國立高雄大學圖資館 |

Applications of linear and nonlinear modelsfixed effects, random effects, and total least squares /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Applications of linear and nonlinear modelsby Erik W. Grafarend, Silvelyn Zwanzig, Joseph L. Awange.
Reminder of title:	fixed effects, random effects, and total least squares /
Author:	Grafarend, Erik W.
other author:	Zwanzig, Silvelyn.
Published:	Cham :Springer International Publishing :2022.
Description:	xxv, 1113 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	GeophysicsMathematical models.
Online resource:	https://doi.org/10.1007/978-3-030-94598-5
ISBN:	9783030945985$q(electronic bk.)

Applications of linear and nonlinear modelsfixed effects, random effects, and total least squares /
Grafarend, Erik W.

Applications of linear and nonlinear modelsfixed effects, random effects, and total least squares /[electronic resource] :by Erik W. Grafarend, Silvelyn Zwanzig, Joseph L. Awange. - Second edition. - Cham :Springer International Publishing :2022. - xxv, 1113 p. :ill., digital ;24 cm. - Springer geophysics,2364-9127. - Springer geophysics..

The First Problem of Algebraic Regression -- The First problem of probabilistic regression - the bias problem -- The second problem of algebraic regression - inconsistent system of linear observational equations -- The second problem of probabilistic regression- special Gauss-Markov model without datum defect - Setup of BLUUE for the moments of first order and of BIQUUE for the central moment of second order -- The third problem of probabilistic regression - special Gauss - Markov model with datum problem -Setup of BLUMBE and BLE for the moments of first order and of BIQUUE and BIQE for the central moment of second order.

This book provides numerous examples of linear and nonlinear model applications. 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 and 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 overdetermined 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-Plucker coordinates, criterion matrices of type Taylor-Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overjet. This second edition adds three new chapters: (1) Chapter on integer least squares that covers (i) model for positioning as a mixed integer linear model which includes integer parameters. (ii) The general integer least squares problem is formulated, and the optimality of the least squares solution is shown. (iii) The relation to the closest vector problem is considered, and the notion of reduced lattice basis is introduced. (iv) The famous LLL algorithm for generating a Lovasz reduced basis is explained. (2) Bayes methods that covers (i) general principle of Bayesian modeling. Explain the notion of prior distribution and posterior distribution. Choose the pragmatic approach for exploring the advantages of iterative Bayesian calculations and hierarchical modeling. (ii) Present the Bayes methods for linear models with normal distributed errors, including noninformative priors, conjugate priors, normal gamma distributions and (iii) short outview to modern application of Bayesian modeling. Useful in case of nonlinear models or linear models with no normal distribution: Monte Carlo (MC), Markov chain Monte Carlo (MCMC), approximative Bayesian computation (ABC) methods. (3) Error-in-variables models, which cover: (i) Introduce the error-in-variables (EIV) model, discuss the difference to least squares estimators (LSE), (ii) calculate the total least squares (TLS) estimator. Summarize the properties of TLS, (iii) explain the idea of simulation extrapolation (SIMEX) estimators, (iv) introduce the symmetrized SIMEX (SYMEX) estimator and its relation to TLS, and (v) short outview to nonlinear EIV models. The chapter on algebraic solution of nonlinear system of equations has also been updated in line with the new emerging field of hybrid numeric-symbolic solutions to systems of nonlinear equations, ermined system of nonlinear equations on curved manifolds. The von Mises-Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter 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.

ISBN: 9783030945985$q(electronic bk.)

