Nonlinear dynamics of the Great Salt Lake: system identification and prediction

Transcription

1 Climate Dynamics (1996) 12: q Springer-Verlag 1996 Nonlinear dynamics of the Great Salt Lake: system identification and prediction Henry D. I. Abarbanel 1, Upmanu Lall 2 1 Department of Physics and Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego, Mail Code 0402, La Jolla, CA , USA 2 Utah Water Research Laboratory and Department of Civil and Environmental Engineering, Utah State University, UMC 82, Logan, Utah , USA Received: 5 April 1995 / Accepted: 2 October 1995 Abstract. A case study of the application of recent methods of nonlinear time series analysis is presented. The biweekly time series of the Great Salt Lake (GSL) volume is analyzed for evidence of low dimensional dynamics and predictability. The spectrum of Lyapunov exponents indicates that the average predictability of the GSL is a few hundred days. Use of the false nearest neighbor statistic shows that the dynamics of the GSL can be described in time delay coordinates by four dimensional vectors with components lagged by about half a year. Local linear maps are used in this embedding of the data and their skill in forecasting is tested in split sample mode for a variety of GSL conditions: lake average volume, near the beginning of a drought, near the end of a drought, prior to a period of rapid lake rise. Implications for modeling low frequency components of the hydro-climate system are discussed. 1 Introduction Advances in understanding the dynamics of nonlinear physical systems have provided a new perspective for the analysis of time series. New directions are suggested for forecasting natural systems as well as for reconciling the dynamics shown by physical models and by data. The work of Lorenz (1963) suggested that irregular or random behavior in the climate system may arise from a purely deterministic dynamics each of whose trajectories is unstable. Takens (1981) showed how essential aspects of the dynamics of a system governed by ordinary differential equations or discrete time maps may be recovered from the time series of a single state variable of the system. This has led to research aimed at efficient recovery and characterization of dynamics from experimental data. A recent review of developments is found in Abarbanel et al. (1993). We present a case study of the dynamics of the Great 1 Institute for Nonlinear Science Correspondence to: H. D. I. Abarbanel Salt Lake (GSL) volume time series using the tools discussed in the review. Background information on the techniques is in the review, so here we focus on an interpretation of the GSL data. More detailed presentations that focus on particular aspects of the inferred dynamics are found in the PhD thesis of Sangoyomi (1993). The importance to society of improved understanding and forecasting of low frequency climatic variability is obvious. Closed basin lakes, that is lakes with no surface outlet, exist in many arid regions of the world. A necessary condition for the existence of such lakes is that the average annual evaporation rate is higher than the average annual precipitation rate. Such lakes are very sensitive to long term fluctuations in regional climate, and if they have a large enough drainage area they damp out high frequency climatic variability. Analysis of the time history of such lakes can consequently provide insights into climatic fluctuations over long time scales. The Great Salt Lake (GSL) of Utah, located at approximately 407 to 42 7N and 1107 to 112 7W, is the fourth largest, perennial, closed basin, saline lake in the world. It drains an area of km 2. The GSL s immediate atmospheric catchment is the Pacific Ocean, and its fluctuations exhibit high spectral coherence with the Pacific North American Circulation Index, and the Southern Oscillation index at interannual time scales. These indices measure the fluctuations in the Northern Pacific atmospheric flow and tropical Pacific atmospheric variability related to the El Niño Southern Oscillation, respectively. Its fluctuations have been related by Mann, Lall, and Saltzman (personal communication) to those in the hemispheric surface temperature and sea level pressure records. The levels of the GSL have been recorded by various methods since From these data a biweekly time series of GSL volumes has been compiled by Sangoyomi (1993) which shows substantial variation both on an annual time scale as well as on interannual and interdecadal time scale. Lall and Mann (personal communication) speculate that the low frequency oscilla-

2 288 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction tions found on spectral analysis of this and related hydroclimatic data sets may represent unstable oscillations that arise from the nonlinearity of the climate dynamics, rather than stable periodic motion. This paper analyzes this data set from the point of view of nonlinear dynamics using tools for that analysis which have proven enormously useful in the study of physical and biological systems (Abarbanel et al. 1993). This perspective differs from traditional time series analysis methods that consider the data to be the outcome of an underlying stochastic process. Here one presumes that the data represents measurements of the state of a physical dynamical system. The measurements may not be precise. If the essential behavior of the dynamical system can be described in terms of a few state variables, a finite data set may be used to characterize the dynamics. Conversely, if the dynamics is high dimensional, characterization as a stochastic process may be necessary. The predictability of weather (the