Cross-Sectional Finite Mixture Models Sample Clauses

Cross-Sectional Finite Mixture Models. 2.1.1 Overview n Finite mixture models offer a way to model heterogeneity in a cluster analysis con- text and to accommodate situations in which a single parametric family is unable to satisfactorily model local variations in observed data. As described in McLachlan and Peel(2000) [54], let Y1, . . . , Yn denote a random sample of size n where Yi is a J-dimensional random vector with probability density function f (yi) on XX . For cross- sectional applications, Yi is a vector of random variables corresponding to J measure- ments taken on the ith subject. Note that, although this notation assumes that there are J measurements on each subject, the methodology described below can be nat- urally extended to accommodate unbalanced data. Then, let Y = (YT , . . . , YT )T represent the entire sample. It follows that the probability density for Yi under a C-component mixture model is ∑ C f (yi; ψ) = πgfg(yi; θg), g=1 where fg(yi; θg) are component densities and πg are mixing proportions or weights satisfying ∑ C g=1 πg = 1 for g = 1, . . . , C. The vector ψ = (π1, . . . , πC−1, θ1, . . . , θC) is the vector containing all unknown parameters. Assume that x0, . . . , xX are distinct and let x = (x0, . . . , xX). In this framework, a C-component mixture model can be viewed as arising when Yi is drawn from a population consisting of C subgroups, G1, . . . , GC, in proportions π1, . . . , πC. To clarify this interpretation of finite mixture models, let Zi be a C- dimensional component-label vector, where the gth element of Zi is defined to be one if the origin of Yi is the gth mixture component and zero otherwise. In this framework, Zi is distributed according to a multinomial distribution consisting of one draw on C categories with probabilities π1, . . . , πC, i.e. Zi ∼ MultC(1, π). Further, by Bayes’ Rule, the posterior probability that subject i belongs to the gth component of the mixture model given observation yi can be expressed as τig = τig(yi; ψ) = Pr(Zig = 1|yi) ∑ πgfg(yi; θg) = , d=1 πdfd(yi; θd) where g = 1, . . . , C and i = 1, . . . , n Note that the mixing proportions may also be modeled as functions of a P × 1 vector of covariates, xi[19]. In this situation, the mixing proportions are subject- specific and can be modeled using a polytomous logistic regression model. Specifically, for the ith subject with observed feature variables yi and covariates xi, πig = πg (xi; α) g = exp(αT xi) 1 + ∑C−1 exp(αT xi) g = 1, . . . , C, ( ) C C−1 where α = 0 and α = (αT , ....