Overview
- Numerous geodetic examples and various test computations.
- The treatment of both linear and nonlinear geodetic problems side by side as done in the present book is rare to come by
- The polynomial methods adopting Groeber basis and resultants techniques to solve more complicated nonlinear problems.
- Includes supplementary material: sn.pub/extras
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Table of contents (15 chapters)
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Front Matter
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Back Matter
Keywords
About this book
Here we present a nearly complete treatment of the Grand Universe of linear and weakly nonlinear regression models within the first 8 chapters. Our point of view is both an algebraic view as well as a stochastic one. For example, there is an equivalent lemma between a best, linear uniformly unbiased estimation (BLUUE) in a Gauss-Markov model and a least squares solution (LESS) in a system of linear equations. While BLUUE is a stochastic regression model, LESS is an algebraic solution. In the first six chapters we concentrate on underdetermined and overdeterimined linear systems as well as systems with a datum defect. We review estimators/algebraic solutions of type MINOLESS, BLIMBE, BLUMBE, BLUUE, BIQUE, BLE, BIQUE and Total Least Squares. The highlight is the simultaneous determination of the first moment and the second central moment of a probability distribution in an inhomogeneous multilinear estimation by the so called E-D correspondence as well as its Bayes design. In addition, we discuss continuous networks versus discrete networks, use of Grassmann-Pluecker coordinates, criterion matrices of type Taylor-Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overdetermined system of nonlinear equations on curved manifolds. The von Mises-Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter eight is devoted to probabilistic regression, the special Gauss-Markov model with random effects leading to estimators of type BLIP and VIP including Bayesian estimation.
A great part of the work is presented in four Appendices. Appendix A is a treatment, of tensor algebra, namely linear algebra, matrix algebra and multilinear algebra. Appendix B is devoted to sampling distributions and their use in terms of confidence intervals and confidence regions. Appendix C reviews the elementary notions of statistics, namely random events and stochastic processes. Appendix D introduces the basics of Groebner basis algebra, its careful definition, the Buchberger Algorithm, especially the C. F. Gauss combinatorial algorithm.
Reviews
From the book reviews:
“It is a great book, not only because of its huge volume, but also because of the overwhelming span of topics covered that mainly consider statistical modeling problems from a mathematical point of view. … The book can be especially useful for researchers, scientists, and engineers who apply various kinds of regression modeling to solve theoretical and practical problems.” (Stan Lipovetsky, Technometrics, Vol. 55 (2), May, 2013)
Authors and Affiliations
Bibliographic Information
Book Title: Applications of Linear and Nonlinear Models
Book Subtitle: Fixed Effects, Random Effects, and Total Least Squares
Authors: Erik Grafarend, Joseph Awange
DOI: https://doi.org/10.1007/978-3-642-22241-2
Publisher: Springer Berlin, Heidelberg
eBook Packages: Earth and Environmental Science, Earth and Environmental Science (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2012
Softcover ISBN: 978-3-662-50813-8Published: 23 August 2016
eBook ISBN: 978-3-642-22241-2Published: 15 August 2012
Edition Number: 1
Number of Pages: XXI, 1016
Number of Illustrations: 103 b/w illustrations, 8 illustrations in colour
Topics: Geophysics/Geodesy, Linear and Multilinear Algebras, Matrix Theory, Statistical Theory and Methods