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.

[Spatial Analysis] is a splendid text on spatial statistics written by two eminent scholars in the field who have beautifully presented a wide range of topics. The text begins with some very interesting examples of spatially oriented data and their features and is followed by some superbly compiled expository chapters. What is especially appealing, in my opinion, is the attention paid by the authors to the theoretical developments and exposition of seemingly abstruse topics. The book is compactly written while retaining mathematical rigor. Specifically, the chapters on different flavours of spatial random fields and that on conditional autoregression models stand out in terms of their clarity of presentation. Inference primarily focuses on likelihood based methods and kriging, while appearing somewhat late in the book (Chapter 7 out of eight chapters), receives a very detailed treatment that includes Bayesian prediction methods as well. In summary, this elegant text will serve students, researchers and scholars invested in spatial statistics very well as a source of reference as well as a text to build courses from. I congratulate the authors' on this wonderful accomplishment.  — Sudipto Banerjee, PhD, Professor and Chair, Dept. of Biostatistics, UCLA Fielding School of Public HealthThis book, Spatial Analysis, by John Kent and Kanti Mardia is a must-read by all statisticians whose interests include spatial statistics, and it would make an excellent graduate-level text. The authors find the sweet spot at the juncture of intuition, methodology, and applications, and they show us in many ways how the validity of scientific inferences and its software has its genesis in the construction of valid statistical models.  — Noel Cressie, Distinguished Professor and Director, Centre for Environmental Informatics, University of Wollongong, AustraliaThe modern world of big data is often accompanied by little understanding. Often this data comes in a spatial form, and critically our understanding emerges from spatial analysis. It’s not possible to imagine two better guides to this domain than John Kent and Kanti Mardia. In Spatial Analysis Kent and Mardia provide a comprehensive guide to modern thinking that is classically grounded. This book is a must-read for those who are taking understanding seriously as part of handling modern spatial data sets across the domains of machine learning, statistics and data science.— Neil Lawrence, DeepMind Professor of Machine Learning, University of Cambridge[Spatial Analysis] is a delightful and authoritative book on the subject of spatial statistical analysis by two of the world’s most eminent researchers in the field of spatial statistics and shape analysis. The book eloquently discusses most of the topics in spatial analysis in a wonderfully organised manner. For example, there are two chapters on random fields, two chapters on estimation methods, one on modelling and another large chapter on kriging. With another chapter on additional topics such as Co-kriging, Bayesian hierarchical modeling, spatio-temporal modeling, and thin plate splines this book covers most of the concepts researchers need to know in this area. The main emphasis of the book is on theoretical aspects but it does not lose sight of applications. The chapter one itself motivates the theory with several example data sets which include fingerprint of the famous statistician Sir R A Fisher. The book does justice to the theory by presenting and explaining it in an accessible format for all – graduate students and researchers. The book also provides enjoyable to read personal historical notes and anecdotes regarding the course of development of the theory of spatial analysis.— Sujit Sahu, Professor of Mathematical Sciences, University of Southampton

