In 1960, R. E. Kalman published his celebrated paper on recursive min­ imum variance estimation in dynamical systems [14]. This paper, which introduced an algorithm that has since been known as the discrete Kalman filter, produced a virtual revolution in the field of systems engineering. Today, Kalman filters are used in such diverse areas as navigation, guid­ ance, oil drilling, water and air quality, and geodetic surveys. In addition, Kalman's work led to a multitude of books and papers on minimum vari­ ance estimation in dynamical systems, including one by Kalman and Bucy on continuous time systems [15]. Most of this work was done outside of the mathematics and statistics communities and, in the spirit of true academic parochialism, was, with a few notable exceptions, ignored by them. This text is my effort toward closing that chasm. For mathematics students, the Kalman filtering theorem is a beautiful illustration of functional analysis in action; Hilbert spaces being used to solve an extremely important problem in applied mathematics. For statistics students, the Kalman filter is a vivid example of Bayesian statistics in action. The present text grew out of a series of graduate courses given by me in the past decade. Most of these courses were given at the University of Mas­ sachusetts at Amherst.

Inhalt

1 Basic Probability.- 1.1. Definitions.- 1.2. Probability Distributions and Densities.- 1.3. Expected Value, Covariance.- 1.4. Independence.- 1.5. The Radon-Nikodym Theorem.- 1.6. Continuously Distributed Random Vectors.- 1.7. The Matrix Inversion Lemma.- 1.8. The Multivariate Normal Distribution.- 1.9. Conditional Expectation.- 1.10. Exercises.- 2 Minimum Variance Estimation-How the Theory Fits.- 2.1. Theory Versus Practice-Some General Observations.- 2.2. The Genesis of Minimum Variance Estimation.- 2.3. The Minimum Variance Estimation Problem.- 2.4. Calculating the Minimum Variance Estimator.- 2.5. Exercises.- 3 The Maximum Entropy Principle.- 3.1. Introduction.- 3.2. The Notion of Entropy.- 3.3. The Maximum Entropy Principle.- 3.4. The Prior Covariance Problem.- 3.5. Minimum Variance Estimation with Prior Covariance.- 3.6. Some Criticisms and Conclusions.- 3.7. Exercises.- 4 Adjoints, Projections, Pseudoinverses.- 4.1. Adjoints.- 4.2. Projections.- 4.3. Pseudoinverses.- 4.4. Calculating the Pseudoinverse in Finite Dimensions.- 4.5. The Grammian.- 4.6. Exercises.- 5 Linear Minimum Variance Estimation.- 5.1. Reformulation.- 5.2. Linear Minimum Variance Estimation.- 5.3. Unbiased Estimators, Affine Estimators.- 5.4. Exercises.- 6 Recursive Linear Estimation (Bayesian Estimation).- 6.1. Introduction.- 6.2. The Recursive Linear Estimator.- 6.3. Exercises.- 7 The Discrete Kalman Filter.- 7.1. Discrete Linear Dynamical Systems.- 7.2. The Kalman Filter.- 7.3. Initialization, Fisher Estimation.- 7.4. Fisher Estimation with Singular Measurement Noise.- 7.5. Exercises.- 8 The Linear Quadratic Tracking Problem.- 8.1. Control of Deterministic Systems.- 8.2. Stochastic Control with Perfect Observations.- 8.3. Stochastic Control with Imperfect Measurement.- 8.4. Exercises.- 9 Fixed Interval Smoothing.- 9.1. Introduction.- 9.2. The Rauch, Tung, Streibel Smoother.- 9.3. The Two-Filter Form of the Smoother.- 9.4. Exercises.- Appendix A Construction Measures.- Appendix B Two Examples from Measure Theory.- Appendix C Measurable Functions.- Appendix D Integration.- Appendix E Introduction to Hilbert Space.- Appendix F The Uniform Boundedness Principle and Invertibility of Operators.