Random fields for spatial data modeling

Why Spatial Data Modeling?......Page 7
 Why Random Fields?......Page 8
 Spatial Data Modeling and Statistical Physics......Page 9
 What Is a Random Field?......Page 10
 How to Read This Book......Page 11
Acknowledgments......Page 16
Contents......Page 18
Acronyms......Page 28
 1.1 Preliminary Remarks......Page 30
 1.2 Why Random Fields?......Page 32
  1.2.1 Random Fields, Trends, Fluctuations, Noise......Page 33
  1.2.3 Inductive Versus Empirical Modeling......Page 38
  1.2.4 Random Fields and Stochastic Systems......Page 40
  1.2.5 Connections with Nonlinear Systems......Page 41
  1.3.1 Notation......Page 43
  1.3.2 Spatial Random Fields......Page 46
  1.3.3 Spatial Domain......Page 49
  1.3.4 Categories of Random Fields......Page 50
  1.3.5 Random Field Model of Spatial Data......Page 53
  1.4.1 Noise......Page 55
  1.4.3 Gaussian White Noise......Page 57
  1.4.4 Wiener Process (Brownian Motion)......Page 58
   1.4.5.1 Experimental Errors......Page 60
   1.4.5.2 Numerical errors......Page 62
   1.4.5.3 Modeling Errors......Page 63
   1.4.5.4 Heavy Tails and Outliers......Page 64
  1.5.1 Sampling Set......Page 65
  1.5.2 Prediction......Page 66
 1.6 A Personal Selection of Relevant Books......Page 67
2 Trend Models and Estimation......Page 70
 2.2 Regression Analysis......Page 71
  2.2.1 Ordinary Least Squares......Page 73
  2.2.2 Weighted Least Squares......Page 74
  2.2.4 Ridge Regression......Page 75
  2.2.5 LASSO......Page 77
  2.2.6 Goodness of Fit......Page 79
  2.3.1 Linear Spatial Dependence......Page 80
  2.3.3 Periodic Spatial Dependence......Page 81
  2.3.4 Multiple Linear Regression......Page 82
  2.4.1 Moving Average (MA)......Page 86
  2.4.2 Savitzky-Golay Filters......Page 89
  2.4.3 Kernel Smoothing......Page 91
  2.4.4 Locally Weighted Regression (LWR)......Page 96
 2.5 Trend Estimation Based on Physical Information......Page 100
 2.6 Trend Model Based on the Laplace Equation......Page 102
 3.1 Introduction......Page 111
 3.2 Single-Point Description......Page 112
  3.2.1 Ensemble Moments......Page 115
   3.2.1.1 Marginal Moments of Integer Order......Page 116
   3.2.1.2 Useful Nonlinear Moments......Page 118
  3.2.2 The Moment Generating Function......Page 119
  3.2.3 The Characteristic Function......Page 120
  3.2.4 The Cumulant Generating Function......Page 122
  3.3.1 Joint Cumulative Distribution Function......Page 124
  3.3.2 Conditional Probability Function......Page 125
  3.3.4 Two-Point Correlation Functions......Page 126
 3.4 Stationarity and Statistical Homogeneity......Page 128
 3.5 Variogram Versus Covariance......Page 130
 3.6 Permissibility of Covariance Functions......Page 132
  3.6.1 Positive Definite Functions......Page 133
  3.6.2 Fourier Transforms in a Nutshell......Page 135
  3.6.3 Bochner's Theorem......Page 139
  3.6.4 Wiener-Khinchin Theorem......Page 142
  3.6.5 Covariance Function Composition......Page 144
 3.7 Permissibility of Variogram Functions......Page 149
  3.7.1 Variogram Spectral Density......Page 152
 4.1 Ergodicity......Page 154
 4.2 Statistical Isotropy......Page 157
  4.2.1 Spectral Representation......Page 158
  4.2.2 Isotropic Correlation Models......Page 160
  4.2.3 Radon Transform......Page 170
  4.3.1 Physical Anisotropy......Page 172
  4.3.2 Statistical Anisotropy......Page 173
  4.3.3 Anisotropy and Scale......Page 181
  4.3.4 Range Anisotropy Versus Elliptical Anisotropy......Page 182
  4.3.5 Zonal Anisotropy......Page 183
 4.4 Anisotropic Spectral Densities......Page 185
  4.4.1 Anisotropy in Planar Domains......Page 186
