Bayesian Spectrum Analysis and Parameter Estimation (Lecture Notes in Statistics (48))

This work is essentially an extensive revision of my Ph.D. dissertation, [1J. It 1S primarily a research document on the application of probability theory to the parameter estimation problem. The people who will be interested in this material are physicists, economists, and engineers who have to deal with data on a daily basis; consequently, we have included a great deal of introductory and tutorial material. Any person with the equivalent of the mathematics background required for the graduate­ level study of physics should be able to follow the material contained in this book, though not without eIfort. From the time the dissertation was written until now (approximately one year) our understanding of the parameter estimation problem has changed extensively. We have tried to incorporate what we have learned into this book. I am indebted to a number of people who have aided me in preparing this docu­ ment: Dr. C. Ray Smith, Steve Finney, Juana Sunchez, Matthew Self, and Dr. Pat Gibbons who acted as readers and editors. In addition, I must extend my deepest thanks to Dr. Joseph Ackerman for his support during the time this manuscript was being prepared. 1. Complex Lie Groups -- 1.1. Analytic Manifolds -- 1.2. Complex Lie Groups -- 1.3. Examples Of Complex Lie Groups -- 1.4. The Automorphism Group -- 1.5. Universal Complexification -- 2. Representative Functions Of Lie Groups -- 2.1. Basic Definitions -- 2.2. Algebras Of Representative Functions -- 2.3. Proper Automorphism Groups -- 2.4. Analytic Representative Functions -- 2.5. Universal Algebraic Hull -- 2.6. Relative Algebras -- 2.7. Unipotent Hull -- 3. Extensions Of Representations -- 3.1. Some Examples -- 3.2. Decomposition Theorem -- 3.3. Main Lemma -- 3.4. Extensions Of Representations -- 3.5. Unipotent Analytic Groups -- 3.6. Application To Solvable Groups -- 4. The Structure Of Complex Lie Groups -- 4.1. Abelian Complex Analytic Groups -- 4.2. Decomposition Of The Adjoint Group -- 4.3. Semisimple Complex Analytic Groups -- 4.4. Reductive Complex Analytic Groups -- 4.5. Compact Subgroups Of Reductive Groups -- 4.6. Representation Radical -- 4.7. Faithfully Representable Groups -- 4.8. Conjugacy Of Reductive Subgroups -- 4.9. Unipotent Hull Of Complex Lie Groups -- 5. Algebraic Subgroups In Lie Groups -- 5.1. Affine Algebraic Structure In Lie Groups -- 5.2. Extension Lemma -- 5.3. Affine Algebraic Structure On Reductive Groups -- 5.4. The Maximum Algebraic Subgroup -- 5.5. Further Properties Of Reductive Groups -- 6. Observability In Complex Lie Groups -- 6.1. Pro-affine Groups And Observability -- 6.2. Observability In Algebraic Groups -- 6.3. Extension Of Representative Functions -- 6.4. Structure Of Observable Subgroups -- A. Elementary Theory Of Lie Algebras -- B. Pro-affine Algebraic Groups. Dong Hoon Lee. Includes Bibliographical References (p. 211-213) And Index. This Book Is Primarily A Research Document On The Application Of Probability Theory To The Parameter Estimation Problem. The People Who Will Be Interested In This Material Are Physicists, Chemists, Economists, And Engineers Who Have To Deal With Data On A Daily Basis; Consequently, We Have Included A Great Deal Of Introductory And Tutorial Material. Any Person With The Equivalent Of The Mathematics Background Required For The Graduate-level Study Of Physics Should Be Able To Follow The Material Contained In This Book, Though Not Without Effort. In This Work We Apply Probability Theory To The Problem Of Estimating Parameters In Rather General Models. In Particular When The Model Consists Of A Single Stationary Sinusoid We Show That The Direct Application Of Probability Theory Will Yield Frequency Estimates An Order Of Magnitude Better Than A Discrete Fourier Transform In Signal-to-noise Of One. Latter, We Generalize The Problem And Show That Probability Theory Can Separate Two Close Frequencies Long After The Peaks In A Discrete Fourier Transform Have Merged. Contents: Introduction -- Single Stationary Sinusoid Plus Noise -- The General Model Equation Plus Noise -- Estimating The Parameters -- Model Selection -- Spectral Estimation -- Applications -- Summary And Conclusions -- Appendices A-e. G. Larry Bretthorst. Includes Index. Bibliography: P. [203]-206. This work is essentially an extensive revision of my Ph. D. dissertation, [1J. It 1S primarily a research document on the application of probability theory to the parameter estimation problem. The people who will be interested in this material are physicists, economists, and engineers who have to deal with data on a daily basis; consequently, we have included a great deal of introductory and tutorial material. Any person with the equivalent of the mathematics background required for the graduateƯ level study of physics should be able to follow the material contained in this book, though not without eIfort. From the time the dissertation was written until now (approximately one year) our understanding of the parameter estimation problem has changed extensively. We have tried to incorporate what we have learned into this book. I am indebted to a number of people who have aided me in preparing this docuƯ ment: Dr. C. Ray Smith, Steve Finney, Juana Sunchez, Matthew Self, and Dr. Pat Gibbons who acted as readers and editors. In addition, I must extend my deepest thanks to Dr. Joseph Ackerman for his support during the time this manuscript was being prepared Front Matter....Pages I-XII Introduction....Pages 1-11 Single Stationary Sinusoid Plus Noise....Pages 13-30 The General Model Equation Plus Noise....Pages 31-41 Estimating the Parameters....Pages 43-53 Model Selection....Pages 55-67 Spectral Estimation....Pages 69-115 Applications....Pages 117-177 Summary and Conclusions....Pages 179-181 Back Matter....Pages 183-209 Complex Lie groups have often been used as auxiliaries in the study of real Lie groups in areas such as differential geometry and representation theory. To date, however, no book has fully explored and developed their structural aspects.The Structure of Complex Lie Groups addresses this need. Self-contained, it begins with general concepts In this chapter the basic notion of complex Lie groups is introduced, and some of the essential tools that will be used in the remaining chapters are developed ([2], [8], [22]).