Linear Innovations State Space Models

In Chap. 2, state space models were introduced for all 15 exponential smoothing methods. Six of these involved only linear relationships, and so are “linear innovations state space models”. In this chapter, we consider linear innovations state space models, including the six linear models of Chap. 2, but also any other models of the same form. The advantage of working with the general framework is that estimation and prediction methods for the general model automatically apply to the six special cases in Chap. 2 and other cases conforming to its structure. There is no need to derive these results on a case by case basis.

The general linear innovations state spacemodel is introduced in Sect. 3.1. Section 3.2 provides a simple algorithm for computing the one-step prediction errors (or innovations); it is this algorithm which makes innovations state space models so appealing. Some of the properties of the models, including stationarity and stability, are discussed in Sect. 3.3. In Sect. 3.4 we discuss some basic innovations state space models that were introduced briefly in Chap. 2. Interesting variations on these models are considered in Sect. 3.5.