arXiv subjects

Claudia Czado

Publications and source records attributed to Claudia Czado.

At least 55 records · Page 3Linked to original sources

arXiv2017

Growing simplified vine copula trees: improving Dißmann's algorithm

Vine copulas are pair-copula constructions enabling multivariate dependence modeling in terms of bivariate building blocks. One of the main tasks of fitting a vine copula is the selection of a suitable tree structure. For this the prevalent method is a heuristic called Dißmann's algorithm. It sequentially constructs the vine's trees by maximizing dependence at each tree level, where dependence is measured in terms of absolute Kendall's $τ$. However, the algorithm disregards any implications of the tree structure on the simplifying assumption that is usually made for vine copulas to keep inference tractable. We develop two new algorithms that select tree structures focused on producing simplified vine copulas for which the simplifying assumption is violated as little as possible. For this we make use of a recently developed statistical test of the simplifying assumption. In a simulation study we show that our proposed methods outperform the benchmark given by Dißmann's algorithm by a great margin. Several real data applications emphasize their practical relevance.

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arXiv2016

Representing sparse Gaussian DAGs as sparse R-vines allowing for non-Gaussian dependence

Modeling dependence in high dimensional systems has become an increasingly important topic. Most approaches rely on the assumption of a multivariate Gaussian distribution such as statistical models on directed acyclic graphs (DAGs). They are based on modeling conditional independencies and are scalable to high dimensions. In contrast, vine copula models accommodate more elaborate features like tail dependence and asymmetry, as well as independent modeling of the marginals. This flexibility comes however at the cost of exponentially increasing complexity for model selection and estimation. We show a novel connection between DAGs with limited number of parents and truncated vine copulas under sufficient conditions. This motivates a more general procedure exploiting the fast model selection and estimation of sparse DAGs while allowing for non-Gaussian dependence using vine copulas. We demonstrate in a simulation study and using a high dimensional data application that our approach outperforms standard methods for vine structure estimation.

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arXiv2016

D-vine copula based quantile regression

Quantile regression, that is the prediction of conditional quantiles, has steadily gained importance in statistical modeling and financial applications. The authors 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 by maximizing the conditional log-likelihood. Further, typical issues of quantile regression such as quantile crossing or transformations, interactions and collinearity of variables are automatically taken care of. 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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arXiv2016

Examination and visualisation of the simplifying assumption for vine copulas in three dimensions

Vine copulas are a highly flexible class of dependence models, which are based on the decomposition of the density into bivariate building blocks. For applications one usually makes the simplifying assumption that copulas of conditional distributions are independent of the variables on which they are conditioned. However this assumption has been criticised for being too restrictive. We examine both simplified and non-simplified vine copulas in three dimensions and investigate conceptual differences. We show and compare contour surfaces of three-dimensional vine copula models, which prove to be much more informative than the contour lines of the bivariate marginals. Our investigation shows that non-simplified vine copulas can exhibit arbitrarily irregular shapes, whereas simplified vine copulas appear to be smooth extrapolations of their bivariate margins to three dimensions. In addition to a variety of constructed examples, we also investigate a three-dimensional subset of the well-known uranium data set and visually detect that a non-simplified vine copula is necessary to capture its complex dependence structure.

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arXiv2016

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

Practical applications of nonparametric density estimators in more than three dimensions suffer a great deal from the well-known curse of dimensionality: convergence slows down as dimension increases. We show that one can evade the curse of dimensionality by assuming a simplified vine copula model for the dependence between variables. We formulate a general nonparametric estimator for such a model and show under high-level assumptions that the speed of convergence is independent of dimension. We further discuss a particular implementation for which we validate the high-level assumptions and establish its asymptotic normality. Simulation experiments illustrate a large gain in finite sample performance when the simplifying assumption is at least approximately true. But even when it is severely violated, the vine copula based approach proves advantageous as soon as more than a few variables are involved. Lastly, we give an application of the estimator to a classification problem from astrophysics.

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arXiv2016

Model distances for vine copulas in high dimensions

Vine copulas are a flexible class of dependence models consisting of bivariate building blocks and have proven to be particularly useful in high dimensions. Classical model distance measures require multivariate integration and thus suffer from the curse of dimensionality. In this paper we provide numerically tractable methods to measure the distance between two vine copulas even in high dimensions. For this purpose, we consecutively develop three new distance measures based on the Kullback-Leibler distance, using the result that it can be expressed as the sum over expectations of KL distances between univariate conditional densities, which can be easily obtained for vine copulas. To reduce numerical calculations we approximate these expectations on adequately designed grids, outperforming Monte Carlo-integration with respect to computational time. In numerous examples and applications we illustrate the strengths and weaknesses of the developed distance measures.

