Spatial Statistics 2013, S2.2 6 th June Institute for Geoinformatics University of Münster.

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1 Spatial Statistics 2013, S2.2 6 th June 2013 Institute for Geoinformatics University of Münster Vine Vine 1

2 Spatial/spatio-temporal data Typically, spatial/spatio-temporal data is given at a set of discrete locations s i S ( time steps t j T ). We desire a full spatial/spatio-temporal rom field Z modelling the process at any location s in space/(s, t) in space time. We will look at daily mean fine dust concentrations across Europe (PM 10 ). In the following, we will refer to Z as a spatio-temporal rom field (including the spatial case). Vine Vine 2

3 describe the dependence structure between the margins of a multivariate distribution. Sklar s Theorem states: H(x 1,..., x d ) = C ( F 1 (x 1 ),..., F d (x d ) ) where H is any multivariate CDF, F 1,..., F d are the corresponding marginal univariate CDFs C is a suitable copula (uniquely determined in a continuous setting). Since F i (X i ) U(0, 1), copulas can be thought of as CDFs on the unit (hyper-)cube. Vine Vine 3

4 Beyond Gaussian dependence structure Vine Vine 4

5 Vine copulas copulas are pretty well understood rather easy to estimate. Unfortunately, most bivariate families do not nicely extend to a multivariate setting or lack the necessary flexibility. Vine allow to approximate multivariate copulas by mixing (conditional) bivariate copulas following a vine decomposition [ACFB09, BC02]. Vine Vine 5

6 The spatio-temporal neighbourhood - the first tree Vine Vine 6

7 Spatio-temporal vine copulas The distribution of local neighbourhoods is decomposed into marginal distributions F i a spatio-temporal vine copula: The first tree is modelled as bivariate spatio-temporal copulas accounting for spatial temporal distances. Remaining trees are modelled from a wide set of classical bivariate copulas as a vine or truncated vine. Vine Vine 7

8 Accounting for spatial distance Thinking of pairs of spatio-temporal locations ( (s1, t 1 ), (s 2, t 2 ) ) we assume... distance has a strong influence on the strength of dependence dependence structure is identical for all neighbours, but may change with distance stationarity build k bins by distance to estimate a bivariate copula c j,h (u, v) for all spatial bins [0, l 1 ),[l 1, l 2 ),..., [l k 1, l k ) per temporal lag. Vine Vine 8

9 Density of the spatial spatio-temporal copula The density of the bivariate spatial copula is then given by a convex combination of bivariate copula densities: c h (u, v) :=. where λ j := h lj 1 l j l j 1. c 1,h (u, v), 0 h < l 1 (1 λ 2 )c 1,h (u, v) + λ 2c 2,h (u, v), l 1 h < l 2 (1 λ k )c k 1,h (u, v) + λ k 1, l k 1 h < l k 1, l k h The density of the bivariate spatio-temporal copula c h, (u, v) is then given by a convex combination of bivariate spatial copula densities in an analogous manner.. Vine Vine 9

10 The full density The remaining copulas c j,j+i 0,...j 1 are estimated over the conditional sample. We get the full (here 10-dim) spatio-temporal vine copula density as a product of all involved bivariate densities: c h, (u 0,..., u 9 ) j = c h, (u 0, u i ) c j,j+i 0,...,j 1 (u j 0,...,j 1, u j+i 0,...,j 1 ) i=1 j=1 i=1 Vine Vine 10

11 Spatio-temporal vine copula interpolation The estimate can be obtained as the expected value Ẑ m (s 0, t 0 ) = F 1 ( ) (u) c h, u u1,..., u d du [0,1] or by calculating any percentile p (i.e. the median) Ẑ p (s 0, t 0 ) = F 1( C 1 h, (p u 1,..., u d ) ) with the conditional density c h, (u u 1,..., u d ) := c h, (u, u 1,..., u d ) 1 0 c h, (v, u 1,..., u d )dv u i = F i ( Z(si, t i ) ). Vine Vine 11

12 R-package spcopula The developed methods are implemented in R are available as package spcopula at R-Forge. The package spcopula extends combines the R-packages VineCopula, spacetime copula. Vine Vine 12