high frequency component of climate) is rather limited and it is unlikely that a few state variables can adequately describe its evolution. On the other hand, there may be some hope that the record of large closed basin lakes such as the Great Salt Lake essentially filters out such components and may reflect a few key modes of climatic variability. While long term (e.g., over paleo time scales) climatic dynamics and its manifestation in the record of the Great Salt Lake may still be complex, it may be possible to model century long fluctuations using nonlinear time series analysis. We will discuss this application in some detail below. The main conclusion of our analysis is that an excellent representation of the data is as a sequence of multivariate states from a dynamical system with dimension four. This is established in a number of ways. A geometrical test based on false nearest neighbors (Abarbanel et al. 1993) suggests that the dynamics can be recovered from an embedding formed by using 4 lagged copies (formed with a suitable delay) of the biweekly GSL time series. We then evaluate the Lyapunov exponents of the GSL from the volume time series data working in dp4, and learn that 1. the largest exponent is l 1 ;100 days P1, and 2. the second exponent is zero. The first piece of information tells us that on the average, the horizon for accurate predictions is a few times a hundred days or about a year while the second fact tells us that a set of differential equations is appropriate for modeling the data (Abarbanel et al. 1993), Eckmann and Ruelle (1985). We discuss methods for modeling the GSL volume in a four dimensional space using local maps in state space, and demonstrate their utility in predicting the GSL volume data. 2 Nonlinear Analysis Our first look at the data is the time series in Figure 1. We see clear annual variation and a much longer period associated with other dynamics. The Fourier power spectrum of the time series is shown in Figure 2. Here Fig. 1. The time series of the Great Salt Lake volume in units of 10 7 acre feet. The time index begins in 1847 and continues every 15 days Fig. 2. The Fourier power spectrum of the GSL time series. The units on the frequency axis are 1/85 years we see rather clear annual and semi-annual cycles along with an apparently broad spectral background which is the main focus of this paper. Lall and Mann (personal communication) found that the spectral power in selected interannual and interdecadal bands was significant using the multi-taper method of Thomson (1982) of spectral analysis. However, upon bandpassing the time series at these frequencies, the associated oscillatory modes were found to have variable amplitude and phase discontinuities over the record. We will show that this broad spectral background contributes a small number of active degrees of freedom to the dynamics underlying the GSL volume.

3 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction 289 When analyzing the time series of the GSL volume, we are studying the dynamics of the total system (composed of among other things, solar radiation, atmospheric circulation, ocean-atmosphere interaction, local precipitation and evaporation) that determines the time evolution of the lake s volume. We use knowledge of only the volume time series, and are unable to address directly which and how many of these other factors directly come into play. We expect from physical principles that the interactions among these dynamical variables factors are nonlinear. The analysis presented here has as its goal identification of the number of dynamical variables or coordinates that are needed to adequately describe the dynamics of the GSL volume. These variables, the volume itself and its time lags, are not directly interpretable in terms of other physical variables, at least not interpretable absent a reasonably well developed physical theory of the processes involved. The analysis here gives constraints on the number of such dynamical variables which would appear in a dynamical description of the processes underlying the variation of GSL volume without requiring those dynamical equations to be known. When a dynamical model is proposed and analyzed, the invariant quantities we evaluate, such as Lyapunov exponents, will be tested against that model. While we expect to see such dynamical models emerge in the future, this paper has as its focus extracting from the data using tools of nonlinear time series analysis information which is in the data and will guide subsequent model construction and testing. In those models identifiable physical variables will replace GSL volume and its time lags though these are (unknown) nonlinear combinations of those eventual other variables. One of the key lessons in the analysis of nonlinear systems is that from a one dimensional time series such as the GSL volume v(t) pv(t 0 cnt s )pv(n), where t 0 is the time of the initial measurement and t s is the sampling time (here t s p15 days), we can reconstruct the multivariate space in which the dynamics unfolds. This is done by creating vectors in dimension d out of the measurements v(t) and its time lags as in Abarbanel et al. (1993): y(n)p[v(t 0 cnt s ), v(t 0 c(nct)t s... v(t 0 c(nc(dp1) T) t s )] p[v(n), v(nct),..., v(nc(dp1) T)]. (1) The first issue we address is the determination of an appropriate time lag T and after that we look at the required number of dimensions d of the multivariate vectors y(n). The tools we use for this are the average mutual information among the elements of the time series v(n) which appear in y(n) and a geometric statement about the fact that orbits of a dynamical system cannot intersect in an appropriate state space. Noise, of course, can contaminate this property of orbits as it consists of very high dimensional additions to the signal. However, the method of false nearest neighbors which we describe and utilize below is quite robust against noise and degrades gracefully in its presence as shown in Abarbanel et al. 1993). 