Standard No.: 10.1007/978-3-030-94598-5doiSubjects--Topical Terms:

485148
Geophysics
--Mathematical models.

LC Class. No.: QA278.2

Dewey Class. No.: 550.015118

Applications of linear and nonlinear modelsfixed effects, random effects, and total least squares /
LDR:05771nmm a2200349 a 4500 001 630699
003 DE-He213
005 20221001123608.0
006 m d
007 cr nn 008maaau
008 230329s2022 sz s 0 eng d
020 $a 9783030945985$q(electronic bk.)
020 $a 9783030945978$q(paper)
024 7 $a 10.1007/978-3-030-94598-5 $2 doi
035 $a 978-3-030-94598-5
040 $a GP $c GP
041 0 $a eng
050 4 $a QA278.2
072 7 $a RBG $2 bicssc
072 7 $a SCI032000 $2 bisacsh
072 7 $a RBG $2 thema
082 0 4 $a 550.015118 $2 23
090 $a QA278.2 $b .G736 2022
100 1 $a Grafarend, Erik W. $3 243706
245 1 0 $a Applications of linear and nonlinear models $h [electronic resource] : $b fixed effects, random effects, and total least squares / $c by Erik W. Grafarend, Silvelyn Zwanzig, Joseph L. Awange.
250 $a Second edition.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2022.
300 $a xxv, 1113 p. : $b ill., digital ; $c 24 cm.
490 1 $a Springer geophysics, $x 2364-9127
505 0 $a The First Problem of Algebraic Regression -- The First problem of probabilistic regression - the bias problem -- The second problem of algebraic regression - inconsistent system of linear observational equations -- The second problem of probabilistic regression- special Gauss-Markov model without datum defect - Setup of BLUUE for the moments of first order and of BIQUUE for the central moment of second order -- The third problem of probabilistic regression - special Gauss - Markov model with datum problem -Setup of BLUMBE and BLE for the moments of first order and of BIQUUE and BIQE for the central moment of second order.
520 $a This book provides numerous examples of linear and nonlinear model applications. 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 and 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 overdetermined 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-Plucker coordinates, criterion matrices of type Taylor-Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overjet. This second edition adds three new chapters: (1) Chapter on integer least squares that covers (i) model for positioning as a mixed integer linear model which includes integer parameters. (ii) The general integer least squares problem is formulated, and the optimality of the least squares solution is shown. (iii) The relation to the closest vector problem is considered, and the notion of reduced lattice basis is introduced. (iv) The famous LLL algorithm for generating a Lovasz reduced basis is explained. (2) Bayes methods that covers (i) general principle of Bayesian modeling. Explain the notion of prior distribution and posterior distribution. Choose the pragmatic approach for exploring the advantages of iterative Bayesian calculations and hierarchical modeling. (ii) Present the Bayes methods for linear models with normal distributed errors, including noninformative priors, conjugate priors, normal gamma distributions and (iii) short outview to modern application of Bayesian modeling. Useful in case of nonlinear models or linear models with no normal distribution: Monte Carlo (MC), Markov chain Monte Carlo (MCMC), approximative Bayesian computation (ABC) methods. (3) Error-in-variables models, which cover: (i) Introduce the error-in-variables (EIV) model, discuss the difference to least squares estimators (LSE), (ii) calculate the total least squares (TLS) estimator. Summarize the properties of TLS, (iii) explain the idea of simulation extrapolation (SIMEX) estimators, (iv) introduce the symmetrized SIMEX (SYMEX) estimator and its relation to TLS, and (v) short outview to nonlinear EIV models. The chapter on algebraic solution of nonlinear system of equations has also been updated in line with the new emerging field of hybrid numeric-symbolic solutions to systems of nonlinear equations, ermined system of nonlinear equations on curved manifolds. The von Mises-Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter 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.
650 0 $a Geophysics $x Mathematical models. $3 485148
650 0 $a Regression analysis. $3 181872
650 0 $a Linear models (Statistics) $3 181866
650 1 4 $a Geology. $3 196077
650 2 4 $a Linear Algebra. $3 821215
650 2 4 $a Statistical Theory and Methods. $3 274054
650 2 4 $a Surveying. $3 204054
700 1 $a Zwanzig, Silvelyn. $3 936031
700 1 $a Awange, Joseph L. $3 485080
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
830 0 $a Springer geophysics. $3 573083
856 4 0 $u https://doi.org/10.1007/978-3-030-94598-5
950 $a Mathematics and Statistics (SpringerNature-11649)