  4.4.2 Anisotropy in Three-Dimensional Domains......Page 188
  4.5.2 Cumulative Joint Probability Function......Page 191
  4.5.3 Statistical Moments......Page 192
  4.5.4 Characteristic Function......Page 193
  4.5.5 Cumulant and Moment Generating Functions......Page 196
 5.1 Local Properties......Page 199
  5.1.1 Stochastic Convergence......Page 202
  5.1.2 Random Field Continuity......Page 205
   5.1.2.2 Mean-Square Continuity......Page 206
  5.1.3 Sample-Path Continuity......Page 208
  5.1.5 Differentiability in the Mean-Square Sense......Page 209
   5.1.5.2 Homogeneous Random Fields......Page 210
  5.2.1 CHI for Two-Dimensional Random Fields......Page 218
 5.3 Spectral Moments......Page 222
  5.3.1 Radial Spectral Densities......Page 223
  5.3.2 Second-Order Spectral Moments......Page 224
  5.3.3 Variance of Random Field Gradient and Curvature......Page 226
 5.4 Length Scales of Random Fields......Page 229
  5.4.1 Practical Range......Page 230
  5.4.2 Integral Range......Page 231
  5.4.3 Correlation Radius......Page 232
  5.4.4 Turbulence Microscale......Page 233
  5.4.5 Correlation Spectrum......Page 234
 5.5 Fractal Dimension......Page 236
  5.5.1 Fractal Dimension and Variogram Function......Page 238
 5.6 Long-Range Dependence......Page 240
 5.7 Intrinsic Random Fields......Page 245
  5.7.1 Random Fields with Stationary Increments......Page 248
  5.7.2 Higher-Order Stationary Increments......Page 252
 5.8 Fractional Brownian Motion......Page 255
  5.8.1 Properties of fBm Fields......Page 256
  5.8.2 Spectral Representation......Page 258
  5.8.3 Long-Range Dependence......Page 260
  5.8.4 Random Walk Model......Page 261
  5.8.5 Applications of Fractional Brownian Motion......Page 263
  5.8.6 Roughness of Random Field Surfaces......Page 264
  5.9.1 Classification Based on Joint Probability Density Function......Page 266
  5.9.2 Classification Based on Statistical Homogeneity......Page 267
  5.9.3 Classification Based on the Type of Correlations......Page 268
  5.9.4 Desired Properties of Random Field Models......Page 269
6 Gaussian Random Fields......Page 271
 6.1 Multivariate Normal Distribution......Page 272
  6.1.1 Boltzmann-Gibbs Representation......Page 274
  6.1.2 Gaussian Second-Order Cumulant......Page 275
  6.1.3 Two-Dimensional Joint Gaussian pdf......Page 277
  6.1.4 Conditional Probability Density Functions......Page 281
 6.2 Field Integral Formulation......Page 288
  6.2.1 A Detour in Functional Derivatives......Page 293
  6.2.2 Moment Generating Functional......Page 295
  6.2.3 Cumulant Generating Functional and Moments......Page 297
  6.3.1 Isserlis-Wick Theorem......Page 299
  6.3.2 Novikov-Furutsu-Donsker Theorem......Page 302
   6.3.2.2 Continuum-Space Formulation......Page 303
  6.3.3 Gaussian Moments......Page 305
  6.3.4 The Cumulant Expansion......Page 306
  6.3.5 The Lognormal Distribution......Page 307
 6.4 Perturbation Theory for Non-Gaussian Probability Densities......Page 308
  6.4.1 Perturbation Expansion......Page 309
  6.4.2 The Cumulant Expansion......Page 313
  6.4.3 Non-stationarity Caused by Non-uniform Perturbations......Page 320
  6.4.4 Non-stationarity Caused by LocalizedPerturbations......Page 322
  6.4.5 The Variational Method......Page 324
 6.5 Non-stationary Covariance Functions......Page 331
7 Random Fields Based on Local Interactions......Page 334
  7.1.1 Overture......Page 335
  7.1.2 The Fluctuation-Gradient-Curvature EnergyFunction......Page 336
  7.1.3 SSRF Model Parameters......Page 338
  7.1.4 More on Nomenclature......Page 339
  7.1.5 Are SSRFs Gaussian Random Fields?......Page 340
