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Z measured at N sites Incorporates spatial dependencies Matrix of weights System of neighborhoods:WW$n  <Z(t): N1, t=1,& ,T W(l): NN 2 Weighting Matrices W(l)$Fixed beforehand Reflect physical features Regular spaced systems: equal weights Irregularly spaced systems inversely proportional to distance j+ZZ$Z+$  Modelling procedure   Identification Identification-Space-time analogue of Yule-Walker equations .-  EstimationAssumption: orders are known Conditional maximum likelihood estimation F=(f)kl ,, Q=(q)kl, s2 maximize: fiG ,L  Estimation IssuesUNonlinear nature of the problem with STMA terms Initial estimates Order determination)Initial Estimates and Order Determination*)$Hannan and Rissanen(1982) - ARMA Extension to STARMA High order STAR(k1,& ,1) Y-W P55$? Diagnostic CheckingF95% confidence intervals for the residuals space-time autocorrelationsGF7 Test the Hypotheses G=D$ Sample covariance matrix of the residuals Test statistics: H0: G=s2I - H0: G=D ^<< ,@   )Data#Monthly mean temperatures 1951-1966+Data ,Identification 0Identification  1Identification  .Estimation Results ZAssumption : G = s2I STAR(24) spatial order 1D 2 ( X     2Diagnostic Checking  3Diagnostic Checking   4Identification Assumption : general G $  5Diagnostic Checking  /Summary and Extensions "Stationarity Gaussianity Linearity*  6#$    &Periodically correlated spatial time series Non-gaussian  Pareto distribution Subba Rao and Wong(1997) Non-linear  Bilinear Dai and Billard(1998)LOOj0      -   /9:;=>ABCD E F G H IJKLMNOPQRST  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> .&(    6s P  T Click to edit Master title style! !  0v   RClick to edit Master text styles Second level Third level Fourth level F