Basics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsScallops catch sites, NY/NJ coast, USAScallops sites with log-catch contoursImage plot with log-catch contoursStationarity Stationarity Notes on Stationarity Notes on Stationarity Notes on Stationarity VariogramsVariogramsVariogramsRelationship between $C(h )$ and $gamma (h )$Relationship between $C(h )$and $gamma (h )$Relationship between $C(h )$and $gamma (h )$Relationship between $C(h )$ and $gamma (h )$Relationship between $C(h )$and $gamma (h )$Relationship between $C(h )$and $gamma (h )$Relationship between $C(h )$and $gamma (h )$Relationship between $C(h )$and $gamma (h )$IsotropyIsotropyIsotropyIsotropySome common isotropic covariogramsSome common isotropic variogramsExample: Spherical semivariogramExample: Spherical semivariogramExample: Spherical semivariogram3 common semivariogram models3 common semivariogram modelsNotes on the exponential modelNotes on the exponential modelNotes on the exponential modelNotes on the exponential modelNotes on the exponential modelThe Mat`{e}rn Correlation FunctionThe Mat`{e}rn Correlation FunctionThe Mat`{e}rn Correlation FunctionThe Mat`{e}rn Correlation FunctionSome covariance function theorySome covariance function theorySome covariance function theorySome covariance function theorySome covariance function theoryConstructing valid covariance functionsConstructing valid covariance functionsConstructing valid covariance functionsVariogram model fittingVariogram model fittingEmpirical variogram: scallops dataVariogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)Variogram model fitting (cont'd)AnisotropyAnisotropyAnisotropyAnisotropyAnisotropyAnisotropy (cont'd)Anisotropy (cont'd)Anisotropy (cont'd)Anisotropy (cont'd)Exploration of Spatial DataExploration of Spatial DataExploration of Spatial DataExploration of Spatial DataExploration of Spatial DataExploration of Spatial DataFirst Law of GeostatisticsArrangement of plotsDrop-line scatter plotSurface plotImage-contour plotEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEDA for assessing anisotropyEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsEmpirical semivariogram contour plotsESC plot of the 1993 scallop dataESC plot of the 1993 scallop dataESC plot of the 1993 scallop dataClassical spatial prediction (Kriging)Classical spatial prediction (Kriging)Classical spatial prediction (Kriging)Classical spatial prediction (Kriging)Classical spatial prediction (Kriging)DifficultiesDifficultiesDifficultiesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesKriging with Gaussian processesBasics of Point-Referenced Data ModelsBasic tool is a spatial process, {Y (s), s ∈ D}, whereD ⊂ ℜrChapter 2: Basics of Point-Referenced Data Models – p. 1/45Basics of Point-Referenced Data ModelsBasic tool is a spatial process, {Y (s), s ∈ D}, whereD ⊂ ℜrNote that time series follows this approach with r = 1;we will usually have r = 2 or 3Chapter 2: Basics of Point-Referenced Data Models – p. 1/45Basics of Point-Referenced Data ModelsBasic tool is a spatial process, {Y (s), s ∈ D}, whereD ⊂ ℜrNote that time series follows this approach with r = 1;we will usually have r = 2 or 3We begin with essentials of point-level data modeling,including stationarity, isotropy, and variograms – keyelements of the "Matheron school"Chapter 2: Basics of Point-Referenced Data Models – p. 1/45Basics of Point-Referenced Data ModelsBasic tool is a spatial process, {Y (s), s ∈ D}, whereD ⊂ ℜrNote that time series follows this approach with r = 1;we will usually have r = 2 or 3We begin with essentials of point-level data modeling,including stationarity, isotropy, and variograms – keyelements of the "Matheron school"Then we add the spatial (typically Gaussian) processmodeling that enables likelihood (and Bayesian)inference in these settings.Chapter 2: Basics of Point-Referenced Data Models – p. 1/45Scallops catch sites, NY/NJ coast, USA••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••• •••••••••••• ••••••••Chapter 2: Basics of Point-Referenced Data Models – p. 2/45Scallops sites with log-catch contours••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••• •••••••••••• ••••••••222444444666666Chapter 2: Basics of Point-Referenced Data Models – p. 3/45Image plot with log-catch contours−73.5 −73.0 −72.5 −72.039.0 39.5 40.0 40.5LongitudeLatitudeChapter 2: Basics of Point-Referenced Data Models – p. 4/45StationaritySuppose our spatial process has a mean, µ (s) = E (Y (s)),and that the variance of Y (s) exists for all s ∈ D.The process is said to be strictly