  7.1.6 Spectral Representation......Page 344
 7.2 Two-Point Functions and Realizations......Page 348
  7.2.1 SSRF Length Scales......Page 349
  7.2.2 One-Dimensional SSRFs......Page 350
  7.2.3 The Role of Rigidity......Page 355
  7.2.4 Two-Dimensional SSRFs......Page 359
  7.2.5 Three-Dimensional SSRFs......Page 364
  7.2.6 On Mean-Square Differentiability......Page 367
 7.3 Statistical and Geometric Properties......Page 369
  7.3.1 SSRF Variance......Page 370
  7.3.3 Large Rigidity......Page 371
  7.3.4 SSRF Correlation Spectrum......Page 374
 7.4 Random Fields with Bessel-Lommel Covariance......Page 381
8 Lattice Representations of Spartan Random Fields......Page 389
 8.1 Introduction to Gauss-Markov Random Fields (GMRFs)......Page 390
  8.1.1 Conditional Independence......Page 392
  8.1.3 GMRF Joint Probability Distribution......Page 394
 8.2 From SSRFs to Gauss-Markov Random Fields......Page 396
  8.2.2 Lattice SSRF with Isotropic Structure......Page 397
 8.3 Lattice Spectral Density......Page 399
 8.4 SSRF Lattice Moments......Page 404
 8.5 SSRF Inverse Covariance Operator on Lattices......Page 407
  8.5.1 Low-Order Discretization Schemes......Page 408
  8.5.2 Higher-Order Discretization Schemes......Page 411
9 Spartan Random Fields and Langevin Equations......Page 417
 9.1 Introduction to Stochastic Differential Equations (SPDEs)......Page 418
 9.2 Classical Harmonic Oscillator......Page 421
 9.3 Stochastic Partial Differential Equations......Page 426
  9.3.1 Linear SPDEs with Constant Coefficients......Page 427
  9.3.2 Covariance Equation in Real Space......Page 428
  9.3.3 Spectral Density from Linear SPDEs......Page 430
  9.3.4 Polynomials of the Diffusion Operator......Page 431
 9.4 Spartan Random Fields and SPDEs......Page 433
  9.4.1 Partial Differential Equation for SSRF Covariance Functions......Page 434
  9.4.2 Spatial Harmonic Oscillators......Page 435
 9.5 Covariances and Green's Functions......Page 438
 9.6 Whittle-Matérn Stochastic Partial Differential Equation......Page 439
  9.7.1 Brief Overview of Linear Time Series Modeling......Page 441
  9.7.2 Properties of ARMA Models......Page 443
  9.7.3 Autoregressive Formulation of SSRFs......Page 449
10 Spatial Prediction Fundamentals......Page 457
 10.2 Deterministic Interpolation......Page 462
  10.2.2 Inverse Distance Weighting......Page 463
  10.2.3 Minimum Curvature Interpolation......Page 465
  10.2.4 Natural Neighbor Interpolation......Page 471
 10.3 Stochastic Methods......Page 472
 10.4 Simple Kriging (SK)......Page 475
  10.4.1 Compact Form of SK Equations......Page 478
  10.4.2 Properties of the SK Predictor......Page 479
  10.4.3 Examples of Simple Kriging......Page 481
  10.4.4 Impact of the Nugget Term......Page 485
  10.4.5 Properties of the Kriging Error......Page 486
 10.5 Ordinary Kriging (OK)......Page 490
  10.5.1 Compact Form of Ordinary Kriging Equations......Page 493
  10.5.2 Kriging Variance......Page 494
  10.5.3 How the Nugget Term Affects the KrigingEquations......Page 495
  10.5.4 Ordinary Kriging Examples......Page 497
 10.6 Properties of the Kriging Predictor......Page 501
 10.7 Topics Related to the Application of Kriging......Page 502
  10.7.2 Impact of Anisotropy on Kriging Weights......Page 504
 10.8 Evaluating Model Performance......Page 506
11 More on Spatial Prediction......Page 509
  11.1.1 Ordinary Kriging with Intrinsic Random Fields......Page 510
  11.1.3 Universal Kriging......Page 511
  11.1.4 Cokriging......Page 513
  11.1.5 Functional Kriging......Page 518
 11.2 Nonlinear Extensions of Kriging......Page 519
