Hidden Markov models (HMMs) have wide application “since they emerged as the key technology for speech recognition.” This text provides an introduction to HMMs for the dynamical systems community.

The book is organized into six chapters. Chapter 1 introduces state-space models and discrete HMMs, with suitable examples. Chapter 2 presents basic algorithms: forward, backward, Viterbi, Baum-Welch, and expectation-maximization (EM), for fitting and using the probability distributions when both the possible states and the set of possible observations are discrete. Chapter 3 describes some model families with Gaussian observations using a novel regularization approach. In chapter 4, the forward and backward algorithms are presented concisely for continuous states and observations using Kalman filters. Chapter 5 fits models to data from chaotic dynamical systems using Lyapunov exponent calculations. Chapter 6 describes the algorithm for decoding sequences of classification and presents an application to experimental measurements of electrocardiograms.

An undergraduate background in engineering, mathematics, or science that includes work in probability, linear algebra, and differential equations is a prerequisite for the book. The book presents algorithms for using HMMs and explains the derivation of those algorithms. Although algorithms are given in pseudocode in the text, a working implementation is available on the accompanying Web site (http://www.siam.org/books/ot107/). This text provides essential introductory material, as well as the theory behind basic algorithms, so that the readers can use it as a guide to developing their own variants.

This well-organized book will certainly be a valuable enhancement to the existing literature. Every chapter contains all needed definitions and formulas, with deep discussions of their meanings, proofs, and examples. Bravo to Fraser for writing a truly fine book that is really needed in this area. I strongly recommend this book to those who want to appreciate the interplay between HMMs and dynamical systems.