Gradient Radial Basis Function Based Varying-Coefficient Autoregressive Model for Nonlinear and Nonstationary Time Series

Abstract

We propose a gradient radial basis function based varying-coefficient autoregressive (GRBF-AR) model for modeling and predicting time series that exhibit nonlinearity and homogeneous nonstationarity. This GRBF-AR model is a synthesis of the gradient RBF and the functional-coefficient autoregressive (FAR) model. The gradient RBFs, which react to the gradient of the series, are used to construct varying coefficients of the FAR model. The Mackey-Glass chaotic time series are used to evaluate the performance of the proposed method. It is shown that the GRBF-AR model not only achieves much more parsimonious structure but also much better prediction performance than that of GRBF network.Index Terms-Functional-coefficient autoregressive model, gradient radial basis function, nonlinear and nonstationary time series, separable nonlinear least squares.

Modeling and Control Approach to Coupled Tanks Liquid Level System Based on Function‐Type Weight RBF‐ARX Model

“…Note that to calculate the system matrices ̅ A t , ̅ B t , ̅ C, and ̅ Γ t in model (15), knowledge of the working state prediction w tþjjt ð Þ(j = 1, 2, ⋯ , N p À 1) is required. If the working state prediction is not available at time t, one may replacê w tþjjt ð Þ with the current working state w(t) to compute ̅ A t , ̅ B t , ̅ C , and ̅ Γ t in (15)(16)(17)(18)(19). Once the working-point w tþjjt ð Þ is fixed as w(t), the coefficient matrixes in (15) at time t can be obtained without using future system information.…”

“…According to the multi-step-ahead prediction model (11)(12)(13)(14)(15)(16)(17)(18)(19), the output prediction can be derived aŝ whereŶ t ð Þ is the estimated model predictive output vector along the horizon N p , G t is the matrix of the dynamics, and Y 0 (t) is the free response vector. Defining the control increment sequence ΔÛ t ð Þ and the desired output sequenceŶ r t ð Þ as follows…”

Highly computationally efficient state filter based on the delta operator

“…Parameter estimation and state filtering are basic for system identification [27][28][29] and system analysis and design, [30][31][32][33] and can be applied to many areas. [34][35][36][37][38][39] The Kalman filter (KF) is known as the optimal state filter for linear systems under the Gaussian white noise and has been extended to study the parameter estimation for bilinear systems.…”

Recursive methods for estimating the radial basis function‐based state‐dependent autoregressive model

“…By using an analytical expression of the Jacobian matrix instead of finite differences, Chen et al developed the VP algorithm for the RBF‐AR models based on the modified Gram‐Schmidt method . The idea of the VP method was also employed to identify the GRBF‐AR model which can handle a class of nonlinear nonstationary time series …”

“…6 The idea of the VP method was also employed to identify the GRBF-AR model which can handle a class of nonlinear nonstationary time series. 10 System identification is the theory and method of establishing the mathematical models of systems. 11 Many identification methods have been developed for linear systems, bilinear systems, 12,13 multivariable systems, 14 and nonlinear systems.…”