A Kalman Filter Primer
Series Editor: William Shucany
Chapman and Hall/CRC – 2005 – 200 pages
System state estimation in the presence of noise is critical for control systems, signal processing, and many other applications in a variety of fields. Developed decades ago, the Kalman filter remains an important, powerful tool for estimating the variables in a system in the presence of noise. However, when inundated with theory and vast notations, learning just how the Kalman filter works can be a daunting task.
With its mathematically rigorous, “no frills” approach to the basic discrete-time Kalman filter, A Kalman Filter Primer builds a thorough understanding of the inner workings and basic concepts of Kalman filter recursions from first principles. Instead of the typical Bayesian perspective, the author develops the topic via least-squares and classical matrix methods using the Cholesky decomposition to distill the essence of the Kalman filter and reveal the motivations behind the choice of the initializing state vector. He supplies pseudo-code algorithms for the various recursions, enabling code development to implement the filter in practice. The book thoroughly studies the development of modern smoothing algorithms and methods for determining initial states, along with a comprehensive development of the “diffuse” Kalman filter.
Using a tiered presentation that builds on simple discussions to more complex and thorough treatments, A Kalman Filter Primer is the perfect introduction to quickly and effectively using the Kalman filter in practice.
“We strongly recommend Eubank’s book. It is a masterpiece of exposition, in a class by itself (there are no similar books at a truly introductory level), which makes basic understanding and applying the Kalman filter as simple as possible. It is written in a way that motivates the readers’ interest, a pleasure (enjoyable) to read; it provides in an interesting and concise way the right amount of detail.
“…One of the strengths of the book is the author’s love of and enthusiasm for the subject. It clearly shows the author’s concern with making as easy as possible the reader’s path to learning.”
— Prof. Emanuel Parzen, Texas A&M University
”…develops the algorithmic aspects of Kalman filtering for state space models. The Basic tool is best linear unbiased prediction, which is immediately interpreted in terms of the Cholesky factorization of the relevant variance-covariance matrices. This also leads to an easy presentation of the ‘backward’ filtering step when one wishes to solve smoothing problems…
“…The author gives a comprehensive (and comprehensible) account of how the Kalman filter may be adapted to handle [diffuse priors]. This is no mean feat, as the literature on diffuse priors is scattered and algorithms are often presented with little motivation, other than that they work.
“…This is a do-it-yourself text. It treats Kalman filtering for two fundamental examples in detail: ARMA models for time-series and Brownian motion in white noise. The complete algorithms are presented, ready for computer implementation, and more importantly, ready for modification. This will give graduate students and researchers a flying start for treating their own applications.
“The author has given us a rare glimpse of what happens when someone wants to get to the bottom of things: one senses his wonder about these beautiful and beautifully efficient algorithms.”
—Paul Eggermont, Department of Food & Resource Economics, University of Delaware
“… In general, the book has a strong focus on algorithms, … there are many examples of pseudocode summaries , and corresponding Java code is downloadable from the author’s Web pages. …This short book is a pleasure to read. The arguments are clear and will be much appreciated by the many users of the Kalman filter. …”
— Byron J. T. Morgan, University of Kent, Canterbury, in Journal of the Royal Statistical Society
“… it provides “a self-contained”, ‘no frills’, mathematically rigorous derivation of all the basic [discrete] Kalman filter recursions from first principles”. … every major result is proved in detail … There are two appendices … recommend this book to all concerned with the Kalman filter …”
— A. F. Gualtierotti, in Mathematical Reviews, 2007
The Prediction Problem
What Lies Ahead
The Fundamental Covariance Structure
Some Tools of the Trade
State and Innovation Covariances
Recursions for L and L−1
Recursions for L
Recursions for L−1
Computing the Innovations
State and Signal Prediction
Fixed Interval Smoothing
Diffuseness and Least-Squares Estimation
A General State-Space Model
Estimation of β
Appendix A: The Cholesky Decomposition
Appendix B: Notation Guide