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Applying Finite Mixture Models - University of Queensland：应用有限混合模型-昆士兰大学
Topics Introduction Application of EM algorithm Examples of normal mixtures Robust mixture modeling Number of components in a mixture model Number of nonnormal components Mixture models for failure-time data Mixture software Astronomy Biology Economics Engineering 1.2 Initial Approach to Mixture Analysis Classic paper of Pearson (1894) Figure 1: Plot of forehead to body length data on 1000 crabs and of the fitted one-component (dashed line) and two-component (solid line) normal mixture models. 1.3 Basic Definition We let Y1,…. Yn denote a random sample of size n where Yj is a p-dimensional random vector with probability density function f (yj) where the f i(yj) are densities and the pi are nonnegative quantities that sum to one. 1.4 Interpretation of Mixture Models An obvious way of generating a random vector Yj with the g-component mixture density f (Yj), given by (1), is as follows. Let Zj be a categorical random variable taking on the values 1, … ,g with probabilities p1, … pg, respectively, and suppose that the conditional density of Yj given Zj=i is f i(yj) (i=1, … , g). Then the unconditional density of Yj, (that is, its marginal density) is given by f (yj). 1.5 Shapes of Some Univariate Normal Mixtures Consider where denotes the univariate normal density with mean m and variance s2. Figure 2: Plot of a mixture density of two univariate normal components in equal proportions with common variance s2=1 Figure 3: Plot of a mixture density of two univariate normal components in proportions 0.75 and 0.25 with common variance 1.6 Parametric Formulation of Mixture Model In many applications, the component densities fi(yj) are specified to belong to some parametric family. In this case, the component densities fi(yj) are specified as fi(yj;qi), where qi is the vector of unknown parameters in the postulated form for the ith component density in the mixture. The mixture density f (yj) can then be written as 1.6 cont. 1.7 Identifiability of Mixture Di
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