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Schabenberger, Oliver.; Gotway, Carol A.
publisher: CRC Press
subject: Spatial analysis (Statistics)
publication date: 2005
Page ix
Contents
Preface xv
1 Introduction 1
1.1 The Need for Spatial Analysis 1
1.2 Types of Spatial Data 6
1.2.1 Geostatistical Data 7
1.2.2 Lattice Data, Regional Data 8
1.2.3 Point Patterns 11
1.3 Autocorrelation—Concept and Elementary Measures 14
1.3.1 Mantel’s Tests for Clustering 14
1.3.2 Measures on Lattices 18
1.3.3 Localized Indicators of Spatial Autocorrelation 23
1.4 Autocorrelation Functions 25
1.4.1 The Autocorrelation Function of a Time Series 25
1.4.2 Autocorrelation Functions in Space—Covariance and Semivariogram 26
1.4.3 From Mantel’s Statistic to the Semivariogram 29
1.5 The Effects of Autocorrelation on Statistical Inference 31
1.5.1 Effects on Prediction 32
1.5.2 Effects on Precision of Estimators 34
1.6 Chapter Problems 37
2 Some Theory on Random Fields 41
2.1 Stochastic Processes and Samples of Size One 41
2.2 Stationarity, Isotropy, and Heterogeneity 42
2.3 Spatial Continuity and Differentiability 48
2.4 Random Fields in the Spatial Domain 52
2.4.1 Model Representation 53
2.4.2 Convolution Representation 57
2.5 Random Fields in the Frequency Domain 62
2.5.1 Spectral Representation of Deterministic Functions 62
2.5.2 Spectral Representation of Random Processes 65
2.5.3 Covariance and Spectral Density Function 66
2.5.4 Properties of Spectral Distribution Functions
2.5.7 Importance of Spectral Analysis 77
2.6 Chapter Problems 78
3 Mapped Point Patterns 81
3.1 Random, Aggregated, and Regular Patterns 81
3.2 Binomial and Poisson Processes 83
3.2.1 Bernoulli and Binomial Processes 83
3.2.2 Poisson Processes 84
3.2.3 Process Equivalence 85
3.3 Testing for Complete Spatial Randomness 86
3.3.1 Monte Carlo Tests 87
3.3.2 Simulation Envelopes 88
3.3.3 Tests Based on Quadrat Counts 90
3.3.4 Tests Based on Distances 97
3.4 Second-Order Properties of Point Patterns 99
3.4.1 The Reduced Second Moment Measure—The K-Function 101
3.4.2 Estimation of K- and L-Functions 102
3.4.3 Assessing the Relationship between Two Patterns 103
3.5 The Inhomogeneous Poisson Process 107
3.5.1 Estimation of the Intensity Function 110
3.5.2 Estimating the Ratio of Intensity Functions 112
3.5.3 Clustering and Cluster Detection 114
3.6 Marked and Multivariate Point Patterns 118
3.6.1 Extensions 118
3.6.2 Intensities and Moment Measures for Multivariate Point Patterns 120
3.7 Point Process Models 122
3.7.1 Thinning and Clustering 123
3.7.2 Clustered Processes 125
3.7.3 Regular Processes 128
3.8 Chapter Problems 129
4 Semivariogram and Covariance Function Analysis and Estimation 133
4.1 Introduction 133
4.2 Semivariogram and Covariogram 135
4.2.1 Definition and Empirical Counterparts 135
4.2.2 Interpretation as Structural Tools 138
4.3 Covariance and Semivariogram Models 141
4.3.1 Model Validity 141
4.3.2 The Matérn Class of Covariance Functions 143
4.3.3 The Spherical Family of Covariance Functions 145
4.3.4 Isotropic Models Allowing Negative Correlations 146
4.3.5 Basic Models Not Second-Order Stationary 149
4.3.6 Models with Nugget Effects and Nested Models 150
4.3.7 Accommodating Anisotropy 151
4.4 Estimating the Semivariogram 153
4.4.1 Matheron’s Estimator 153
4.4.2 The Cressie-Hawkins Robust Estimator 159
4.4.3 Estimators Based on Order Statistics and Quantiles 161
4.5 Parametric Modeling 163
4.5.1 Least Squares and the Semivariogram 164
4.5.2 Maximum and Restricted Maximum Likelihood 166
4.5.3 Composite Likelihood and Generalized Estimating Equations 169
4.5.4 Comparisons 172
4.6 Nonparametric Estimation and Modeling 178
4.6.1 The Spectral Approach 179
4.6.2 The Moving-Average Approach 183
4.6.3 Incorporating a Nugget Effect 186
4.7 Estimation and Inference in the Frequency Domain 188
4.7.1 The Periodogram on a Rectangular Lattice 190
4.7.2 Spectral Density Functions 198
4.7.3 Analysis of Point Patterns 200
4.8 On the Use of Non-Euclidean Distances in Geostatistics 204
4.8.1 Distance Metrics and Isotropic Covariance Functions 205
4.8.2 Multidimensional Scaling 206
4.9 Supplement: Bessel Functions 210
4.9.1 Bessel Function of the First Kind 210
4.9.2 Modified Bessel Functions of the First and Second Kind 210
4.10 Chapter Problems 211
5 Spatial Prediction and Kriging 215
5.1 Optimal Prediction in Random Fields 215
5.2 Linear Prediction—Simple and Ordinary Kriging 221
5.2.1 The Mean Is Known—Simple Kriging 223
5.2.2 The Mean Is Unknown and Constant—Ordinary Kriging 226
5.2.3 Effects of Nugget, Sill, and Range 228
5.3 Linear Prediction with a Spatially Varying Mean 232
5.3.1 Trend Surface Models 234
5.3.2 Localized Estimation 238
5.3.3 Universal Kriging 241
5.4 Kriging in Practice 243
5.4.1 On the Uniqueness of the Decomposition 243
5.4.2 Local Versus Global Kriging 244
5.4.3 Filtering and Smoothing 248
5.5 Estimating Covariance Parameters 254
5.5.1 Least Squares Estimation 256
5.5.2 Maximum Likelihood 259
5.5.3 Restricted Maximum Likelihood
7.4 Simulating from Convolutions 413
7.5 Simulating Point Processes 418
7.5.1 Homogeneous Poisson Process on the Rectangle (0, 0)× (a, b) with Intensity λ 418
7.5.2 Inhomogeneous Poisson Process with Intensity λ(s) 419
7.6 Chapter Problems 419
8 Non-Stationary Covariance 421
8.1 Types of Non-Stationarity 421
8.2 Global Modeling Approaches 422
8.2.1 Parametric Models 422
8.2.2 Space Deformation 423
8.3 Local Stationarity 425
8.3.1 Moving Windows 425
8.3.2 Convolution Methods 426
8.3.3 Weighted Stationary Processes 428
9 Spatio-Temporal Processes 431
9.1 A New Dimension 431
9.2 Separable Covariance Functions 434
9.3 Non-Separable Covariance Functions 435
9.3.1 Monotone Function Approach 436
9.3.2 Spectral Approach 436
9.3.3 Mixture Approach 438
9.3.4 Differential Equation Approach 439
9.4 The Spatio-Temporal Semivariogram 440
9.5 Spatio-Temporal Point Processes 442
9.5.1 Types of Processes 442
9.5.2 Intensity Measures 443
9.5.3 Stationarity and Complete Randomness 444
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