13 Daily mean PM 10 concentrations across Europe Daily mean PM 10 concentrations observed at 194 rural background stations across Europe for the year Density Vine Vine daily mean PM 10 concentrations [µg/m 3 ] 13

14 The spatio-temporal neighbourhood Vine Vine 14

15 The bivariate spatio-temporal copula I > stbins <- calcstbins(eu_rb_2005,"rtpm10", nbins=40, + t.lags=-(0:2), instances=na, + cor.method="fasttau", plot=f) > calcktau <- fitcorfun(stbins,c(3,3,3)) correlation [Kendall's tau] same day one day two days Vine Vine distance [km] 15

16 The bivariate spatio-temporal copula II > logliktau <- list() > for(j in 1:3) { + tmpbins <-... # j-th subset of stbins + logliktau[[j]] <- loglikbylags(tmpbins, families, + calcktau[[j]]) + } The following families achieve the highest log-likelihood: distance = 0 t F t F F F F F F F F F F F F = 1 F F F F F F F F F F F F F F A = 2 F F F F F F A A A A A A A A A Vine Vine 16

17 The bivariate spatio-temporal copula III Pick the best fitting copulas: > stconvcop <- stcopula(components = listcops, + distances = listdists, + t.lags=c(0,-1,-2)) Strength of dependence on copula scale the same day [km] 250 [km] 500 [km] Vine Vine

18 The bivariate spatio-temporal copula IV Strength of dependence on copula scale one day difference [km] 250 [km] 500 [km] Strength of dependence on copula scale two days difference [km] 250 [km] 500 [km] Vine Vine

19 The spatio-temporal vine copula I Build the spatio-temporal neighbourhood fit the upper vine: > stneigh <- getstneighbours(eu_rb_2005, var="rtpm10", + spsize=4, t.lags=-(0:2), + timesteps=90, min.dist=10) > stvinefit <- fitcopula(stvinecopula(stconvcop, + vinecopula(9l)), + stneigh, method="indeptest") > stvinefit@loglik [1] > stvine <- stvinefit@copula > stvine Spatio-temporal vine copula family with 1 spatio-temporal tree. Dimension: 10 Vine Vine 19

20 Interpolation of the air qualities Predictions can be obtained through > predneigh <- getstneighbours(eu_rb_2005, targetgeom, + "rtpm10", spsize=4, prediction=t) > stvinepred <- stcoppredict(predneigh, stvine, + list(q=qfun), "quantile") Vine Vine 20

21 Cross validation results for the daily mean PM 10 interpolation RMSE bias MAE expected value Ẑm median Ẑ metric cov. kriging metric cov. res. kriging Table: Cross validation results for the expected value median estimates following the vine copula approach two methods from a recent comparison study [GGP12] on spatio-temporal kriging approaches in PM 10 mapping. Vine Vine 21

22 Benefits of the spatial/spatio-temporal vine copulas richer flexibility due to the various dependence structures asymmetric dependence structures become possible (temporal direction) probabilistic advantage flexible uncertainty analysis The predictive CDFs from the joker data set: prediction CDF pure spatial copula residual kriging joker data set Vine Vine radiation [nsv/h] 22

23 Further extensions including covariates (e.g. altitude, population,... ) complex neighbourhoods (e.g. by spatial direction,... ) introduce several spatio-temporal trees in the vine (as in the spatial version, see poster: Modelling Extremes with the Spatial Vine Copula) include further copula families improve performance Vine Vine 23

24 Kjersti Aas, Claudia Czado, Arnoldo Frigessi, Henrik Bakken, Pair-copula constructions of multiple dependence, Insurance: Mathematics Economics 44 (2009), Tim Bedford Roger M. Cooke, Vines a new graphical model for dependent rom variables, Annals of Statistics 30 (2002), no. 4, B. Gräler, L. E. Gerharz, E. Pebesma, Spatio-temporal analysis interpolation of PM10 measurements in Europe, Tech. report, ETC/ACM, Roger B. Nelsen, An introduction to copulas, second ed., Springer Science+Buisness, New York, Vine Vine 24