Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Prior Information
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Date
2012-12
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
This dissertation is about total least-squares (TLS) adjustments within the errorsin-
variables (EIV) model. In particular, it deals with symmetric positive-(semi)definite
cofactor matrices that are otherwise quite arbitrary, including the case of crosscorrelation
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 contribution
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 independent
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 derivations
are given for various TLS estimators and predictors within these models. Algorithms
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 presented
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 showing
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.
Description
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 acknowledgment
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.