Computing the volume of a high-dimensional semi-unsupervised hierarchical copula

Abstract

We propose an algorithm for the computation of the volume of a multivariate copula function (and the probability distribution of the counting variable linked to this multidimensional copula function), which is very complex for large dimensions. As is common practice for large dimensional problem, we restrict ourselves to positive orthant dependence and we construct a Hierarchical copula which describes the joint distribution of random variables accounting for dependence among them. This approach approximates a multivariate distribution function of heterogenous variables with a distribution of a fixed number of homogenous clusters, organized through a semi-unsupervised clustering method. These clusters, representing the second-level sectors of hierarchical copula function, are characterized by an into-sector dependence parameter determined by a method which is very similar to the Diversity Score method. The algorithm, implemented in MatLab^™ code, is particularly efficient allowing us to treat cases with a large number of variables, as can be seen in our scalability analysis. As an application, we study the problem of valuing the risk exposure of an insurance company, given the marginals i.e. the risks of each policy.

Keywords:

• copula volume
• hierarchical copula
• clustering unsupervised method
• high dimensional problem
• MatLab^TM code

2000 AMS Subject Classifications :

• 65C50
• 91B30

Acknowledgements

The authors would like to thank their colleagues and friends Umberto Cherubini and Sabrina Mulinacci for useful comments and support and two anonymous referees for pointing out many useful remarks and suggestions.

Notes

The coordinates of the box S are computed as and , where the function sets orderly all the columns of a matrix into the same column.

We refer to D(i, k) as the number of the ways in which one can distribute the integer i into k groups taking into account the order into the groups (o.c.d.’s for short). If we also take into account the cardinalities of the groups, we refer to D ^c (i, k) as the number of the ordered compatible (for groups’ cardinalities) combinatorial distributions.

We refer to [Dcirc](i, k) as the number of the ways in which one can distribute the integer i into k groups without taking into account the order of the groups (c.d.’s for short). If we take into account the groups’ cardinalities, then we refer to as the number of the compatible (for cardinalities’ groups) combinatorial distributions.

The rule which explains how the coordinates are generated, is presented in the following Corollary.

We assume to know the estimations of the real parameters within and between the groups which may be estimated by an econometric methodology, like the maximum likelihood method, possibly simplifying the problem by considering the limiting distributions proposed in Schönbucher 19 for Archimedean copulas with granularity adjustment, and then compensated to account for the homogeneous approximation as suggested in Definition 1.

We must recall the representation of the Survival copula function proposed in Corollary 7. The terms represented in Figure 3 are those inside the first sum in the corollary.

Following this calling order, the c.c.d. considered here is the second one, i.e. j=2.