Mixtures of Mixtures Based on Multiresolution Analysis Theory

Abstract

In this article, the approaches for exploiting mixtures of mixtures are expanded by using the Multiresolution family of probability density functions (MR pdf). The flexibility and the properties of local analysis of the MR pdf facilitate the location of subpopulations into a given population. In order to do this, two algorithms are provided.

The MR model is more flexible in adapting to the different subpopulations than the traditional mixtures. In addition, the problems of identification of mixtures distributions and the label-switching do not appear in the MR pdf context.

Keywords:

• Cubic box spline
• Location of subpopulations
• Mixture models
• Mixtures of mixtures
• Multiresolution analysis

Mathematics Subject Classification:

• 62G07

Acknowledgments

We gratefully acknowledge the anonymous referees for their valuable comments.

Notes

Nonetheless, recent publications (see, for instance, Lee and Mclachlan, 2011; Lin et al., 2007a; Lin et al., 2007b, Pyne et al., 2009) have approached this problem using skew t distributions. This research shows that the use of these distributions is very appropriate in situations where the clusters are skewed and contain outliers.

Other authors such as Dias and Garcia (2006) used B-Spline to estimate densities in which the number of basis functions acts as the smoothing parameter (which is equivalent to our level of resolution).

This is a translation of four convolutions of 1 _{[0, 1]} with itself, centered at t = 0. Its Fourier transform is 

Note that is a regular grid of points over equally spaced at distance .

This proposition, for simplifying notation, is formulated by using the h last component of mixture without loss of generality because the order of addends in a mixture is not relevant.

In addition, a mixture of three Normal pdfs and a mixture of three Gamma densities have been fitted. The figures have been omitted to save on space because they provide worse results than the previous ones.