[Report] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost...

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E441, Fall 2008

Nonlinear Dynamics of the Great Salt Lake: Short Term

Forecasting and Probabilistic Cost Estimation

Cameron W. BrackenDepartment of Environmental Resources Engineering, Humboldt State University

Abstract. A methodology has been presented which uses ideas from chaos theory toreconstruct and subsequently forecast the water surface level in the Great Salt Lake. Lo-cally weighted polynomial (LWP) models are used to model the phase space. This method-ology includes the generation of ensemble forecasts from a suite of models with differ-ent parameter combinations. The generalized cross validation statistic is minimized byvarying the embedding dimension dE, characteristic time lag T, the local polynomial de-gree p and, neighborhood size . Techniques from attractor reconstruction theory are usedto select appropriate search ranges for the parameters. The utility of this method is shownin its ability to accurately blind forecast within the expected forecast time horizon (1year). In the blind forecast beginning in 1985, the method accurately blind forecasts theobserved peak. The ability to generate ensemble forecasts enables the generation of prob-abilistic cost estimates at every future time step. This type of information enables pol-icy makers to asses risk and make informed decisions.

1. Introduction

Theoretically, any natural system could be modeled deter-ministically with physical equations derived from first princi-pals. In practice, this is not possible due to lack of data, lackof precision and accuracy in the data itself, lack of comput-ing resources and possibly a lack of physical equations whichadequately describe the system. Additionally, Lorenz [1963]described a type of system, known as dynamical or chaotic,in which long term prediction is inherently impossible. Inthese types of systems, a simple deterministic relationshipsmay appear as randomness. In fact, a high order dynamicalsystem appears as complete randomness [Abarbanel et al.,1993]. Traditional statistical methods explain a time series

as a realization of a random process resulting from a sys-tem with many degree of freedom [Regonda et al., 2005].An alternate explanation is that the system governed maybe governed by low order deterministic chaos. In general achaos is (1)aperiodic long-term behavior in a (2)determin-istic system that exhibits (3)sensitive dependence on initialconditions [Strogatz, 1994]. A consequence of (3) is thatlong term forecasting of these systems is inherently impos-sible. Short and mid term forecasting is still possible andmay be improved upon by incorporating concepts from chaostheory.

The Great Salt Lake (GSL) is a terminal or closed basinlake located in north western Utah between 4042N and110112W (Figure 1). A terminal lake is one in which nosurface outflow occurs. That is, the only hydrologic lossesoccur from evaporation and subsurface flow. The GSL is sonamed because of its high salt content which is typical of ter-minal lakes. Terminal lakes with large drainage areas (suchas the GSL) effectively filter out high frequency atmosphericfluctuations and tend to be strongly influenced by long termatmospheric and climatic fluctuations [Arbarbanel and Lall,1996]. This gives confidence in the to the theory that thesystem may be described a low order dynamical system.

The Great Salt Lake stage has fluctuated wildly in thepast 150 years of historical record. Sharp rises and dropsin lake stage are of particular interest to humans becauseof potential for property damage or loss of business [ James

Engineering 441, Fall 2008.

et al., 1979]. Of particular interest is the sharp rise in wa-ter level from 19831987. Short and mid term forecasts areimportant for property and business owners near the GSL.In 1986 the Great Salt Lake Pumping project was initiated,for which accurate forecasts would have been particularlyuseful [UDNR, 2007]. By producing ensemble forecasts, aprobabilistic estimate of monetary damage may be made.

Sangoyomi [1993] compiled a biweekly record of GSL vol-ume (ac-ft) beginning in 1848. Original measured data isstage (feet above mean sea level) from which volume is cal-culated via a stage-volume relationship such as the the onegiven in [James et al., 1979]. The variable used here willbe standardized stage which is identical to the standardizedvolume (Figure 3).

2. Literature Review

Chaos or dynamics in a deterministic system was firstdescribed by Lorenz [1963] where the famous Lorenz equa-tions were derived. Aside from revealing that fairly simplesystems can exhibit chaotic behavior, this discovery revealedthe possibility that long term prediction of chaotic physicalsystems may be inherently impossible.

