Vine Copulas. Spatial Copula Workshop 2014 September 22, Institute for Geoinformatics University of Münster.

Transcription

1 Spatial Workshop 2014 September 22, 2014 Institute for Geoinformatics University of Münster 1

2 spatio-temporal data Typically, spatio-temporal data is given at a set of discrete locations s i S and time steps t j T. We desire a bivariate spatio-temporal random field (Z, Y ) : Ω S T R 2 modelling the process at any location (s, t) in space and time. Here, we look at daily fine dust concentrations across Europe ( ) measured at stations (Z) and modelled (Y) through the European Monitoring and Evaluation Programme (EMEP). 2

3 basic set-up & assumptions We assume 1 that the marginal distribution can be parametrized by location s S: F s and G s 2 stationarity on the copula scale 3 that the dependence does not change within the study area. This allows us to consider lag classes by distances in space and time on the copula scale instead of single locations in space. 3

4 The classical geostatistical approach A multivariate Gaussian distribution is assumed where a variogram function can be used to parametrize the (large) covariance matrix by distance 4

5 The classical geostatistical approach A multivariate Gaussian distribution is assumed where a variogram function can be used to parametrize the (large) covariance matrix by distance the mean vector is set to the observed values 4

6 Our approach using the full distribution of the observed phenomenon of a local neighbourhood 5

7 Our approach using the full distribution of the observed phenomenon of a local neighbourhood each observed location (s 0, t 0 ) (S, T ), is grouped with its nine strongest correlated neighbours. 5

8 Our approach using the full distribution of the observed phenomenon of a local neighbourhood each observed location (s 0, t 0 ) (S, T ), is grouped with its nine strongest correlated neighbours. an estimate is calculated from the conditional distribution at an unobserved location conditioned under the values of its spatio-temporal neighbourhood and covariate yielding an elven dimensional distribution. 5

9 A metric spatio-temporal neighbourhood 6

10 The spatio-temporal covariate vine-copula We decompose the eleven dimensional distribution into it s marginal distribution F (identical for all 10 margins), the marginal distribution of the covariate G and a vine copula: On the first tree, we use a spatio-temporal bivariate copula accounting for spatial and temporal distance and the copula relating the variable of interest and its covariate. The following trees are modelled from a wide set of classical bivariate copulas. 7

11 Accounting for spatial distance Thinking of pairs of locations (s 1, t 1 ), (s 2, t 2 ) we assume... distance has a strong influence on the strength of dependence 8

12 Accounting for spatial distance Thinking of pairs of locations (s 1, 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 might change with distance 8

13 Accounting for spatial distance Thinking of pairs of locations (s 1, 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 might change with distance stationarity on the copula scale and build k lag classes by spatial distance for each temporal distance and estimate a bivariate copula c j (u, v) for all lag classes { [0, l 1 ),[l 1, l 2 ),..., [l k 1, l k ) } { 0, 1, 2 } 8

14 Density of the spatial and 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 2 c 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.. 9

15 Adding the Covariate 10

16 The full density I We get the full 11-dim copula density as a product of all involved bivariate densities: c h (u 0, v 0, u 1,..., u d ) d =c ZY (u 0, v 0 ) c h(0,i) (u 0, u i ) d j i=1 d c Y,i 0 (u Y 0, u i 0 ) i=1 d 1 c j,j+i Y,0,...,j 1 (u j Y,0,...,j 1, u j+i Y,0,...,j 1 ) j=1 i=1 where v 0 = G 0 ( Y (s0, t 0 ) ) with G 0, u i = F i ( Z(sq, t r ) ) for 0 i d with (s q, t r ) denoting the i-th strongest correlated neighbour of (s 0, t 0 ) with F i = F q,r and... 11

