1 caracas, september 28 2006 universität stuttgart lehrstuhl für hydrologie und geohydrologie the...

1Caracas, September 28 2006

Universität StuttgartLehrstuhl für Hydrologie und Geohydrologie

The use of copulas for the description of the spatial variability of environmental variables

András Bárdossy

Universität Stuttgart

Spatial patterns are not necessarily symmetrical



Paul Klee: Copula



Dependence

Goal: dependence between two variables: • To recognize• To quantify

Correlation (Signifikance Normal distribution)

Linear Regression



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8

12

16



-4 -2 0 2 4

-4

-2

0

2

4



Dependence changes through a Transformation of the Marginal distribution

Correlations between 0.4 and 0.8

Idea: take the same marginal distribution

Uniform in [0,1]

Transformation using the pdf



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Dependence Marginal distributions



Bivariate Distributions and Copulas

Bivariate Copula = bivariate pdf with uniform marginals

))(),((),( 21

211

121 uFuFFuuC



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Variable A

0

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1

Var

iabl

e B



Copulas are a new way of modelling the correlation structure between variables.

They dissociate the correlation structure from the marginal distributions of the individual variables.



Entropy = measure of Information (Shannon)

Differential entropy:

Conditional Differential entropy

Interesting for „extreme“ (v large)

II

c dudvvucvucE ),()),(ln(

I

c dvvucvucuE ),()),(ln()(



Multivariate Distributions and Copulas

Sklar (1959) all F pdf can be written in this form and C is unique if F is continuous

Measures of the dependence • Differential Entropy• Rank correlation (Spearman)• Kendalls tau

))(,),((),,( 111 KKK xFxFCxxF



Copula density

n

nn

n xx

xxCxxc

...

),...,(),...,(

1

11



Multivariate normal copula

Copula density:



Normal copula

Can copulas be used for the description of spatial variability ?

Do we need this ?



Geostatistics

Z(x) Random function – Realisation z(xi)

Assumption – „uniform continuity“

No differences are known a-priori

Independent of the location – depends only on h

(Semi)Variogramm Covariance function

2

1

2

1

2uZhuZEuZhuZVarh

muZE



Experimental Variogramm EC

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50

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400

0 500 1000 1500 2000 2500 3000



Spatial copulas

Assumption:• Multivariate copula exists for any

number of points

• The bi-variate marginal copulas corresponding to pairs separated by a vector h are translation invariant

• How to find such copulas ?

vZFuZFPvu zz ))((,))((),,( xhxhCS



Empirical copulas

Set of pdf pairs corresponding to points separated by the vector h

Generalization of the variogram

Empirical density using kernel smoothing

hxxxxh iijnin ZFZFS |))(()),(()(



Nitrat und Phosphat



Copula Nitrate GW 5 km

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Q uantile Punkt x

Qu

an

tile

Pu

nkt

y



Entropy: Nitrat 3000 m und 30000m

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0.22

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0.26



Cl Variogramm



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Chloride concentration (mg/l)

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600



Copula pH groundwater

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00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220



pH Variogramm



5.2 5.6 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6

0

400

800

1200



n-dimensional Chi-square copula



Chi-Quadrat Copulas



Testing the multivariate copulas

Analytically difficult due to the dependence of the pairs

Simulation and Bootstrap to compare bi-variate marginals to theoretiacal

Summary and conclusions

Copulas offer an interesting alternative

Many natural variables show a non-Gaussian spatial dependence

Thank you !

PS: I have another 253 slides to show – maybe next time !



Indikator Variablen

Indikator variablen

sonst

CuuIC 0

falls1)(

)( falls0

)( falls1)(

uZ

uZuI

Indikator variogram

huu

ji

ji

uIuIhN

h 2* ))()(()(2

1)(



Spatial dependence – 5 kmEvent 70

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Precip ita tion am ount location A

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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85




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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85




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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85



Radarniederschlag29. Dezember 2001 11:20-13:20

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05101520253035404550556065707580859095100105110115120



Copula Radarniederschlag29. Dezember 2001 11:20-13:20

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00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220



Gauss – Chi-square

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-4.5-4.1-3.7-3.3-2.9-2.5-2.1-1.7-1.3-0.9-0.5-0.10.30.71.11.51.92.32.73.13.53.94.3

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-4.5-4.1-3.7-3.3-2.9-2.5-2.1-1.7-1.3-0.9-0.5-0.10.30.71.11.51.92.32.73.13.53.94.3



