The pair-copula construction in space and time: a new approach to model spatio-temporal dependencies
Zusammenfassung der Projektergebnisse
Continuously spreading environmental processes are often only observed at a distinct set of locations and a continuous model is adopted to capture the phenomenon across space. Temperature measurements recorded at weather stations that are used to produce smooth maps is one of the many examples. Hence, an interpolation has to take place to derive values at unobserved locations. Besides deterministic approaches, kriging is a widely used probabilistic interpolation technique. At times, certain properties of the underlying multivariate Gaussian distribution are not desired and a more flexible probabilistic representation of the phenomenon is needed. Copulas have proven to be a useful tool to build up non-Gaussian distributions. Vine copulas allow to flexibly combine bivariate copulas to multivariate copulas leading to distributions of higher dimensions. The research carried out in this project has led to a new approach that allows to build vine copulas that are aware of separating distances across space and time. To achieve this, the building blocks of the vine copula are composed out of convex combinations of bivariate copulas. The weight of the convex combination as well as the copulas’ parameters are defined by distance over space and time. Different use cases are considered to asses power and quality of this new probabilistic modelling approach. A prototypical implementation is available as R package. The implementations in R have been made in conjunction with the research to empirically support this new development. While improvements of the interpolation in terms of cross-validation statistics depend on the application scenario, the obtained confidence bands have desirable properties. This is partly due to the conditional predictive distribution being able to take any shape for each prediction location. Additionally, the freedom to choose any marginal distribution ensures that the confidence intervals are within the range of the modelled distribution. Furthermore, the confidence intervals depend on the predicted value and the layout of the local neighbourhood. Therefore, the spatial vine copula approach is assumed to provide in general a more realistic view of the uncertainties than the kriging variance. While this approach is still in its infancy, its potential to improve especially the modelling of heavily skewed spatial random fields becomes apparent.
Projektbezogene Publikationen (Auswahl)
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(2012): Modelling Dependence in Space and Time with Vine Copulas. Extended abstract for Geostats 2012, Oslo, Norway, 11-15 June 2012
Gräler, B. & E. Pebesma
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(2013): Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation. Hydrology and Earth System Sciences, 17, 1281-1296. HESS-17-1281-2013
Gräler, B., M. J. van den Berg, S. Vandenberghe, A. Petroselli, S. Grimaldi, B. De Baets, & N. E. C. Verhoest
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(2014): Modelling Skewed Spatial Random Fields through the Spatial Vine Copula. Spatial Statistics, Elsevier, 10, 87 - 102
Gräler, B.
