José-María Montero and Gema Fernández-Avilés, Department of Statistics, University of Castilla-La Mancha, SpainJorge Mateu,Department of Mathematics, University Jaume I of Castellon, Spain

• List of figures xiList of tables xviiForeword xixPreface xxiThe companion website xxiii1 From classical statistics to geostatistics 11.1 Not all spatial data are geostatistical data 11.2 The limits of classical statistics 51.3 A real geostatistical dataset: data on carbon monoxide in Madrid, Spain 72 Geostatistics: preliminaries 102.1 Regionalized variables 102.2 Random functions 112.3 Stationary and intrinsic hypotheses 132.3.1 Stationarity 132.3.2 Stationary random functions in the strict sense 142.3.3 Second-order stationary random functions 152.3.4 Intrinsically stationary random functions 162.3.5 Non-stationary random functions 182.4 Support 193 Structural analysis 203.1 Introduction 203.2 Covariance function 213.2.1 Definition and properties 213.2.2 Some theoretical isotropic covariance functions 233.3 Empirical covariogram 263.4 Semivariogram 273.4.1 Definition and properties 273.4.2 Behavior at intermediate and large distances 303.4.3 Behavior near the origin 313.4.4 A discontinuity at the origin 333.5 Theoretical semivariogram models 353.5.1 Semivariograms with a sill 363.5.2 Semivariograms with a hole effect 463.5.3 Semivariograms without a sill 473.5.4 Combining semivariogram models 503.6 Empirical semivariogram 523.7 Anisotropy 643.8 Fitting a semivariogram model 693.8.1 Manual fitting 703.8.2 Automatic fitting 714 Spatial prediction and kriging 804.1 Introduction 804.2 Neighborhood 834.3 Ordinary kriging 844.3.1 Point observation support and point predictor 844.3.2 Effects of a change in the model parameters 904.3.3 Point observation support and block predictor 994.3.4 Block observation support and block predictor 1104.4 Simple kriging: the special case of known mean 1134.5 Simple kriging with an estimated mean 1154.6 Universal kriging 1164.6.1 Point observation support and point predictor 1164.6.2 Point observation support and block predictor 1214.6.3 Block observation support and block predictor 1214.6.4 Kriging and exact interpolation 1224.7 Residual kriging 1224.7.1 Direct residual kriging 1234.7.2 Iterative residual kriging 1244.7.3 Modified iterative residual kriging 1254.8 Median-Polish kriging 1254.9 Cross-validation 1344.10 Non-linear kriging 1384.10.1 Disjunctive kriging 1384.10.2 Indicator kriging 1425 Geostatistics and spatio-temporal random functions 1455.1 Spatio-temporal geostatistics 1455.2 Spatio-temporal continuity 1465.3 Relevant spatio-temporal concepts 1475.4 Properties of the spatio-temporal covariance and semivariogram 1576 Spatio-temporal structural analysis (I): empirical semivariogramand covariogram estimation and model fitting 1626.1 Introduction 1626.2 The empirical spatio-temporal semivariogram and covariogram 1636.3 Fitting spatio-temporal semivariogram and covariogram models 1706.4 Validation and comparison of spatio-temporal semivariogram and covariogram models 1747 Spatio-temporal structural analysis (II): theoretical covariance models 1787.1 Introduction 1787.2 Combined distance or metric model 1807.3 Sum model 1837.4 Combined metric-sum model 1847.5 Product model 1877.6 Product-sum model 1917.7 Porcu and Mateu mixture-based models 1927.8 General product-sum model 1947.9 Integrated product and product-sum models 1987.10 Models proposed by Cressie and Huang 2017.11 Models proposed by Gneiting 2077.12 Mixture models proposed by Ma 2117.12.1 Covariance functions generated by scale mixtures 2117.12.2 Covariance functions generated by positive power mixtures 2127.13 Models generated by linear combinations proposed by Ma 2157.14 Models proposed by Stein 2227.15 Construction of covariance functions using copulas and completely monotonic functions 2237.16 Generalized product-sum model 2237.17 Models that are not fully symmetric 2367.18 Mixture-based Bernstein zonally anisotropic covariance functions 2377.19 Non-stationary models 2417.19.1 Mixture of locally orthogonal stationary processes 2417.19.2 Non-stationary models proposed by Ma 2427.19.3 Non-stationary models proposed by Porcu and Mateu 2467.20 Anisotropic covariance functions by Porcu and Mateu 2477.20.1 