Statistische Tests

Problem: Paare sind nicht unabhängig

Deshalb klassische Tests nicht verwendbar

Lösung: Bootstrap



Zusammenfassung

•Zusammenhänge können mit Copulas „einheitlich“ quantifiziert werden

•Viele natürliche Parameter zeigen assymetrische Zusammenhänge

•Diese Eigenschaft kann bei der räumlichen Betrachtung berücksichtigt werden



Simulation results

0 5 1 0 1 5 2 0 2 5 3 00

5

1 0

1 5

2 0

2 5

- 101234567891 01 11 21 31 41 51 61 71 81 92 02 12 22 32 42 5

0 5 1 0 1 5 2 0 2 5 3 00

5

1 0

1 5

2 0

2 5

- 101234567891 01 11 21 31 41 51 61 71 81 92 02 12 22 32 42 5

0 5 1 0 1 5 2 0 2 5 3 00

5

1 0

1 5

2 0

2 5

- 101234567891 01 11 21 31 41 51 61 71 81 92 02 12 22 32 42 5



Räumlicher Zusammenhang – 5 kmEreignis 71

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M oisture flux

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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85




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M oisture flux

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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85




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M oisture flux

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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85



Asymmetrie des Zusammenhanges

Hohe Werte zeigen einen anderen Zusammenhang wie die niedrigen

u u

u u

dxdxyxc

dxdxyxc

uuC

uuCuA

0 0

1

1

1

1

),(

),(

),(

)1,1(21



Empirische Copuladichten

Nitrat pH



Radarniederschlag29. Dezember 2001 8:20-9:20

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05101520253035404550556065707580859095100105110115120



Niederschlag

Geeignete Copulas finden• Bessere Interpolation (HW relevant)• Theoretische

Gebietsabminderungsfunktionen• Simulationsmodelle mit beliebigen

Randverteilungen• Realistische Extreme über räumliche

und zeitliche Skalen ( Fraktale Modelle)



Neckar Einzugsgebiet



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00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220



Korrelation – Zusammenhang der Extreme (99,5%)

0.6 0.7 0.8 0.9 1Korrelation

8.4

8.6

8.8

9B

ed

ing

te D

iffe

ren

tia

len

tro

pie



Zusammenfassung

Zusammenhaänge können mit Copulas beschrieben werden

Viele der Zusammenhänge in der Hydrologie sind asymmetrisch

Extreme sind of stärker abhängig als mittlere

Korrelation ist hierfür kein gutes Maß



Introduction - ModellingHydrological modeling is necessary• Design

– Changes– Climate change– Land use change

– Unobserved catchments (PUB)• Forecasts• In combination with

– Quality– Ecology

• For understanding



Introduction - VariabilityVariability due to natural conditions• Weather

– Annual cycle– Random variability

• Catchment reaction– State– Output - discharge



The Upper Neckar Catchment



Dependence between discharge series

Cross correlations (Pearson)

Cross rank correlations (Spearman)

Copulas



Dependence results

Cross correlations

0.66 – 0.95 for all pairs

>0.89 for the best pair for each site

Cross rank correlations

0.65 – 0.98 for all pairs

>0.88 for the best pair for each site



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Variable A

0

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0.8

1

Var

iabl

e B



Dependence structure

Dependence between Quantiles instead of Variable values

Copula – Dependence separated from the Marginal distributions

),,(),,(

ondistributi m Unifor)1,0(,,

111

1

KKK

K

uUuUPuuC

UU

U



Copula density of the pair C8 and C9

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00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220




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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85




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00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85



Simulation models

Multivariate normal copulas

Non Gaussian copulas (non-central chi-square copulas)

Correct marginal distribution



Copula density of a bivariate non-central chi-square distribution

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n-dimensional Chi-square copula



Summary

Hydrological modeling is necessary

and difficult

Discharge series are similar

Events are different

Hydrological models small differences

Spatial resolution is not the answer

Spatial variability is partly responsible

Rainfall variability is “asymmetrical”

6100 6150 6200 6250 6300 6350

Tim e (days)

0

100

200

300

400

500

Dis

char

ge (

m3 /

s)



Interpolation



Radar Measurement



Future work

Compare different sets of downscaled discharge extremes (3 versions)• Calculate spatial indices• Find appropriate copulas

Assess scenario dependent extremes

Develop new methodology for spatial extremes



Variability of discharge

Distribution of discharge for a single site • Depends on the aggregation• Skewed distribution• Few maxima – due to precipitation

(snow melt)

Natural Variability

Discharge series are similar because of the spatial extent of rainfall events.



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1 caracas, september 28 2006 universität stuttgart lehrstuhl für hydrologie und geohydrologie the...

Documents

pdf folie

variogramm folie

symmetrical folie

uniform marginals folie

kernel smoothing folie

extreme v large folie

copulas bivariate copula

multivariate copula