• List of Figures xiiiList of Tables xviiPreface xixList of Notation and Terminology xxv1 Introduction 11.1 Spatial Analysis 11.2 Presentation of the Data 21.3 Objectives 91.4 The Covariance Function and Semivariogram 111.4.1 General Properties 111.4.2 Regularly Spaced Data 131.4.3 Irregularly Spaced Data 141.5 Behavior of the Sample Semivariogram 161.6 Some Special Features of Spatial Analysis 22Exercises 272 Stationary Random Fields 312.1 Introduction 312.2 Second Moment Properties 322.3 Positive Definiteness and the Spectral Representation 342.4 Isotropic Stationary Random Fields 362.5 Construction of Stationary Covariance Functions 412.6 Matérn Scheme 432.7 Other Examples of Isotropic Stationary Covariance Functions 452.8 Construction of Nonstationary Random Fields 482.8.1 Random Drift 482.8.2 Conditioning 492.9 Smoothness 492.10 Regularization 512.11 Lattice Random Fields 532.12 Torus Models 562.12.1 Models on the Continuous Torus 562.12.2 Models on the Lattice Torus 572.13 Long-range Correlation 582.14 Simulation 612.14.1 General Points 612.14.2 The Direct Approach 612.14.3 Spectral Methods 622.14.4 Circulant Methods 66Exercises 673 Intrinsic and Generalized Random Fields 733.1 Introduction 733.2 Intrinsic Random Fields of Order k = 0 743.3 Characterizations of Semivariograms 803.4 Higher Order Intrinsic Random Fields 833.5 Registration of Higher Order Intrinsic Random Fields 863.6 Generalized Random Fields 873.7 Generalized Intrinsic Random Fields of Intrinsic Order k ≥ 0 913.8 Spectral Theory for Intrinsic and Generalized Processes 913.9 Regularization for Intrinsic and Generalized Processes 953.10 Self-Similarity 963.11 Simulation 1003.11.1 General Points 1003.11.2 The Direct Method 1013.11.3 Spectral Methods 1013.12 Dispersion Variance 102Exercises 1044 Autoregression and Related Models 1154.1 Introduction 1154.2 Background 1184.3 Moving Averages 1204.3.1 Lattice Case 1204.3.2 Continuously Indexed Case 1214.4 Finite Symmetric Neighborhoods of the Origin in Z d 1224.5 Simultaneous Autoregressions (SARs) 1244.5.1 Lattice Case 1244.5.2 Continuously Indexed Random Fields 1254.6 Conditional Autoregressions (CARs) 1274.6.1 Stationary CARs 1284.6.2 Iterated SARs and CARs 1304.6.3 Intrinsic CARs 1314.6.4 CARs on a Lattice Torus 1324.6.5 Finite Regions 1324.7 Limits of CAR Models Under Fine Lattice Spacing 1344.8 Unilateral Autoregressions for Lattice Random Fields 1354.8.1 Half-spaces in Z d 1354.8.2 Unilateral Models 1364.8.3 Quadrant Autoregressions 1394.9 Markov Random Fields (MRFs) 1404.9.1 The Spatial Markov Property 1404.9.2 The Subset Expansion of the Negative Potential Function 1434.9.3 Characterization of Markov Random Fields in Terms of Cliques 1454.9.4 Auto-models 1474.10 Markov Mesh Models 1494.10.1 Validity 1494.10.2 Marginalization 1504.10.3 Markov Random Fields 1504.10.4 Usefulness 151Exercises 1515 Estimation of Spatial Structure 1595.1 Introduction 1595.2 Patterns of Behavior 1605.2.1 One-dimensional Case 1605.2.2 Two-dimensional Case 1615.2.3 Nugget Effect 1625.3 Preliminaries 1645.3.1 Domain of the Spatial Process 1645.3.2 Model Specification 1645.3.3 Spacing of Data 1655.4 Exploratory and Graphical Methods 1665.5 Maximum Likelihood for Stationary Models 1685.5.1 Maximum Likelihood Estimates – Known Mean 1695.5.2 Maximum Likelihood Estimates – Unknown Mean 1715.5.3 Fisher Information Matrix and Outfill Asymptotics 1725.6 Parameterization Issues for the Matérn Scheme 1735.7 Maximum Likelihood Examples for Stationary Models 1745.8 Restricted Maximum Likelihood (REML) 1795.9 Vecchia’s Composite Likelihood 1805.10 REML Revisited with Composite Likelihood 1825.11 Spatial Linear Model 1855.11.1 MLEs 1865.11.2 Outfill Asymptotics for the Spatial Linear Model 1885.12 REML for the Spatial Linear Model 1885.13 Intrinsic Random Fields 1895.14 Infill Asymptotics and Fractal Dimension 192Exercises 1956 Estimation for Lattice Models 2016.1 Introduction 2016.2 Sample Moments 2036.3 The AR(1) Process on Z 2056.4 Moment Methods for Lattice Data 2086.4.1 Moment Methods for Unilateral Autoregressions (UARs) 2096.4.2 Moment Estimators for Conditional Autoregression (CAR) Models 2106.5 Approximate Likelihoods for Lattice Data 2126.6 Accuracy of the Maximum Likelihood Estimator 2156.7 The Moment Estimator for a CAR Model 218Exercises 2197 Kriging 2317.1 