Constructing temporally symmetric and spatially anisotropic covariance functions 2477.20.2 Generalizing the class of spatio-temporal covariance functions proposed by Gneiting 2487.20.3 Differentiation and integration operators acting on classes of anisotropic covariance functions on the basis of isotropic components: ‘La descente étendue’ 2517.21 Spatio-temporal constructions based on quasi-arithmetic means of covariance functions 2537.21.1 Multivariate quasi-arithmetic compositions 2557.21.2 Permissibility criteria for quasi-arithmetic means of covariance functions in ℝd 2567.21.3 The use of quasi-arithmetic functionals to build non-separable, stationary, spatio-temporal covariance functions 2597.21.4 Quasi-arithmeticity and non-stationarity in space 2648 Spatio-temporal prediction and kriging 2668.1 Spatio-temporal kriging 2668.2 Spatio-temporal kriging equations 2679 An introduction to functional geostatistics 2749.1 Functional data analysis 2749.2 Functional geostatistics: The parametric vs. the non-parametric approach 2799.3 Functional ordinary kriging 2839.3.1 Preliminaries 2839.3.2 Functional ordinary kriging equations 2849.3.3 Estimating the trace-semivariogram 2889.3.4 Functional cross-validation 289A Spectral representations 295B Probabilistic aspects of Uij = Z(si)−Z(sj) 300C Basic theory on restricted maximum likelihood 302D Most relevant proofs 304Bibliography and further reading 327Index 351

José-María Montero and Gema Fernández-Avilés, Department of Statistics, University of Castilla-La Mancha, SpainJorge Mateu,Department of Mathematics, University Jaume I of Castellon, Spain

• List of figures xiList of tables xviiForeword xixPreface xxiThe companion website xxiii1 From classical statistics to geostatistics 11.1 Not all spatial data are geostatistical data 11.2 The limits of classical statistics 51.3 A real geostatistical dataset: data on carbon monoxide in Madrid, Spain 72 Geostatistics: preliminaries 102.1 Regionalized variables 102.2 Random functions 112.3 Stationary and intrinsic hypotheses 132.3.1 Stationarity 132.3.2 Stationary random functions in the strict sense 142.3.3 Second-order stationary random functions 152.3.4 Intrinsically stationary random functions 162.3.5 Non-stationary random functions 182.4 Support 193 Structural analysis 203.1 Introduction 203.2 Covariance function 213.2.1 Definition and properties 213.2.2 Some theoretical isotropic covariance functions 233.3 Empirical covariogram 263.4 Semivariogram 273.4.1 Definition and properties 273.4.2 Behavior at intermediate and large distances 303.4.3 Behavior near the origin 313.4.4 A discontinuity at the origin 333.5 Theoretical semivariogram models 353.5.1 Semivariograms with a sill 363.5.2 Semivariograms with a hole effect 463.5.3 Semivariograms without a sill 473.5.4 Combining semivariogram models 503.6 Empirical semivariogram 523.7 Anisotropy 643.8 Fitting a semivariogram model 693.8.1 Manual fitting 703.8.2 Automatic fitting 714 Spatial prediction and kriging 804.1 Introduction 804.2 Neighborhood 834.3 Ordinary kriging 844.3.1 Point observation support and point predictor 844.3.2 Effects of a change in the model parameters 904.3.3 Point observation support and block predictor 994.3.4 Block observation support and block predictor 1104.4 Simple kriging: the special case of known mean 1134.5 Simple kriging with an estimated mean 1154.6 Universal kriging 1164.6.1 Point observation support and point predictor 1164.6.2 Point observation support and block predictor 1214.6.3 Block observation support and block predictor 1214.6.4 Kriging and exact interpolation 1224.7 Residual kriging 1224.7.1 Direct residual kriging 1234.7.2 Iterative residual kriging 1244.7.3 Modified iterative residual kriging 1254.8 Median-Polish kriging 1254.9 Cross-validation 1344.10 Non-linear kriging 1384.10.1 Disjunctive kriging 1384.10.2 Indicator kriging 1425 Geostatistics and spatio-temporal random functions 1455.1 Spatio-temporal geostatistics 1455.2 Spatio-temporal continuity 1465.3 Relevant spatio-temporal concepts 1475.4 Properties of the spatio-temporal covariance and semivariogram 1576 Spatio-temporal structural analysis (I): empirical semivariogramand covariogram estimation and model fitting 1626.1 Introduction 1626.2 The empirical