Prediction of chaotic time series based on Nyström Cauchy kernel conjugate gradient algorithm

Abstract

Chaotic time series can well reflect the nonlinearity and non-stationarity of real environment changes. The traditional kernel adaptive filter (KAF) with second-order statistical characteristics suffers performance degeneration dramatically for predicting chaotic time series containing noises and outliers. In order to improve the robustness of adaptive filters in the presence of impulsive noise, a nonlinear similarity measure named Cauchy kernel loss (CKL) is proposed, and the global convexity of CKL is guaranteed by the half-quadratic (HQ) method. To improve the convergence rate of stochastic gradient descent and avoid a local optimum simultaneously, the conjugate gradient (CG) method is used to optimize CKL. Furthermore, to address the issue of kernel matrix network growth, the Nyström sparse strategy is adopted to approximate the kernel matrix and then the probability density rank-based quantization (PRQ) is used to improve the approximation accuracy. To this end, a novel Nyström Cauchy kernel conjugate gradient with PRQ (NCKCG-PRQ) algorithm is proposed for the prediction of chaotic time series in this paper. Simulations on prediction of synthetic and real-world chaotic time series validate the advantages of the proposed algorithm in terms of filtering accuracy, robustness, and computational storage complexity.