Introduction 2317.2 The Prediction Problem 2337.3 Simple Kriging 2367.4 Ordinary Kriging 2387.5 Universal Kriging 2407.6 Further Details for the Universal Kriging Predictor 2417.6.1 Transfer Matrices 2417.6.2 Projection Representation of the Transfer Matrices 2427.6.3 Second Derivation of the Universal Kriging Predictor 2447.6.4 A Bordered Matrix Equation for the Transfer Matrices 2457.6.5 The Augmented Matrix Representation of the Universal Kriging Predictor 2457.6.6 Summary 2477.7 Stationary Examples 2487.8 Intrinsic Random Fields 2537.8.1 Formulas for the Kriging Predictor and Kriging Variance 2537.8.2 Conditionally Positive Definite Matrices 2547.9 Intrinsic Examples 2567.10 Square Example 2587.11 Kriging with Derivative Information 2597.12 Bayesian Kriging 2627.12.1 Overview 2627.12.2 Details for Simple Bayesian Kriging 2647.12.3 Details for Bayesian Kriging with Drift 2647.13 Kriging and Machine Learning 2667.14 The Link Between Kriging and Splines 2697.14.1 Nonparametric Regression 2697.14.2 Interpolating Splines 2717.14.3 Comments on Interpolating Splines 2737.14.4 Smoothing Splines 2747.15 Reproducing Kernel Hilbert Spaces 2747.16 Deformations 275Exercises 2778 Additional Topics 2838.1 Introduction 2838.2 Log-normal Random Fields 2848.3 Generalized Linear Spatial Mixed Models (GLSMMs) 2858.4 Bayesian Hierarchical Modeling and Inference 2868.5 Co-kriging 2878.6 Spatial–temporal Models 2918.6.1 General Considerations 2918.6.2 Examples 2928.7 Clamped Plate Splines 2948.8 Gaussian Markov Random Field Approximations 2958.9 Designing a Monitoring Network 296Exercises 298Appendix A Mathematical Background 303A. 1 Domains for Sequences and Functions 303A. 2 Classes of Sequences and Functions 305A.2. 1 Functions on the Domain Rd 305A.2. 2 Sequences on the Domain Zd 305A.2. 3 Classes of Functions on the Domain S d1 306A.2 4 Classes of Sequences on the Domain ZNd, Where N = (n[1], .,n[d]) 306A. 3 Matrix Algebra 306A.3. 1 The Spectral Decomposition Theorem 306A.3. 2 Moore–Penrose Generalized Inverse 307A.3. 3 Orthogonal Projection Matrices 308A.3. 4 Partitioned Matrices 308A.3. 5 Schur Product 309A.3. 6 Woodbury Formula for a Matrix Inverse 310A.3. 7 Quadratic Forms 311A.3. 8 Toeplitz and Circulant Matrices 311A.3. 9 Tensor Product Matrices 312A.3. 10 The Spectral Decomposition and Tensor Products 313A.3. 11 Matrix Derivatives 313A. 4 Fourier Transforms 313A. 5 Properties of the Fourier Transform 315A. 6 Generalizations of the Fourier Transform 318A. 7 Discrete Fourier Transform and Matrix Algebra 318A.7. 1 DFT in d = 1Dimension 318A.7. 2 Properties of the Unitary Matrix G, d = 1 319A.7. 3 Circulant Matrices and the DFT, d = 1 320A.7. 4 The Case d > 1 321A.7. 5 The Periodogram 322A. 8 Discrete Cosine Transform (DCT) 322A.8. 1 One-dimensional Case 322A.8. 2 The Case d > 1 323A.8. 3 Indexing for the Discrete Fourier and Cosine Transforms 323A. 9 Periodic Approximations to Sequences 324A. 10 Structured Matrices in d = 1Dimension 325A. 11 Matrix Approximations for an Inverse Covariance Matrix 327A.1. 1 The Inverse Covariance Function 328A.11. 2 The Toeplitz Approximation to Σ − 1 330A.11. 3 The Circulant Approximation to Σ − 1 330A.11. 4 The Folded Circulant Approximation to Σ − 1 330A.11. 5 Comments on the Approximations 331A.11. 6 Sparsity 332A. 12 Maximum Likelihood Estimation 332A.2. 1 General Considerations 332A.1. 2 The Multivariate Normal Distribution and the Spatial Linear Model 333A.12. 3 Change of Variables 335A.12. 4 Profile Log-likelihood 335A.12. 5 Confidence Intervals 336A.12. 6 Linked Parameterization 337A.12. 7 Model Choice 338A. 13 Bias in Maximum Likelihood Estimation 338A.3. 1 A General Result 338A.13. 2 The Spatial Linear Model 340Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach 347B.1 Introduction 347B.2 Matheron and Watson 348B.3 Geostatistics at Leeds 1977–1987 349B.3.1 Courses, Publications, Early Dissemination 349B.3.2 Numerical Problems with Maximum Likelihood 351B.4 Frequentist vs. Bayesian Inference 352References and Author Index 355Index 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.

