Applications of linear and nonlinear models : fixed effects, random effects, and total least squares

Cover......Page 1
Applications of Linear and Nonlinear Models
......Page 4
References......Page 6
Index......Page 8
Contents......Page 14
1 The First Problem of Algebraic Regression......Page 23
1-1 Introduction......Page 26
1-12 The Front Page Example: Matrix Algebra......Page 27
1-13 The Front Page Example: MINOS, Horizontal Rank Partitioning......Page 30
1-14 The Range R(f) and the Kernel N(f)......Page 32
1-15 The Interpretation of MINOS......Page 34
1-2 Minimum Norm Solution (MINOS)......Page 38
1-21 A Discussion of the Metric of the Parameter Space X......Page 43
1-23 Gx-MINOS and Its Generalized Inverse......Page 44
1-24 Eigenvalue Decomposition of Gx-MINOS: Canonical MINOS......Page 46
1-3 Case Study......Page 59
1-31 Fourier Series......Page 60
1-32 Fourier–Legendre Series......Page 71
1-33 Nyquist Frequency for Spherical Data......Page 84
1-41 Taylor Polynomials, Generalized Newton Iteration......Page 85
1-42 Linearized Models with Datum Defect......Page 91
1-5 Notes......Page 100
2 The First Problem of Probabilistic Regression:The Bias Problem......Page 103
2-1 Linear Uniformly Minimum Bias Estimator (LUMBE)......Page 106
2-2 The Equivalence Theorem of Gx-MINOS and S-LUMBE......Page 109
2-3 Example......Page 110
3 The Second Problem of Algebraic Regression......Page 111
3-11 The Front Page Example......Page 114
3-12 The Front Page Example in Matrix Algebra......Page 115
3-13 Least Squares Solution of the Front Page Example by Means of Vertical Rank Partitioning......Page 117
3-14 The Range R(f) and the Kernel N(f), Interpretation of ``LESS'' by Three Partitionings......Page 120
3-2 The Least Squares Solution: ``LESS''......Page 127
3-21 A Discussion of the Metric of the Parameter Space X......Page 134
3-22 Alternative Choices of the Metric of the Observation Y......Page 135
3-221 Optimal Choice of Weight Matrix: SOD......Page 136
3-222 The Taylor Karman Criterion Matrix......Page 140
3-223 Optimal Choice of the Weight Matrix: The Space R( A ) and R(A)......Page 141
3-224 Fuzzy Sets......Page 144
3-23 Gx -LESS and Its Generalized Inverse......Page 151
3-24 Eigenvalue Decomposition of Gy -LESS: Canonical LESS......Page 152
3-3 Case Study......Page 163
3-31 Canonical Analysis of the Hat Matrix, Partial Redundancies, High Leverage Points......Page 164
3-32 Multilinear Algebra, ``Join'' and ``Meet'', the Hodge Star Operator......Page 172
3-33 From A to B: Latent Restrictions, Grassmann Coordinates, Plücker Coordinates......Page 178
3-34 From B to A: Latent Parametric Equations, Dual Grassmann Coordinates, Dual Plücker Coordinates......Page 190
3-35 Break Points......Page 194
3-5 A Historical Note on C.F. Gauss and A.M. Legendre......Page 202
4 The Second Problem of Probabilistic Regression......Page 205
4-1 Introduction......Page 209
4-11 The Front Page Example......Page 210
4-12 Estimators of Type BLUUE and BIQUUE of the Front Page Example......Page 211
4-13 BLUUE and BIQUUE of the Front Page Example, Sample Median, MedianAbsolute Deviation......Page 220
4-14 Alternative Estimation MaximumLikelihood (MALE)......Page 224
4-2 Setup of the Best Linear Uniformly Unbiased Estimator......Page 227
4-21 The Best Linear Uniformly Unbiased Estimation  of ξ: y-BLUUE......Page 228
4-22 The Equivalence Theorem of Gy -LESS and y -BLUUE......Page 235
4-3 Setup of the Best Invariant Quadratic Uniformly Unbiased Estimator......Page 236
4-31 Block Partitioning of the Dispersion Matrix and Linear Space Generated by Variance-Covariance Components......Page 237
4-32 Invariant Quadratic Estimation of Variance-Covariance Components of Type IQE......Page 242
4-33 Invariant Quadratic Uniformly Unbiased Estimations of Variance-Covariance Components of Type IQUUE......Page 246
4-34 Invariant Quadratic Uniformly Unbiased Estimations of One Variance Component (IQUUE) from y-BLUUE: HIQUUE......Page 250
4-35 Invariant Quadratic Uniformly Unbiased Estimators of Variance Covariance Components of Helmert Type: HIQUUE Versus HIQE......Page 252
4-36 Best Quadratic Uniformly Unbiased Estimations of One Variance Component: BIQUUE......Page 256
4-37 Simultaneous Determination of First Moment and the Second Central Moment, Inhomogeneous Multilinear Estimation, the E-D Correspondence, Bayes Design with Moment Estimations......Page 263
5 The Third Problem of Algebraic Regression......Page 284
5-1 Introduction......Page 286
5-12 The Front Page Example in Matrix Algebra......Page 287
5-13 Minimum Norm: Least Squares Solution of the Front Page Example by Means of Additive Rank Partitioning......Page 289