The field of dynamics has evolved significantly findingbroad applications in many fields including Chemistry, Bi-ology, Medicine and Engineering [Strogatz, 1994]. One re-markable discovery in the field of dynamics is the techniqueof attractor reconstruction. Lorenz himself described thetechnique as the most surprising development in nonlineardynamics [Strogatz, 1994]. Embedding theory, also known

as attractor reconstruction, describes how the dynamics ofa system can be recovered by samples taken from a singlestate (phase) space coordinate Takens [1981]. This remark-able technique has become a valuable tool in time seriesmodeling [Abarbanel et al., 1993]. These methods involve re-constructing the chaotic attractor in dimension d with somecharacteristic time lag T. Models are constructed of the ddimensional phase (state) space which describe how a sys-tem evolves from an initial point x. The form of the modelis for I time steps in the future is

xt+I = f(xI, xtI, xt2I,..., xt+(d1)I); f : Rd R (1)

where f may be local or global. In fact if I = 1 and f isa linear function then the model is a linear autoregressive

1


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2 Bracken: Nonlinear Dynamics of the Great Salt Lake

(AR) model [Lall et al., 1996]. The model can be thought ofas a nonlinear autoregressive process. Attractor reconstruc-tion was first used by Packard et al. [1980] to reconstructthe time derivatives of a time series, st = st0+ns where s isthe sampling interval and n = 1, ...,N. The theory behindthis technique was first described mathematically by Takens[1981]. The overview of this method provided by Abarbanelet al. [1993] is strongly recommended.

Attractor reconstruction has been used in forecasting

of natural systems [Bordignon and Lisi, 2000; Islam andSivakumar, 2002; Coulibaly and Baldwinb, 2005], particu-larly with success in the case of the GSL [Sangoyomi, 1993;Abarbanel et al., 1995; Arbarbanel and Lall, 1996; Regondaet al., 2005; Lall et al., 2006]. Arbarbanel and Lall [1996]used a local linear model to reconstruct the phase space.Their single step forecasts were amazingly accurate though

Nevada

Idaho

Outline of GreatSalt Lake Basin

N

Wyo

Utah

Great

Salt

Lake

Figure 1. Great Salt Lake and drainage basin (Adaptedfrom James et al. [1979]).

1880 1920 1960 2000

4195

4205

Stage(ft.aboveMS

L)

Figure 2. Standardized Great Salt Lake Stage.

no confidence intervals on the forecast were given. The de-crease in forecast skill was apparent using 10 step forecast,that is the forecasted value at every point is blind forecasted10 days earlier. Regonda et al. [2005] focused on blind fore-casts only. Arbarbanel and Lall [1996] calculated liapunovexponents to show that the forecast time horizon is about 1year. Lall et al. [1996] modeled the phase space with mul-tivariate adaptive regression splines and was able to showsignificant forecast skill over traditional linear AR models;

AR lags of up to 73 were used.In many cases when using statistical models with param-

eters , one is faced with competing models that, basedon some criteria C(), are very similar. This criteria maybe a measure of goodness of fit or parsimony (Coefficientof determination, Mallows Cp, Akaike information criterion)or predictive risk (Cross validation, generalized cross vali-dation [Craven and Wahba, 1979]). Traditional regressionwould choose a single the optimal parameter set ignor-ing any other models. The multi-model method claims thata sufficiently different parameter set, say with a similarcriteria value, C(), contains valuable information aboutthe system that is lost if it is not included in some overall orensemble model. The multi-model method been used withsuccess in the forecasting skill in many hydrologic systems[Krishnamurti et al., 1999, 2000; Rajagopalan et al., 2002;Grantz et al., 2005; Hagedorn et al., 2005; Regonda et al.,2005, 2006].