spatio-temporal semivariogram and covariogram 1636.3 Fitting spatio-temporal semivariogram and covariogram models 1706.4 Validation and comparison of spatio-temporal semivariogram and covariogram models 1747 Spatio-temporal structural analysis (II): theoretical covariance models 1787.1 Introduction 1787.2 Combined distance or metric model 1807.3 Sum model 1837.4 Combined metric-sum model 1847.5 Product model 1877.6 Product-sum model 1917.7 Porcu and Mateu mixture-based models 1927.8 General product-sum model 1947.9 Integrated product and product-sum models 1987.10 Models proposed by Cressie and Huang 2017.11 Models proposed by Gneiting 2077.12 Mixture models proposed by Ma 2117.12.1 Covariance functions generated by scale mixtures 2117.12.2 Covariance functions generated by positive power mixtures 2127.13 Models generated by linear combinations proposed by Ma 2157.14 Models proposed by Stein 2227.15 Construction of covariance functions using copulas and completely monotonic functions 2237.16 Generalized product-sum model 2237.17 Models that are not fully symmetric 2367.18 Mixture-based Bernstein zonally anisotropic covariance functions 2377.19 Non-stationary models 2417.19.1 Mixture of locally orthogonal stationary processes 2417.19.2 Non-stationary models proposed by Ma 2427.19.3 Non-stationary models proposed by Porcu and Mateu 2467.20 Anisotropic covariance functions by Porcu and Mateu 2477.20.1 Constructing temporally symmetric and spatially anisotropic covariance functions 2477.20.2 Generalizing the class of spatio-temporal covariance functions proposed by Gneiting 2487.20.3 Differentiation and integration operators acting on classes of anisotropic covariance functions on the basis of isotropic components: ‘La descente étendue’ 2517.21 Spatio-temporal constructions based on quasi-arithmetic means of covariance functions 2537.21.1 Multivariate quasi-arithmetic compositions 2557.21.2 Permissibility criteria for quasi-arithmetic means of covariance functions in ℝd 2567.21.3 The use of quasi-arithmetic functionals to build non-separable, stationary, spatio-temporal covariance functions 2597.21.4 Quasi-arithmeticity and non-stationarity in space 2648 Spatio-temporal prediction and kriging 2668.1 Spatio-temporal kriging 2668.2 Spatio-temporal kriging equations 2679 An introduction to functional geostatistics 2749.1 Functional data analysis 2749.2 Functional geostatistics: The parametric vs. the non-parametric approach 2799.3 Functional ordinary kriging 2839.3.1 Preliminaries 2839.3.2 Functional ordinary kriging equations 2849.3.3 Estimating the trace-semivariogram 2889.3.4 Functional cross-validation 289A Spectral representations 295B Probabilistic aspects of Uij = Z(si)−Z(sj) 300C Basic theory on restricted maximum likelihood 302D Most relevant proofs 304Bibliography and further reading 327Index 351

José-María Montero and Gema Fernández-Avilés, Department of Statistics, University of Castilla-La Mancha, SpainJorge Mateu,Department of Mathematics, University Jaume I of Castellon, Spain

• List of figures xiList of tables xviiForeword xixPreface xxiThe companion website xxiii1 From classical statistics to geostatistics 11.1 Not all spatial data are geostatistical data 11.2 The limits of classical statistics 51.3 A real geostatistical dataset: data on carbon monoxide in Madrid, Spain 72 Geostatistics: preliminaries 102.1 Regionalized variables 102.2 Random functions 112.3 Stationary and intrinsic hypotheses 132.3.1 Stationarity 132.3.2 Stationary random functions in the strict sense 142.3.3 Second-order stationary random functions 152.3.4 Intrinsically stationary random functions 162.3.5 Non-stationary random functions 182.4 Support 193 Structural analysis 203.1 Introduction 203.2 Covariance function 213.2.1 Definition and properties 213.2.2 Some theoretical isotropic covariance functions 233.3 Empirical covariogram 263.4 Semivariogram 273.4.1 Definition and properties 273.4.2 Behavior at intermediate and large distances 303.4.3 Behavior near the origin 313.4.4 A discontinuity at the origin 333.5 Theoretical semivariogram models 353.5.1 Semivariograms with a sill 363.5.2 Semivariograms with a hole effect 463.5.3 Semivariograms without a sill 473.5.4 Combining semivariogram models 503.6 Empirical