2.1 Time Lag: Mutual Information The choice of time lag T is prescriptive, not algorithmic or optimal, and is made to assure that the coordinates v(n) and v(nct) are rather independent of each other. If T is too small, these two measurements are essentially the same, as the system has not had a chance to evolve into another independent state. If T is too large, then v(n) and v(nct) are numerically independent since nonlinear systems are intrinsically unstable (Abarbanel et al. 1993). To choose the lag T we use the first minimum of the average mutual information (Fraser et al. (1986)) between the sets of measurements v(n) and v(nct). Again we emphasize that this is prescriptive as the topological idea behind using time lagged coordinates is not dependent on the value of T. However, it presumes an infinite amount of infinitely accurate data. Since this is not available, a prescription based on measures of information generation in nonlinear systems becomes necessary. For a set of measurements Apa(1), a(2),..., a(k) and another set of measurements Bpb(1), b(2),..., b(m) the amount one learns in bits, on the average over all measurements about A measurements from B measurements is I AB p kpk;mpm A P AB (a(k), b(m)) kp1; mp1 log 2 3 P AB(a(k), b(m)) P A (a(k)) P B (b(m))4, (2) where P AB (I, I) is the joint probability distribution of the A and B measurements, and P A (I) and P B (I) are the marginal probability distributions of A or B. The quantities a(k) and b(m) need not be drawn from a random set. Where the time series arise from deterministic dynamics, these probabilities can be considered as relative frequencies of occurrence of the measured values. Clearly I AB pi BA. Choosing a(k)pv(k) and b(m)pv(mct) we can construct I(T), the average mutual information between measurements at time n and time nct. To evaluate this quantity from the observations v(n) we first make a histogram of values of the volume v(n). When normalized to have unit area this histogram is P A (a(k)). This underlines the fact that the variable v(n) does not have to be random to have a distribution of values. Similarly for the values of v(mct) for various T we construct the normalized histogram P B (v(mct). If the data set is long and stationary, then this histogram is the same as P A (v(n)). Finally we construct the two dimensional joint histogram of values of v(n) and v(mct). Normalized this defines P AB (v(n),v(mct)). These are the ingredients required to determine I(T). This is shown in Figure 3 for the GSL volume data. We see a broad first minimum in the region 12^T^17 or so. We will use Tp12 throughout much of this paper. This choice resolves the annual cycle since we have twenty four observations per year. Results for Tp17 differ little, and occasionally we may utilize that value for the time lag. Sim-

4 290 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction choose the first zero crossing of this autocorrelation at T;400 for an appropriate time lag. This is a time of nearly twenty years and is likely to be unconnected with the dynamics observed here. If we changed the usual rule for selecting the first zero crossing of the autocorrelation function as the desired lag and selected instead the time delay at the first minimum of the autocorrelation function, we would actually again get Tp12. However, the interpretation of the first minimum of the autocorrelation function is more like that we have been giving to the average mutual information rather than the usual interpretation of the first zero crossing as giving linearly independent coordinates. It is rare in our experience to see such a striking distinction between the timelag suggested by autocorrelation methods and mutual information methods, but the difference underlines that these are quite different statistics on the attractor associated with the data. Fig. 3. The average mutual information for the GSL time series. It has a broad first minimum in the region of T;12PP17 in units of 15 days. As a prescription we choose a time lag in the region of the first minimum for use in reconstructing a multivariate state space using time lagged version of the GSL volume Fig. 4. The autocorrelation function for the GSL volume. The first zero crossing which indicates coordinates v(n) and v(nct) which are linearly independent is T;400. The first local minimum of the autocorrelation function happens to be near the first minimum of the average mutual information ilar choices for T result if more sophisticated, nonparametric probability density estimators are used to estimate I(T). In striking comparison to this characteristic nonlinear measure of correlation between measurements, we display in Figure 4 the traditional autocorrelation of the GSL volume signal. Were we to adhere to linear methods and focus on the linear independence of the coordinates v(n) and v(nct), we would be led to 2.2 Global Embedding Dimension To choose the dimension d appropriate for describing the data in multivariate state space, we use the method of global false nearest neighbors discussed in Abarbanel et al. (1993). This examines, in dimension d, the nearest neighbor of every vector y(n) as it behaves in dimension dc1. If the nearest neighbor moves far away from y(n) as we move up in dimension, then it is declared a