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E-raamat: Spatial Analysis [Wiley Online]

(University of Leeds, UK), (University of Leeds, UK)
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Spatial AnalysisSuurem pilt
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Raamatu kodulehekülg: https://onlinelibrary.wiley.com/doi/book/10.1002/9781118763551
SPATIAL ANALYSIS Explore the foundations and latest developments in spatial statistical analysis

In Spatial Analysis, two distinguished authors deliver a practical and insightful exploration of the statistical investigation of the interdependence of random variables as a function of their spatial proximity. The book expertly blends theory and application, offering numerous worked examples and exercises at the end of each chapter.

Increasingly relevant to fields as diverse as epidemiology, geography, geology, image analysis, and machine learning, spatial statistics is becoming more important to a wide range of specialists and professionals. The book includes:





Thorough introduction to stationary random fields, intrinsic and generalized random fields, and stochastic models Comprehensive exploration of the estimation of spatial structure Practical discussion of kriging and the spatial linear model

Spatial Analysis is an invaluable resource for advanced undergraduate and postgraduate students in statistics, data science, digital imaging, geostatistics, and agriculture. Its also an accessible reference for professionals who are required to use spatial models in their work.
List of Figures
xiii
List of Tables
xvii
Preface xix
List of Notation and Terminology
xxv
1 Introduction
1(30)
1.1 Spatial Analysis
1(1)
1.2 Presentation of the Data
2(7)
1.3 Objectives
9(2)
1.4 The Covariance Function and Semivariogram
11(5)
1.4.1 General Properties
11(2)
1.4.2 Regularly Spaced Data
13(1)
1.4.3 Irregularly Spaced Data
14(2)
1.5 Behavior of the Sample Semivariogram
16(6)
1.6 Some Special Features of Spatial Analysis
22(9)
Exercises
27(4)
2 Stationary Random Fields
31(42)
2.1 Introduction
31(1)
2.2 Second Moment Properties
32(2)
2.3 Positive Definiteness and the Spectral Representation
34(2)
2.4 Isotropic Stationary Random Fields
36(5)
2.5 Construction of Stationary Covariance Functions
41(2)
2.6 Matern Scheme
43(2)
2.7 Other Examples of Isotropic Stationary Covariance Functions
45(3)
2.8 Construction of Nonstationary Random Fields
48(1)
2.8.1 Random Drift
48(1)
2.8.2 Conditioning
49(1)
2.9 Smoothness
49(2)
2.10 Regularization
51(2)
2.11 Lattice Random Fields
53(3)
2.12 Torus Models
56(2)
2.12.1 Models on the Continuous Torus
56(1)
2.12.2 Models on the Lattice Torus
57(1)
2.13 Long-range Correlation
58(3)
2.14 Simulation
61(12)
2.14.1 General Points
61(1)
2.14.2 The Direct Approach
61(1)
2.14.3 Spectral Methods
62(4)
2.14.4 Circulant Methods
66(1)
Exercises
67(6)
3 Intrinsic and Generalized Random Fields
73(42)
3.1 Introduction
73(1)
3.2 Intrinsic Random Fields of Order k = 0
74(6)
3.3 Characterizations of Semivariograms
80(3)
3.4 Higher Order Intrinsic Random Fields
83(3)
3.5 Registration of Higher Order Intrinsic Random Fields
86(1)
3.6 Generalized Random Fields
87(4)
3.7 Generalized Intrinsic Random Fields of Intrinsic Order k ≥ 0
91(1)
3.8 Spectral Theory for Intrinsic and Generalized Processes
91(4)
3.9 Regularization for Intrinsic and Generalized Processes
95(1)
3.10 Self-Similarity
96(4)
3.11 Simulation
100(2)
3.11.1 General Points
100(1)
3.11.2 The Direct Method
101(1)
3.11.3 Spectral Methods
101(1)
3.12 Dispersion Variance
102(13)
Exercises
104(11)
4 Autoregression and Related Models
115(44)
4.1 Introduction
115(3)
4.2 Background
118(2)
4.3 Moving Averages
120(2)
4.3.1 Lattice Case
120(1)
4.3.2 Continuously Indexed Case
121(1)
4.4 Finite Symmetric Neighborhoods of the Origin in Zd
122(2)
4.5 Simultaneous Autoregressions (SARs)
124(3)
4.5.1 Lattice Case
124(1)
4.5.2 Continuously Indexed Random Fields
125(2)
4.6 Conditional Autoregressions (CARs)
127(7)
4.6.1 Stationary CARs
128(2)
