Non-Gaussian State—Space Modeling of Nonstationary Time Series

Abstract

A non-Gaussian state—space approach to the modeling of nonstationary time series is shown. The model is expressed in state—space form, where the system noise and the observational noise are not necessarily Gaussian. Recursive formulas of prediction, filtering, and smoothing for the state estimation and identification of the non-Gaussian state—space model are given. Also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas. Significant merits of non-Gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples. A typical application of this non-Gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes. Simple Gaussian state—space modeling is not adequate for this situation. Here the model with small system noise variance cannot detect jump, whereas the one with large system noise variance yields unfavorable wiggle. To work out this problem within the ordinary linear Gaussian model framework, sophisticated treatment of outliers is required. But by the use of an appropriate non-Gaussian model for system noise, it is possible to reproduce both abrupt and gradual change of the mean without any special treatment. Nonstandard observations such as the ones distributed as non-Gaussian distribution can be easily treated by the direct modeling of an observational scheme. Smoothing of a transformed series such as a log periodogram can be treated by this method. Outliers in the observations can be treated as well by using heavy-tailed distribution for observational noise density. The algorithms herein can be easily extended to a wider class of models. As an example, the smoothing of nonhomogeneous binomial mean function is shown, where the observation is distributed according to a discrete random variable. Extension to a nonlinear system is also straightforward.