Factorization Methods for Discrete Sequential Estimation

by Bierman, Gerald J.

• ISBN13: 9780486449814
• ISBN10: 0486449815
• Format: Paperback
• Copyright: 2006-05-26
• Publisher: Dover Publications

Summary

This estimation reference text thoroughly describes matrix factorization methods successfully employed by numerical analysts, familiarizing readers with the techniques that lead to efficient, economical, reliable, and flexible estimation algorithms. Topics include Householder orthogonal transformations; sequential square root data processing; mapping effects and process noise; biases and correlated process noise; covariance and SRIF error analysis of effects; and square root information smoothing. Geared toward advanced undergraduates and graduate students, this pragmatically oriented and detailed presentation is also a useful reference, featuring numerous helpful appendixes throughout the text. 1977 ed.

Table of Contents

Preface	ix
Acknowledgments	xi
List of Symbols	xiii
I Introduction
I.1 Introduction	1	(1)
I.2 Prerequisites	2	(1)
I.3 Scope and Objectives	3	(2)
I.4 Historical Perspectives	5	(4)
I.5 Chapter Synopses	9	(2)
References	11	(2)
II Review of Least Squares Data Processing and the Kalman Filter Algorithm
II.1 Introduction	13	(1)
II.2 Linear Least Squares	14	(1)
II.3 Statistical Interpretation of the Least Squares Solution	15	(1)
II.4 Inclusion of a Priori Statistics	16	(1)
II.5 Recursions for the Least Squares Information Processor	17	(2)
II.6 Kalman Filter Data Processing	19	(3)
II.7 Potter's Mechanization of the Kalman Algorithm	22	(2)
II.8 Computational Considerations Associated with Covariance Data Processing	24	(1)
Appendix II.A Proof that an Overdetermingd System with Full Rank Has a Nonsingular Normal Matrix	25	(1)
Appendix II.B A Matrix Inversion Lemma	26	(1)
Appendix II.C Data Processing Using the Information Matrix	27	(1)
Appendix II.D Data Processing Using the Kalman Algorithm	28	(2)
Appendix II.E Data Processing Using the Potter Algorithm	30	(1)
References	31	(2)
III Positive Definite Matrices, the Cholesky Decomposition, and Some Applications
III.1 Positive Definite Matrices	33	(1)
III.2 Properties of PD Matrices	34	(3)
III.3 Matrix Square Roots and the Cholesky Decomposition Algorithm	37	(7)
III.4 Rank One Modification of the Cholesky Factorization	44	(3)
III.5 Whitening Observation Errors	47	(2)
III.6 Observation Errors That Are Pairwise Correlated	49	(2)
III.7 Construction of Random Samples Having a Given Covariance	51	(1)
Appendix III.A Upper Triangular Matrix Factorization Algorithm	51	(3)
Appendix III.B FORTRAN Mechanization of the Lower Triangular Cholesky Factorization	54	(1)
Appendix III.C FORTRAN Mechanization of the UDUT Update	55	(1)
References	55	(2)
IV Householder Orthogonal Transformations
IV.1 Review of Orthogonal Transformations	57	(1)
IV.2 Application of Orthogonal Matrices to the Least Squares Problem	58	(1)
IV.3 The Householder Transformation	59	(4)
Appendix IV. A Annihilating the First Column of a Matrix Using the Householder Transformation	63	(1)
Appendix IV.B Solution of the Triangular System Rx = y and In-version of a Triangular Matrix	64	(2)
References	66	(2)
V Sequential Square Root Data Processing
V.1 Introduction	68	(1)
V.2 The SRIF Data Processing Algorithm	69	(7)
V.3 Data Processing Using the U—D Covariance Factorization	76	(6)
V.4 Sequential Data Processing Algorithm Computation Counts and Comparisons	82	(8)
V.5 Filter Algorithm Numerical Deterioration; Some Examples	90	(10)
Appendix V.A U—D and Upper Triangular P½ FORTRAN Mechanizations	100	(3)
Appendix V.B Arithmetic Operation Counts for Various Data Processing Algorithms	103	(9)
References	112	(1)
VI Inclusion of Mapping Effects and Process Noise
VI.1 Introduction	113	(2)
VI.2 Mapping and the Inclusion of Process Noise into the SRIF	115	(7)
VI.3 Mapping and the Inclusion of Process Noise into the Kalman Filter	122	(2)
VI.4 Mapping and the Inclusion of Process Noise into the U-D Covariance Filter	124	(5)
VI.5 Duality Relationships between Information and Covariance Algorithms	129	(2)
Appendix VI.A FORTRAN Mechanization of the MWG-S Algorithm for Time Updating of the U-D Factors	131	(2)
References	133	(2)
VII Treatment of Biases and Correlated Process Noise
VII.1 Introduction	135	(1)
VII.2 SRIF Data Processing for a System Composed Partly of Biases and Partly of Correlated Process Noise	136	(5)
VII.3 SRIF Mapping Mechanization Including the Effects of Correlated Process Noise	141	(4)
VII.4 U-D Mapping Mechanization to Account for Bias Parameters and Correlated Process Noise	145	(4)
VII.5 Computational Considerations	149	(1)
Appendix VII.A Exponentially Correlated Process Noise	150	(2)
Appendix VII.B A FORTRAN Mechanization of the Epoch State SRIF	152	(9)
References	161	(1)
VIII Covariance Analysis of Effects Due to Mismodeled Variables and incorrect Filter a Priori Statistics
VIII.1 Introduction	162	(3)
VIII.2 Consider Analyses for the Batch (Constant Parameter) Filter	165	(6)
VIII.3 Considering the Effects of Unestimated Parameters in the Kalman Filter	171	(7)
VIII.4 State Variable Augmentation to Evaluate the Effects of Mismodeled Colored Noise	178	(4)
References	182	(1)
IX SRIF Error Analysis of Effects Due to Mismodeled Variables and Incorrect Filter a Priori Statistics
IX.1 Introduction	183	(2)
IX.2 Evaluating the Effects of Incorrect Filter a Priori Statistics	185	(6)
IX.3 Considering the Effects of Unmodeled Biases	191	(11)
Appendix IX.A Processing of the Augmented Information Array Columns	202	(2)
Appendix IX.B The Augmented Array SRIF Error Analysis Algorithm	204	(6)
References	210	(1)
X Square Root Information Smoothing
X.1 Introduction	211	(3)
X.2 Square Root and Non-Square Root Smoother Recursions	214	(3)
X.3 The Treatment of Biases in Square Root Smoothing	217	(14)
Appendix X.A Kalman Filter Related Smoothers	220	(5)
Appendix X.B Smoother Recursions Corresponding to the Epoch State SRIF	225	(6)
References	231	(2)
Bibliography	233	(4)
Index	237