false neighbor as it arrived in the neighborhood of y(n) in dimension d by projection from a distant part of the attractor. Moving far away has been determined in earlier work by Kennel, Brown and Abarbanel (1992) through numerical experimentation with the operational criterion being that the rule should be independent of the number of data and the threshold for far away. This occurs when the added Euclidian length of the distance relative to the original distance is between ten and fifty or so. We typically use about 20 as the criterion. As soon as the number of data is such that the attractor has been visited a few times, the resulting percentage of false nearest neighbors is independent of that. When the percentage of these false neighbors drops to zero, the geometric structure of the attractor has been unfolded and the orbits of the system through state space are now unambiguous and do not cross. The crossing of orbits is forbidden if the data is the outcome of a set of autonomous differential equations or discrete time maps. It is the apparent crossing of orbits in dimensions too low to make the orbit unambiguous which gives rise to false neighbors. Figure 5 displays the percentage of false nearest neighbors in dimensions 1^d^8 for this data set. Clearly the number of false neighbors drops to zero by dp4, and, as it should, remains zero thereafter. This tells us that globally we require a four dimensional space to unfold the attractor for the GSL data. It is in this space that we can make predictions. If we attempt predictions in spaces with d~4, we would encounter locations in the data where the orbits, supposed to be

5 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction 291 Fig. 5. Global false nearest neighbors for the GSL volume. At d E p4 the percentage of false neighbors goes to zero and stays there. This figure uses Tp12 for multivariate state space vectors; using Tp17 does not change the conclusion unique everywhere in state space, actually cross. This makes prediction an uncertain possibility as we cannot know in that lower dimension which orbit is going where, and then cannot predict ahead. Sangoyomi, Lall, and Abarbanel (personal communication) report that the Grassberger-Procaccia and the Nearest Neighbor Dimension algorithms, that analyze the scaling properties on the attractor estimate the dimension of the system as approximately 3.5. This is consistent with the estimate of embedding dimension 4 by the false neighbors method. Indeed, Ding et al. (1993) have shown that for evaluating the correlation dimension one only requires an integer dimension for data vectors y(n) just larger than the fractional correlation dimension. In other words, one does not require a full unfolding of the attractor for that dimension calculation though for the forecasting we do later, a full unfolding is needed. As it happens d E p4 works here for both purposes. 2.3 Local Dimension for the Data There is still a question to be resolved for data such as these: how many of the global dimensions required to unfold the observations represent dynamical activity? It can happen that the particular coordinate system used for looking at the data can nonlinearly distort the attractor in such a way that a larger global dimension can be required than the local dimension of the dynamics. To address this we examine the local dimension using the method of local false neighbors by Abarbanel and Kennel (1993). This method goes to the neighborhood of a state space point y(n), and computing all distances in the global dimension d E established by the previous global Fig. 6. Local false nearest neighbors for the GSL volume. The percentage of bad predictions is given as a function of the local dimension d L and the number of neighbors N B used in making the neighborhood to neighborhood predictions. We show results for N B p10, 25, 50, 75. The predictions are deemed bad if they deviate by a fraction bp0.25 of the size of the attractor in time T. As one varies b, the curve of bad predictions shifts up or down in P K, but still becomes independent of N B and d L at d L p4 false neighbor algorithm, determines which are the nearest N B neighbors of this point. Then we ask if we can build a good prediction rule for these N B c1 points to arrive at the location of the known N B neighbors of the point y(nc1). This predictability is examined for local evolution rules from neighborhood to neighborhood using local polynomial maps in dimension d L p1, 2,..., d E, and we establish in which d L^d E the predictability becomes independent of d and of N B. When this predictability becomes independent of these quantities, we have established the local dimension appropriate for a dynamical description of the processes involved. This algorithm injects another notion, namely predictability, into the formulation of the dynamical description of the data source and goes beyond the purely geometrical ideas of global false neighbors. In Figure 6 we show the results of this local false nearest neighbor calculation for the GSL data. It is clear here that in d L p4 we have captured all the local dynamics on the attractor. We conclude that working in four dimensions is what is dictated by these volume data. Indeed, we have determined that the attractor can be unfolded in d E p4, and that the local dynamics on the attractor is d L p4. This means that when we evaluate the Lyapunov exponents, as we do just below, we should work in a state space of dimension d E p4 and construct local maps of dimension d L p4 to determine exactly four exponents. Further when we make predictive models we need to use the same rules for local evolution.

