A method for calculating higher-order spectra of a nonlinear autoregressive model with low memory requirements

In order to capture the characteristics of non-Gaussian time series observe d in biological signals, it is necessary to study higher-order spectra, suc h as the bispectrum, in addition to the power spectrum. In order to obtain estimates with little statistical variations in the estimation of higher-or der spectra, establishment of a parametric estimation method is desired. Re cently, a method has been proposed to numerically derive the higher-order s pectra of nonlinear autoregressive models, such as neural net autoregressiv e models, by means of the repetitive integral transform. However, in effort s to describe the probability density function simply in a discrete manner, there is the problem of an exponentially increasing number of data points to be computed with increasing lag order. In this paper, in order to resolv e this problem, a numerical method is proposed for description of the proba bility density function and integral kernel by means of a tensor product ex pansion. The tensor product expansion is a representation in terms of the s um of the tensor products of characteristic one-variable functions dependin g on the given multivariable function. There is a possibility of reducing t he memory capacity needed for the calculations. Numerical examples of accur ate calculations with fewer data points in the proposed method than in the conventional method are presented. Finally, an example of parametric estima tion of the power spectrum and bispectrum of Wolf's sunspot numbers by the proposed method is given. (C) 1999 Scripta Technica, Electron Comm Jpn Pt 3 , 82(11): 47-57, 1999.