Spatial modelsISet upIClassifying spatial dataIModeling areal dataIModeling point-referenced dataISpatial interpolationPage 1 of 31Set upAnalysis of spatially distributed data is an area that has beenheavily used in practical applications, specially over the lastdecades:IRipley (1981) - 252 pagesICressie (1993) - 900 pagesThe models start by considering a region with sites or pixels{s1, . . . , sd}.For each site, a variable of interest yiis observed with mean h(θi),for some function hE(yi) = h(θi)Page 2 of 31Typically, it is assumed that there is an underlying pattern formedby the values of θ = (θ1, . . . , θd)0associated with the observedscene.This pattern is corrupted by some random observation mechanism.The main effort of the inference is to remove this observationalnoise and recover the underlying, unobserved scene.Most models commonly assume observational independenceconditional on the unobserved image θ , ie.l(θ)=f (y1, . . . , yd|θ1, . . . , θd)=Yif (yi|θi)Other effects:IExplanatory variables - fixed variation;IUnstructured pixelwise random effects to account forunspecified data heterogeneity.Page 3 of 31Classifying spatial dataIPoint-referenced datay(s) is a random vector at location s, which variescontinuously over fixed region D.IAreal dataFixed (regular or irregular) region D partitioned intoa finite number of areal units with well definedboundaries.IPoint process dataRandom locations in D.Page 4 of 31Point-referenced dataFigure 1.1 from Banerjee et al. (2004) - Map of particulate matter less than 2.5microns in diameter (PM2.5, measured in ppb) sampling (air pollution monitoring)sites over three midwestern U.S. states: plotting character indicates range of averagemonitored PM2.5 level over the year 2001.Page 5 of 31Point process dataFigure 1.1 from Bognar (2005) - Location of 124 Norway spruce trees in a 56 × 38meter field. Character size is proportional to the trunk diameter.Page 6 of 31Areal dataNobre, Schmidt and Lopes (2005) Spatio-temporal models for mapping the incidenceof malaria in Par´a, northern Brazil.Relative RiskExcluded0 - 22 - 55 - 1010 - 90Page 7 of 31US continental statesPage 8 of 31Adjacency matrixPage 9 of 31Modeling areal dataThe spatial structure is specified through the prior p(θ).Besag, York and Molli´e (1991) suggest a pairwise difference (PD)prior formp(θ) ∝ exp−Xi<jwijh(θi− θj),where the proportionality constant is not uniquely defined due toits degeneracy.They consider h(x) = x2/2W leading to a singular normaldistributionθ ∼ PD(w, W )where the proportionality constant now includes the term W−d/2.Page 10 of 31Local dependenceIt is easy to see that, for i = 1, . . . , d,θi|θ−i, W ∼ N(ai, Ri)whereai=1nidXj=1wijθjand Ri=1niW ,where ni=Pdj=1wij.θis depend on their spatial neighbours just like in a temporalautoregressive structure.PD forms are usually referred to as conditional autoregressive(CAR) models.Page 11 of 31Choosing wijA typical choice of weights iswij= I (sj∈ Ni)where Nidefines a neighbourhood of si, i = 1, . . . , d.Other choices of wijpossibly depending on unknownhyperparameters may also be used.Gamerman and Moreira (2004) consideredwij∝1dτijwhere dijis some distance between pixels i and j and τ is anunknown quantity measuring the strength of the spatialdependence.The distribution ab ove can also be identified with Gaussian Markovrandom fields (GMRF).Page 12 of 31Gaussian random fields (GRF)A space-varying process θ(·) taking values in a region S follows aGRF if θ = (θ(s1), . . . , θ(sd)) possesses a d-variate normaldistribution for any d and any given set of d locations s1, . . . , sdinS.The identity θ(si) = θi, for all i, is frequently used, especially whenthe locations sirefer to areas contained in S rather than pointlocations belonging to S.If the dependence between them is provided by the non-zeroweights, a Markovian structure in space is defined.Page 13 of 31Markov random fields (MRF)A collection {θ1, θ2, . . .} is a MRF if the full conditionaldistributions of θidepend only on θjfor j ∈ Ni, the set ofneighbors of i, i = 1, 2, . . .Page 14 of 31Hierarchical structureA common choice for observation model is given by the normaldistribution.It provides suitable representation (possibly after sometransformation) of (i) crop output; (ii) economic indices; orclimatic indicators.IFirst level: data structurey|θ, σ2∼ N(θ, σ2Id)ISecond level: spatial structureθ|W ∼ CAR(w, W )IThird level: hyperparametersσ2∼ IGµnσ2,nσSσ2¶W ∼ IGµnW2,nWSW2¶Page 15 of 31Full conditionalsIs is straightforward to see thatσ2∼ IG(n∗σ/2, n∗σS∗σ/2)W ∼ IG (n∗W/2, n∗WS∗W/2)θi∼ N(mi, Ci)for i = 1, . . . , d, withn∗σ= nσ+ dn∗σS∗σ= nσSσ+X(yi− θi)2n∗W= nW+ dn∗WS∗W= nWSW+Xwij(θi− θj)2mi= Ci(σ−2yi+ W−1Xwijθj)C−1i= σ−2+ niW−1for i = 1, . . . , d.Page 16 of 31Computational issuesIn practical applications the value of d is usually large, makingdirect sampling of θ very slow.Rue (2001) suggested numerical techniques to take advantage ofthe sparseness (presence of many 0s) of C−1(θ’s precision matrix)and hence improve sampling considerably.These techniques basically involve appropriate reordering of thepixels to ensure a minimal diagonal band structure to C−1followedby a numerically efficient Cholesky decomposition that takesadvantage of this band diagonalization.See Gamerman, Moreira and Rue (2003) and Knorr-Held and Rue(2002) for further discussion on blocking advantages anddisadvantages.Page 17 of 31Page 18 of 31ExtensionsHigdon and Besag (1999) considered the inclusion of a regressionterm X β in the mean structure of the observations:y|θ, σ2∼ N(X β + θ, σ2Id) ;Congdon (1997) introduced the idea of allowing spatial variationalso to regression coefficients β;Assun¸c˜ao, Gamerman and Assun¸c˜ao (1999) used the multivariateformp(β) ∝ exp−Xi<jwij(βi− βj)0W−1(βi− βj)for jointly modelling the prior for space-varying regressioncoefficients β = (β1, . . . , βd).Observations in the exponential family of distributions (Banerjee,Carlin and Gelfand, 2004, and Rue and Held, 2005).Page 19 of 31Modeling point-referenced dataUnder the usual simplifying assumptions of common mean,homoscedasticity (common variance) and isotropy (correlationdepends only on distance),θ ∼ N(µ1d, τ2Rλ)whereRλ= (ρij)withρij= ρλ(|si− sj|)for some suitably defined correlation function
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