Parametric estimation of higher‐order spectra by tensor product expansion with coordinate transformation
Yoshitomo Nishiguchi, Naohiro Toda, Shiro Usui · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 2003
Abstract When the characteristics of non‐Gaussian time series, such as biological signals, are described, higher‐order spectra such as the bispectrum must be referenced in addition to the power spectrum. To obtain estimates having few statistical fluctuations, parametric estimation methods of these spectra should be established. In recent years, a simple discretization method (SDM) has been proposed as a numerical solution based on iterated integral transform of the stationary joint density probability function and the higher‐order spectra of nonlinear autoregressive models like a neural network autoregressive model. However, the SDM requires an amount of calculation memory that increases exponentially with respect to the lag order of corresponding models. In order to overcome this drawback, the tensor product expansion method (TPEM) where the joint probability density function is expanded as the sum of single‐variable functions was proposed. This method can suppress the increase in the required memory capacity to a linear increase. A numerical example has shown where the TPEM calculated the bispectrum more accurately than the SDM with an equivalent amount of memory. However, the relationship between the performance of tensor product expansion and the dependence properties of the target time series has not as yet been investigated. In this paper, we study the property dependence of the time series in the TPEM and point out the problem of worsening calculation accuracy for a time series in which values at each time instant highly depend on each other. As the solution, we propose a new method that introduces a linear coordinate transform in state space where the joint probability density function undergoes the tensor product expansion. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(1): 19–28, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/ecjc.10103