LEADER 07068nam 22007695 450 001 9910616381603321 005 20251118150156.0 010 $a9783030945985 010 $a3030945987 024 7 $a10.1007/978-3-030-94598-5 035 $a(MiAaPQ)EBC7105501 035 $a(Au-PeEL)EBL7105501 035 $a(CKB)24978817000041 035 $a(PPN)26585783X 035 $a(DE-He213)978-3-030-94598-5 035 $a(EXLCZ)9924978817000041 100 $a20221001d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aApplications of Linear and Nonlinear Models $eFixed Effects, Random Effects, and Total Least Squares /$fby Erik W. Grafarend, Silvelyn Zwanzig, Joseph L. Awange 205 $a2nd ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (1127 pages) 225 1 $aSpringer Geophysics,$x2364-9127 311 08$aPrint version: Grafarend, Erik W. Applications of Linear and Nonlinear Models Cham : Springer International Publishing AG,c2022 9783030945978 320 $aIncludes bibliographical references and index. 327 $aThe 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. 330 $aThis 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 estimationby 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. 410 0$aSpringer Geophysics,$x2364-9127 606 $aGeology 606 $aAlgebras, Linear 606 $aStatistics 606 $aSurveying 606 $aGeology 606 $aLinear Algebra 606 $aStatistical Theory and Methods 606 $aSurveying 606 $aGeofísica$2thub 606 $aModels lineals (Estadística)$2thub 606 $aModels matemàtics$2thub 608 $aLlibres electrònics$2thub 615 0$aGeology. 615 0$aAlgebras, Linear. 615 0$aStatistics. 615 0$aSurveying. 615 14$aGeology. 615 24$aLinear Algebra. 615 24$aStatistical Theory and Methods. 615 24$aSurveying. 615 7$aGeofísica 615 7$aModels lineals (Estadística) 615 7$aModels matemàtics 676 $a550 676 $a550.015118 700 $aAwange$b Joseph L.$f1969-$01229364 702 $aGrafarend$b Erik W. 702 $aZwanzig$b Silvelyn 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910616381603321 996 $aApplications of linear and nonlinear models$93041552 997 $aUNINA