Here we use the forecast algorithm ofRegonda et al. [2005]which uses a locally weighted p olynomial (LWP) model.LWP models have been applied successfully in many fieldsof hydrologic modeling including ensemble stream flow fore-casting and simulation [Grantz et al., 2005; Regonda et al.,2006; Prairie, 2006; Opitz-Stapleton et al., 2007] rainfall andclimate forecasting [Krishnamurti et al., 2000; Singhrattnaet al., 2005] and the GSL itself [Regonda et al., 2005; Lallet al., 2006]. One of the benefits of local regression is itsability to capture arbitrary local nonlinearities which areexpected to be prevalent in this setting.

3. MethodologyThis section contains the details of the methods outlined

in Sections 1 and 2. Consider a dynamical system with an at-tractor dimension dA from which the time series sn = st0+nshas been sampled at the interval s. According to the theory,the dynamics of a system can be reconstructed by creatinga dE dimensional vector of lagged or embedded time series.

y(n) = [st0+ns , st0+(n+T)s ,

st0+(n+2T)s ,..., st0+(n+(dE1)T)s ]

= [sn, sn+T, sn+2T,..., sn+(dE1)T]

(2)

where T is an unknown integer and n = 1, ...,N, known as

the characteristic lag Abarbanel et al. [1993]. Viewing theseembedded time series as the coordinates of a dynamical sys-tem is the basis of attractor reconstruction. The embeddingtheory ofTakens [1981] guarantees that a dA dimensional at-tractor can be reconstructed (unfolded) unambiguously froms(n) in a dimension dE > 2dA. The parameter dE is said tobe the embedding dimension of s(n). That is any point xiin the phase space lies on one or zero trajectories of the dy-namical system. The notion that no two trajectories cancross in phase space is an important result from chaos the-ory [Strogatz, 1994]. Reconstructing the phase space in thisfashion the evolution of the system to be examined from anyinitial condition x0.

In practice there are actually a large number of embeddedtime series that depend on how much data is available prior


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Bracken: Nonlinear Dynamics of the Great Salt Lake 3

the time of the forecast. If we want to forecast from timestep Ionward then there are nE embedded time series wherenE = I (dE 1)T 2. First, placing each embedded timeseries in a vector sn+T = [sn, sn+T, sn+2T,..., sn+(dE1)T]. Allthe embedded time series are gathered into an nE dE ma-trix S where

S =

s1+T

s2+T...

sI1(dE1)T1

=

s1 s1+T s1+(dE1)Ts2 s2+T s2+(dE1)T...

......

...sI2(dE1) sI2(dE1)T+T sI2(dE1)T+(dE1)T

.

(3)

The bottom rightmost element has been written out explic-itly but simplifies to s(I 2), that is two time steps beforethe first forecast time. Generally each i, j entry ofP is givenby

Si,j+1 = si+jT (4)

where i = 1,...nE and j = 0,1, ...,dE 1. In addition torecording the conditions in the phase space in the the ma-trix S the next successive lagged value of the time series isrecorded in a vector r of length nE where

r =

s1+(dE1)T+1s2+(dE1)T+1

...sI2(dE1)T+(dE1)T+1

(5)

The last entry in r simplifies to sI1 which is the last timebefore the start the forecast. The vector r is the key to re-

lating a state of the system in the phase space to a value ofthe time series.A pseudo code for constructing the phase space calibra-

tion model looks like:

DO i from 0 to nE 1DO j from 1 to dE

Si,j = si+jTri = sdET+i+1

END DOEND DO

For even further clarification a numeric example from theGSL. Say the forecast time step is I = 3000 and sn havedata starts from n = 1 and dE = 4 and T= 15. There arenE = I 2 (dE = 1)T= 3000 2 (4 1) 15 = 2953 em-bedded time series! It should be mentioned that this method

is computationally intensive, especially with large historicalrecords.

3.1. Locally Weighted Polynomial Phase Space

Model and Ensemble Generation

As mentioned in Section 2 local polynomial models(LWP) are used in this analysis to fit the phase space.These models are have to a local polynomial degree p andare fitted to k nearest neighbors in the phase space wherek = N, (0, 1] and N is the number of data points. If = p = 1 the local regression collapses to multiple linearregression and thus is a generalization which is is more flexi-ble. The local regression is preformed with the LOCFIT pack-age [Loader, 1999]. The generalized cross validation (GCV)statistic is used here as a measure of predictive risk [Craven

and Wahba, 1979] and is indirectly dependent on all of theparameters of our model. The GCV is

GCV() =

1N

N

i=1

e2i

(1 mN)2

(6)

where = [dE, T, , p] is a vector of the parameters, ei isthe model residual and m is the number of parameters.

