Filtering and Prediction - A Primer
This book is mainly intended as an introduction to the problem of filtering and prediction in time homogeneous Markov chains and the Wiener process. The problem of how to estimate an unobservable signal based on an available observation of response (a signal corrupted by noise) is known as filtering. Prediction means estimating the future of a random process based on its history.
The first two chapters of the book are introductory and deal with basic probability concepts and discrete Markov chains. In the third chapter, the discrete Markov chains are used as the simplest model for an introduction of the filtering problem. In chapter 4, the conditional expectation is introduced as a necessary tool for continuous (in space and/or time) processes. The continuous space Markov chains are then discussed in chapter 5; the famous Kalman filter is described and the chapter closes with linear filtering. Chapter 6 covers filtering for the Wiener process and the continuous time Kalman filter is introduced in a classical setting. The last two chapters deal with the prediction problem in stationary random sequences.
The book is written in an elementary way (no special knowledge of probability and statistics is needed to read the book) but it is still mathematically rigorous. The book can be recommended to all students interested in stochastic models. Since the book contains many problems and remarks, it can also be recommended as the basis of a one semester introductory course; almost all theorems are proved and problems may be used as homework assignments although some of the problems are more challenging than others.