benedikt gr aler time with vine copulasmodelling dependence in space and time with vine copulas...

ModellingDependence in

Space and Timewith Vine Copulas

Benedikt Graler

Problem

Solution

Vine-Copulas

Bivariate SpatialCopula

Spatio-TemporalVine-Copulainterpolation

Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

1

Modelling Dependence in Space and

Time with Vine Copulas

Ninth International Geostatistics Congress15.06.2012

Benedikt GralerInstitute for Geoinformatics

University of Munsterhttp://ifgi.uni-muenster.de/graeler

http://ifgi.uni-muenster.de/graeler



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

2

spatio-temporal data

Typically, spatio-temporal data is given at a set of discretelocations si ∈ S and time steps tj ∈ T .

We desire a full spatio-temporal random field Z modellingthe process at any location (s, t) in space and time.

Here, we look at daily fine dust concentrations acrossEurope (PM10).



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

3

basic set-up & assumptions

We assume a stationary and isotropic process across theregion S and the time frame T .

This allows us to consider lag classes by distances in spaceand time instead of single locations in space.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

4

Our approach

using the full distribution of the observed phenomenonof a local neighbourhood

each observed location (s0, t0) ∈ (S, T ), is grouped withits three nearest spatial neighbours {s1, s2, s3} ⊂ S, thecurrent (t0) and two preceding time steps (t−1,t−2) ofthis neighbourhood generating a 10 dimensional sample:(Xs0,t0 ,Xs1,t0 , . . . ,Xs3,t−2)

an estimate is calculated from the conditionaldistribution at an unobserved location conditionedunder the values of its spatio-temporal neighbourhood



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

5

The spatio-temporal neighbourhood

Figure: 10-dim spatio-temporal neighbourhood



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

6

Copulas

Copulas describe the dependence structure between themargins of a multivariate distribution. Sklar’s Theoremstates:

H(x1, . . . , xd) = C(F1(x1), . . . , Fd(xd)

)where H is any multivariate CDF, F1, . . . , Fd are thecorresponding marginal univariate CDFs and C is a suitablecopula (uniquely determined in a continuous setting).

Since Fi(Xi) ∼ U(0, 1), copulas can be thought of as CDFson the unit (hyper-)cube.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

7

Beyond Gaussian dependence structure

Figure: Copula density plots with rank correlation τ = −0.3;density on Z-axis ∼ strength of dependence.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

8

Vine-Copula

Bivariate copulas are pretty well understood and rather easyto estimate.

Unfortunately, most bivariate families do not nicely extend toa multivariate setting or lack the necessary flexibility.

Vine-Copulas allow to approximate multivariate copulas bymixing (conditional) bivariate copulas following a vinedecomposition.

Figure: A 4-dim canonical vine decomposition.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

9

Our spatio-temporal vine-copula

We decompose the ten dimensional distribution into itsmarginal distribution F (identical for all 10 margins) and avine copula:

On the first tree, we use a spatio-temporal bivariate copulaaccounting for spatial and temporal distance.

The following trees are modelled from a wide set of”classical” bivariate copulas.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

10

Accounting for distance

Thinking of pairs of locations (s1, t1), (s2, t2) we assume . . .

distance has a strong influence on the strength ofdependence

dependence structure is identical for all neighbours, butmight change with distance

stationarity and build k lag classes by spatial distance foreach temporal distance ∆ and estimate abivariate copula c∆

j (u, v) for all lag classes{[0, l1),[l1, l2), . . . , [lk−1, lk)

}×{

0, 1, 2}



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

11

Density of the spatial and spatio-temporal copula

The density of the bivariate spatial copula is then given by aconvex combination of bivariate copula densities:

c∆h (u, v) :=

c∆1 (u, v) , 0 ≤ h < l1(1− λ2)c∆1 (u, v) + λ2c

∆2 (u, v) , l1 ≤ h < l2

......

(1− λk)c∆k−1(u, v) + λk · 1 , lk−1 ≤ h < lk1 , lk ≤ h

where λj :=h−lj−1

lj−lj−1.

The density of the bivariate spatio-temporal copula ch,∆(u, v) is

then given by a convex combination of bivariate spatial copula

densities in a analogous manner.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

12

The full density

The remaining copulas cj,j+i|0,...j−1 with integers1 ≤ j ≤ 9, 1 ≤ i ≤ 9− j are estimated over the ninedimensional conditional sample(Fh,∆(Xs1,t0 |Xs0,t0), . . . , Fh,∆(Xs3,t−2 |Xs0,t0)

).