Once constructed for a given parameter set , S and rdo not change though they will be different for any param-eter combination of which there are quite a lot. Followingthe multi-model method all models with GCV values within10% of the top model are included and used in the ensem-ble forecast. This suite of models is then used to obtainan ensemble of predictions. This ensemble of predictionsalso gives an idea of the forecast uncertainty, i.e. the widervariation in ensemble members, the more uncertain the pre-diction.

The model for predicting K time steps into the future hasthe form

sI+K = f(z(I)) + I (7)

where K = 0 is considered the first future point and z(I)is called the feature vector which is simply the most recent

embedded time series ending at I 1 where

z(I) =

sI(dE1)T1sI(dE2)T1

...sI1

. (8)

For every new time step into the future z(I) is reconstructedby including the latest forecasted point at the end of the ob-served time series and repeating as if the observed serieswas one point longer. Forecasts are assumed to become suc-cessively worse for larger Ks as more forecasted points areincluded. By preforming the forecast with each model inthe ensemble, an estimate of the forecast uncertainty can be

obtained by observing the spread in the ensemble members.

3.2. Probabilistic Damage Estimates

Rising water levels cause damage to businesses and prop-erty which reside on the GSL lakefront. In the case of theGSL pumping project costs are related to increased pump-ing. From such information a stage versus cost relationshipcan be developed. Here a hypothetical relationship is used to

4180 4190 4200 4210

2.3

35

2.34

5

Cost(Million$)

Stage (ft. above MSL)

Figure 3. Hypothetical stage-cost curve.


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demonstrate the utility of ensemble forecasts for generatingcost estimates.

3.3. Methods for Choosing Time Delay

Until now the lag Thas been an unknown integer, thoughthe choice of its value is important. One of the benefits ofthe multi-model ensemble method is that the pressure istaken off of choosing a single best value; the choice of an

acceptable range is nonetheless important. If T is chosentoo small, each point in phase space will be essentially thesame as the last and if T is chosen too large then informa-tion about the system will be lost. Two common methodsare using the linear autocorrelation function (ACF) and theaverage mutual information criteria.

A somewhat naive approach to determining T is to ex-amine is linear dependence of s(n) and s(n + T) form theACF. One would examine the ACF as a function of T andfind the first zero crossing. Unfortunately this method doesnot take into account the nonlinear dependence of the vari-ables and leads to a gross overestimation ofTwhen used inthe traditional way [Abarbanel et al., 1993]. Fortunately theACF may still be used to estimate Tby finding its first localminimum [Arbarbanel and Lall, 1996].

The average mutual information (AMI) criteria in gen-

eral is used to describe the amount of information one gainson average about an observation ai from an observation bkwhere ai and bk are measurements from systems A and Brespectively. In terms of the time series s(n) the AMI cri-teria describes the about of information gained about sn+Thaving observed sn. The AMI is

AMI(T) N

n=1

P(sn, sn+T) log2

P(sn, sn+T)

P(sn)P(sn+T)

(9)

where P() denotes probability, P(, ) denotes joint prob-ability and AMI(T) 0 [Abarbanel et al., 1993]. Abarbanelet al. [1993] suggests that the first local minimum ofAMI(T)gives a good estimate ofT.

3.4. Methods for Choosing Time Delay

Embedding theory guarantees that an embedding dimen-sion dE > 2dA is sufficient to unfold the attractor but notnecessary. In practice a dE 2dA may sufficiently unfoldthe attractor. A reasons for not simply choosing a large em-bedding dimension are (1) that computational costs becomelarger and (2) relationships between variables in higher di-mensions may simply be noise in observed data [Abarbanelet al., 1993]. Two common techniques for determining theembedding dimension are the false nearest neighbor (FNN)technique and the saturation of system invariants or the cor-relation integral method.

