Fitting finite mixtures of multivariate t-distributions via the em algorithm.

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Mulenga, Francis.M.
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The University of Zambia
Finite mixture models are an important tool in modelling the distribution of a wide class of statistical problems. Mathematically, a g−component finite mixture model of multivariate t-distributions is specified by a convex combination (mixture distribution) of g multivariate t-distributions, which is given by F(x; Ψ) = Xg j=1 τjf(x; μj ,Σj , νj) where Ψ = �� τ1, · · · , τg−1, θT 1 , · · · , θT g T is the parameter vector containing all the parameters of the mixture model, θj = �� μj ,Σj , νj T is the parameter of the jth component of the mixture model whose location parameter is μj , scale matrix Σj , degrees of freedom νj and the jth component mixing proportion τj , with Pg j=1 τj = 1 for j = 1, 2, · · · , g. In the computation of maximum likelihood estimates of Ψ, the EM algorithm often outshines other iterative estimation procedures. The main shortcoming of the EM process is that convergence to the global mode is not guaranteed due to dependence on the starting point Ψ(0). To address the problem of convergence to a local mode when fitting data to mixture models via the EM algorithm, optimized EM initialization methods such as k-means algorithm have been developed in the selection of Ψ(0), especially when the underlying mixture model features Gaussian distributions. Compared to mixtures of Gaussian distributions however, mixtures of t-distributions have been identified as more robust modelling tools due to their heavier tails. The pitfall is that the later have more parameters to be estimated. This may make common EM initialization methods insufficient at attaining the global mode. Hence the need for more refined initialization methods. In this study, we extend the application of the burn-in techniques to mixtures of multivariate t - distributions. The performance of the burn-in scheme is compared with kmeans algorithm, hierarchical clustering and random start based methods. The implementation of the EM algorithm using the package EMMIXskew in the statistical software R, shows that global convergence percentage is highest with the burn-in scheme initialized EM algorithm. With examples using various data sets, we show that the burn-in scheme is a competitive EM initialization method, even when the underlying mixture model features multivariate t-distributions. Key Words: Finite Mixture models, Mixture distributions, EM algorithm, Multivariate t-distributions, the k-means algorithm, Hierarchical clustering, Random starts, Burn-in scheme, EMMIXskew.
Finite mixture models , Multivariate t-distributions.