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.
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ISBN-13
978-1-4612-4528-5 (9781461245285)
DOI
10.1007/978-1-4612-4528-5
Schweitzer Klassifikation
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.