5-14 Minimum Norm: Least Squares Solution of the Front Page Example by Means of Multiplicative Rank Partitioning......Page 293
5-15 The Range R(f) and the Kernel N(f) Interpretation of ``MINOLESS''by Three Partitionings......Page 297
5-21 The Minimum Norm-Least Squares Solution: ``MINOLESS''......Page 304
5-22 (Gx ,Gy )-MINOS and Its Generalized Inverse......Page 314
5-23 Eigenvalue Decomposition of(Gx ,Gy )-MINOLESS......Page 318
5-24 Notes......Page 322
5-3 The Hybrid Approximation Solution: α-HAPS and Tykhonov–Phillips Regularization......Page 323
6 The Third Problem of Probabilistic Regression......Page 326
6-1 Setup of the Best Linear Minimum Bias Estimator of Type BLUMBE......Page 329
6-11 Definitions, Lemmas and Theorems......Page 331
6-12 The First Example: BLUMBE Versus BLE, BIQUUE Versus BIQE, TriangularLeveling Network......Page 338
6-121 The First Example: I3, I3-BLUMBE......Page 339
6-122 The First Example: V, S-BLUMBE......Page 343
6-123 The First Example: I3, I3-BLE......Page 347
6-124 The First Example: V, S-BLE......Page 349
6-2 Setup of the Best Linear Estimators of Type hom BLE, hom S-BLE and hom a-BLE for Fixed Effects......Page 353
6-3 Continuous Networks......Page 366
6-31 Continuous Networks of Second Derivatives Type......Page 367
6-32 Discrete Versus Continuous Geodetic Networks......Page 378
7 Overdetermined System of Nonlinear Equations on Curved Manifolds......Page 382
7-1 Introduction......Page 383
7-2 Minimal Geodesic Distance: MINGEODISC......Page 386
7-31 A Historical Note of the von Mises Distribution......Page 391
7-32 Oblique Map Projection......Page 393
7-33 A Note on the Angular Metric......Page 396
7-4 Case Study......Page 397
8 The Fourth Problem of Probabilistic Regression......Page 404
8-1 The Random Effect Model......Page 405
8-2 Examples......Page 420
9-1 Gy -LESS of a System of a Inconsistent Homogeneous Conditional Equations......Page 432
9-2 Solving a System of Inconsistent Inhomogeneous Conditional Equations......Page 436
9-3 Examples......Page 437
10 The Fifth Problem of Probabilistic Regression......Page 439
10-1 Inhomogeneous General Linear Gauss–Markov Model Fixed Effects and Random Effects......Page 441
10-2 Explicit Representations of Errors in the General Gauss–Markov Model with Mixed Effects......Page 446
10-3 An Example for Collocation......Page 448
10-4 Comments......Page 458
11 The Sixth Problem of Probabilistic Regression......Page 462
11-1 The Model of Error-in-Variables or Total Least Squares......Page 466
11-2 Algebraic Total Least Squares for the Nonlinear System of the Model ``Error-in-Variables''......Page 467
11-3 Example: The Straight Line Fit......Page 469
11-4 The Models SIMEX and SYMEX......Page 472
11-5 References......Page 478
12 The Nonlinear Problem of the 3d Datum Transformation and the Procrustes Algorithm......Page 479
12-1 The 3d Datum Transformation and the Procrustes Algorithm......Page 481
12-2 The Variance: Covariance Matrix of the Error Matrix E......Page 488
12-21 Case Studies: The 3d Datum Transformation and the Procrustes Algorithm......Page 489
12-3 References......Page 492
13 The Sixth Problem of Generalized Algebraic Regression......Page 494
13-1 Variance-Covariance-Component Estimation in the Linear Model Ax+=y, y R(A)......Page 496
13-2 Variance-Covariance-Component Estimation in the Linear Model B =By - c, By R(A)+c......Page 499
13-3 Variance-Covariance-Component Estimation in the Linear Model Ax++ B=By-c, By R(A)+ c......Page 502
13-4 The Block Structure of Dispersion Matrix D{y}......Page 506
14-1 The Multivariate Gauss–Markov Model: A Special Problem of Probabilistic Regression......Page 509
14-2 n-Way Classification Models......Page 514
14-21 A First Example: 1-Way Classification......Page 515
14-22 A Second Example: 2-Way Classification Without Interaction......Page 519
14-23 A Third Example: 2-Way Classificationwith Interaction......Page 525
14-24 Higher Classifications with Interaction......Page 530
14-3 Dynamical Systems......Page 533
15-1 Introductory Remarks......Page 542
15-2 Background to Algebraic Solutions......Page 543
15-31 Solution of Nonlinear Gauss–Markov Model......Page 547
15-311 Construction and Solution of the Combinatorial Subsets......Page 548
15-312 Groebner Basis Method......Page 549
15-313 Multipolynomial Resultants Method......Page 561
15-32 Adjustment of the combinatorial subsets......Page 567
15-4 Examples......Page 571
15-5 Notes......Page 578
A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra......Page 585
A-1 Multilinear Functions and the Tensor Space Tpq......Page 586