semivariogram 523.7 Anisotropy 643.8 Fitting a semivariogram model 693.8.1 Manual fitting 703.8.2 Automatic fitting 714 Spatial prediction and kriging 804.1 Introduction 804.2 Neighborhood 834.3 Ordinary kriging 844.3.1 Point observation support and point predictor 844.3.2 Effects of a change in the model parameters 904.3.3 Point observation support and block predictor 994.3.4 Block observation support and block predictor 1104.4 Simple kriging: the special case of known mean 1134.5 Simple kriging with an estimated mean 1154.6 Universal kriging 1164.6.1 Point observation support and point predictor 1164.6.2 Point observation support and block predictor 1214.6.3 Block observation support and block predictor 1214.6.4 Kriging and exact interpolation 1224.7 Residual kriging 1224.7.1 Direct residual kriging 1234.7.2 Iterative residual kriging 1244.7.3 Modified iterative residual kriging 1254.8 Median-Polish kriging 1254.9 Cross-validation 1344.10 Non-linear kriging 1384.10.1 Disjunctive kriging 1384.10.2 Indicator kriging 1425 Geostatistics and spatio-temporal random functions 1455.1 Spatio-temporal geostatistics 1455.2 Spatio-temporal continuity 1465.3 Relevant spatio-temporal concepts 1475.4 Properties of the spatio-temporal covariance and semivariogram 1576 Spatio-temporal structural analysis (I): empirical semivariogramand covariogram estimation and model fitting 1626.1 Introduction 1626.2 The empirical spatio-temporal semivariogram and covariogram 1636.3 Fitting spatio-temporal semivariogram and covariogram models 1706.4 Validation and comparison of spatio-temporal semivariogram and covariogram models 1747 Spatio-temporal structural analysis (II): theoretical covariance models 1787.1 Introduction 1787.2 Combined distance or metric model 1807.3 Sum model 1837.4 Combined metric-sum model 1847.5 Product model 1877.6 Product-sum model 1917.7 Porcu and Mateu mixture-based models 1927.8 General product-sum model 1947.9 Integrated product and product-sum models 1987.10 Models proposed by Cressie and Huang 2017.11 Models proposed by Gneiting 2077.12 Mixture models proposed by Ma 2117.12.1 Covariance functions generated by scale mixtures 2117.12.2 Covariance functions generated by positive power mixtures 2127.13 Models generated by linear combinations proposed by Ma 2157.14 Models proposed by Stein 2227.15 Construction of covariance functions using copulas and completely monotonic functions 2237.16 Generalized product-sum model 2237.17 Models that are not fully symmetric 2367.18 Mixture-based Bernstein zonally anisotropic covariance functions 2377.19 Non-stationary models 2417.19.1 Mixture of locally orthogonal stationary processes 2417.19.2 Non-stationary models proposed by Ma 2427.19.3 Non-stationary models proposed by Porcu and Mateu 2467.20 Anisotropic covariance functions by Porcu and Mateu 2477.20.1 Constructing temporally symmetric and spatially anisotropic covariance functions 2477.20.2 Generalizing the class of spatio-temporal covariance functions proposed by Gneiting 2487.20.3 Differentiation and integration operators acting on classes of anisotropic covariance functions on the basis of isotropic components: ‘La descente étendue’ 2517.21 Spatio-temporal constructions based on quasi-arithmetic means of covariance functions 2537.21.1 Multivariate quasi-arithmetic compositions 2557.21.2 Permissibility criteria for quasi-arithmetic means of covariance functions in ℝd 2567.21.3 The use of quasi-arithmetic functionals to build non-separable, stationary, spatio-temporal covariance functions 2597.21.4 Quasi-arithmeticity and non-stationarity in space 2648 Spatio-temporal prediction and kriging 2668.1 Spatio-temporal kriging 2668.2 Spatio-temporal kriging equations 2679 An introduction to functional geostatistics 2749.1 Functional data analysis 2749.2 Functional geostatistics: The parametric vs. the non-parametric approach 2799.3 Functional ordinary kriging 2839.3.1 Preliminaries 2839.3.2 Functional ordinary kriging equations 2849.3.3 Estimating the trace-semivariogram 2889.3.4 Functional cross-validation 289A Spectral representations 295B Probabilistic aspects of Uij = Z(si)−Z(sj) 300C Basic theory on restricted maximum likelihood 302D Most relevant proofs 304Bibliography and further reading 327Index 351