CASPIAN SEA LEVEL PREDICTION USING ARTIFICIAL NEURAL NETWORK AND EMPIRICAL MODE DECOMPOSITION

This paper demonstrates the possibility of using nonlinear modeling for prediction of the Caspian Sea level. Phase space geometry of the of a model can be reconstructed by the embedology methods. Dynamical invariants, such as the Lyapunov exponents, the KaplanYorke dimension, and the prediction horizon were estimated from reconstruction. Fractal and multifractal analyses were carried out for various time intervals of the Caspian Sea level and multifractal spectra were calculated. Then, historical data resolution was improved with the help of fractal approximation. The EMD method was used to reduce noise of the time series. Global nonlinear predictions were made with the help of Artificial Neural Network for combinations of different empirical modes.


INTRODUCTION
The Caspian Sea is the largest intercontinental reservoir without water outflow which exhibits a unique global evolution over an extremely long interval of time.On geological scale, the history of the Caspian Sea is represented by alternations of transgressive and regressive phases reflected clearly by paleodata, reflected in scanty historical records, and shown by instrumental measurements during recent and a relatively short monitoring period.In Holocene, for example, the fluctuations of the Caspian Sea Level (CSL) were caused by climate changes.It is possible that the CSL was influenced not only by water balance, but also by tectonic changes of the seabed, and that these factors did not necessarily coincide in time [Golitsyn, 1995].Up to date, the CSL has being raising for already 18 years.By the beginning of 1996 year, the CSL has reached -26,6 meters and the area of the Caspian Sea has increased by 40 000 square kilometers.The rising sea level and strong winds resulted in multiple problems for economic development.Economics of the regions near the Caspian Sea depended on modern sea fluctuations.Modeling the dynamics of a closed reservoir without outflows may appear straightforward on the first sight.However, simple balance models may not adequately describe the situation.Only nonlinear models may be applied to describe a chaotic dynamics of the CSL [Makarenko et al., 2004;Kozhevnikova & Shveikina, 2008].Thus, in this paper, we applied nonlinear modeling to predict the CSL.This approach was based on reconstruction of phase dynamics from observed time series by means of topological embedding methods.A nonlinear global forecast has been made with the help of Artificial Neural Network (ANN).The CSL time series were constructed by fractal approximation.

EMBEDOLOGY AND NONLINEAR PREDICTION
The nonlinear approach for modeling and prediction of dynamic regimes of sea level can be based on chaotic dynamics [Makarenko, et al., 2004].According to general assumptions about properties of an unknown dynamic model of sea level we can reconstruct the diffeomorphic copy of its attractor in an n-dimensional space.This technique, called embedology, is widely known [Sauer et al., 1991].We used it to create the nonlinear scheme of sea level prediction.Embedology methods assume that an observed variable is typical and contains all characteristic elements of dynamics.However, the CSL data have being measured only since 1830 and they do not contain information about global variations of sea level.Thus, one can hope only for a short-term forecast.In fact, the prediction horizon is determined by the maximum positive value of the Lyapunov exponent [Schaw, 1981]  l = -0.838176for the instrumental time series.The Kaplan-Yorke dimension is 2.18876 that is close to the embedding dimension.In our case, the length is N = 1955 and the prediction horizon is Т ≈ 12-36 months [Makarenko, et al., 2004].
The reconstruction of the copy of the attractor into Euclidian space R m gives possibility to obtain the following predictor [Sauer et al., 1991, Makarenko, 2003]: We used m = 27 and τ = 37 to construct delay vectors y.Unknown function F-predictor is nonlinear and continuous function of m vectors of the reconstruction.Their best approximation is found by ANN [Poggio & Girosi, 1989;Bishop, 2006].ANN training was carried out using a set from available values of the CSL data.In the case when a lag of τ ≠ 1, one can obtain τ predicted points simultaneously, i.e. can construct a vector prediction.The lower estimate of the embedding dimension m > 8 was obtained with the help of the False Nearest Neighbors method.The prediction horizon is limited by the time series length and rate of divergence of close reconstructed trajectories.The prediction horizon of instrumental monthly CSL data was estimated at 12-36 months, as mentioned above.