6 292 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction 3 Lyapunov Exponents: Local and Global The evolution of small changes to an orbit of a dynamical system are governed by the Lyapunov exponents, locally and globally in state space. These exponents provide a measure of the predictability of the system through estimates of the rate with which nearby trajectories come approach each other (e.g., near a stable equilibrium solution) or separate (e.g., near states that are unstable). A small perturbation w(n) to an orbit y(n) evolves as w(nc1)pdf(y(n))7w(n), (3) where DF ab (x)p if a(x), (4) ix b is the Jacobian matrix of the y(n)]y(nc1) dynamics y(nc1)pf(y(n)). (5) The singular values of the Jacobian matrices DF(y(n)) composed along the orbit are the same as the eigenvalues of the Oseledec matrix discussed by Oseledec (1968) OSE(x,L)p{DF L (x) T 7DF L (x)} 1/2L, (6) where DF L (x) is the composition of L Jacobian matrices along the orbit starting from the state space point x. Evaluating these Jacobian matrices from observed one dimensional data and then diagonalizing them using a recursive QR decomposition is described in Abarbanel et al. (1993). The logarithm of the eigenvalues of OSE(x,L) we call l 1 (x,l), l 2 (x,l),..., l d (x,l) in d-dimensional state space. When L]e, or in practice becomes large enough that the orbit has visited the whole of the attractor, these exponents become independent of x within the basin of the attractor being studied. We report the average of the l a (x,l) over the state space variable x weighted by the natural measure on the attractor which weights regions of state space by the relative frequency of visits to those regions. This average local Lyapunov exponent l a(l) is independent of initial conditions on the orbit and converges as a power of L to the global Lyapunov exponent l a ; ap1, 2,..., d. The largest global Lyapunov exponent l 1 determines the horizon of global predictability for the system. If one has two points close to one another in the state space at some time t, their distance at time tb`t is the original distance multiplied by e l 1 (tbpt), (7) and when this quantity times the original distance is as large as the attractor, predictability in the global sense is lost. The local Lyapunov exponents l a (x,l) tell us how small perturbations to the orbit made at location x grow or shrink after L time steps. These local exponents vary substantially at different locations on the attractor. It can be that some regions of the attractor are much more predictable than the global average dictated by the value of l 1. Local predictability is very important for geophysical processes such as the GSL volume. We have not explored this matter for the present paper, but we shall report on it in the near future. For the GSL data we estimated the local Jacobian matrices by making local cubic neighborhood-to-neighborhood maps in dp4 state space and picking off the linear part of these local maps. These maps can be thought of as regressions of the volume v(nc1) on the vector defined by the embedding, [v(n), v(npt),..., v(np(dp1)t)]. These estimated local Jacobians were then multiplied to give the Oseledec matrix and diagonalized as indicated. In Figure 7 we display the average local Lyapunov exponents as a function of number of steps 2 (LP1) along the orbit for Lp0, 1, The average exponents have been averaged over 750 different starting locations on the attractor. These locations are spaced approximately equally in time from the center of the measured data set. Had one chosen them randomly from the data set or essentially any other way, when the number is large enough, the results will be the same as implied by the multiplicative ergodic theorem by Oseledec (1968). The exponents converge to the approximate global values: l 1 p0.17, l 2 p0, l 3 pp0.13, and l 4 pp0.68. These estimates used a reconstructed state space with time lag Tp12. For comparison we show in Figure 8 the same quantities evaluated with Tp17. It is important that there is essentially no difference in the computed values. In principle, they are independent of the lag T. We conclude two important things from these values for the l a. First, one of the exponents is zero within Fig. 7. The average local Lyapunov exponents evaluated in d E pd L p4 using a time delay of Tp12. The local exponents are evaluated at 750 locations around the attractor for 2 (LP1) steps forward from that location; we use Lp1, 2,..., 12. The large L limit of these average exponents are the global Lyapunov exponents. Here we have l 1`0 which is a critical indicator of chaos, and l 2 p0.0 which tells us that this system should be modeled by a set of differential equations

7 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction 293 Fig. 8. The average local Lyapunov exponents evaluated in d E pd L p4 using a time delay of Tp17. The local exponents are evaluated at 750 locations around the attractor for 2 (LP1) steps forward from that location; we use Lp1, 2,..., 12. The large L limit of these average exponents are the global Lyapunov exponents. Here we have l 1`0 which is a critical indicator of chaos, and l 2 p0.0 which tells us that this system should be modeled by a set of differential equations. Compare this Figure 7 which has Tp12 numerical accuracy. This means that the underlying dynamics comes from a differential equation as a zero global exponent is always associated with a flow in state space. Secondly, the value of the largest exponent is l 1 ;1/5.5, in units of 1/t s. Errors along the orbit or in initial conditions grow as exp[t/5.5t s ], so when the time after some perturbation or initial state is order a few times 5.5t s or a few hundreds of days or a year or so, we lose the ability to predict the behavior of