The false nearest neighbor (FNN) method attempts bygeometrical criteria to determine weather two points in di-mension d are neighbors due to the projection of the attrac-

tor into dimension that is insufficiently low.The correlation integral is another geometric methodwhich starts by placing a ball of radius enclosing and aver-age fraction C() of the points in dimension d. The numberof points enclosed is expected to grow like

C() d, (10)

therefore the dimension is the slope of the line

logC() = a + d log . (11)

In practice this linearity only holds for some middle region ofthe range of because the ball eventually encloses all of thepoints [Strogatz, 1994]. The dimension in which the slope

of the linear part of the correlation integral curve ceases tochange as the dimension increases. This corresponds the theattractor being completely unfolded.

3.5. Forecast Algorithm

The forecast algorithm as a whole proceeds as follows:1. Using the appropriate methods (ACF, AMI, FNN) ob-

tain a range of values for dE and T

2. Construct S and r for a particular combination of, p,de, and T.

3. Fit a LWP model to S and r and calculate the GCVvalue for the particular fit.

4. Repeat steps 1-2 for all combinations of , p, de, andT.

5. Collect the parameter values for all models within 10%of the model with the lowest GCV value.

-2 -1 0 1 2 3

-2

-1

0

1

2

3

-2-1

01

2

e(t)e(t+ T)

e(t+

2T)

Figure 4. Reconstructed GSL attractor viewed in R3

(i.e. dE = 3), unfolded with T= 4.

0 50 100 150 200

0.

5

0.

6

0.

7

0.

8

0.

9

1.

0

T

ACF

Figure 5. Autocorrelation function.


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6. Form an ensemble by making a forecasts at time stepI from each of the models for K time steps into the future.

4. Application

The GSL water level time series was compiled by San-goyomi [1993]. The analysis is done on the standardizedtime series with a mean 4201 feet and a standard deviation4.4 feet. A reconstructed attractor with d

E= 3 is shown in

figure 4.The autocorrelation function (ACF) is shown in figure 5.

The first local minimum occurs at T= 15. As a side note thecriteria of a linear ACF would result in a time lag T> 400!The Average mutual information (AMI) function is shownin figure 6. The first local minimum occurs at T= 16 whichis in good agreement with the lag suggested form the ACF .This leads to a search range for T of1319.

The correlation integrals for the GSL time series areshown in Figure 7. The slope of linear part is independent

0 50 100 150 200

0.

5

1.

0

1.

5

2.

0

2.

5

T

AMI(T)

Figure 6. Average Mutual information criteria.

0.5 1.0 2.0 5.0 10.0

0.

2

0.

4

0.

6

1.

0

C(

)

dE = 1

dE = 2

dE = 3

dE = 4

dE = 5

dE = 6

dE = 7

Figure 7. Correlation Integrals for dE = 1,...,7.

of dimension at around dE = 4. Figure 8 shows the fractionof the false nearest neighbors as a function of embeddingdimension. The FNN method suggests an embedding di-mension dE = 4 which agree with the observation of thecorrelation integrals. This prompts the search for dE in therange 35.

5. Results

Blind forecasts beginning in 1979, before the sharp rise inthe GSL water level and for 1985, after 2 years of dramaticwater level rise and just before the implementation of theGSL pumping project. In addition single step forecasts arepresented beginning at both times. And an example of aprobabilistic cost estimate.

1 2 3 4 5 6 7

0.4

0.5

0.

6

0.7

Embedding dimension

Fraction

offalse

nearestneighbors

Figure 8. Correlation Integrals for dE = 1,...,7.

1979 1980 1981 1982 1983

4195

4196

419

7

4198

4199

4200

Stage(f

t.aboveMSL)

Figure 9. Blind forecast beginning January 1st, 1979.The dashed line is the median forecast and the dottedlines are the 5th and 95th percentiles of the forecast. Cir-cles represent measured data.