[Spatial Analysis] is a splendid text on spatial statistics written by two eminent scholars in the field who have beautifully presented a wide range of topics. The text begins with some very interesting examples of spatially oriented data and their features and is followed by some superbly compiled expository chapters. What is especially appealing, in my opinion, is the attention paid by the authors to the theoretical developments and exposition of seemingly abstruse topics. The book is compactly written while retaining mathematical rigor. Specifically, the chapters on different flavours of spatial random fields and that on conditional autoregression models stand out in terms of their clarity of presentation. Inference primarily focuses on likelihood based methods and kriging, while appearing somewhat late in the book (Chapter 7 out of eight chapters), receives a very detailed treatment that includes Bayesian prediction methods as well. In summary, this elegant text will serve students, researchers and scholars invested in spatial statistics very well as a source of reference as well as a text to build courses from. I congratulate the authors' on this wonderful accomplishment.  — Sudipto Banerjee, PhD, Professor and Chair, Dept. of Biostatistics, UCLA Fielding School of Public HealthThis book, Spatial Analysis, by John Kent and Kanti Mardia is a must-read by all statisticians whose interests include spatial statistics, and it would make an excellent graduate-level text. The authors find the sweet spot at the juncture of intuition, methodology, and applications, and they show us in many ways how the validity of scientific inferences and its software has its genesis in the construction of valid statistical models.  — Noel Cressie, Distinguished Professor and Director, Centre for Environmental Informatics, University of Wollongong, AustraliaThe modern world of big data is often accompanied by little understanding. Often this data comes in a spatial form, and critically our understanding emerges from spatial analysis. It’s not possible to imagine two better guides to this domain than John Kent and Kanti Mardia. In Spatial Analysis Kent and Mardia provide a comprehensive guide to modern thinking that is classically grounded. This book is a must-read for those who are taking understanding seriously as part of handling modern spatial data sets across the domains of machine learning, statistics and data science.— Neil Lawrence, DeepMind Professor of Machine Learning, University of Cambridge[Spatial Analysis] is a delightful and authoritative book on the subject of spatial statistical analysis by two of the world’s most eminent researchers in the field of spatial statistics and shape analysis. The book eloquently discusses most of the topics in spatial analysis in a wonderfully organised manner. For example, there are two chapters on random fields, two chapters on estimation methods, one on modelling and another large chapter on kriging. With another chapter on additional topics such as Co-kriging, Bayesian hierarchical modeling, spatio-temporal modeling, and thin plate splines this book covers most of the concepts researchers need to know in this area. The main emphasis of the book is on theoretical aspects but it does not lose sight of applications. The chapter one itself motivates the theory with several example data sets which include fingerprint of the famous statistician Sir R A Fisher. The book does justice to the theory by presenting and explaining it in an accessible format for all – graduate students and researchers. The book also provides enjoyable to read personal historical notes and anecdotes regarding the course of development of the theory of spatial analysis.— Sujit Sahu, Professor of Mathematical Sciences, University of Southampton