the volume of the GSL. These estimates for the Lyapunov exponents can be used to estimate the fractal dimension of the attractor using the idea of Lyapunov dimension discussed in Kaplan and Yorke (1979) which has been conjectured to be the same as the information dimension. This Lyapunov dimension is evaluated by finding at which K the sum K A ap1 l a, (8) is positive, but the sum Kc1 A l a, (9) ap1 is negative. The Lyapunov dimension is defined as K A ap1 l a D L pk c hl Kc1 h. (10) For the GSL data we find D L ;3.05. This means that we should be able to get a good idea of the structure of Fig. 9. The attractor for the GSL volume as seen in time delay [v(t), v(tct), v(tc2t)] space with Tp12. The fractal dimension (Lyapunov dimension) of the GSL volume attractor is D L ;3.05, so it is nearly unfolded in this picture the attractor by looking at it in three dimensions with Tp12. This is displayed in Figure 9. The three dimensional view of the attractor suggests that the annual cycle is approximately motion around the smaller radius of the spool on which the attractor lies while the longer term motion which has larger amplitude moves the orbits along the longer axis of this spool. Such motions may correspond to the spectral power identified by Lall and Mann (personal communication) at certain interannual and interdecadal frequencies. Interestingly, such structure persists even if larger delays (T) that obscure the annual cycle are used. 4 Prediction on the GSL Attractor We do not have a set of four differential equations to model the GSL dynamics, but we can nonetheless use our knowledge of the dynamics in dp4 to model and predict the evolution of the GSL volume. A number of methods are available for approximating the map relating a four dimensional state vector to the state vector at a future point in time. One can use global or local polynomial predictors, for example. Also available are smoothing or regression splines, and of course there are many other basis functions, from which one can choose. Lall, Sangoyomi and Abarbanel (personal communication) successfully used Multiple Adaptive Regression Splines (MARS) to predict the GSL volumes during some critical periods. Here we confine ourselves to the use of local polynomial neighborhood to neighborhood maps as they are somewhat easier to explain. We establish the map by building local models of the evolution on the attractor from one neighborhood to another. This means that in the neighborhood of a point y(n) on the attractor, we locate N B neighbors y (r) (n)rp1, 2,..., N B in state space. We then determine where these N B c1 points go in one time step:

8 294 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction Fig. 10. Predictions using a local linear predictor for the GSL volume along with observed values. These predictions have been made one time step or 15 days ahead. The local linear maps for each region of the attractor were learned from the points 0 to 3263 of the data. Predictions were then made from points 3301 through 3400 Fig. 11. Predictions using a local linear predictor for the GSL volume along with observed values. These predictions have been made ten time steps or 150 days ahead. The local linear maps for each region of the attractor were learned from the points 0 to 3050 of the data. Predictions were then made from points 3248 through 3398 y (r) (n)]y(r, nc1), rp0,1,..., N B ; y (0) (n)py(n); y(0, nc1)py(nc1). Note that the point y(r,nc1) to which y (r) (n) goes in one time step may not be the r th nearest neighbor of y(nc1). We make a local polynomial model of this evolution. In the local linear case y(r,nc1)pa(n)cb(n)7y (r) (n), (11) and we use twice the number of neighbors as parameters in the polynomial. Higher order polynomials or other basis functions are also possible, but this local linear method works rather well and is simple to implement. Cross validatory choices of the polynomial order and the number of neighbors to use are desirable if one has adequate computational resources and accurate predictions are desired. In the expository setting here, we have stayed with the prescriptive choices indicated above. The local parameters in the vector A(n) and the matrix B(n) are determined by minimizing N B A rp0 hy(r, nc1)pa(n)pb(n)7y (r) (n)h 2, (12) and then recorded for future use. When presented with a new point in state space, say z, we search for the nearest neighbor to z, say y(q), and the predicted value on the orbit of the point to which z evolves would be A(Q)cB(Q)7z. We can use this method to predict ahead one step at a time using these local polynomial maps, or we can learn from the outset how to go forward N steps by fitting the map y (r) (n)]y(r,ncn). Both have their advantages. For this paper we remain with the latter method which we call the direct prediction method. Fig. 12. Predictions using a local linear predictor for the GSL volume along with observed values. These predictions have been made ten time steps or 150 days ahead. The local linear maps for each region of the attractor were learned from the points 0 to 3200 of the data. Predictions were then made from points 3348 through 3398 In Figure 10 we display the result of predicting one step ahead for the 100 time points after time index 3301; April 24, 1985 to June 2, The data used to determine the local linear maps was taken from the beginning of the time series up to point It is clear that the ability to predict one step (15 days) ahead is very high. Next we determined the local linear maps

