Spatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsEmpirical Orthogonal FunctionsEmpirical Orthogonal FunctionsEmpirical Orthogonal FunctionsEmpirical Orthogonal FunctionsSpatio-temporal ModelsSpatio-temporal ModelsSpatio-temporal ModelsAreal unit dataAreal unit dataAreal unit dataAreal unit dataNeighbors in time and spaceTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaTraffic density and pediatric asthmaLarge datasetsLarge datasetsLarge datasetsLarge datasetsLarge datasetsSpatio-temporal ModelsAgain point-referenced vs. areal unit dataChapter 8: Spatiotemporal Modeling – p. 1/13Spatio-temporal ModelsAgain point-referenced vs. areal unit dataContinuous time vs. discretized timeChapter 8: Spatiotemporal Modeling – p. 1/13Spatio-temporal ModelsAgain point-referenced vs. areal unit dataContinuous time vs. discretized timeAssociation in space vs. association in time!For point-referenced data, t continuous, Gaussian data,Y (s, t) = µ(s, t) + w(s, t) + ǫ(s, t)Chapter 8: Spatiotemporal Modeling – p. 1/13Spatio-temporal ModelsAgain point-referenced vs. areal unit dataContinuous time vs. discretized timeAssociation in space vs. association in time!For point-referenced data, t continuous, Gaussian data,Y (s, t) = µ(s, t) + w(s, t) + ǫ(s, t)For non-Gaussian data, instead use appropriatelikelihood with link g(E(Y (s, t))) = µ(s, t) + w(s, t)Chapter 8: Spatiotemporal Modeling – p. 1/13Spatio-temporal ModelsAgain point-referenced vs. areal unit dataContinuous time vs. discretized timeAssociation in space vs. association in time!For point-referenced data, t continuous, Gaussian data,Y (s, t) = µ(s, t) + w(s, t) + ǫ(s, t)For non-Gaussian data, instead use appropriatelikelihood with link g(E(Y (s, t))) = µ(s, t) + w(s, t)Don’t treat time as a third coordinate – scale issue!sensible: Cov(Y (s, t), Y (s′, t′)) = C(s − s′, t − t′)NOT sensible: Cov(Y (s, t), Y (s′, t′)) = C((s, t) − (s′, t′))Chapter 8: Spatiotemporal Modeling – p. 1/13Spatio-temporal ModelsSeparable form:C(s − s′, t − t′) = σ2ρ1(s − s′; φ1)ρ2(t − t′; φ2)Chapter 8: Spatiotemporal Modeling – p. 2/13Spatio-temporal ModelsSeparable form:C(s − s′, t − t′) = σ2ρ1(s − s′; φ1)ρ2(t − t′; φ2)Nonseparable form:Sum of independent separable processesMixing of separable covariance functionsSpectral domain approachesChapter 8: Spatiotemporal Modeling – p. 2/13Spatio-temporal ModelsNow suppose time is discretized, i.e. data areYt(s), t = 1, . . . , TChapter 8: Spatiotemporal Modeling – p. 3/13Spatio-temporal ModelsNow suppose time is discretized, i.e. data areYt(s), t = 1, . . . , TType of data: time series versus cross-sectional (e.g.,real estate sales)Chapter 8: Spatiotemporal Modeling – p. 3/13Spatio-temporal ModelsNow suppose time is discretized, i.e. data areYt(s), t = 1, . . . , TType of data: time series versus cross-sectional (e.g.,real estate sales)For time series data, exploratory analysis:Arrange into an n × T matrix Y with entries Yt(si)Center by row averages of Y yields YrowsCenter by column averages of Y yields Ycolssample spatial covariance matrix:1TYrowsYTrowssample autocorrelation matrix:1nYTcolsYcolsE, residuals matrix after a regression fittingChapter 8: Spatiotemporal Modeling – p. 3/13Empirical Orthogonal FunctionsCan understand the structure of Y, Yrows, Ycols, E usingempirical orthogonal functions:Chapter 8: Spatiotemporal Modeling – p. 4/13Empirical Orthogonal FunctionsCan understand the structure of Y, Yrows, Ycols, E usingempirical orthogonal functions:Say for Y and T < n, use singular value decomposition,Y = UDVT=TXj=1djujvTj,where U is n × n orthogonal, V is T × T orthogonal andD is a T × T diagonal matrix augmented with n − Trows of 0’sChapter 8: Spatiotemporal Modeling – p. 4/13Empirical Orthogonal FunctionsCan understand the structure of Y, Yrows, Ycols, E usingempirical orthogonal functions:Say for Y and T < n, use singular value decomposition,Y = UDVT=TXj=1djujvTj,where U is n × n orthogonal, V is T × T orthogonal andD is a T × T diagonal matrix augmented with n − Trows of 0’sIf we arrange the djin decreasing order then ujvTjisthe jth empirical orthogonal functionChapter 8: Spatiotemporal Modeling – p. 4/13Empirical Orthogonal FunctionsCan understand the structure of Y, Yrows, Ycols, E usingempirical orthogonal functions:Say for Y and T < n, use singular value decomposition,Y = UDVT=TXj=1djujvTj,where U is n × n orthogonal, V is T × T orthogonal andD is a T × T diagonal matrix augmented with n − Trows of 0’sIf we arrange the djin decreasing order then ujvTjisthe jth empirical orthogonal functionTypically, we only need a few terms in the sum to wellapproximate Y . With just the first term it would suggestapproximating Y (s, t) by d1u1(s)v1(t).Chapter 8: Spatiotemporal Modeling – p. 4/13Spatio-temporal ModelsModeling: Yt(s) = µt(s) + wt(s) + ǫt(s),or perhaps g(E(Yt(s)) = µt(s) + wt(s)Chapter 8: Spatiotemporal Modeling – p. 5/13Spatio-temporal ModelsModeling: Yt(s) = µt(s) + wt(s) + ǫt(s),or perhaps g(E(Yt(s)) = µt(s) + wt(s)For ǫt(s), independent N(0, τ2t)Chapter 8: Spatiotemporal Modeling – p. 5/13Spatio-temporal ModelsModeling: Yt(s) = µt(s) + wt(s) + ǫt(s),or perhaps g(E(Yt(s)) = µt(s) + wt(s)For ǫt(s), independent N(0, τ2t)For wt(s)wt(s) = αt+ w(s)wt(s) independent for each twt(s) = wt−1(s) + ηt(s), independent spatial processinnovationsChapter 8: Spatiotemporal Modeling – p. 5/13Areal unit dataYi(t), temporal process for each unit (rare!)Yit, a time series for each unit (and occasionally, Yijt), ismore commonChapter 8: Spatiotemporal Modeling – p. 6/13Areal unit dataYi(t), temporal process for each unit (rare!)Yit, a time series for each unit (and occasionally, Yijt), ismore