17 The full density II and u Y 0 = F Y 0 (v 0 u 0 ) = C Z,Y (u 0, v 0 ) u 0 u i 0 = F i 0 (u i u 0 ) = C h(0,i) (u 0, u i ) u 0 u j+i Y,0,...,j 1 = F j+i Y,0,...j 1 (u j+i v 0, u 0,..., u j 1 ) = C j 1,j+i Y,0,...j 2(u j 1 Y,0,...j 2, u j+i Y,0,...j 2 ) u j 1 Y,0,...j 2 for 1 j < d and 0 i d j. 12

18 interpolation The estimate can be obtained as the expected value Ẑ m (s 0 ) = = R [0,1] z f h (z y 0, z 1,..., z d ) dz F0 1 (u) c ( ) h u v0, u 1,..., u d du or by calculating any percentile p (i.e. the median) Ẑ p (s 0 ) = F0 1 ( C 1 h (p v0, u 1,..., u d ) ) 13

19 R-package spcopula The developed methods are implemented as R-scripts and are bundled in the package spcopula available at R-Forge (briefly presented later today). The package spcopula extends and combines the R-packages Vine, spacetime and copula. 14

20 across Europe We applied our method to daily mean concentrations observed at 194 rural background stations for the year 2005 (70810 obs.). The data is hosted by the European Environmental Agency (EEA) originally provided by the member states and freely available at As covariate, daily mean PM 10 concentrations derived from the EMEP model are included. 15

21 The marginal distributions We fit extreme value distributions for each location s S based on the time series leading to margins F s and G s. For the interpolation scenario we use 1 a linear model incorporating the locations coordinates and altitude followed by an inverse distance weighted interpolation of the residuals... 2 inverse distance weighted interpolation... of the local neighbourhood s marginal parameters. Density Histogram of daily mean PM 10 measurements PM 10 [ µ g m 3 ] 16

22 The covariate copula Kendall's tau correlation structure of PM10 and EMEP over time Joe Frank Gumbel Clayton Student Gauss copula family day in

23 A look inside the spatio-temporal copula I correlation [Kendall's tau] Spatio Temporal Dependence Structure same day 1 day before 2 days before 3 days before 4 days before distance [km] 18

24 A look inside the spatio-temporal copula II 19

25 A look inside the spatio-temporal copula III 20

26 A look inside the spatio-temporal copula IV Spatial lag ID mean dist. [km] = 0 t G t... t G... G F N = 1 G... G F... F N = 2 G... G N... = 3 G... G = 4 G... G J... J G G 21

27 Cross validation Dependence model Margin RMSE MAE ME COR STCV Ẑm local GEV Gaussian STCV Ẑm local GEV STCV Ẑm lm+idw GEV STCV Ẑm IDW GEV metric res. kriging log linear reg sp.-temp. vine Ẑm global GEV NA 22

28 Box Plots of marginal reproduction PM 10 [ µ g m 3 ] obs. local GEV STCV FI00351 Gauss lm+idw STCV IDW STCV Kriging PM 10 [ µ g m 3 ] obs. local GEV STCV DENW081 Gauss lm+idw STCV IDW STCV Kriging 23

29 A station in Finland FI00351 PM 10 [ µ g m 3 ] Apr Apr Apr May May local GEV lm+idw GEV IDW GEV Jun observed Kriging Jun Jun

30 Conditional distribution functions prediction CDF daily mean PM 10 concentrations [µg/m 3 ] : STCV : STCV lm+idw : Gauss. STCV : STCV : STCV lm+idw : Gauss. STCV 25

31 Benefits richer flexibility due to the various dependence structures 26

32 Benefits richer flexibility due to the various dependence structures asymmetric dependence structures become possible (temporal direction) 26

33 Benefits richer flexibility due to the various dependence structures asymmetric dependence structures become possible (temporal direction) probabilistic advantage sophisticated uncertainty analysis, drawing random samples,... 26

34 Further extensions larger neighbourhoods possibly using vine truncation techniques 27

35 Further extensions larger neighbourhoods possibly using vine truncation techniques include further copula families 27

36 Further extensions larger neighbourhoods possibly using vine truncation techniques include further copula families improve performance 27