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arXiv2016

Regime switching vine copula models for global equity and volatility indices

For nearly every major stock market there exist equity and implied volatility indices. These play important roles within finance: be it as a benchmark, a measure of general uncertainty or a way of investing or hedging. It is well known in the academic literature, that correlations and higher moments between different indices tend to vary in time. However, to the best of our knowledge, no one has yet considered a global setup including both, equity and implied volatility indices of various continents, and allowing for a changing dependence structure. We aim to close this gap by applying Markov-switching $R$-vine models to investigate the existence of different, global dependence regimes. In particular, we identify times of "normal" and "abnormal" states within a data set consisting of North-American, European and Asian indices. Our results confirm the existence of joint points in time at which global regime switching takes place.

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arXiv2015

Sequential Bayesian Model Selection of Regular Vine Copulas

Regular vine copulas can describe a wider array of dependency patterns than the multivariate Gaussian copula or the multivariate Student's t copula. This paper presents two contributions related to model selection of regular vine copulas. First, our pair copula family selection procedure extends existing Bayesian family selection methods by allowing pair families to be chosen from an arbitrary set of candidate families. Second, our method represents the first Bayesian model selection approach to include the regular vine density construction in its scope of inference. The merits of our approach are established in a simulation study that benchmarks against methods suggested in current literature. A real data example about forecasting of portfolio asset returns for risk measurement and investment allocation illustrates the viability and relevance of the proposed scheme.

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arXiv2015

Standardized drought indices: A novel uni- and multivariate approach

As drought is among the natural hazards which affects people and economies worldwide and often results in huge monetary losses sophisticated methods for drought monitoring and decision making are needed. Several different approaches to quantify drought have been developed during past decades. However, most of these drought indices suffer from different shortcomings and do not account for the multiple driving factors which promote drought conditions and their inter-dependencies. We provide a novel methodology for the calculation of (multivariate) drought indices, which combines the advantages of existing approaches and omits their disadvantages. Moreover, our approach benefits from the flexibility of vine copulas in modeling multivariate non-Gaussian inter-variable dependence structures. A three-variate data example is used in order to investigate drought conditions in Europe and to illustrate and reason the different modeling steps. The data analysis shows the appropriateness of the described methodology. Comparison to well-established drought indices shows the benefits of our multivariate approach. The validity of the new methodology is verified by comparing the spatial extent of historic drought events based on different drought indices. Further, we show that the assumption of non-Gaussian dependence structures is well-grounded in this real-world application.

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arXiv2015

Block-Maxima of Vines

We examine the dependence structure of finite block-maxima of multivariate distributions. We provide a closed form expression for the copula density of the vector of the block-maxima. Further, we show how partial derivatives of three-dimensional vine copulas can be obtained by only one-dimensional integration. Combining these results allows the numerical treatment of the block-maxima of any three-dimensional vine copula for finite block-sizes. We look at certain vine copula specifications and examine how the density of the block-maxima behaves for different block-sizes. Additionally, a real data example from hydrology is considered. In extreme-value theory for multivariate normal distributions, a certain scaling of each variable and the correlation matrix is necessary to obtain a non-trivial limiting distribution when the block-size goes to infinity. This scaling is applied to different three-dimensional vine copula specifications.

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arXiv2014

Spatial composite likelihood inference using local C-vines

We present a vine copula based composite likelihood approach to model spatial dependencies, which allows to perform prediction at arbitrary locations. This approach combines established methods to model (spatial) dependencies. On the one hand the geostatistical concept utilizing spatial differences between the variable locations to model the extend of spatial dependencies is applied. On the other hand the flexible class of C-vine copulas is utilized to model the spatial dependency structure locally. These local C-vine copulas are parametrized jointly, exploiting an existing relationship between the copula parameters and the respective spatial distances and elevation differences, and are combined in a composite likelihood approach. The new methodology called spatial local C-vine composite likelihood (S-LCVCL) method benefits from the fact that it is able to capture non-Gaussian dependency structures. The development and validation of the new methodology is illustrated using a data set of daily mean temperatures observed at 73 observation stations spread over Germany. For validation continuous ranked probability scores are utilized. Comparison with two other approaches of spatial dependency modeling introduced in yet unpublished work of Erhardt, Czado and Schepsmeier (2014) shows a preference for the local C-vine composite likelihood approach.

