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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Great Salt Lake
Nevada
Idaho
Outline of GreatSalt Lake Basin
N
Wyo
Utah
Great
Salt
Lake
Terminal Lake No Surface Outflow Only outflows are
subsurface andevaporation
High Salt Content
Drainage Area 90,000 km2
Lake acts as a low passfilter. Reflects long term
atmospheric trends.
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Typical Terminal Lake Behavior
Erratic and drastic fluctuations in volume and stage.
Su
rfaceElevation
(ft.)
1850 1900 1950 2000
4195
4200
4205
4210
GSL 1983-1987: Lake gained 1.5 MAF and rose 10 ft. Peak of4211.60 ft.
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Past efforts
Water balance model which require a TON of data (1979).
Correlate with other atmospheric data (2008).
ARMA model directly on the time series? Data is not stationary. ARMA models with huge ( 70) lags have failed. Really need a nonlinear autoregressive model.
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Past efforts
Water balance model which require a TON of data (1979).
Correlate with other atmospheric data (2008).
ARMA model directly on the time series? 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 lowdimensional chaotic system.
I B M R C
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
I B M R C
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
Lorenz System:
x = (y x)
y = rx y xz
z = xy bz
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
Lorenz System:
x = (y x)
y = rx y xz
z = xy bz
Nonlinear
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
Lorenz System:
x = (y x)
y = rx y xz
z = xy bz
Nonlinear t = 0
Two indistinguishable
initial conditions
thorizon
Prediction
fails outhere
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
10/23
Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
Lorenz System:
x = (y x)
y = rx y xz
z = xy bz
Nonlinear
x
y
z
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
11/23
Introduction Background Methodology Results Conclusion
Chaotic Systems
Chaos is:
(1) Aperiodic long-term behavior in a (2)deterministic system that exhibits (3) sensitivedependence on initial conditions. [Strogatz, 1994]
Lorenz System:
x = (y x)
y = rx y xz
z = xy bz
Nonlinear
x
y
z
Statistically, high dimensional chaos = randomness.
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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ro uc o c grou o o og R su s Co c us o
Attractor Reconstruction
Technique of geometrically reconstructing an attractor fromsample of a single coordinate of a dynamical system (just a timeseries)!
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Attractor Reconstruction
Technique of geometrically reconstructing an attractor fromsample of a single coordinate of a dynamical system (just a timeseries)!
First define the time series st in terms of indexes:
sn+T = st0+(n+T)s
Then construct a vector of lagged or embedded time series:
yn = [sn, sn+T, sn+2T, ..., sn+(dE1)T]
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Attractor Reconstruction
Technique of geometrically reconstructing an attractor fromsample of a single coordinate of a dynamical system (just a timeseries)!
zn
zn+3
zn+6
en
en+3
en+6
Lorenz reconstructed GSL Reconstructed
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Forecast Model
For a forecast starting at index I, the water surface elevation Ksteps in the future is a function of the current state of thesystem:
sI+K = f(yI) + I
where
yI = [sI(dE1)T1, ..., sI(dE2)T1, sI1]
Any regression model can be used to construct f. If f is linearand T = 1 then this is a linear AR model.
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Forecast Results
Time horizon for GSL 1 year. 1985 event.
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Forecast Results
Time horizon for GSL 1 year. 1985 event.
1985 1986 1987 1988 1989
4206
42084210
4212
Stage(ft.aboveMSL)
1985 1986 1987 1988 1989
4206
4208
4
210
4212
Blind Forecast10-step Forecast.
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Forecast Results
Time horizon for GSL 1 year. 1985 event.
1985 1986 1987 1988 1989
4206
42084210
4212
Stage(ft.aboveMSL)
1985 1986 1987 1988 1989
4206
4208
4
210
4212
5-step Forecast.1-step Forecast.
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Probabilistic Cost Estimate
Cost ($)
ProbabilityD
ensity
Real densityfunction forFebruary 1987,cost function ishypothetical.
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Conclusion
Nonlinear time series methods are powerful generalizations
to linear models. Able to blind forecast accurately within time horizon.
Ensemble for probabilistic cost forecast.
The End
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Constructing a Phase Space Model
Use a series of tests (Average mutual information, False nearestneighbor, etc.) to determine appropriate ranges of dE and T(GSL dE = 35, T = 1318). For a forecast starting at I
S =
s1 s1+T s1+(dE1)Ts2 s2+T s2+(dE1)T...
.... . .
...sI2(dE1) sI2(dE1)T+T sI2(dE1)T+(dE1)T
Introduction Background Methodology Results Conclusion
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8/14/2019 [Presentation] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost Estimation
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Constructing a Phase Space Model
Use a series of tests (Average mutual information, False nearestneighbor, etc.) to determine appropriate ranges of dE and T(GSL dE = 35, T = 1318). For a forecast starting at I
S =
s1 s1+T s1+(dE1)Ts2 s2+T s2+(dE1)T...
.... . .
...sI2(dE1) sI2(dE1)T+T sI2(dE1)T+(dE1)T
r =
s1+(dE1)T+1s2+(dE1)T+1
...sI2(dE1)T+(dE1)T+1
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