Applications of Linear and Nonlinear Models

Fixed Effects, Random Effects, and Total Least Squares

Authors: Grafarend, Erik, Awange, Joseph

  • 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.
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eBook $229.00
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  • ISBN 978-3-642-22241-2
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  • Immediate eBook download after purchase
Hardcover $299.00
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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)


Table of contents (15 chapters)

  • The First Problem of Algebraic Regression

    Grafarend, Erik W. (et al.)

    Pages 1-80

  • The First Problem of Probabilistic Regression: The Bias Problem

    Grafarend, Erik W. (et al.)

    Pages 81-88

  • The Second Problem of Algebraic Regression

    Grafarend, Erik W. (et al.)

    Pages 89-182

  • The Second Problem of Probabilistic Regression

    Grafarend, Erik W. (et al.)

    Pages 183-261

  • The Third Problem of Algebraic Regression

    Grafarend, Erik W. (et al.)

    Pages 263-304

Buy this book

eBook $229.00
price for USA (gross)
  • ISBN 978-3-642-22241-2
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $299.00
price for USA
  • ISBN 978-3-642-22240-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Applications of Linear and Nonlinear Models
Book Subtitle
Fixed Effects, Random Effects, and Total Least Squares
Authors
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-22241-2
DOI
10.1007/978-3-642-22241-2
Hardcover ISBN
978-3-642-22240-5
Edition Number
1
Number of Pages
XXI, 1016
Number of Illustrations and Tables
103 b/w illustrations, 8 illustrations in colour
Topics