ABSTRACT
Both the theory and practice of statistics frequently involves consideration of stochastic models evolving with respect to a discrete, time-like, index variable. This book is concerned with an important special class of such stochastic processes that can be described using state-space models. In this and subsequent chapters we will develop a mathematical framework that can be used to understand the computational algorithms that are widely used in conducting statistical inference about these processes. State-space models are examples of signal-plus-noise
models and, accordingly, we will begin the development in Section 1.2 in the more general signal-plus-noise environment. This provides us with a broader view of prediction problems while also allowing us to appreciate the benefits that can be realized when a state-space framework is appropriate. In particular, we will see that for state-space models it is possible to develop algorithms for computing predictions, parameter estimators, etc., that are orders of magnitude faster than for the general signal-plus-noise setting. These algorithms are usually referred to under the titular umbrella of the Kalman filter as a result of the pioneering work in Kalman (1960) and Kalman and Bucy (1961).
