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

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