Basics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsBasics of Point-Referenced Data ModelsStationarity 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 exponential modelNotes on exponential modelNotes on exponential modelNotes on exponential model (cont'd)Notes on exponential model (cont'd)Notes on exponential model (cont'd)Variogram model fittingVariogram model fittingVariogram 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)EDA for point-referenced dataEDA for point-referenced dataEDA for point-referenced dataEDA 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)Kriging 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 that of a spatial process, {Y (s), s ∈ D},where D ⊂ 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. 1/34StationaritySuppose 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 stationary (also calledstrongly stationary) if, for any given n ≥ 1, any set of nsites {s1, . . . , sn} and any h ∈ ℜr,the distribution of(Y (s1) , . . . , Y (sn))is the same as that of(Y (s1+ h) , . . . , Y (sn+ h)).A less restrictive condition is given by weak stationarity(also called second-order stationarity): A process isweakly stationary if µ (s) ≡ µ andCov (Y (s) , Y (s + h)) = C (h) for all h ∈ ℜrsuch that sand s + h both lie within D.Chapter 2: Basics of Point-Referenced Data Models – p. 2/34Notes on StationarityWeak stationarity implies that the covariancerelationship between the values of the process at anytwo locations can be summarized by a covariancefunction C (h) (sometimes called a covariogram), andthis function dependsonly on the separation vector h.Note that with all variances assumed to exist, strongstationarity implies weak stationarity.The converse is not true in general, but it does hold forGaussian processesChapter 2: Basics of Point-Referenced Data Models – p. 3/34VariogramsSuppose we assume E[Y (s + h) − Y (s)] = 0 and defineE[Y (s + h) − Y (s)]2= V ar (Y (s + h) − Y (s)) = 2γ (h) .This expression makes sense only if the left-hand sidedepends only on h (so that the right-hand side can bewritten at all), and not the particular choice of s. If this isthe case, we say the process isintrinsically stationary.The function 2γ (h) is then called the variogram, andγ (h) is called thesemivariogram.Note that intrinsic stationarity definesonly the first andsecond moments of the differences Y (s + h) − Y (s). It saysnothing about thejoint distribution of a collection ofvariables Y (s1), . . . , Y (sn), and thus providesno likelihood.Chapter 2: Basics of Point-Referenced Data Models – p. 4/34Relationship between C(h) and γ(h)2γ(h) = V ar (Y (s + h) − Y (s))= V ar(Y (s + h)) + V ar(Y (s)) − 2Cov(Y (s + h), Y (s))= C(0) + C(0) − 2C (h)= 2 [C (0) − C (h)] .Thus,γ (h) = C (0) − C (h) .So given C, we are able to recover γ easily.But what about the converse: in general, can we recover Cfrom γ?...Chapter 2: Basics of Point-Referenced Data Models – p. 5/34Relationship between C(h) and γ(h)If the spatial process is ergodic, then C (h) → 0 as||h|| → ∞, where ||h|| denotes the length of the h vector.Taking the limit of both sides of γ (h) = C (0) − C (h) as||h|| → ∞, we then have thatlim||h||→∞γ (h) = C (0).Thus, using the dummy variable u to avoid confusion,C (h) = C(0) − γ(h) = lim||u||→∞γ (u) − γ (h) .Thus if the limit on the right-hand side exists, the process isweakly (second-order) stationary with C (h) as given above.We then have a way to determine C from γ.Thus weak stationarity implies intrinsic stationarity, but theconverse isnot true in generalChapter 2: Basics of Point-Referenced Data Models – p.