Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model

132 Pages·2012·0.88 MB·English

Topics in Total Least-Squares Adjustment within the
Errors-In-Variables Model: Singular Cofactor Matrices and
Prior Information
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy in the Graduate School of The Ohio State
University
By
Kyle Snow, B.S., M.S.
Graduate Program in Geodetic Science and Surveying
School of Earth Sciences
The Ohio State University
2012
Dissertation Committee:
Burkhard Schaffrin, Advisor
Michael G. Bevis
Michael Durand
(cid:13)c Copyright by
Kyle Snow
2012
This document was typeset by the author with LATEX2ε.
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
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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-
inghowtoupdatethemodelproperlyduringiterationinordertoavoidcertainpitfalls
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.
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This dissertation is dedicated to my wife of 28 years, Karla Jean (Huwe) Snow.
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Acknowledgments
I could not have written this dissertation without the encouragement and support
of a number of individuals, for whom I am very grateful to have as friends, family
members,colleagues,andmentors. Thoughmanyfulfilledthisrole,hereIwillmention
a few that are particularly appreciated.
Amongst my friends, I am very thankful for a group of people I have met with for
several years at the home of Al and Robin Schmidt, where we have shared our lives
and experiences with one another in common fellowship as followers of Christ. Jim
and Sue Vagnier and Jon and Jenifer Mullineaux have also been wonderful friends of
my family for several years and were an encouragement to me during my extended
period of PhD studies.
It would take pages for me to adequately express my thankfulness for the love and
support of my wife Karla and daughters Kyla and Kate Snow. Karla has a long list
of great qualities, of which unselfishness stands out and is demonstrated in countless
ways, not the least of which has been her unwavering support of me through my
many years of academic study. She has been a constant friend and companion to me,
always encouraging, always optimistic. I am grateful for our 28 years of marriage and
look forward to spending many more years together.
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During my PhD candidacy period, both of my daughters made me very proud (as
they have many times before) by graduating with honors from The Ohio State Uni-
versity, both in foreign languages. They are both a great source of joy and inspiration
to my life.
In terms of mentors, I am very privileged to have worked with Dr. Clyde Goad
for several years at Topcon Positioning Systems before his retirement. Not only did
I benefit from Dr. Goad’s immense knowledge of GPS theory but also from his many
clever ideas for turning theory into working algorithms. I appreciate Dr. Goad’s
interest in my PhD work and am thankful for the support and encouragement that
he has provided to me. I am also grateful to Dr. John Bossler for his interest in my
work and for his practical advice and encouragement over the past few years.
IwanttoacknowledgeProfessorFrankNeitzel, fromtheBerlinUniversityofTech-
nology, for providing the data for the 2-D transformation problem solved in Chapter
6 of this dissertation and for checking my solution to that problem. I also thank
Dr. Michael Bevis and Dr. Michael Durand for their willingness to serve on my dis-
sertation committee. I appreciate their review of my dissertation and their critique
of my work during my oral examination. In this regard, I also thank Dr. Normand
St-Pierre, professor of animal sciences, for serving as the Graduate Faculty Represen-
tative during my examination.
Finally, I am indebted to my PhD advisor Professor Burkhard Schaffrin. I could
not have reached this point without his advice and guidance.
Ithasbeensaidthateveryonegetsaneducationinlife,whetherformalorinformal;
what matters is the quality of that education. Of course, no student will receive a
good education without making a significant effort, but to have good teachers is
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critical, too. Those who have had the opportunity to learn from and be mentored by
one who is a master in his/her field will certainly consider it a tremendous privilege.
I am honored to be so privileged.
Professor Schaffrin is undoubtedly a master in the field of mathematical and sta-
tistical methods in geodesy. Learning from him has been a great opportunity for me.
The experience was always challenging, never easy, and frequently rewarding — I
gained deeper insights into problems each time I studied one of Professor Schaffrin’s
publications or engaged in direct discussions with him. I am thankful for the many
Friday-afternoon meetings in his office, from which I usually left with some new in-
sight or some problem or material to mull over on my own so that my thinking and
understanding would be stretched. Though my formal studies with Professor Schaf-
frin have now drawn to a close, I recognize that there is still more that I can learn
from him, and I appreciate that he has given me a good foundation to work from.
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Vita
August 4, 1962 .............................Born–Springfield, Missouri
1981-1985 .................................. United States Air Force
1987 ........................................A.A., A.S., Antelope Valley College,
Lancaster, California
1990 ........................................Obtained land surveying license, State
of California
1997 ........................................B.S. Surveying Engineering, Minor in
Mathematics, California State Univer-
sity, Fresno
2002 ........................................M.S. Geodetic Science, The Ohio State
University
2002-present ................................Geodesy Team Leader,
Topcon Positioning Systems, Inc.
Publications
Research Publications
B.Schaffrin, K.Snow, andF.Neitzel. OntheErrors-In-VariablesModelwithsingular
covariancematrices. 21stIntl.WorkshoponMatricesandStatistics,Be¸dlewo,Poland,
Jul. 2012.
K. Snow and B. Schaffrin. Weighted Total Least-Squares Collocation with Geodetic
Applications. SIAM Conf. in Applied Linear Algebra, Valencia, Spain, Jun. 2012.
B. Schaffrin and K. Snow. Total Least-Squares regularization of Tykhonov type
and an ancient racetrack in Corinth. Linear Algebra and its Applications, 432(8):
2061–2076, Apr. 2010.
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B. Schaffrin and K. Snow. On a quasi-optimal regularization parameter for TLS
estimation within an EIV-Model. 57th Session of the Intl. Statist. Inst., Durban,
South Africa, Aug. 2009.
K. Snow and B. Schaffrin. GPS-Network Analysis with BLIMPBE: An Alternative to
Least-SquaresAdjustmentforBetterBiasControl. Journal of Surveying Engineering,
133(3):114–122, Aug. 2007.
K. B. Snow and B. Schaffrin. Three-dimensional outlier detection for GPS networks
and their densification via the BLIMPBE approach. GPS Solutions, 7(2):130–139,
Aug. 2003.
K. B. Snow. Applications of Parameter Estimation and Hypothesis Testing to GPS
Network Adjustments. Technical Report 465, Dept. of Civil and Environmental
Engineering and Geodetic Science, The Ohio State University, Columbus, Ohio, 2002.
Instructional Publications
K. Snow, editor, Class notes for Adjustment Computations, Geodetic Science 650
and 651, Columbus, Ohio, 2010.
K. Snow, editor, Class notes for Advanced Adjustment Computations, Geodetic
Science 762, Columbus, Ohio, 2010.
Fields of Study
Major Field: Geodetic Science
Studies in Mathematical and Statistical Methods in Geodetic Science: Burkhard
Schaffrin
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