  11.2.2 Spin-Based Indicator Models......Page 520
  11.2.3 Lognormal Kriging......Page 521
 11.3 Connections with Gaussian Process Regression......Page 522
 11.4 Bayesian Kriging......Page 524
 11.5 Continuum Formulation of Linear Prediction......Page 526
  11.5.1 Continuum Prediction for Spartan Random Fields......Page 527
 11.6 The ``Local-Interaction'' Approach......Page 528
  11.6.1 Lattice Site Indexing......Page 529
  11.6.2 Local Interactions on Regular Grids......Page 531
 11.7 Big Spatial Data......Page 538
12 Basic Concepts and Methods of Estimation......Page 540
 12.1 Estimator Properties......Page 541
 12.2 Estimating the Mean with Ordinary Kriging......Page 545
  12.2.1 A Synthetic Example......Page 547
  12.3.1 Types of Estimation Methods......Page 549
  12.3.2 Method of Moments (MoM)......Page 550
  12.3.3 Properties of MoM Estimator......Page 552
  12.3.4 Method of Moments on Regular Grids......Page 553
  12.3.5 Fit to Theoretical Variogram Model......Page 554
  12.3.6 Non-parametric Variogram Estimation......Page 555
  12.3.7 Practical Issues of Variogram Estimation......Page 556
 12.4 Maximum Likelihood Estimation (MLE)......Page 558
  12.4.1 Basic Steps of MLE......Page 560
  12.4.2 Fisher Information Matrix and ParameterUncertainty......Page 565
  12.4.3 MLE Properties......Page 566
 12.5 Cross Validation......Page 568
  12.5.1 Splitting the Data......Page 569
  12.5.2 Cross Validation Measures......Page 570
  12.5.3 Parameter Estimation via Cross-validation......Page 572
13 More on Estimation......Page 574
 13.1 Method of Normalized Correlations (MoNC)......Page 575
  13.1.1 MoNC Constraints for Data on Regular Grids......Page 576
  13.1.2 MoNC Constraints for Data on Irregular Grids......Page 577
  13.1.3 Parameter Inference with MoNC......Page 580
  13.1.4 MoNC Constraints for Time Series......Page 582
  13.2.1 Introduction to Entropy......Page 584
  13.2.3 Formulation of the Maximum Entropy Method......Page 588
  13.2.4 Maximum Entropy Formulation of Spartan Random Fields......Page 593
 13.3 Stochastic Local Interactions (SLI)......Page 594
  13.3.1 Energy Function......Page 595
  13.3.2 Nadaraya-Watson Average......Page 596
  13.3.3 Precision Matrix Formulation......Page 598
  13.3.4 Mode Prediction......Page 600
  13.3.5 Parameter Estimation......Page 604
 13.4 Measuring Ergodicity......Page 607
  13.4.1 Ergodic Index......Page 608
  13.4.2 Improved Ergodic Index......Page 610
  13.4.3 Directional Ergodic Index......Page 611
14 Beyond the Gaussian Models......Page 613
 14.1 Trans-Gaussian Random Fields......Page 615
  14.1.1 Joint Density of Trans-Gaussian Random Fields......Page 618
 14.2 Gaussian Anamorphosis......Page 619
  14.2.2 Johnson's Hyperbolic Sine Transformation......Page 620
  14.2.3 Box-Cox Transformation......Page 621
 14.3 Tukey g-h Random Fields......Page 625
 14.4 Transformations Based on Kaniadakis Exponential......Page 626
  14.4.1 Properties of Kaniadakis Exponential and Logarithm Functions......Page 627
  14.4.2 Kaniadakis Entropy......Page 635
  14.4.3 κ-Lognormal Random Fields......Page 636
 14.5 Hermite Polynomials......Page 638
  14.5.1 Hermite Polynomial Expansions......Page 640
  14.5.2 Practical Use of Hermite Expansions......Page 642
  14.6.1 Univariate Student's t-Distribution......Page 644
  14.6.2 Connection with the Tsallis Distribution......Page 646
  14.6.3 Multivariate Student's t-Distributions......Page 647
  14.6.4 Student's t-Distributed Random Fields......Page 650
  14.6.5 Hierarchical Representations......Page 652
  14.6.6 Log-Student's t-Random Field......Page 654