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Recall the estimated forecast time horizon was 1 year.The blind forecast in Figure 9 starting in 1979 verifies this.The blind forecast captures the observations well until afteralmost exactly a year the observed data begins to veer offfrom the forecast. For this forecast 27 models were selectedwithin 10% of the top model. The parameter ranges forembedding dimension was 45, time lag 1415 (units of 15days), local polynomial degree 12, and neighborhood size.1.6. The one step forecast shown in Figure 10 does amaz-

ingly well, though the utility of a 15 day forecast is limited.The Blind forecast shown in Figure 11 beginning in 1985

does well at capturing the peak and not overshooting. Themost important piece of information at that point in timewas weather the lake would continue to rise at its currentrate or start to decline. The medina of the ensemble fore-casts would have forecasted that the peak lake level to gono higher than it actually did over two years in advance.

1979 1980 1981 1982 1983

4196

4197

4198

41

99

4200

Stage(ft.aboveMSL)

Figure 10. Same as figure 9 but one step forecast be-ginning January 1st, 1979.

1985 1986 1987 1988 1989

4202

4204

4206

42

08

4210

4212

Stage(ft.ab

oveMSL)

Figure 11. Same as figure 9 but blind forecast beginningJanuary 1st, 1985.

For this forecast 25 models were selected within 10% of thetop model. Surprisingly enough the parameter ranges wereidentical to those selected for the 1979 blind forecast. Theseranges agree with those reported by Regonda et al. [2006].The one step forecast does exceptionally well again for thistime interval (Figure 13).

Figure 12 is an example of the probabilistic cost forecastthat can be produced at any time future time step. Hereinlies the utility in an ensemble forecast. This type of informa-

tion allows policy makers to asses risk and make decisionsaccordingly.

6. Conclusion

A methodology has been presented which uses ideas fromchaos theory to reconstruct and subsequently forecast thewater surface level in the Great Salt Lake. Locally weighted

5020 5040 5060 5080 5100 51200e+00

2e-05

4e-05

6e-05

8e-05

Cost (Ten Thousand $)

ProbabilityDensity

Figure 12. 1985 probability density function at 50 stepsahead.

1985 1986 1987 1988 1989

4206

4207

4208

4209

4210

4211

Stage(ft.ab

oveMSL)

Figure 13. Same as figure 10 but one step forecast be-ginning January 1st, 1985.


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polynomial models are used to model the phase space. Thismethodology includes the generation of ensemble forecastsfrom a suite of models with different parameter combina-tions. The generalized cross validation statistic is minimizedby varying the embedding dimension dE, characteristic timelag T, the local polynomial degree p and, neighborhood size. Techniques from attractor reconstruction theory are usedto select appropriate search ranges for the parameters.

The utility of this method is shown in its ability to accu-

rately blind forecast within the expected forecast time hori-zon (1 year). In the blind forecast beginning in 1985, themethod accurately blind forecasts the observed peak. Theability to generate ensemble forecasts enables the generationof probabilistic cost estimates at every future time step. Thistype of information enables policy makers to asses risk andmake informed decisions.

One obvious downside of this type of model is its inabil-ity to generate long term series. Though a portion of thephase space and attractor within it are reconstructed, themodels do not cover the entire phase space (granted that thereconstructed portion is of the most interest). As a resultwhen generating long term series, the model fails as soonas a point is generated where there is no data. So, whilethis method is useful for short term forecasting and decisionmaking, its uses in long term planning are limited. That is,

unless a method to correct for the long term instability in themodel is devised. On that note one way might be to use themedian forecast of all the ensemble members as the updateto all the models in the next time step. This would admit-tedly destroy the benefit of producing an ensemble forecastbut could enable the generation of long term series by stabi-lizing the forecasts. Another extension might b e to weightthe forecasts of each model according to their relative GCVscore.

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C. W. Bracken, Environmental Resources Engineering, Hum-boldt State University, Arcata, CA, USA. ([email protected])

[Report] Nonlinear Dynamics of the Great Salt Lake: Short Term Forecasting and Probabilistic Cost...

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