András Bárdossy IWS Universität Stuttgart Copulas (2) András Bárdossy IWS Universität Stuttgart

Sampling only at a number of locations What is between ? 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

Experimental Variogramm EC

Point kriging Unbiasedness

Estimation variance using the variogram

Kriging equations using variogram

Estimation variance is an index of spatial configuration 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

Digital elevation models – water dominated regions Contaminations 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 Indicator variogram

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?

How to find such copulas ? 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 ?

Empirical copulas Set of pdf pairs corresponding to points separated by the vector h Generalization of the variogram Empirical density using kernel smoothing

Empirical copula density chloride h=5000m

Empirical copula density chloride h=30000m

Empirical copula density nitrate h=5000m

Empirical copula density pH h=5000m

Cl Variogramm

pH Variogramm

Conditional Entropy: Nitrate 3000 m and 30000m

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

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 km Event 70

Spatial dependence – 5 km Event 347

Spatial dependence – 5 km Event 159

Radarniederschlag 29. Dezember 2001 11:20-13:20

Copula Radarniederschlag 29. Dezember 2001 11:20-13:20

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

Multivariate normal copula Derived multivariate copulas 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

Normal copula Correlation = 0.85

Chi square Non central chi-square distribution

Chi square Multivariate case

n-dimensional Chi-square copula Density

Chi-Square Copulas

Gauss – Chi-square

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

K=2, m=1

Empirical copula density chloride h=5000m

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

For n+1 points the joint distribution is known 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

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

Example interpolation – conditional densities normal copula

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

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Nitrat und Phosphat