A numerical approach for estimating higher order spectra using neural network autoregressive model

A method for parametric estimation of higher order spectra of time series using a nonlinear autoregressive model based on multi-layered neural networks (NNAR model) is presented. In real world problems there exist signals that can not be described sufficiently by linear time series models such as AR or ARMA models. In order to characterize such signals, several nonlinear time series models have been investigated in recent years. However, in contrast with the case of linear models, there are a few parametric approaches that estimate the higher order statistical characteristics of observed time series using such nonlinear time series models. It is very difficult to derive analytically explicit formulations of higher order spectra from the expressions of such nonlinear time series models. In this study, employing numerical techniques, the authors construct a parametric estimator of higher order spectra. It consists of the following steps: 1. training an NNAR model on the given time series, 2. iteration of numerical integrals for solving the joint probability density function, 3. calculation of higher order cumulant functions by renewal equations based on the joint probability density function solved in 2., and 4. multidimensional discrete Fourier transforms of higher order cumulant functions calculated in 3. The authors also show that any NNAR model with finite valued weights satisfies a sufficient condition of convergence.