FRACTAL APPROXIMATION OF THE CSL HISTORICAL DATA
In order to use a nonlinear context and obtain a nonlinear prediction for historical data from 600 BC, we applied fractal approximation [Barnsly, 2000;Karimova et al., 2003;Makarenko et al. 2004].The latter could enhance the historical data that are poor in accuracy and have low timeresolution (a point per ten years).The usage of historical data in the prediction task is very important, because its accurate instrumental measurements reflect short time variations of the CSL and does not trace its global evolution.The main ideas of fractal approximation is as follows.The interpolation problem deals with a set of input pairs ( ) x y where the 0 = x 0 < x 1 < ... < x N = 1 are nodes and As a rule, in the case of smooth data, the input points are interpolated by a single-degree N polynomial, or by piecewise interpolations with a lowdegree polynomial.Recent research has provided an alternative assumption that the interpolation function Fis self-similar, and typically not smooth, but fractal.We note that a function F:[0, 1] → R is well defined by its graph, and use the same symbol to denote the set of points in its graph.Hence a point (x, y) ∈ F if and only if.We also use the notation x , where c i < 1 is given as a parameter controlling the roughness of the function, and a i , b i , d i and e i are determined either by the constraints or the "reflected" constraints Given a metric d((t Here N δ denotes the number of cubes of size δ with the coarse Hölder exponent roughly equal to α.Let μ(N δ ) be a measure contained in a δ-cube, then a = logμ(N δ )/logδ.
We computed the large deviation spectrum f G (α) and the Legendre spectrum f L (α) using FracLab software (http://fraclab.saclay.inria.fr/).Both Legendre and large deviation spectra of the time series are presented in Fig. 1.The multifractal spectra were computed for the fragment of the annual data constructed by fractal approximation using 1500-yr long decennial data.Fig. 1 shows that these spectra have similar maxima corresponding approximately to the Hölder exponent equal to 0.8.

EMPIRICAL MODE DECOMPOSITION (EMD) FOR THE CSL
The main difficulty for construction of the global nonlinear prediction is an uncertainty associated with monthly instrumental time series.To avoid this difficulty we apply the well known Empirical Mode (EMD) decomposition [Huang et al., 1998] and approximate time series by the sum of smooth empirical modes.According to this method, it is possible to use a coarse-grained approximation of a signal, excluding high-frequency details, without breaking global structure of the signal (Fig. 2).
Thus, the method of decomposition of a signal by means of empirical modes [Huang et al., 1998;Flandrin et al., 2003] represents the signal as a set of functions corresponding to various oscillations in observed signal.A basic operation in EMD is the estimation of the upper and lower "envelopes" as interpolated curves between extremes.The nature of the chosen interpolation plays an important role, and our experiments tend to confirm (and are in agreement with) what is recommended in [Huang et al., 1998], specifically, that cubic splines are to be preferred.Other types of interpolation (linear or polynomial) tend to increase the required number of sifting iterations and to "over-decompose" signals by spreading out their components over adjacent modes.
Given a signal x(t), the effective algorithm of EMD can be summarized as follows [Flandrin, et al., 2004]: 1. identify all extremes e min , e max of signal x(t).
2. interpolate between minima and maxima, ending up with some envelope e min (t), e max (t).

iterate on the residual m(t).
In practice, the above procedure has to be refined by a sifting process which amounts to first iterating steps 1 to 4 upon the detail signal d(t), until this latter can be considered as zero-mean according to some stopping criterion.Once this is achieved, the detail is referred to as an Intrinsic Mode Function (IMF), the corresponding residual is computed and step 5 is applied.By construction, the number of extremes is decreased when going from one residual to the next, and the whole decomposition is guaranteed to be completed with a finite number of modes.For calculation of empirical components software batch http://perso.enslyon.fr/patrick.flandrin/emd.html was applied.

CONCLUSION
We have discussed the method of the CSLs predictions based on the reconstruction of a dynamical system with the help of embedding time series in Euclidian space of an appropriate dimension.This approach makes it possible to construct vector prediction, i.e. to forecast a consecutive set at the same time.Therefore, one can avoid an exponential increase of errors inherent in a step-by-step prediction.
The prediction is realized by means of the ANNs, that are optimal approximating tool for an unknown continuous and multivariable function, i.e. a nonlinear predictor.

Fig. 1 .Fig. 2 .
Fig. 1.The large deviation f G (α) (1) and the Legendre f L (α) (2) spectra for the fragment of the annual data constructed by fractal approximation using 1500-yr long decennial data The ANN is trained by transforming input examples into the outputs as the known answer of the training set constructed by the known records from the past.To use the decennial historical data together with the annual instrumental data, we constructed fractal approximation of the decennial data that allowed increasing the time series.The necessary property of statistical scale invariance was verified by multifractal spectra.Additional improvement of the data was implemented with the help of EMD technique that allowed delicate noise filtration without disturbance of correlation of the time series.The global nonlinear predictions were made with the help of ANN for combinations of different empirical modes.Our experiments based on the approaches mentioned above have shown possibility to forecast the CSLs at 1-3 yrs intervals, which is extremely useful in practice.It is necessary to point out that a nonlinear prediction represents an ill-posed problem as it produces a great number of possible variants.Selection of the most probable variant from a set of predictions remains an open challenge.