پیش بینی محلی از سری های زمانی غیر خطی با استفاده از رگرسیون بردار پشتیبانی

Prediction on complex time series has received much attention during the last decade. This paper reviews least square and radial basis function based predictors and proposes a support vector regression (SVR) based local predictor to improve phase space prediction of chaotic time series by combining the strength of SVR and the reconstruction properties of chaotic dynamics. The proposed method is applied to Hénon map and Lorenz flow with and without additive noise, and also to Sunspots time series. The method provides a relatively better long term prediction performance in comparison with the others.

The study of prediction has been influenced, for a long time, by statistical methods such as the ARMA model [1] for linear stationary time series. Recent developments in non-linear and/or non-stationary time series analysis can be found in Refs. [2] and [3]. Another approach to predicting time series using neural networks has been investigated [4]. All of these methods are known as global time series prediction in which only one function is engaged for all available data. Local prediction uses more than one function to fit the data. This approach is also known as lazy learning [5]. Support vector regression (SVR) [6] and [7] and radial basis function (RBF) networks [8] and [9] are two different approaches to non-linear regression problems, but both methods are known as single layer networks. The relationship between the SVR and RBF networks can be found in Ref. [7]. SVR has been applied to such as drug discovery [10], Travel time prediction [11] and computational vision [12]. Recently, SVR has been used for predicting chaotic time series [13]. The RBF has not only the ability of approximating scattered data without using any mesh. Therefore it is a good solution for the multivariate interpolation problems. It is also a method of turning an ill-posed problem into a well posed problem by regularization [14], [15] and [16]. RBF has been applied to such as 3D object recognition [17] and facial expressions [18]. A chaotic attractor is obtained by measuring a chaotic time series. The properties of the chaotic attractor can be retained through a reconstruction procedure. This procedure is known as the delay coordinate embedding [19] resulting in a reconstructed state space which contains a reconstructed chaotic attractor preserving both geometrical and dynamical properties of the original chaotic attractor. In this paper we propose the SVR as a local predictor. The local predictor chooses a set of nearest neighbours which evolves similarly in the reconstructed chaotic attractor. This approach to predicting the chaotic time series has interested many researchers such as Farmer and Sidorowich [20], Casdagli [21] and Sauer [22]. Our approach is different from the work done by Mukherjee et al. [13] and Casdagli [21]. We combine the strength of SVR and local predictor to achieve a better prediction result. The resulting predictor is referred as SVR based local predictor (SVRLP). The proposed algorithm is applied to two benchmark problems of chaotic time series, known as the noisy Hénon time series [23] and the noisy Lorenz time series [24], respectively. The benchmark problems are mainly concerned with chaotic dynamics which is difficult to predict. Then the proposed algorithm is further applied to Sunspots series. Through the simulation study presented in this paper, based on these three benchmark problems, it is demonstrated that the prediction performance of SVRLP is better than RBF based local predictor (RBFLP) in most situations. In this paper Section 2 describes the delay coordinate embedding methodology which shows an attractor can be reconstructed from an univariate time series. Section 3 reviews the existing method of local prediction. The SVR is introduced in Section 4. The comparison of the prediction performances of SVRLP, RBFLP and least square local predictor (LSLP) is given in Section 5. The conclusion is given in Section 6.

We have presented the SVRLP for phase space prediction of chaotic time series. The proposed algorithm is evaluated on the Hénon and Lorenz time series with and without additive noise, respectively. We find that the SVRLP can achieve long term prediction with a higher prediction accuracy than both of the LSLP and RBFLP. We also test the SVRLP using Sunspots time series. We find again the SVRLP has better prediction performance than RBFLP. It is because that the RBPLP is a multi-dimensional interpolation method not suitable for noisy data while the SVRLP is designed to adapting noise via the ɛɛ-insensitive loss function (16). VC-dimension is a measure of the capacity of the function space containing the true function f [32]. When the capacity is increased, the training error is decreased but the generalization error is increased. The SVRLP minimizes a bound depending on the VC-dimension and the number of training errors at the same time [6] and [7], therefore the generalization error can be smaller. The advantages of the SVRLP come with a greater computational cost, therefore the training time of SVRLP is generally longer than both of the LSLP and RBFLP. However our previous work has shown the potential of reducing the computation of SVR [33], which can be applied to SVRLP.