An approach to structure determination and estimation of hierarchical Archimedean Copulas and its application to Bayesian classification

Abstract

Copulas are distribution functions with standard uniform univariate marginals. Copulas are widely used for studying dependence among continuously distributed random variables, with applications in finance and quantitative risk management; see, e.g., the pricing of collateralized debt obligations (Hofert and Scherer, Quantitative Finance, 11(5), 775-787, 2011). The ability to model complex dependence structures among variables has recently become increasingly popular in the realm of statistics, one example being data mining (e.g., cluster analysis, evolutionary algorithms or classification). The present work considers an estimator for both the structure and the parameters of hierarchical Archimedean copulas. Such copulas have recently become popular alternatives to the widely used Gaussian copulas. The proposed estimator is based on a pairwise inversion of Kendall's tau estimator recently considered in the literature but can be based on other estimators as well, such as likelihood-based. A simple algorithm implementing the proposed estimator is provided. Its performance is investigated in several experiments including a comparison to other available estimators. The results show that the proposed estimator can be a suitable J Intell Inf Syst alternative in the terms of goodness-of-fit and computational efficiency. Additionally, an application of the estimator to copula-based Bayesian classification is presented. A set of new Archimedean and hierarchical Archimedean copula-based Bayesian classifiers is compared with other commonly known classifiers in terms of accuracy on several well-known datasets. The results show that the hierarchical Archimedean copula-based Bayesian classifiers are, despite their limited applicability for high-dimensional data due to expensive time consumption, similar to highly-accurate classifiers like support vector machines or ensemble methods on low-dimensional data in terms of accuracy while keeping the produced models rather comprehensible.

Kendall’s tau and agglomerative clustering for structure determination of hierarchical Archimedean copulas

“…However note that if Algorithm 1 is transformed to an algorithm for HAC estimation, as, e.g, has been shown in [2], see Section 4 therein, the choice of g becomes essential for the t of the HAC estimates, as is also experimentally shown in the cited article.…”

“…As has already been outlined in the introduction, kt_kagg merges the structure estimator from [2], which delivers a binary structure, with a collapsing procedure from [14], which, if needed, turns the binary structure to a non-binary one. The binary structure estimator from [2] stems from the approach introduced in [4], see Algorithm 2 therein, which, given a Kendall correlation matrix 1 , applies agglomerative clustering in order to recover the underlying structure. Here, this approach is recalled in Algorithm 1.…”

“…We can observe intensive research in this direction, e.g, see [2,5,10,14,17,18,20], which provide answers to the question of how to estimate the structure of a HAC based on a data sample. These answers involve di erent approaches to the problem.…”

“…In [10], Kendall's tau used in the previous two approaches is replaced by a more general τ-Euclidean metric. Finally in [2,5], the authors use an agglomerative clustering technique in which the similarity between two random variables is directly the corresponding Kendall's tau and consider the single-, complete-and average-linkage type of clustering.…”

“…There, the approach denoted kt_kagg, originally proposed in [2] and merged with an approach to collapsing of HAC structures from [14], has shown the best results in the ability to estimate the true structure of a HAC among the 11 estimators considered. Also note that in [3], the kt_kagg approach was improved by a new approach to collapsing HAC structures and re-estimation of the parameters, which is con rmed in the simulation study reported therein.…”