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Filtering and prediction: A primer. (English) Zbl 1165.62070

Student Mathematical Library 38. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4333-8/pbk). xi, 252 p. (2007).
This book provides an introduction to the concepts and tools arising in filtering and prediction. The underlying problem is concerned with the need to obtain information about the dynamic behaviour of some moving objects, when only observations of the behaviour subject to perturbations by noise are available. Filtering is then a technique to extract the desired information as precise as possible by ‘filtering out’ the noisy perturbations. Based on information on the past behaviour of the object one would then like to make predictions about the next or the next several movements.
The first chapter of the book, “Preliminaries”, provides some background material on series, probability concepts and and conditional expectations in the discrete case. The material of the following seven chapters may be divided into three parts, roughly relating to filtering of discrete Markov chains, filtering of continuous-space Markov chains and Wiener processes, and filtering and prediction of stationary sequences.
The first part consists of Chapters 2 and 3. In Chapter 2, random walks and Markov chains in discrete time and space are introduced, whereas Chapter 3 treats filtering for discrete Markov chains, including hidden Markov models, parameter estimation in a Bayesian setting for discrete Markov chains, interpolation or smoothing after obtaining more information for the discrete Markov chains, as well as prediction of the evolution of discrete Markov chains.
The second part, going over three chapters, starts with an introduction of the notion of conditional expectations in terms of \(L_2\)-theory. Chapter 5 treats filtering for Markov chains with state space \({\mathbb R}^d\), discussing discrete Kalman filters and linear filtering, in particular also for hidden Markov models. Chapter 6 is concerned with continuous time filtering in the case of Wiener processes. The latter are introduced and basic properties as well as integration against Wiener processes is discussed. The main part of the chapter is devoted to Kalman filters in continuous time.
In the last part of the book, the authors introduce stationary sequences, basic properties and spectral densities and then treat filtering for these in chapter 7. The last chapter is then devoted to prediction, dealing with autoregressive and autoregressive-moving average sequences.
The book is intended as a textbook for an undergraduate, or beginning graduate course. A large part of the material is developed in the form of problems given throughout the text and many examples are used to illustrate the concepts. The book is well-written and provides a very nice basis for lecturing about this topic.

MSC:

62M20 Inference from stochastic processes and prediction
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
60G35 Signal detection and filtering (aspects of stochastic processes)
60G25 Prediction theory (aspects of stochastic processes)
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
60J60 Diffusion processes
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