State-Space Models and the Kalman Recursions

Abstract

In recent years, state-space representations and the associated Kalman recursions have had a profound impact on time series analysis and many related areas. The techniques were originally developed in connection with the control of linear systems (for accounts of this subject, see the books of Davis and Vinter (1985) and Hannan and Deistler (1988)). The general form of the state-space model needed for the applications in this chapter is defined in Section 12.1, where some illustrative examples are also given. The Kalman recursions are developed in Section 12.2 and applied in Section 12.3 to the analysis of ARMA and ARIMA processes with missing values. In Section 12.4 we examine the fundamental concepts of controllability and observability and their relevance to the determination of the minimal dimension of a state-space representation. Section 12.5 deals with recursive Bayesian state estimation, which can be used (at least in principle) to compute conditional expectations for a large class of not necessarily Gaussian processes. Further applications of the Bayesian approach can be found in the papers of Sorenson and Alspach (1971), Kitagawa (1987) and Grunwald, Raftery and Guttorp (1989).