Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Prior Information by Kyle Snow Report No. 502 Geodetic Science The Ohio State University Columbus, Ohio 43210 December 2012 Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Prior Information by Kyle Snow Report No. 502 Geodetic Science Ohio State University Columbus, Ohio 43210 December 2012 c Copyright by Kyle Snow 2012 This document was typeset by the author with LATEX 2ε. Abstract This dissertation is about total least-squares (TLS) adjustments within the errors- in-variables (EIV) model. In particular, it deals with symmetric positive-(semi)def- inite cofactor matrices that are otherwise quite arbitrary, including the case of cross- correlation between cofactor matrices for the observation vector and the coefficient matrix and also the case of singular cofactor matrices. The former case has been addressed already in a recent dissertation by Fang[2011], whereas the latter case has not been treated until very recently in a presentation by Schaffrin et al.[2012b], which was developed in conjunction with this dissertation. The second primary con- tribution of this work is the introduction of prior information on the parameters to the EIV model, thereby resulting in an errors-in-variables with random effects model (EIV-REM)[Snow and Schaffrin, 2012]. The (total) least-squares predictor within this model is herein called weighted total least-squares collocation (WTLSC), which was introduced just a few years ago by Schaffrin[2009] as TLSC for the case of in- dependent and identically distributed (iid) data. Here the restriction of iid data is removed. The EIV models treated in this work are presented in detail, and thorough deriva- tions are given for various TLS estimators and predictors within these models. Algo- rithms for their use are also presented. In order to demonstrate the usefulness of the presented algorithms, basic geodetic problems in 2-D line-fitting and 2-D similarity transformations are solved numerically. The new extensions to the EIV model pre- sented here will allow the model to be used by both researchers and practitioners to solve a wider range of problems than was hitherto feasible. In addition, the Gauss-Helmert model (GHM) is reviewed, including details show- ing how to update the model properly during iteration in order to avoid certain pitfalls pointed out by Pope[1972]. After this, some connections between the GHM and the EIV model are explored. Though the dissertation is written with a certain bent towards geodetic science, it is hoped that the work will be of benefit to those researching and working in other branches of applied science as well. Likewise, an important motivation of this work is to highlight the classical EIV model, and its recent extensions, within the geodetic science community, as it seems to have received little attention in this community until a few years ago when Professor Burkhard Schaffrin began publishing papers on the topic in both geodetic and applied mathematics publications. ii Preface This report is substantially the same as a dissertation that was prepared for and submitted to the Graduate School of The Ohio State University for the PhD degree. Except for the omission of some pages from the front matter, a different acknowl- edgment page, and a change from double-space to single-space format, this report is identical to the dissertation, which contains 15 pages with Roman numerals and 116 pages with Arabic numerals. iii Acknowledgments I thank Professor Frank Neitzel, of the Berlin University of Technology, for pro- viding the data for the 2-D transformation problem solved in Chapter 6 of this report and for checking my solution to that problem. I am indebted to my adviser, Professor Burkhard Schaffrin, not only for his advice and guidance but also for his critical review of my dissertation, which resulted in many improvements to the original manuscript. Any remaining errors or deficiencies are solely the responsibility of the author. iv Contents Page Abstract....................................... ii Preface....................................... iii Acknowledgments.................................. iv 1. Introduction..................................1 1.1 Contributions of this dissertation...................1 1.2 A brief introduction to the EIV model and TLS adjustment....2 1.2.1 Some historic connections of geodesy to linear algebra, least- squares, and statistical estimation theory..........2 1.2.2 The notion of the total least-squares principle........3 1.2.3 The development of TLS adjustment within the EIV model7 1.2.4 A progression of weighting schemes within the EIV model.7 1.3 Comments on the notation....................... 