Stochastic unobserved component models for adaptive signal extraction and forecasting

This paper is concerned with adaptive, off-line signal processing and forecasting for nonstationary signals described by the unobserved component model y_t=T_t+S_t+f(u_t)+N_t +e_t, for e_t~N{O, σ²}, where y_t is the observed time series, T_t is a trend or low-frequency component, S_t is a periodic component possibly exhibiting temporal changes in both amplitude and phase, f(u_t) captures the influence of a vector of exogenous variables u_t, if necessary including stochastic, nonlinear static or dynamic relationships, N_t is a stochastic perturbation component, and e_t is an irregular component, normally defined for analytical convenience as a normally distributed Gaussian sequence with zero mean value and variance σ² (i.e. discrete-time white noise). In order to allow for nonstationarity in the time series y_t, the various components, including T _t, are characterised by stochastic time-varying parameters each of which is defined as a nonstationary stochastic variable. The paper describes a new and flexible approach to off-line signal processing based on a dynamic harmonic regression (DHR) model of the unobserved components (UC) type and illustrates its performance via a typical problem of audio signal restoration