9 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction 295 from data starting at the beginning of the data set and using points up to index In Figure 11 we display the result of predicting 10 time steps ahead, that is 150 days or about a half year, again starting with time index In Figure 12 we learned local linear maps from the first 3200 points of the data set, and then predicted ahead ten steps from time index These predictions are actually rather good, and, as we can see, the prediction ability degrades with the number of steps ahead one wishes to predict, as it is expected to given that the largest Lyapunov exponent is positive, implying that the system is chaotic. The main message in these predictions is that we can accurately follow the features of the data without elaborate computations by working in dimension four as our systematic analysis would suggest. Of course, it is possible to devise even better predictors using more elaborate methods. MARS based predictions by Lall, Sangoyomi and Abarbanel (personal communication) were able to predict the dramatic rise and fall of the GSL over a 4 year period ( ), using only data up to October However, the prediction horizon is considerably smaller (a few hundred days) if the predictions are started in It is interesting that predictability is high during the extreme event, even though the transition to the extreme event can not be forecast a priori. 5 Summary and Conclusions This paper has examined the biweekly data for the volume of the Great Salt Lake using recently developed methods of nonlinear time series analysis that have been used extensively in earlier work on observations of physical and biological systems. The methods, described in the text and in more detail in the review by Abarbanel et al. (1993), are founded on geometric ideas about the attractor and are couched entirely in time domain language. They actually establish from the outset if there is an attractor to begin with and, if so, what integer embedding dimension d E is required to capture the structure of that geometric object. Within that embedding dimension one can evaluate invariant quantities for the dynamics such as Lyapunov exponents and fractal dimensions. Further the initial exercise establishes the space within which one can expect accurate predictions to be possible. If one tries to predict in dimensions less than d E, one is assured of running into inaccuracies. This arises because d E is established as that minimum dimension where orbits of the observed dynamical system do not cross each other. When orbits cross, prediction is foiled by ignorance of which orbit is going where. This ambiguity is resolved in higher dimensions, and our methods determine which to use. The GSL data clearly show that we can unfold its attractor in d E p4, and that this is rather insensitive to the value of the time lag T used for constructing state space vectors y(n)p[v(n), v(nct), v(nc2t), v(nc3t)]. (13) We used Tp12 in much of our work, but we checked that the use of values of Tp9 and Tp17 had little effect on our results. These values sensibly bracket the nominal time lag of Tp12 we utilized the most. It is actually quite important that little depends on this choice of time lag, within sensible bounds, as the basic embedding theorem should work regardless of the choice. Using the notion of local false nearest neighbors we also established that the local dynamical dimension for these data is d L p4, and then we evaluated four Lyapunov exponents for the underlying processes. These exponents indicated that the system is predictable for a few times 5.5t s, where t s is the sampling time of fifteen days for these data. From the Lyapunov exponents we established the Lyapunov dimension of D L ;3.05, and we then displayed the attractor in three dimensions knowing some orbit overlaps were inevitable. The display was actually rather clean; perhaps not a surprise given the closeness of D L to three. Finally working in dimension four, we constructed local neighborhood to neighborhood polynomial maps for the process. These worked as we expected being excellent for short term predictions within the time horizon of ;5.5t s and degrading gracefully for longer time. Many other formats for the local maps are possible, and certainly one may anticipate that some of them will work much better than the rather crude linear polynomials we used here. Our main goal was not to find the very best predictions using elaborate basis functions. That is interesting in itself, and an analysis of other basis functions in d E pd L p4 for these data is well worthwhile. Here we simply wished to demonstrate the systematic fashion in which one could establish the space to work with and how to proceed in at least one way in that space. The identification that GSL dynamics can be resolved well through reconstruction in 4 dimensions does not suggest that the dimension of the climatic attractor is 4. Lorenz (1991) points out that while a number of analyses of climate and weather data find low dimensional attractors, the climatic attractor is likely to be of much higher dimension. Lorenz suggests that if the coupling between two components of a system is weak, the dimension computed from an analysis of the time series of a state variable from one of the components will not reflect the true dimension of the system. We suspect that the GSL is indeed weakly coupled to a large number of sources of climatic variability, in particular components that have a very high frequency character. However, if our interest is in modeling the evolution of the GSL, this filtering effect may actually be a plus. One aspect of this plus is that one has a well contained, well measured low dimensional geophysical system coupled to climatic dynamics. This provides a baseline of climatic behavior over a period before and during substantial human intervention in climate effects. A quantitative measure of these anthropomorphic effects may possibly be extracted from this record and then tracked in the future.

