Examination and visualization of the simplifying assumption for three-dimensional vine copulas

Members: Matthias Killiches and Daniel Kraus

Description: When using vine copula models, one usually makes the so-called simplifying assumption in order to facilitaten estimation and inference at low computational costs. It is assumed that the bivariate copulas of conditional distributions are independent of the corresponding conditioning variables. We try to gain insight which impact the simplifying assumption has on the resulting model. For this purpose we consider three-dimensional vine copula densities where the influence of the simplifying assumption is easily interpretable. We compare simplified and non-simplified vine copula densities by considering their contour surfaces in three dimensions in order to find characteristics of non-simplified models.

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Vine copula based modeling of spatial dependencies

Members: Tobias Erhardt, Claudia Czado, Ulf Schepsmeier

Description: Classical approaches for the modeling of spatial dependency mostly assume Gaussian dependency structures. This assumption may simplify the modeling process, however it is not always met, when we face real world problems. We investigate and develop new approaches for spatial dependency modeling, based on the flexible class of vine copula distributions, which are able to capture non-Gaussian spatial dependencies. One approach enhances the regular vine copula for spatial applications, another is a composite likelihood based inference technique combining local canonical vine copulas. The proposed "spatial R-vine copula model" and the "spatial local C-vine composite likelihood approach" combine the flexibility of vine copulas with the geostatistical idea of modeling the extent of spatial dependencies using the distances between the variable locations. We develop maximum likelihood estimation techniques for parameter estimation, as well as methods for spatial prediction at unobserved locations.

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Drought modeling and monitoring by novel statistical and analytical methods

Members: Upasana Bhuyan, Claudia Czado, Tobias Erhardt, Michael Matiu, Annette Menzel, Christian Zang

Description: Droughts are weather-related disasters which affect the majority of the world’s population. They are caused by recurring climate extremes which lack easily identifiable onset and termination dates. The increasing occurrence and intensity of droughts, aggravated by climate change, land-use change and other factors, requires mitigation-based drought management approaches. Such approaches demand a consistent multivariate spatio-temporal framework for defining, characterizing, backcasting and monitoring drought. The recent progress towards understanding drought needs further improvement in terms of multiscalar, temporal and spatial characterization of drought events.
This project aims to develop and apply vine copula based methodologies to the joint modeling of meteorological variables and to validate the model with impacts of past events on the biosphere. The spatial and temporal dependency structures of drought relevant variables will be modeled with a stepwise approach. The multivariate dependencies of the different variables will be modeled using a new extension of vine copulas supporting non-symmetric dependence. The models will provide novel drought indices, which are supposed to surpass existing approaches of drought characterization. These indices will be validated by responses of different biosphere systems to historical drought events. The proxy data for the validation process consists of tree ring data, data on atmospheric isotopic composition and archived impact data from various sources.

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Modified Kullback-Leibler divergence for vine copulas

Members: Matthias Killiches, Daniel Kraus, Claudia Czado

Description: Classical model distance measures include multivariate integration and thus suffer from the curse of dimensionality. We provide numerically tractable methods to measure the distance between two vine copulas even in high dimensions. For this purpose, we develop a distance measure based on the Kullback-Leibler distance. To reduce numerical calculations we focus only on crucial spots. For inference and model selection of vines one usually makes the simplifying assumption that the copulas of conditional distributions are independent of their conditioning variables. We present a hypothesis test for this simplifying assumption based on parametric bootstrapping and our distance measure and empirically show the test to have a high power.

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Block-maxima of vines

Members: Matthias Killiches, Claudia Czado

Description: We are interested in examining the dependence structure of finite componentwise block-maxima of multivariate distributions. We provide a general closed form expression for the copula density of the vector of the block-maxima using partial derivatives of the corresponding copula cdf. For three-dimensional vine copulas partial derivatives can be obtained by only one-dimensional integration allowing the numerical treatment of the copula density of the block-maxima of any three-dimensional vine for finite block-sizes. As a link to extreme-value theory the convergence behavior of the copula of the (properly scaled) maxima is considered empirically for increasing block-size.

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Sparse vine distributions

Members: Dominik Müller, Claudia Czado

Description: Regular vines (R-vines) have been introduced and further explored as they constitute a flexible graphical model to describe dependencies using only pair copulas as building blocks. Unfortunately, the complexity of these models strongly increases in larger dimensions. Hence, methods have to be developed and explored in order to handle these models in such settings, as present in financial (risk) management or engineering. Thus, we consider dimensionality reduction techniques, e.g. by application of DAG (directed acyclic graph) models to infer (conditional) independences between random variables, which can be used in order to yield (maximally) truncated vines. In these truncated vines, we constrain ourselves to model only the most statistically significant pairwise correlations and assume independence otherwise. In a first step, this work is carried out for Gaussian copulas and will be subsequently extended to the non-Gaussian case. In applications, these vine copula models are linked to methods from data mining and multivariate statistics.

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Nonparametric estimation of pair copula constructions

Members: Thomas Nagler, Claudia Czado

Description: Pair copula constructions are a flexible framework to model multivariate dependency, since they allow to model each (conditioned) bivariate dependence by a separate bivariate copula. Although the number of bivariate copula families is large, it might happen, that no parametric copula provides a reasonable fit to one or more of the (conditional) bivariate dependencies considered. In this case it might be helpful to be able to choose a nonparametric copula for the respective pairs. We want to extend the inference of R-vine PCCs by a nonparametric choice for the bivariate copulas in the R-vine. Further, a model selection procedure shall be established, that allows to automatically decide, if a nonparametric estimation of the bivariate copula is necessary for a certain pair in the R-vine.

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D-vine Copula Based Quantile Regression

Members: Daniel Kraus, Claudia Czado

Description: Quantile regression, that is the prediction of a random variable's quantiles conditioned on other random variables taking on certain values, has perpetually gained importance in statistical modeling and financial applications. We introduce a new semiparametric quantile regression method based on sequentially fitting a likelihood optimal D-vine copula to given data resulting in highly flexible models with easily extractable conditional quantiles. As a subclass of regular vine copulas, D-vines enable the modeling of multivariate copulas in terms of bivariate building blocks, a so-called pair-copula construction (PCC). The proposed algorithm works fast and accurate even in high dimensions and incorporates an automatic variable selection. In a simulation study the improved accuracy and saved computational time of the approach in comparison with established quantile regression methods is highlighted. An extensive financial application to international credit default swap (CDS) data including stress testing and Value at Risk (VaR) prediction demonstrates the usefulness of the proposed method.

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