• List of Figures xiiiList of Tables xviiPreface xixList of Notation and Terminology xxv1 Introduction 11.1 Spatial Analysis 11.2 Presentation of the Data 21.3 Objectives 91.4 The Covariance Function and Semivariogram 111.4.1 General Properties 111.4.2 Regularly Spaced Data 131.4.3 Irregularly Spaced Data 141.5 Behavior of the Sample Semivariogram 161.6 Some Special Features of Spatial Analysis 22Exercises 272 Stationary Random Fields 312.1 Introduction 312.2 Second Moment Properties 322.3 Positive Definiteness and the Spectral Representation 342.4 Isotropic Stationary Random Fields 362.5 Construction of Stationary Covariance Functions 412.6 Matérn Scheme 432.7 Other Examples of Isotropic Stationary Covariance Functions 452.8 Construction of Nonstationary Random Fields 482.8.1 Random Drift 482.8.2 Conditioning 492.9 Smoothness 492.10 Regularization 512.11 Lattice Random Fields 532.12 Torus Models 562.12.1 Models on the Continuous Torus 562.12.2 Models on the Lattice Torus 572.13 Long-range Correlation 582.14 Simulation 612.14.1 General Points 612.14.2 The Direct Approach 612.14.3 Spectral Methods 622.14.4 Circulant Methods 66Exercises 673 Intrinsic and Generalized Random Fields 733.1 Introduction 733.2 Intrinsic Random Fields of Order k = 0 743.3 Characterizations of Semivariograms 803.4 Higher Order Intrinsic Random Fields 833.5 Registration of Higher Order Intrinsic Random Fields 863.6 Generalized Random Fields 873.7 Generalized Intrinsic Random Fields of Intrinsic Order k ≥ 0 913.8 Spectral Theory for Intrinsic and Generalized Processes 913.9 Regularization for Intrinsic and Generalized Processes 953.10 Self-Similarity 963.11 Simulation 1003.11.1 General Points 1003.11.2 The Direct Method 1013.11.3 Spectral Methods 1013.12 Dispersion Variance 102Exercises 1044 Autoregression and Related Models 1154.1 Introduction 1154.2 Background 1184.3 Moving Averages 1204.3.1 Lattice Case 1204.3.2 Continuously Indexed Case 1214.4 Finite Symmetric Neighborhoods of the Origin in Z d 1224.5 Simultaneous Autoregressions (SARs) 1244.5.1 Lattice Case 1244.5.2 Continuously Indexed Random Fields 1254.6 Conditional Autoregressions (CARs) 1274.6.1 Stationary CARs 1284.6.2 Iterated SARs and CARs 1304.6.3 Intrinsic CARs 1314.6.4 CARs on a Lattice Torus 1324.6.5 Finite Regions 1324.7 Limits of CAR Models Under Fine Lattice Spacing 1344.8 Unilateral Autoregressions for Lattice Random Fields 1354.8.1 Half-spaces in Z d 1354.8.2 Unilateral Models 1364.8.3 Quadrant Autoregressions 1394.9 Markov Random Fields (MRFs) 1404.9.1 The Spatial Markov Property 1404.9.2 The Subset Expansion of the Negative Potential Function 1434.9.3 Characterization of Markov Random Fields in Terms of Cliques 1454.9.4 Auto-models 1474.10 Markov Mesh Models 1494.10.1 Validity 1494.10.2 Marginalization 1504.10.3 Markov Random Fields 1504.10.4 Usefulness 151Exercises 1515 Estimation of Spatial Structure 1595.1 Introduction 1595.2 Patterns of Behavior 1605.2.1 One-dimensional Case 1605.2.2 Two-dimensional Case 1615.2.3 Nugget Effect 1625.3 Preliminaries 1645.3.1 Domain of the Spatial Process 1645.3.2 Model Specification 1645.3.3 Spacing of Data 1655.4 Exploratory and Graphical Methods 1665.5 Maximum Likelihood for Stationary Models 1685.5.1 Maximum Likelihood Estimates – Known Mean 1695.5.2 Maximum Likelihood Estimates – Unknown Mean 1715.5.3 Fisher Information Matrix and Outfill Asymptotics 1725.6 Parameterization Issues for the Matérn Scheme 1735.7 Maximum Likelihood Examples for Stationary Models 1745.8 Restricted Maximum Likelihood (REML) 1795.9 Vecchia’s Composite Likelihood 1805.10 REML Revisited with Composite Likelihood 1825.11 Spatial Linear Model 1855.11.1 MLEs 1865.11.2 Outfill Asymptotics for the Spatial Linear Model 1885.12 REML for the Spatial Linear Model 1885.13 Intrinsic Random Fields 1895.14 Infill Asymptotics and Fractal Dimension 192Exercises 1956 Estimation for Lattice Models 2016.1 Introduction 2016.2 Sample Moments 2036.3 The AR(1) Process on Z 2056.4 Moment Methods for Lattice Data 2086.4.1 Moment Methods for Unilateral Autoregressions (UARs) 2096.4.2 Moment Estimators for Conditional Autoregression (CAR) Models 2106.5 Approximate Likelihoods for Lattice Data 2126.6 Accuracy of the Maximum Likelihood Estimator 2156.7 The Moment Estimator for a CAR Model 218Exercises 2197 Kriging 2317.1 