10 296 Abarbanel and Lall: Nonlinear dynamics of the Great Salt Lake: system identification and prediction Not treated at all here is a possible set of four dynamical equations which would capture the behavior of the GSL data that we have revealed. For some purposes, such as simply predicting accurately, this may not be necessary. However, for understanding the processes underlying the data, it is quite important to pursue this line of inquiry. We are doing that as a separate research topic with the constraints of the results here being born in mind. These constraints include, of course, the knowledge that working in four dynamical dimensions is enough. Also they include the knowledge of the statistical invariants, the spectrum of Lyapunov exponents, we have determined from the data. Any model must reproduce them. If we wanted further information about the structure of the attractor, we could easily evaluate other invariants such as the many fractal dimensions and require the modeling to reproduce them as well. It is important to note that while the methods we have used do allow significant improvement in the analysis of chaotic, spectrally broadband data, they do not pretend to replace good scientific intuition in determining what underlying equations will describe the dynamics. Tsonis, Triantafyllou and Elsner (1994) and Tsonis et al. (1994) have looked into the feasibility of predicting meteorological variables using similar methods, and have met with varied success depending on the underlying model. An important question that arises here is this: When, in general, can one expect such methods to work, and how sophisticated do the actual map reconstruction methods need to be? We suspect that there are no obvious answers to such questions for natural processes, where stationarity and ergodicity of the observed trajectories can not be rigorously established. Likewise, one cannot conclusively answer the question of how much data is needed to conduct such analyses. While some analysts suggest tens of thousands or more data points, others seem to have success with as little as a few hundred. In our experience, it is not the sheer number of data points, but whether the attractor (i.e., the collection of feasible states of the system) has been adequately explored. It is clear that even if one collected a million observations over one day, there is little information on the evolution of climate. However, if such a day occurred again, we may be well placed to forecast conditions within that day. On the other hand, if we had data points that were representative of climatic variations over the last millennium, the possibility of exploiting them for extended forecasts exists. In summary, a record is useful as long as the climate does not undergo a shift into a different stable or unstable mode. Were we to believe that the near term climate will bear some similarity to events that have occurred in the past 100 years or so, exercises such as the one presented here would be useful. Note that such an assumption is not at all foreign to stochastic time series analyses. In a number of ways, analyzing systems that have lower frequency dynamics seems more interesting and promising. Our preliminary analyses suggest that analyses similar to the one presented here may work well for larger rivers (e.g., Mississippi), whereas small streams may be more difficult to analyze, either because of increased complexity in the dynamics or because of a relatively poor signal to noise ratio in the data. Nevertheless we feel that the nonlinear analyses may provide better extended range predictions than corresponding linear time series methods. Future work on classifying streamflow processes and their predictability is planned. Mann and Park (1993, 1994), Dettinger and Ghil (1991) and Diaz and Pulwarty (1994) have recently recognized structure in climate at interannual, interdecadal and longer time scales. Conceptual models that aim to explain such variations usually consider atmosphere-ocean interactions with feedbacks, as well as feedbacks from land surface hydrologic processes. The selection of multiple state variables for attractor reconstruction and subsequent analysis and prediction of a univariate or multivariate state vector, e.g., GSL volume using an atmospheric flow index, are interesting problems. Work on such reconstructions is in progress. It is our hope that methods such as the ones presented here can provide the bridge between data and such conceptual models in ultimately developing an understanding of the physics of climate. Acknowledgements. The work of U. Lall was supported in part by the NSF (EAR ) and the USGS ( G-226). The work of H. Abarbanel was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Engineering and Geosciences, under contract DE-FG03-90ER14138, and in part by the Army Research Office (Contract DAAL03-91-C-052), and by the Office of Naval Research (Contract N C-0125), under subcontract to the Lockheed/ Sanders Corporation. 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