985 1986 1987 1988 1989

Introduction

Background

Methodology

Results

Conclusion

Forecast Results

4206 4208 4210 4212

Stage (ft. above MSL) 4206 4208 4210 4212

Time horizon for GSL ≈ 1 year. 1985 event.

1985 1986 1987 1988 1989

1985 1986 1987 1988 1989 5-step Forecast.1-step Forecast.

Introduction

Background

Methodology

Results

Conclusion

Probability Density

Probabilistic Cost Estimate

Real density function for February 1987, cost function is hypothetical.

Cost ($)

Introduction

Background

Methodology

Results

Conclusion

Conclusion

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Nonlinear time series methods are powerful generalizations to linear models.

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Able to blind forecast accurately within time horizon.

I

Ensemble for probabilistic cost forecast.

The End

Introduction

Background

Methodology

Results

Conclusion

Constructing a Phase Space Model Use a series of tests (Average mutual information, False nearest neighbor, etc.) to determine appropriate ranges of dE and T (GSL dE = 3–5, T = 13–18). For a forecast starting at I    S= 

s1 s2 .. .

s1+T s2+T .. .

sI −2−(dE −1) sI −2−(dE −1)T +T

··· ··· .. .

s1+(dE −1)T s2+(dE −1)T .. .

···

sI −2−(dE −1)T +(dE −1)T

    

Introduction

Background

Methodology

Results

Conclusion

Constructing a Phase Space Model Use a series of tests (Average mutual information, False nearest neighbor, etc.) to determine appropriate ranges of dE and T (GSL dE = 3–5, T = 13–18). For a forecast starting at I    S= 

s1 s2 .. .

s1+T s2+T .. .

sI −2−(dE −1) sI −2−(dE −1)T +T    r= 

··· ··· .. .

s1+(dE −1)T s2+(dE −1)T .. .

···

sI −2−(dE −1)T +(dE −1)T

s1+(dE −1)T +1 s2+(dE −1)T +1 .. . sI −2−(dE −1)T +(dE −1)T +1

    

    