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Präsentation zum Thema: "András Bárdossy IWS Universität Stuttgart"— Präsentation transkript:
1 András Bárdossy IWS Universität Stuttgart Copulas (2)András BárdossyIWSUniversität Stuttgart
2 Sampling only at a number of locations What is between ? Spatial problemsSampling only at a number of locationsWhat is between ?EstimateQuality of estimationSimulate realizationsGeostatistics (Krige, Matheron)Mining applicationsHydro and Environmental sciences
3 GeostatisticsZ(x) Random function – Realisation z(xi)Assumption – „uniform continuity“No differences are known a-prioriIndependent of the location – depends only on h(Semi)Variogramm Covariance function
8 Estimation variance is an index of spatial configuration ProblemsEstimation variance is an index of spatial configurationDoes not depend on the local values“Best” for Gaussian distributionSymmetrical (high and low values not distinguished)Variogram estimation difficultSquared differences – skewed distributionDominated by high valuesIndependence of the pairs not fulfilledStrongly influenced by the marginal distribution
15 Digital elevation models – water dominated regions Contaminations SymmetryDigital elevation models – water dominated regionsMaxima and minimaContaminationsSource vs Background concentrationsKnown but unquantified deterministic processes lead to asymmetry and non-Gaussian dependence
17 Indicator variablesInterpretation as probabilityInterpolation of the indicatorsResult pdf for each locationSimulation restricted to the observed rangeCan copulas be used to overcome some of these problems?
18 How to find such copulas ? Spatial copulasAssumption:Multivariate copula exists for any number of pointsThe bi-variate marginal copulas corresponding to pairs separated by a vector h are translation invariantHow to find such copulas ?
19 Empirical copulasSet of pdf pairs corresponding to points separated by the vector hGeneralization of the variogramEmpirical density using kernel smoothing
33 Copulas and natural processes Natural processes influence high and low values differentlyErosion at high elevationsPollution is spreading not the backgroundWeather relates the high dischargesCopulas of digital elevation models:Spain – eroded old landscapeEcuador – younger but errodedMars – eroded and meteorites
35 Copulas of daily rainfall 601 rainfall stations in the Rhein catchment GermanySize = km2Days with important events with good spatial coverage were selected (400 days of the period )Spatial copulas (densities) for different distances were calculated
39 Radarniederschlag 29. Dezember 2001 11:20-13:20
40 Copula Radarniederschlag 29. Dezember 2001 11:20-13:20
41 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.
42 Definition of a copula from a multivariate distribution:
43 Multivariate normal copula Derived multivariate copulas PossibilitiesMultivariate normal copulaSimple but symmetricalDerived multivariate copulasIf g monotonic – no change of the copulaIf g non monotonic one can get interesting copulas
54 Parameter estimationNon independent pairs – MLFit the rank correlation function and the asymmetryParametric form of the covariance of the original normalFurther work needed
55 For n+1 points the joint distribution is known InterpolationFor n+1 points the joint distribution is knownCalculate the conditional for the unobserved pointFull conditional distribution known – thus confidence intervals can be calculatedExample:4 points – corner of a unit squareA: two of them with F(x)=1 two with F(x)=0B: all with F(x)=0.5
56 Example interpolation – conditional densities m=0, k=1
57 Example interpolation – conditional densities m=1, k=3
58 Example interpolation – conditional densities normal copula
62 Validation of the conditional densities Are the conditional densities OK ?Cross validationCalculation of the frequencies of non exceedence for the observed valuesComparison with the uniformV is much better then normal or Kriging
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