Nonlinear time series modelling with the radial basis function-based state-dependent autoregressive model

Abstract

This paper investigates nonlinear time series modelling using the general state-dependent autoregressive model. To achieve the estimate of the model, an attempt is made to approximate the state-dependent parameter by employing the Gaussian radial basis function for its universal approximation capability. As a result, a radial basis function-based autoregressive (RBF-AR) model is derived which has a form similar to a generalized exponential autoregressive model. To reach the applicability of the RBF-AR model, the evolutionary programming algorithm is employed to select a suitable set of radial basis function centres. By applying the resulting model to some complex data, it is shown that the RBF-AR model can not only reconstruct the dynamics of given nonlinear time series e ectively, but also give much better ® tting to complex time series than the approach of directly RBF neural network modelling. Therefore, as a paradigm combining a statistical model and neural network, the RBF-AR model has better performance than the RBF neural network especially for its problem reduction of the curse of dimensionality which is usually regarded as one of the potential main di culties of RBF neural network modelling.

Reconstruction of gastric slow wave from finger photoplethysmographic signal using radial basis function neural network

“…Particularly, the radial basis function neural network (RBFNN) is known for its strengths of simple topological structure and fast learning [18,28]. Both theoretical and experimental analysis [31,34] indicated that the RBFNN could approximate any functions and identify nonlinear systems by using members of a family of basis functions. The RBFNN is a three-layer feed-forward network that generally uses a linear transfer function for the output units and a nonlinear transfer function (typically the Gaussian function) for the hidden units.…”

A locally linear RBF network-based state-dependent AR model for nonlinear time series modeling

“…The idea behind this definition is that the term exp Àk k max tÀ1 kXðt À 1Þ À Z 0 k k 2 2 n o should approach zero when the state X(t À 1) is far away from the center Z k , so that the linear weights are bounded and stable [29,25]. After selecting an initial nonlinear parameter vector h 0 N , and keeping it fixed, use the LSM to compute the initial linear weights h 0 L :…”

“…For the model order selection, several methods have been proposed, e.g., the Akaike Information Criterion (AIC) [1] and cross validation [18,20,22]. Shi et al [29] and Peng et al [25] suggest the use of the AIC as the RBF-AR model selection criteria. Although the AIC may not be the best model selection method when the data comes from a mathematical model [28], it may give a good synthetic result for modeling the data where an exact mathematical model for the data does not exist.…”

“…An efficient approach to solving the problem is to use the RBF neural networks approximation [25,29,33]. In this paper, a special type of RBF network called a locally linear RBF network (LLRBFN) is proposed to approximate the state-dependent coefficients of model (2).…”

“…Valenzuela et al [34] proposed a hybridization of intelligent techniques such as ANNs, fuzzy systems, and evolutionary algorithms to improve the performance of the hybrid ARIMA-ANN model. Vesin [33] proposed an RBF approach to state-dependent AR modeling and was followed by Shi et al [29] who developed the RBF-AR model, which was further developed by Peng et al [25] to the RBF-ARX model, extending the idea of RBF-AR to the case where there are several external inputs to the system. 0020-0255/$ -see front matter Ó 2010 Elsevier Inc. All rights reserved.…”