10 1.4 Linear algebra references........................ 12 2. The EIV Model with Correlation...................... 13 2.1 WTLS adjustment within the EIV model with correlation..... 14 2.1.1 Alternative solution...................... 18 2.2 Sum of squared residuals (SSR) for the WTLS solution....... 20 2.3 The TLS problem within the classical EIV model.......... 20 3. The EIV Model with Singular Cofactor Matrices.............. 23 3.1 Singular cofactor matrices without cross-correlation......... 23 3.1.1 Alternative solution...................... 25 3.2 Singular cofactor matrices with cross-correlation........... 27 3.2.1 Alternative solution...................... 30 v 4. Analytical Comparisons Between the Least-Squares Solutions within the EIV and Gauss-Helmert Models....................... 33 4.1 Iterative linearization of the Gauss-Helmert model......... 34 4.2 Analytical comparisons between the least-squares solutions..... 36 5. Weighted Total Least-Squares Collocation................. 40 5.1 Brief review of the random effects model (REM) and least-squares collocation............................... 40 5.2 The EIV with random effects model (EIV-REM).......... 42 5.2.1 Total least-squares collocation (TLSC)............ 43 5.2.2 The special case of iid data within the EIV-REM...... 47 6. Numerical Applications............................ 50 6.1 2-D line-fitting............................. 50 6.1.1 Case 1 – nonsingular cofactor matrices............ 52 6.1.2 Case 2 – singular cofactor matrices.............. 55 T 6.1.3 Equivalence between Q1 and BQB ............. 56 6.2 2-D similarity transformation..................... 57 T 6.2.1 Equivalence between Q1 and BQB ............. 59 6.3 2-D similarity transformation with singular cofactor matrices.... 59 6.4 2-D line-fitting by WTLS collocation................. 63 6.4.1 Experiments with scaled heteroscedastic weights...... 64 7. Conclusions and Outlook........................... 69 Appendices 71 A. Cofactor and Correlation Matrices..................... 71 A.1 Correlation coefficients for 2-D line-fitting.............. 71 A.2 Cofactor matrices for the 2-D similarity transformation....... 74 B. Connections between the Work of Pearson and that of Schaffrin..... 76 C. The Neitzel/Schaffrin Criterion for Uniqueness............... 78 vi D. Fang’s WTLS Estimator as a Generalization of Schaffrin and Wieser’s Estimator................................... 79 E. Some Properties of the vec Operator and the Kronecker Product..... 82 References...................................... 84 Author Index.................................... 90 vii Chapter 1: Introduction 1.1 Contributions of this dissertation This dissertation is about total least-squares (TLS) adjustments within the errors- in-variables (EIV) model. In particular, it deals with symmetric positive-(semi)def- inite cofactor matrices that are otherwise quite arbitrary, including the case of cross- correlation between cofactor matrices for the observation vector and the coefficient matrix and also the case of singular cofactor matrices. The former case has been addressed already in a recent dissertation by Fang[2011], whereas the latter case has not been treated until very recently in a presentation by Schaffrin et al.[2012b], which was developed in conjunction with this dissertation. The second primary con- tribution of this work is the introduction of prior information on the parameters to the EIV model, thereby resulting in an errors-in-variables with random effects model (EIV-REM)[Snow and Schaffrin, 2012]. The (total) least-squares predictor within this model is herein called weighted total least-squares collocation (WTLSC), which was introduced just a few years ago by Schaffrin[2009] as TLSC for the case of in- dependent and identically distributed (iid) data. Here the restriction of iid data is removed. The EIV models treated in this work are presented in detail, and thorough deriva- tions are given for various TLS estimators and predictors within these models. Algo- rithms for their use are also presented. In order to demonstrate the usefulness of the presented algorithms, basic geodetic problems in 2-D line-fitting and 2-D similarity transformations are solved numerically. The new extensions to the EIV model pre- sented here will allow the model to be used by both researchers and practitioners to solve a wider range of problems than was hitherto feasible. In addition, the Gauss-Helmert model (GHM) is reviewed, including details show- ing how to update the model properly during iteration in order to avoid certain pitfalls pointed out by Pope[1972]. After this, some connections between the GHM and the EIV model are explored. Though the dissertation is written with a certain bent towards geodetic science,