Introduction 2317.2 The Prediction Problem 2337.3 Simple Kriging 2367.4 Ordinary Kriging 2387.5 Universal Kriging 2407.6 Further Details for the Universal Kriging Predictor 2417.6.1 Transfer Matrices 2417.6.2 Projection Representation of the Transfer Matrices 2427.6.3 Second Derivation of the Universal Kriging Predictor 2447.6.4 A Bordered Matrix Equation for the Transfer Matrices 2457.6.5 The Augmented Matrix Representation of the Universal Kriging Predictor 2457.6.6 Summary 2477.7 Stationary Examples 2487.8 Intrinsic Random Fields 2537.8.1 Formulas for the Kriging Predictor and Kriging Variance 2537.8.2 Conditionally Positive Definite Matrices 2547.9 Intrinsic Examples 2567.10 Square Example 2587.11 Kriging with Derivative Information 2597.12 Bayesian Kriging 2627.12.1 Overview 2627.12.2 Details for Simple Bayesian Kriging 2647.12.3 Details for Bayesian Kriging with Drift 2647.13 Kriging and Machine Learning 2667.14 The Link Between Kriging and Splines 2697.14.1 Nonparametric Regression 2697.14.2 Interpolating Splines 2717.14.3 Comments on Interpolating Splines 2737.14.4 Smoothing Splines 2747.15 Reproducing Kernel Hilbert Spaces 2747.16 Deformations 275Exercises 2778 Additional Topics 2838.1 Introduction 2838.2 Log-normal Random Fields 2848.3 Generalized Linear Spatial Mixed Models (GLSMMs) 2858.4 Bayesian Hierarchical Modeling and Inference 2868.5 Co-kriging 2878.6 Spatial–temporal Models 2918.6.1 General Considerations 2918.6.2 Examples 2928.7 Clamped Plate Splines 2948.8 Gaussian Markov Random Field Approximations 2958.9 Designing a Monitoring Network 296Exercises 298Appendix A Mathematical Background 303A. 1 Domains for Sequences and Functions 303A. 2 Classes of Sequences and Functions 305A.2. 1 Functions on the Domain Rd 305A.2. 2 Sequences on the Domain Zd 305A.2. 3 Classes of Functions on the Domain S d1 306A.2 4 Classes of Sequences on the Domain ZNd, Where N = (n[1], .,n[d]) 306A. 3 Matrix Algebra 306A.3. 1 The Spectral Decomposition Theorem 306A.3. 2 Moore–Penrose Generalized Inverse 307A.3. 3 Orthogonal Projection Matrices 308A.3. 4 Partitioned Matrices 308A.3. 5 Schur Product 309A.3. 6 Woodbury Formula for a Matrix Inverse 310A.3. 7 Quadratic Forms 311A.3. 8 Toeplitz and Circulant Matrices 311A.3. 9 Tensor Product Matrices 312A.3. 10 The Spectral Decomposition and Tensor Products 313A.3. 11 Matrix Derivatives 313A. 4 Fourier Transforms 313A. 5 Properties of the Fourier Transform 315A. 6 Generalizations of the Fourier Transform 318A. 7 Discrete Fourier Transform and Matrix Algebra 318A.7. 1 DFT in d = 1Dimension 318A.7. 2 Properties of the Unitary Matrix G, d = 1 319A.7. 3 Circulant Matrices and the DFT, d = 1 320A.7. 4 The Case d > 1 321A.7. 5 The Periodogram 322A. 8 Discrete Cosine Transform (DCT) 322A.8. 1 One-dimensional Case 322A.8. 2 The Case d > 1 323A.8. 3 Indexing for the Discrete Fourier and Cosine Transforms 323A. 9 Periodic Approximations to Sequences 324A. 10 Structured Matrices in d = 1Dimension 325A. 11 Matrix Approximations for an Inverse Covariance Matrix 327A.1. 1 The Inverse Covariance Function 328A.11. 2 The Toeplitz Approximation to Σ − 1 330A.11. 3 The Circulant Approximation to Σ − 1 330A.11. 4 The Folded Circulant Approximation to Σ − 1 330A.11. 5 Comments on the Approximations 331A.11. 6 Sparsity 332A. 12 Maximum Likelihood Estimation 332A.2. 1 General Considerations 332A.1. 2 The Multivariate Normal Distribution and the Spatial Linear Model 333A.12. 3 Change of Variables 335A.12. 4 Profile Log-likelihood 335A.12. 5 Confidence Intervals 336A.12. 6 Linked Parameterization 337A.12. 7 Model Choice 338A. 13 Bias in Maximum Likelihood Estimation 338A.3. 1 A General Result 338A.13. 2 The Spatial Linear Model 340Appendix B A Brief History of the Spatial Linear Model and the Gaussian Process Approach 347B.1 Introduction 347B.2 Matheron and Watson 348B.3 Geostatistics at Leeds 1977–1987 349B.3.1 Courses, Publications, Early Dissemination 349B.3.2 Numerical Problems with Maximum Likelihood 351B.4 Frequentist vs. Bayesian Inference 352References and Author Index 355Index 367