master thesis.pdf - Atrium - University of Guelph

graphical diagnostics and proposed a hierarchical method to merge Gaussian componentsbased on the dip test for unimodality (Hartigan and Hartigan, 1985). Li (2005)proposed a method to fit a multilayer mixture (mixture of Gaussian mixtures). Itassumes the number of clusters k is known in advance, and then applies k-meansclustering to the s component means, where s is the estimated number of Gaussianmixture components based on BIC. However, the methods suggested by Wang andRaftery (2002) and Li (2005) shared the same drawback in that both are based onthe means of the clusters but do not take account of clusters’ shape. Besides, thatthe number of clusters is known a priori may be questionable in many real worldapplications.Baudry et al. (2008) proposed fitting a Gaussian mixture model to the data, thenselecting the total number of Gaussian mixture components using BIC, and combiningthem hierarchically according to an entropy criterion. Herein, we propose a newmethod of merging the Gaussian mixture components. In this work (cf., Chapter 3),we use the adjusted Rand index (ARI; Rand, 1971; Hubert and Arabie, 1985) as ourmerging criterion, while Baudry et al. (2008) utilize entropy criteria, and Wang andRaftery (2002) and Li (2005) employ the distance between the means of the clusters.The idea is to create an automated procedure to merge G-component Gaussian mixturedensity models to a multi-layer Gaussian mixture model with the desired numberof clusters (less than G). Multi-layer Gaussian mixture models are an extension tothe Gaussian mixture models in a sense that the density of each cluster is a singleGaussian density or a mixture of Gaussian densities. To illustrate the idea above,consider two examples shown in Figure 1 and Figure 2.12