Rudolf Kalman, now a professor at the Swiss Federal Institute of Technology in Zurich, published his groundbreaking paper on what we now call Kalman filters in March 1960. Suffice it to say that, without the Kalman filter, we wouldn’t be so successful in going to space, or in flying an aircraft safely with such an amazing level of stability. Other application domains for Kalman filters include manufacturing technology, weather forecasting, demographics, computer vision, and economics. The accomplishments of Kalman were recognized by his receipt of the first Kyoto Prize (considered to be a Nobel Prize for technology) in 1984, the same year it was awarded to Claude Shannon.

What is a Kalman filter? Stated briefly, it is an algorithm for extracting signals from noise. More accurately, it is an optimal linear recursive estimator. It is optimal because it provides the best estimate of the signal under certain conditions. It is recursive because (unlike other estimators, which use the entire previous set of signal values in every iteration) it uses only the immediate previous and current values of variables to calculate the estimate. It is linear because it assumes a linear model of the dynamics of the system and measurement device. In addition, it uses a statistical model of the noise and uncertainty, and the initial state of the variables involved.

There have been several books on the Kalman filter published since its introduction [1,2,3]. This book is written by a statistician; its primary goal is to provide self-contained “mathematically rigorous derivations of all the basic Kalman filter recursions from first principles.” As such, it is more theoretically oriented than any of the books mentioned above, and is more of a primer for theoreticians, rather than practitioners.

The first chapter introduces the reader to the basic prediction problem, with the state-space model as a special case. Cholesky decomposition is also introduced here, since the author considers the Kalman filter to be a modified Cholesky algorithm, with some additional structures derived from state-space models. The next four chapters constitute the core of the book, gradually deriving the forward and backward Kalman recursions. Chapter 6 spends time on selecting the initial state vector, and chapters 7 and 8 discuss some special cases and extensions of the state-space model.

Overall, the book is a tight and mathematically sound discourse on the Kalman filter’s internal workings. It may be useful in expanding and deepening the reader’s knowledge of this important technique. However, since the book uses a very small number of examples, it is less suitable for newcomers. On the practical side, it refers readers to the author’s Web site (http://math.asu.edu/~eubank) for Kalman filter software written in Java. Personally, based on the title, I expected this book to be an engineering primer, some kind of pedestrian guide, and after reading the preface I was initially very skeptical about its contents. However, the book surprised me nicely with its clear and mathematically rigorous derivations of all of the concepts.