Linear and nonlinear models : fixed effects, random effects, and mixed models

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. 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 This monograph contains a thorough treatment of methods for solving over- and underdetermined systems of equations, e.g. the minimum norm solution method with respect to weighted norms. The considered equations can be nonlinear or linear, and deterministic models as well as probabilistic ones are considered. An extensive appendix provides all necessary prerequisites like matrix algebra, matrix analysis and Lagrange multipliers, and a long list of references is also included. Contains a treatment of methods for solving over- and under-determined systems of equations, such as the minimum norm solution method with respect to weighted norms. This work features an appendix that provides necessary prerequisites like matrix algebra, matrix analysis and Lagrange multipliers, and includes a long list of reference.