4.6.2 Iterated SARs and CARs
130(1)
4.6.3 Intrinsic CARs
131(1)
4.6.4 CARs on a Lattice Torus
132(1)
4.6.5 Finite Regions
132(2)
4.7 Limits of CAR Models Under Fine Lattice Spacing
134(1)
4.8 Unilateral Autoregressions for Lattice Random Fields
135(5)
4.8.1 Half-spaces in Zd
135(1)
4.8.2 Unilateral Models
136(3)
4.8.3 Quadrant Autoregressions
139(1)
4.9 Markov Random Fields (MRFs)
140(9)
4.9.1 The Spatial Markov Property
140(3)
4.9.2 The Subset Expansion of the Negative Potential Function
143(2)
4.9.3 Characterization of Markov Random Fields in Terms of Cliques
145(2)
4.9.4 Auto-models
147(2)
4.10 Markov Mesh Models
149(10)
4.10.1 Validity
149(1)
4.10.2 Marginalization
150(1)
4.10.3 Markov Random Fields
150(1)
4.10.4 Usefulness
151(1)
Exercises
151(8)
5 Estimation of Spatial Structure
159(1)
5.1 Introduction
159(1)
5.2 Patterns of Behavior
160(4)
5.2.1 One-dimensional Case
160(1)
5.2.2 Two-dimensional Case
161(1)
5.2.3 Nugget Effect
162(2)
5.3 Preliminaries
164(2)
5.3.1 Domain of the Spatial Process
164(1)
5.3.2 Model Specification
164(1)
5.3.3 Spacing of Data
165(1)
5.4 Exploratory and Graphical Methods
166(2)
5.5 Maximum Likelihood for Stationary Models
168(5)
5.5.1 Maximum Likelihood Estimates - Known Mean
169(2)
5.5.2 Maximum Likelihood Estimates-Unknown Mean
171(1)
5.5.3 Fisher Information Matrix and Outfill Asymptotics
172(1)
5.6 Parameterization Issues for the Matern Scheme
173(1)
5.7 Maximum Likelihood Examples for Stationary Models
174(5)
5.8 Restricted Maximum Likelihood (REML)
179(1)
5.9 Vecchia's Composite Likelihood
180(2)
5.10 REML Revisited with Composite Likelihood
182(3)
5.11 Spatial Linear Model
185(3)
5.11.1 MLEs
186(2)
5.11.2 Outfill Asymptotics for the Spatial Linear Model
188(1)
5.12 REML for the Spatial Linear Model
188(1)
5.13 Intrinsic Random Fields
189(3)
5.14 Infill Asymptotics and Fractal Dimension
192(9)
Exercises
195(6)
6 Estimation for Lattice Models
201(30)
6.1 Introduction
201(2)
6.2 Sample Moments
203(2)
6.3 The AR(1) Process on Z
205(3)
6.4 Moment Methods for Lattice Data
208(4)
6.4.1 Moment Methods for Unilateral Autoregressions (UARs)
209(1)
6.4.2 Moment Estimators for Conditional Autoregression (CAR) Models
210(2)
6.5 Approximate Likelihoods for Lattice Data
212(3)
6.6 Accuracy of the Maximum Likelihood Estimator
215(3)
6.7 The Moment Estimator for a CAR Model
218(13)
Exercises
219(12)
7 Kriging
231(52)
7.1 Introduction
231(2)
7.2 The Prediction Problem
233(3)
7.3 Simple Kriging
236(2)
7.4 Ordinary Kriging
238(2)
7.5 Universal Kriging
240(1)
7.6 Further Details for the Universal Kriging Predictor
241(7)
7.6.1 Transfer Matrices
241(1)
7.6.2 Projection Representation of the Transfer Matrices
242(2)
7.6.3 Second Derivation of the Universal Kriging Predictor
244(1)
7.6.4 A Bordered Matrix Equation for the Transfer Matrices
245(1)
7.6.5 The Augmented Matrix Representation of the Universal Kriging Predictor
245(2)
7.6.6 Summary
247(1)
7.7 Stationary Examples
248(5)
7.8 Intrinsic Random Fields
253(3)
7.8.1 Formulas for the Kriging Predictor and Kriging Variance
253(1)
7.8.2 Conditionally Positive Definite Matrices
254(2)
7.9 Intrinsic Examples
256(2)
7.10 Square Example
258(1)
7.11 Kriging with Derivative Information
259(3)
7.12 Bayesian Kriging
262(4)
7.12.1 Overview
262(2)
7.12.2 Details for Simple Bayesian Kriging
264(1)
7.12.3 Details for Bayesian Kriging with Drift
264(2)
7.13 Kriging and Machine Learning
266(3)
7.14 The Link Between Kriging and Splines
269(5)
7.14.1 Nonparametric Regression
269(2)
7.14.2 Interpolating Splines
271(2)
7.14.3 Comments on Interpolating Splines
273(1)
7.14.4 Smoothing Splines
274(1)
7.15 Reproducing Kernel Hilbert Spaces
274(1)
7.16 Deformations
275(8)
Exercises
277(6)
8 Additional Topics
283(20)
8.1 Introduction
283(1)
8.2 Log-normal Random Fields
284(1)
8.3 Generalized Linear Spatial Mixed Models (GLSMMs)
285(1)
8.4 Bayesian Hierarchical Modeling and Inference
286(1)
8.5 Co-kriging
287(4)
8.6 Spatial-temporal Models
291(3)
8.6.1 General Considerations
291(1)
8.6.2 Examples
