estep1: Computing posteriors expected value.

In mixbox: Observed Fisher Information Matrix for Finite Mixture Model

Computing posteriors expected value.

Description

For a finite mixture model with density function

{\cal{M}}(\bold{y}|\bold{\Psi})=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g),

we use method of Basford et al. (1997) for Computing observed Fisher information matrix. Based on above representation for density function of a finite mixture model, we can write

\bold{Y}| {T}, W \sim N_{d}\bigl(\bold{\mu}+\bold{\lambda}{T}, {W} \Sigma\bigr), {T}| W \sim HN({0}, {W}), W \sim f_W(w| \bold{\theta}).

The required posteriors expectations are E\bigl(W^{-1}|\bold{y},\bold{\Psi}\bigr), E\bigl(W^{-1}T|\bold{y},\bold{\Psi}\bigr), and E\bigl(W^{-1}T^2|\bold{y},\bold{\Psi}\bigr).

Usage

estep1(Y, G, weight, mu, sigma, lambda, family, skewness, param, theta, tick, h, N, PDF)

Arguments

Y	an n\times d matrix of observations.
G	the number of components.
weight	a vector of weight parameters (or mixing proportions).
mu	a list of location vectors of G components.
sigma	a list of dispersion matrices of G components.
lambda	a list of skewness vectors of G components.
family	name of the mixing distribution. By default family = "constant" that corresponds to the finite mixture of multivariate normal (or skew normal) distribution. Other candidates for family name are: "bs" (for Birnbaum-Saunders), "burriii" (for Burr type iii), "chisq" (for chi-square), "exp" (for exponential), "f" (for Fisher), "gamma" (for gamma), "gig" (for generalized inverse-Gaussian), "igamma" (for inverse-gamma), "igaussian" (for inverse-Gaussian), "lindley" (for Lindley), "loglog" (for log-logistic), "lognorm" (for log-normal), "lomax" (for Lomax), "ptstable" (for positive \alpha-stable), "ptstable" (for polynomially tilted \alpha-stable), "rayleigh" (for Rayleigh), and "weibull" (for Weibull).
skewness	a logical statement. By default skewness = "FALSE" which means that a symmetric model is fitted to each component (cluster). If skewness = "FALSE", then a skewed model is fitted to each component.
param	name of the elements of {\bold{\theta)}} as the parameter vector of mixing distribution with density function f_W(w|{\bold{\theta)}}.
theta	a list of maximum likelihood estimator for {\bold{\theta)}} across G components.
tick	a binary vector whose length depends on type of family. The elements of tick are either 0 or 1. If element of tick is 0, then the correspoding element of \bold{\theta} is not considered in the formula of f_W(w|{\bold{\theta)}} for computing the required posterior expectations. If element of tick is 1, then the corresponding element of \bold{\theta} is considered in the formula of f_W(w|{\bold{\theta)}}. For instance, if family = "gamma" and either its shape or rate parameter is one, then tick = c(1). This is while, if family = "gamma" and both of the shape and rate parameters are in the formula of f_W(w|{\bold{\theta)}}, then tick = c(1, 1).
h	a positive small value for computing numerical derivative of f_W(w| {\bold{\theta)}}) with respect to {\bold{\theta)}}, that is \partial/ \partial {\bold{\theta)}} f_W(w| {\bold{\theta)}}.
N	a large integer number for computing Monte Carlo approximation of expected value within the E-step of the EM algorithm.
PDF	expression for mixing density function f_W(w| {\bold{\theta)}}.

Value

The required posteriors expectations for approximating the observed Fisher information matrix for restricted finite mixture model. These include \tau_{ig}=E\bigl( Z_{ig}=1 \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr) = {\omega}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr)/\bigl[ \sum_{g=1}^{G} {{\omega}}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr) \bigr], E\bigl( W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( {{U}}W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( U^2 W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), and E\bigl( \partial f_W(w| {\bold{\theta)}})/\partial {\bold{\theta)}} \big \vert {\bold{y}}_{i}, \hat{{\bold{\Psi}}} \bigr).

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- c( 5, -5 )
lambda2 <- c(-5,  5 )
 theta1 <- c( 10 )
 theta2 <- c( 20 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
  param <- c( "a" )
    PDF <- quote( (a/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -a/(2*x) ) )
  tick  <- rep( 1, 2 )
theta10 <- c( 10, 10 )
theta20 <- c( 20, 20 )
 theta0 <- list( theta10 , theta20 )
      Y <- rmix( n, G, weight, model = "restricted", mu, sigma, lambda, family = "igamma",
      theta0)
    estep <- estep1( Y[, 1:2], G, weight, mu, sigma, lambda, family = "igamma",
    skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, PDF)

mixbox documentation built on May 29, 2024, 3:58 a.m.