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arXiv2014

R-vine Models for Spatial Time Series with an Application to Daily Mean Temperature

We introduce an extension of R-vine copula models for the purpose of spatial dependency modeling and model based prediction at unobserved locations. The newly derived spatial R-vine model combines the flexibility of vine copulas with the classical geostatistical idea of modeling spatial dependencies by means of the distances between the variable locations. In particular the model is able to capture non-Gaussian spatial dependencies. For the purpose of model development and as an illustration we consider daily mean temperature data observed at 54 monitoring stations in Germany. We identify a relationship between the vine copula parameters and the station distances and exploit it in order to reduce the huge number of parameters needed to parametrize a 54-dimensional R-vine model needed to fit the data. The new distance based model parametrization results in a distinct reduction in the number of parameters and makes parameter estimation and prediction at unobserved locations feasible. The prediction capabilities are validated using adequate scoring techniques, showing a better performance of the spatial R-vine copula model compared to a Gaussian spatial model.

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arXiv2013

A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

We apply a linear Bayesian model to seismic tomography, a high-dimensional inverse problem in geophysics. The objective is to estimate the three-dimensional structure of the earth's interior from data measured at its surface. Since this typically involves estimating thousands of unknowns or more, it has always been treated as a linear(ized) optimization problem. Here we present a Bayesian hierarchical model to estimate the joint distribution of earth structural and earthquake source parameters. An ellipsoidal spatial prior allows to accommodate the layered nature of the earth's mantle. With our efficient algorithm we can sample the posterior distributions for large-scale linear inverse problems and provide precise uncertainty quantification in terms of parameter distributions and credible intervals given the data. We apply the method to a full-fledged tomography problem, an inversion for upper-mantle structure under western North America that involves more than 11,000 parameters. In studies on simulated and real data, we show that our approach retrieves the major structures of the earth's interior as well as classical least-squares minimization, while additionally providing uncertainty assessments.

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arXiv2012

Pair-copula Bayesian networks

Pair-copula Bayesian networks (PCBNs) are a novel class of multivariate statistical models, which combine the distributional flexibility of pair-copula constructions (PCCs) with the parsimony of conditional independence models associated with directed acyclic graphs (DAG). We are first to provide generic algorithms for random sampling and likelihood inference in arbitrary PCBNs as well as for selecting orderings of the parents of the vertices in the underlying graphs. Model selection of the DAG is facilitated using a version of the well-known PC algorithm which is based on a novel test for conditional independence of random variables tailored to the PCC framework. A simulation study shows the PC algorithm's high aptitude for structure estimation in non-Gaussian PCBNs. The proposed methods are finally applied to modelling financial return data.

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arXiv2012

Total loss estimation using copula-based regression models

We present a joint copula-based model for insurance claims and sizes. It uses bivariate copulae to accommodate for the dependence between these quantities. We derive the general distribution of the policy loss without the restrictive assumption of independence. We illustrate that this distribution tends to be skewed and multi-modal, and that an independence assumption can lead to substantial bias in the estimation of the policy loss. Further, we extend our framework to regression models by combining marginal generalized linear models with a copula. We show that this approach leads to a flexible class of models, and that the parameters can be estimated efficiently using maximum-likelihood. We propose a test procedure for the selection of the optimal copula family. The usefulness of our approach is illustrated in a simulation study and in an analysis of car insurance policies.

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arXiv2012

Simplified Pair Copula Constructions --- Limits and Extensions

So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hobæk Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on.

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arXiv2012

Selecting and estimating regular vine copulae and application to financial returns

Regular vine distributions which constitute a flexible class of multivariate dependence models are discussed. Since multivariate copulae constructed through pair-copula decompositions were introduced to the statistical community, interest in these models has been growing steadily and they are finding successful applications in various fields. Research so far has however been concentrating on so-called canonical and D-vine copulae, which are more restrictive cases of regular vine copulae. It is shown how to evaluate the density of arbitrary regular vine specifications. This opens the vine copula methodology to the flexible modeling of complex dependencies even in larger dimensions. In this regard, a new automated model selection and estimation technique based on graph theoretical considerations is presented. This comprehensive search strategy is evaluated in a large simulation study and applied to a 16-dimensional financial data set of international equity, fixed income and commodity indices which were observed over the last decade, in particular during the recent financial crisis. The analysis provides economically well interpretable results and interesting insights into the dependence structure among these indices.

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arXiv2012

COPAR - Multivariate time series modeling using the COPula AutoRegressive model

Analysis of multivariate time series is a common problem in areas like finance and economics. The classical tool for this purpose are vector autoregressive models. These however are limited to the modeling of linear and symmetric dependence. We propose a novel copula-based model which allows for non-linear and asymmetric modeling of serial as well as between-series dependencies. The model exploits the flexibility of vine copulas which are built up by bivariate copulas only. We describe statistical inference techniques for the new model and demonstrate its usefulness in three relevant applications: We analyze time series of macroeconomic indicators, of electricity load demands and of bond portfolio returns.

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