  14.6.7 Student's t-Distributed Process......Page 655
 14.7 Copula Models......Page 656
  14.7.1 Gaussian Copula......Page 658
  14.7.2 Other Copula Families......Page 659
 14.8 The Replica Method......Page 660
  14.8.1 Applications in Spatial Data Problems......Page 661
  14.8.2 Relevant Literature and Applications......Page 664
15 Binary Random Fields......Page 666
 15.1 Indicator Random Field......Page 667
  15.1.1 Moments of the Indicator Random Field......Page 668
  15.1.2 Level Cuts of Gaussian Random Fields......Page 669
  15.1.3 Connectivity of Extreme Values......Page 672
  15.1.4 Excursion Sets......Page 674
   15.1.4.1 Special Limits of Indicator Covariance......Page 677
  15.1.5 Leveled-Wave Model......Page 679
  15.1.6 Random Porous Morphologies......Page 681
 15.2 Ising Model......Page 685
  15.2.1 Ising Model: Basic Facts......Page 686
  15.2.2 One-Dimensional Ising Model......Page 692
  15.2.3 Mean-Field Theory......Page 695
  15.2.4 What Is the Connection with Spatial DataAnalysis?......Page 698
  15.2.5 Estimation of Missing Spins......Page 699
 15.3 Generalized Linear Models......Page 701
  15.3.1 Logistic Regression......Page 702
  15.3.2 Model-Based Geostatistics......Page 704
  15.3.3 Autologistic Models......Page 708
16 Simulations......Page 710
 16.1 Introduction......Page 712
 16.2 Covariance Matrix Factorization......Page 716
  16.3.1 Fourier Transforms of Random Fields......Page 720
 16.4 Fast-Fourier-Transform Simulation......Page 722
  16.4.1 Discrete Fourier Transforms......Page 723
 16.5 Randomized Spectral Sampling......Page 729
  16.5.1 Simple Sampling Strategies......Page 730
  16.5.3 Importance Sampling for Radial Spectral Densities......Page 731
  16.5.4 Importance Sampling for Specific SpectralDensities......Page 733
  16.5.5 Sampling the Surface of the Unit Sphere......Page 740
  16.5.7 Low-Discrepancy Sequences......Page 742
  16.5.8 Spectral Simulation of Spartan Random Fields......Page 744
 16.6 Conditional Simulation Based on Polarization Method......Page 745
 16.7 Conditional Simulation Based on Covariance Matrix Factorization......Page 752
  16.8.1 Ordinary Monte Carlo Methods......Page 755
  16.8.2 Markov Chain Monte Carlo Methods......Page 757
  16.8.3 Basic Concepts of MCMC Methods......Page 759
  16.8.4 Metropolis-Hastings Sampling......Page 763
  16.8.5 Gibbs Sampling......Page 768
 16.9 Sequential Simulation of Random Fields......Page 770
 16.10 Simulated Annealing......Page 773
 16.11 Karhunen-Loève (KL) Expansion......Page 776
  16.11.1 Definition and Properties of Karhunen-Loève Expansion......Page 778
  16.11.2 Karhunen-Loève Expansion of the Wiener Process......Page 780
  16.11.3 Numerical Expansions......Page 783
 16.12 Karhunen-Loève Expansion of Spartan Random Fields......Page 785
  16.12.2 K-L ODE for Spartan Covariance......Page 786
  16.12.3 K-L Eigenfunctions for Spartan Covariance......Page 787
  16.12.4 K-L Eigenvalues for Spartan Covariance......Page 789
  16.12.5 First Branch of K-L Eigenfunctions......Page 791
  16.12.6 Second Branch of K-L Eigenfunctions......Page 794
  16.12.7 Accuracy of Truncated Karhunen-Loève Expansions......Page 797
  16.12.8 Summary of SSRF Karhunen-Loève Expansion......Page 798
  16.12.9 Examples of K-L SSRF Expansion......Page 800
 16.13 Convergence of Truncated K-L Expansion......Page 803
17 Epilogue......Page 806
A Jacobi's Transformation Theorems......Page 810
B Tables of Spartan Random Field Properties......Page 813
C Linear Algebra Facts......Page 814
D Kolmogorov-Smirnov Test......Page 817
Glossary......Page 821
References......Page 826
Index......Page 862