universität stuttgartinstitut für wasserbau, lehrstuhl für hydrologie und geohydrologie copulas...

Universität Stuttgart Institut für Wasserbau, Lehrstuhl für Hydrologie und Geohydrologie

Copulas (2)

András Bárdossy

IWS

Universität Stuttgart


Spatial problems

• Sampling only at a number of locations• What is between ?

– Estimate– Quality of estimation– Simulate realizations

• Geostatistics (Krige, Matheron)– Mining applications– Hydro and Environmental sciences


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

0

50

100

150

200

250

300

350

400

0 500 1000 1500 2000 2500 3000


Point kriging

in

iii uZuZ

*

DuallformuZE

muZEuZE i

n

iii

*

1

n

iii

Unbiasedness


Estimation variance using the variogram

uuuu

uZuZVaru

i

n

ii

n

jjij

n

ii

11 1

*2

2

)()()(


Kriging equations using variogram

uuuu i

n

jjij

1

i=1,…,n

n

jjijK uuu

1

2 )(

11

n

jj


Problems

• Estimation variance is an index of spatial configuration– Does not depend on the local values– “Best” for Gaussian distribution– Symmetrical (high and low values not distinguished)

• Variogram estimation difficult– Squared differences – skewed distribution– Dominated by high values– Independence of the pairs not fulfilled

Strongly influenced by the marginal distribution


Symmetry

• Digital elevation models – water dominated regions– Maxima and minima

• Contaminations– Source vs Background concentrations

Known but unquantified deterministic processes lead to asymmetry and non-Gaussian dependence


Indicator Variables

Indicator variables

else 0

if1)(

CuuIC

)( if0

)( if1)(

uZ

uZuI

Indicator variogram

huu

ji

ji

uIuIhN

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

1)(


Indicator variables

• Interpretation as probability• Interpolation of the indicators• Result pdf for each location • Simulation restricted to the observed range

Can copulas be used to overcome some of these problems?


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 |))(()),(()(


Empirical copula density chloride h=5000m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

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6

7

8

9

10

11

12

13

14

15



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


Empirical copula density nitrate h=5000m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


Empirical copula density pH h=5000m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


Cl Variogramm


0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Chloride concentration (mg/l)

0

200

400

600


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


Conditional Entropy: Nitrate 3000 m and 30000m

0 5 10 15 20 25

0.22

0.23

0.24

0.25

0.26


Copulas and natural processes

• Natural processes influence high and low values differently– Erosion at high elevations– Pollution is spreading not the background– Weather relates the high discharges

• Copulas of digital elevation models:– Spain – eroded old landscape– Ecuador – younger but erroded– Mars – eroded and meteorites


Copula density of the pair C8 and C9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220


Copulas of daily rainfall

• 601 rainfall stations in the Rhein catchment Germany• Size = 100.000 km2• Days with important events with good spatial coverage

were selected (400 days of the period 1958-2003)• Spatial copulas (densities) for different distances were

calculated


Spatial dependence – 5 kmEvent 70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Precip ita tion am ount location A

0

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

Pre

cip

itatio

n a

mo

unt l

oca

tion

B

00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pre

cipi

tatio

n am

ount

loca

tion

B

00.20.40.60.811.21.41.61.822.22.42.62.833.23.43.63.844.24.44.64.85



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pre

cip

itatio

n a

mo

unt l

oca

tion

B

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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.8

0.9

1

05101520253035404550556065707580859095100105110115120


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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.8

0.9

1

00.81.62.43.244.85.66.47.288.89.610.411.21212.813.614.415.21616.817.618.419.220


Requirements for a spatial copula

1. Stability of the multivariate marginals: which means that any multivariate marginal copula corresponding to a selected set of points should not depend on the set of other selected points used to define the multivariate copula.

2. Wide range of dependence: a geographically close set of points should have an arbitrarily strong dependence structure, while distant points should be independent.

3. Flexible parametrization: the multivariate copula should have a parametrization such that the dependence structure reflects the geometric position of the corresponding set of points.


• Definition of a copula from a multivariate distribution:


Possibilities

• Multivariate normal copula– Simple but symmetrical

• Derived multivariate copulas

• If g monotonic – no change of the copula• If g non monotonic one can get interesting copulas

)( ii YgX


Normal copula

• Correlation = 0.85


Chi square

• Non central chi-square distribution

t

mtmttg

mtmt

tYtPtYPtG

2

)()()(

)()(

)()()(

1

21


Chi square

• Multivariate case

1,0 2

)1(,...,)1(

)()1(

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

1

1

1

12

0

1111

1

k

n

k

kk

nii

in

i

nnnnn

iii

tt

tYttYtPttG

n

n

i

i

ε

mε


n-dimensional Chi-square copula

• Density


Chi-Square Copulas


Gauss – Chi-square

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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1

-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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

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0.5

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0.8

0.9

1

-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


V-transformed copula

• Transformation function:

• Strong dependence of the extremes if shifted to one side and partly to the middle

• If k=1 then it is the chi square copula

else )(

if )()(

mx

mxxmkxg


K=2, m=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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1

0

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15


Parameter estimation

• Non independent pairs – ML• Fit the rank correlation function and the asymmetry • Parametric form of the covariance of the original normal

• Further work needed


Interpolation

• For n+1 points the joint distribution is known– Calculate the conditional for the unobserved point

Full conditional distribution known – thus confidence intervals can be calculated

• Example:

4 points – corner of a unit square

A: two of them with F(x)=1 two with F(x)=0

B: all with F(x)=0.5


Example interpolation – conditional densities m=0, k=1

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025


Example interpolation – conditional densities m=1, k=3

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025


Example interpolation – conditional densities normal copula

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025


Validation of the conditional densities

• Are the conditional densities OK ?• Cross validation

– Calculation of the frequencies of non exceedence for the observed values

– Comparison with the uniform

• V is much better then normal or Kriging


Thank you !

[email protected]


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universität stuttgartinstitut für wasserbau, lehrstuhl für hydrologie und geohydrologie copulas...

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