We get the full 10-dim copula density as a product of allinvolved bivariate densities:

ch,∆(u0, . . . , u9)

=

9∏i=1

ch,∆(u0, ui) ·9−1∏j=1

9−j∏i=1

cj,j+i|0,...,j−1(uj|0,...,j−1, uj+i|0,...,j−1)

Where u0 = F(Z(s0, t0)

), . . . , u9 = F

(Z(s3, t−2)

),

uj|0 = Fh,∆(uj |u0) =∂Ch,∆(u0, uj)

∂u0and

uj+i|0,...,j−1 = Fj+i|0,...,j−1(uj+i|u0, . . . , uj−1)

=∂Cj−1,j+i|0,...j−2(uj−1|0,...j−2, uj+i|0,...j−2)

∂uj−1|0,...j−2



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

13

Spatio-Temporal Vine-Copula interpolation

The estimate can be obtained as the expected value

Zm(s0, t0) =

∫[0,1]

F−1(u) ch,∆(u|u1, . . . , u9

)du

or by calculating any percentile p (i.e. the median)

Zp(s0) = F−1(C−1h,∆(p|u1, . . . , u9)

)with the conditional density

ch,∆(u|u1, . . . , u9) :=ch,∆(u, u1, . . . , u9)∫ 1

0 ch,∆(v, u1, . . . , u9)dv

and u1 = F(Z(s1, t0)

), . . . , u9 = F

(Z(s3, t−2)

)as before.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

14

R-package spcopula

The developed methods are implemented as R-scripts andare bundled in the package spcopula available at R-Forge.

The package spcopula extends and combines the R-packagesCDVine, spacetime and copula.

http://r-forge.r-project.org/projects/spcopula/

http://cran.r-project.org/web/packages/CDVine/index.html

http://cran.r-project.org/web/packages/spacetime/index.html

http://cran.r-project.org/web/packages/copula/index.html



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

15

PM10 across Europe

We applied our method to daily mean PM10 concentrationsobserved 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 freelyavailable at http://www.eea.europa.eu/themes/air/airbase.

http://www.eea.europa.eu/themes/air/airbase



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

16

A look inside the spatio-temporal copula I



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

17

A look inside the spatio-temporal copula II



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

18

A look inside the spatio-temporal copula III



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

19

A look inside the spatio-temporal copula IV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15∆ = 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 A A∆ = −2 F F F F A A A A A A A A A A A

Table: The selected copula families for the first 15 spatial lagclasses (up to ≈ 600 km) and three time instances. (t =Student,F = Frank, A = asymmetric, the vertical lines indicate 100 km,250 km, 500 km breaks)



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

20

comparing log-likelihoods

10-dim copula log-likelihood

spatio-temporal vine-copula 72709. . . backwards in time 63089

Gaussian 53305Gumbel 35292Clayton 30593

Table: Log-likelihoods for different copulafamilies/parametrization.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

21

The margins

The best fit to the marginal distribution could be achievedfor generalized extreme value distribution (GEV):

Figure: Histogram of all PM10 measurements and fitted density.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

22

Cross validation

Leaving out complete time series of single stations one afteranother.

RMSE bias MAE

expected value Zm 11.20 -0.73 6.95

median Z0.5 12.08 1.94 6.87

metric cov. kriging 10.69 -0.29 6.28metric cov. res. kriging 9.84 -0.24 5.66

Table: Cross validation results for the expected value and medianestimates following the vine copula approach and two methodsfrom a recent comparison study [2] on spatio-temporal krigingapproaches in PM10 mapping.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

23

A station in Finland

Figure: A Finish station roughly 600 km apart from any otherstation. The copula approach (magenta) is controlled by the globalmean, kriging (green) reduces almost to a mowing window mean.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

24

A station in Germany

Figure: A German station within a rather dense network. Thecopula approach (magenta) reproduces the observed values (red)while kriging (green) overestimates several daily means.



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

25

Benefits

richer flexibility due to the various dependence structures

asymmetric dependence structures become possible(temporal direction)

probabilistic advantage sophisticated uncertainty analysis,drawing random samples, . . .



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

26

Further extensions

including covariates(e.g. altitude, population, EMEP, . . . )

complex neighbourhoods(e.g. by spatial direction, . . . )

include further copula families

improve performance



Benedikt Graler

Problem

Solution

Vine-Copulas



Software

Application toPM10

Fitment

Goodness of fit

Conclusion &Outlook

References

27

Aas, Kjersti, Claudia Czado, Arnoldo Frigessi & HenrikBakken (2009). Pair-copula constructions of multipledependence. Insurance: Mathematics and Economics,44, 182 - 198.

Graler B, Gerharz LE, Pebesma E (2012)Spatio-temporal analysis and interpolation of PM10measurements in Europe. ETC/ACM Technical Paper2011/10, January 2012

Nelsen, R. (2006). An Introduction to Copulas. 2nd ed.New York: Springer.

benedikt gr aler time with vine copulasmodelling dependence in space and time with vine copulas...

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