292(2)
8.7 Clamped Plate Splines
294(1)
8.8 Gaussian Markov Random Field Approximations
295(1)
8.9 Designing a Monitoring Network
296(7)
Exercises
298(5)
Appendix A Mathematical Background
303(44)
A.1 Domains for Sequences and Functions
303(2)
A.2 Classes of Sequences and Functions
305(1)
A.2.1 Functions on the Domain Rd
305(1)
A.2.2 Sequences on the Domain Zd
305(1)
A.2.3 Classes of Functions on the Domain Sd1
306(1)
A.2.4 Classes of Sequences on the Domain ZdN, Where N = (n[ 1], ..., n[ d])
306(1)
A.3 Matrix Algebra
306(7)
A.3.1 The Spectral Decomposition Theorem
306(1)
A.3.2 Moore--Penrose Generalized Inverse
307(1)
A.3.3 Orthogonal Projection Matrices
308(1)
A.3.4 Partitioned Matrices
308(1)
A.3.5 Schur Product
309(1)
A.3.6 Woodbury Formula for a Matrix Inverse
310(1)
A.3.7 Quadratic Forms
311(1)
A.3.8 Toeplitz and Circulant Matrices
311(1)
A.3.9 Tensor Product Matrices
312(1)
A.3.10 The Spectral Decomposition and Tensor Products
313(1)
A.3.11 Matrix Derivatives
313(1)
A.4 Fourier Transforms
313(2)
A.5 Properties of the Fourier Transform
315(3)
A.6 Generalizations of the Fourier Transform
318(1)
A.7 Discrete Fourier Transform and Matrix Algebra
318(4)
A.7.1 DFT in d = 1 Dimension
318(1)
A.7.2 Properties of the Unitary Matrix G, d = 1
319(1)
A.7.3 Circulant Matrices and the DFT, d = 1
320(1)
A.7.4 The Case d > 1
321(1)
A.7.5 The Periodogram
322(1)
A.8 Discrete Cosine Transform (DCT)
322(2)
A.8.1 One-dimensional Case
322(1)
A.8.2 The Case d > 1
323(1)
A.8.3 Indexing for the Discrete Fourier and Cosine Transforms
323(1)
A.9 Periodic Approximations to Sequences
324(1)
A.10 Structured Matrices in d = 1 Dimension
325(2)
A.11 Matrix Approximations for an Inverse Covariance Matrix
327(5)
A.11.1 The Inverse Covariance Function
328(2)
A.11.2 The Toeplitz Approximation to Σ-1
330(1)
A.11.3 The Circulant Approximation to Σ-1
330(1)
A.11.4 The Folded Circulant Approximation to Σ-1
330(1)
A.11.5 Comments on the Approximations
331(1)
A.11.6 Sparsity
332(1)
A.12 Maximum Likelihood Estimation
332(6)
A.12.1 General Considerations
332(1)
A.12.2 The Multivariate Normal Distribution and the Spatial Linear Model
333(2)
A.12.3 Change of Variables
335(1)
A.12.4 Profile Log-likelihood
335(1)
A.12.5 Confidence Intervals
336(1)
A.12.6 Linked Parameterization
337(1)
A.12.7 Model Choice
338(1)
A.13 Bias in Maximum Likelihood Estimation
338(9)
A.13.1 A General Result
338(2)
A.13.2 The Spatial Linear Model
340(7)
Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach
347(8)
B.1 Introduction
347(1)
B.2 Matheron and Watson
348(1)
B.3 Geostatistics at Leeds 1977--1987
349(3)
B.3.1 Courses, Publications, Early Dissemination
349(2)
B.3.2 Numerical Problems with Maximum Likelihood
351(1)
B.4 Frequentist vs. Bayesian Inference
352(3)
References and Author Index 355(12)
Index 367
John T. Kent is a Professor in the Department of Statistics at the University of Leeds, UK. He began his career as a research fellow at Sidney Sussex College, Cambridge before moving to the University of Leeds. He has published extensively on various aspects of statistics, including infinite divisibility, directional data analysis, multivariate analysis, inference, robustness, shape analysis, image analysis, spatial statistics, and spatial-temporal modelling.

Kanti V. Mardia is a Senior Research Professor and Leverhulme Emeritus Fellow in the Department of Statistics at the University of Leeds, and a Visiting Professor at the University of Oxford. During his career he has received many prestigious honours, including in 2003 the Guy Medal in Silver from the Royal Statistical Society, and in 2013 the Wilks memorial medal from the American Statistical Society. His research interests include bioinformatics, directional statistics, geosciences, image analysis, multivariate analysis, shape analysis, spatial statistics, and spatial-temporal modelling. Kent and Mardia are also joint authors of a well-established monograph on Multivariate Analysis.
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