A-2 Decomposition of Multilinear Functions into Symmetric Multilinear Functions AntisymmetricMulti-linear Functions and Residual Multilinear Functions TTpq= Spq Apq Rpq......Page 592
A-3 Matrix Algebra, Array Algebra, Matrix Normand Inner Product......Page 598
A-4 The Hodge Star Operator, Self Duality......Page 601
A-5 Linear Algebra......Page 606
A-51 Definition of a Linear Algebra......Page 607
A-52 The Diagrams ``Ass'', ``Uni'' and ``Comm''......Page 609
A-53 Ringed Spaces: The Subalgebra``Ring with Identity''......Page 611
A-55 Lie Algebra, Witt Algebra......Page 612
A-56 Definition of a Composition Algebra......Page 613
A-6 Matrix Algebra Revisited, Generalized Inverses......Page 616
A-61 Special Matrices: Helmert Matrix, Hankel Matrix, Vandemonte Matrix......Page 620
A-62 Scalar Measures of Matrices......Page 626
A-63 Three Basic Types of Generalized Inverses......Page 632
A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra, Hurwitz Theorem......Page 633
A-71 Complex Algebra as a Division Algebra as well as a Composition Algebra, Clifford algebra Cl(0,1)......Page 634
A-72 Quaternion Algebra as a Division Algebra as well as a Composition Algebra, Clifford algebra Cl(0,2)......Page 636
A-73 Octanian Algebra as a Non-Associative Algebra as well as a Composition Algebra, Clifford algebra with Respect to HH......Page 643
A-74 Clifford Algebra......Page 647
B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions......Page 651
B-1 A First Vehicle: Transformation of Random Variables......Page 652
B-2 A Second Vehicle: Transformation of Random Variables......Page 656
B-3 A First Confidence Interval of Gauss–Laplace Normally Distributed Observations μ, σ2 Known, the Three Sigma Rule......Page 662
B-31 The Forward Computation of a First Confidence Interval of Gauss–Laplace Normally Distributed Observations: μ,σ2  Known......Page 667
B-32 The Backward Computation of a First Confidence Interval of Gauss–Laplace Normally Distributed Observations: μ,σ2  Known......Page 673
B-4 Sampling from the Gauss–Laplace Normal Distribution: A Second Confidence Interval for the Mean, Variance Known......Page 676
B-41 Sampling Distributions of the Sample Mean μ, σ2  Known, and of the Sample Variance σ2......Page 691
B-42 The Confidence Interval for the Sample Mean, Variance Known......Page 702
B-51 Student's Sampling Distribution of the Random Variable (μ- μ)/σ......Page 706
B-52 The Confidence Interval for the Mean, Variance Unknown......Page 715
B-53 The Uncertainty Principle......Page 721
B-6 Sampling from the Gauss–Laplace Normal Distribution: A Fourth Confidence Interval for the Variance......Page 722
B-61 The Confidence Interval for the Variance......Page 723
B-62 The Uncertainty Principle......Page 729
B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution: The Confidence Region for the Fixed Parameters in the Linear Gauss–Markov Model......Page 731
B-8 Multidimensional Variance Analysis, Sampling from the Multivariate Gauss–Laplace Normal Distribution......Page 753
B-81 Distribution of Sample Meanand Variance-Covariance......Page 754
B-82 Distribution Related to Correlation Coefficients......Page 758
C Statistical Notions, Random Events and Stochastic Processes......Page 766
C-1 Moments of a Probability Distribution, the Gauss–Laplace Normal Distribution and the Quasi-Normal Distribution......Page 767
C-2 Error Propagation......Page 770
C-3 Useful Identities......Page 773
C-4 Scalar – Valued Stochastic Processes of One Parameter......Page 775
C-5 Characteristic of One Parameter Stochastic Processes......Page 778
C-6 Simple Examples of One Parameter Stochastic Processes......Page 782
C-71 Definition of the Wiener Processes......Page 794
C-72 Special Wiener Processes: Ornstein–Uhlenbeck, Wiener Processes with Drift, Integral Wiener Processes......Page 798
C-81 Foundations: Ergodic and Stationary Processes......Page 806
C-82 Processes with Discrete Spectrum......Page 808
C-83 Processes with Continuous Spectrum......Page 811
C-84 Spectral Decomposition of the Mean and Variance-Covariance Function......Page 821
C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes of Multi-Parameter Systems......Page 824
C-91 Characteristic Functional......Page 825
C-92 The Moment Representation of Stochastic Processes for Scalar Valued and Vector Valued Quantities......Page 827
C-93 Tensor-Valued Statistical Homogeneous and Isotropic Field of Multi-Point Systems......Page 831
D-1 Definitions......Page 900
D-21 Mathematica Computation of Groebner Basis......Page 902
D-22 Maple Computation of Groebner Basis......Page 904
D.3 Gauss Combinatorial Formulation......Page 905