Applied Modeling Techniques and Data Analysis 2

10The Double Flexible Dirichlet: A Structured Mixture Model for Compositional Data

Vectors of proportions arise in a great variety of fields: chemistry, economics, medicine, sociology and many others. Supposing that a whole can be split into D mutually exclusive and exhaustive categories, vectors describing the percentage of each category on the total are referred to as compositional data. The latter are subject to a unit-sum constraint and thus their domain is the D-part simplex. A very popular distribution defined on the simplex is the Dirichlet one. This distribution, despite its several mathematical properties, is poorly parameterized and, therefore, it cannot model many dependence patterns. Some authors have proposed alternatives to the Dirichlet, looking for more flexible distributions which still retain some relevant properties for compositional data. Among these is the flexible Dirichlet (FD), introduced by Ongaro and Migliorati (2013), which generalizes the Dirichlet distribution, that is included as an inner point. Thanks to its mixture structure with D components, it exhibits a more suitable modelization of the covariance matrix. Despite its greater flexibility, the FD lacks in allowing for positive covariances, which are plausible in many applications. The aim of this contribution is to present a further generalization of the Dirichlet, called double flexible Dirichlet (DFD), that takes advantage of a finite mixture structure similar to that of the FD (depending ...