Linear and Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models

Grafarend E. W.3110162164, 9783110162165, 9783110199734

This monograph offers a thorough treatment of methods for solving over- and underdetermined systems of equations. The considered problems can be non-linear or linear, and deterministic models as well as statistical effects are discussed. Considered methods include, e.g., minimum norm and least squares solution methods with respect to weighted norms. In-addition, minimum bias and minimum variance methods as well as the Tikhonov-Phillips regularization are considered. In an extensive appendix, all necessary prerequisites like matrix algebra, matrix analysis and Lagrange multipliers are presented. An extended list of references is also provided.

---

Table of contents :
1 The first problem of algebraic regression – consistent system of linear observational equations – underdetermined system of linear equations: [omitted]……Page 22
1-1 Introduction……Page 24
1-2 The minimum norm solution: “MINOS”……Page 38
1-3 Case study: Orthogonal functions, Fourier series versus Fourier-Legendre series, circular harmonic versus spherical harmonic regression……Page 61
1-4 Special nonlinear models……Page 89
1-5 Notes……Page 103
2 The first problem of probabilistic regression – special Gauss-Markov model with datum defect – Setup of the linear uniformly minimum bias estimator of type LUMBE for fixed effects…….Page 106
2-1 Setup of the linear uniformly minimum bias estimator of type LUMBE……Page 107
2-2 The Equivalence Theorem of G[sub(x)] -MINOS and S -LUMBE……Page 111
2-3 Examples……Page 112
3 The second problem of algebraic regression – inconsistent system of linear observational equations – overdetermined system of linear equations: [omitted]……Page 116
3-1 Introduction……Page 118
3-2 The least squares solution: “LESS”……Page 132
3-3 Case study: Partial redundancies, latent conditions, high leverage points versus break points, direct and inverse Grassman coordinates, Plücker coordinates……Page 164
3-4 Special linear and nonlinear models: A family of means for direct observations……Page 205
3-5 A historical note on C.F. Gauss, A.M. Legendre and the inventions of Least Squares and its generalization……Page 206
4 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……Page 208
4-1 Introduction……Page 211
4-2 Setup of the best linear uniformly unbiased estimator of type BLUUE for the moments of first order……Page 229
4-3 Setup of the best invariant quadratic uniformly unbiased estimator of type BIQUUE for the central moments of second order……Page 238
5 The third problem of algebraic regression – inconsistent system of linear observational equations with datum defect: overdetermined- undertermined system of linear equations: [omitted]……Page 264
5-1 Introduction……Page 266
5-2 MINOLESS and related solutions like weighted minimum norm-weighted least squares solutions……Page 284
5-3 The hybrid approximation solution: α-HAPS and Tykhonov-Phillips regularization……Page 303
6 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……Page 306
6-1 Setup of the best linear minimum bias estimator of type BLUMBE……Page 308
6-2 Setup of the best linear estimators of type hom BLE, hom S-BLE and hom α-BLE for fixed effects……Page 333
7 A spherical problem of algebraic representation – inconsistent system of directional observational equations – overdetermined system of nonlinear equations on curved manifolds……Page 348
7-1 Introduction……Page 349
7-2 Minimal geodesic distance: MINGEODISC……Page 352
7-3 Special models: from the circular normal distribution to the oblique normal distribution……Page 356
7-4 Case study……Page 362
8 The fourth problem of probabilistic regression – special Gauss-Markov model with random effects – Setup of BLIP and VIP for the central moments of first order……Page 368
8-1 The random effect model……Page 369
8-2 Examples……Page 383
9 The fifth problem of algebraic regression – the system of conditional equations: homogeneous and inhomogeneous equations – {By = Bi versus -c+By = Bi}……Page 394
9-1 G[sub(y)]-LESS of a system of a inconsistent homogeneous conditional equations……Page 395
9-2 Solving a system of inconsistent inhomogeneous conditional equations……Page 397
9-3 Examples……Page 398
10 The fifth problem of probabilistic regression – general Gauss-Markov model with mixed effects- Setup of BLUUE for the moments of first order (Kolmogorov-Wiener prediction)……Page 400
10-1 Inhomogeneous general linear Gauss-Markov model (fixed effectes and random effects)……Page 401
10-2 Explicit representations of errors in the general Gauss-Markov model with mixed effects……Page 406
10-3 An example for collocation……Page 407
10-4 Comments……Page 418
11 The sixth problem of probabilistic regression – the random effect model – “errors-in-variables”……Page 422
11-1 Solving the nonlinear system of the model “errors-in-variables”……Page 425
11-2 Example: The straight line fit……Page 427
11-3 References……Page 431
12 The sixth problem of generalized algebraic regression – the system of conditional equations with unknowns – (Gauss-Helmert model)……Page 432
12-1 Solving the system of homogeneous condition equations with unknowns……Page 435
12-2 Examples for the generalized algebraic regression problem: homogeneous conditional equations with unknowns……Page 442
12-3 Solving the system of inhomogeneous condition equations with unknowns……Page 445
12-4 Conditional equations with unknowns: from the algebraic approach to the stochastic one……Page 450
13 The nonlinear problem of the 3d datum transformation and the Procrustes Algorithm……Page 452
13-1 The 3d datum transformation and the Procrustes Algorithm……Page 454
13-3 Case studies: The 3d datum transformation and the Procrustes Algorithm……Page 462
13-4 References……Page 465
14 The seventh problem of generalized algebraic regression revisited: The Grand Linear Model: The split level model of conditional equations with unknowns (general Gauss-Helmert model)……Page 466
14-1 Solutions of type W-LESS……Page 467
14-3 Solutions of type R, W-HAPS……Page 471
14-2 Solutions of type R, W-MINOLESS……Page 470
14-4 Review of the various models: the sixth problem……Page 474
15-1 The multivariate Gauss-Markov model – a special problem of probabilistic regression……Page 476
15-2 n-way classification models……Page 481
15-3 Dynamical Systems……Page 497
Appendix A: Matrix Algebra……Page 506
Appendix B: Matrix Analysis……Page 543
Appendix C: Lagrange Multipliers……Page 554
Apendix D: Sampling distributions and their use: confidence intervals and confidence regions……Page 564
Appendix E: Statistical Notions……Page 665
Appendix F: Bibliographic Indexes……Page 676