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## partial derivatives of log complete loglikelihood but using the multivariate
## logit transform for the PIs.
### partial.q. Return a matrix with dimension M * N.
partial.q <- function(x, PI, MU, S, t, logit.PI = TRUE){
##
## t is the posterior classification probability.
## PI is the mixing component proportions
## MU, S are means and variance-covariance matrices of the mixture components
##
K <- length(PI)
N <- dim(x)[1]
p <- dim(x)[2]
mp <- p * (p + 1) / 2
M <- K - 1 + K * p + K * p * (p + 1) / 2
G <- Gmat(p)
Id <- diag(p)
invS <- apply(S, 3, solve)
dim(invS) <- c(p, p, K)
SS <- lapply(1:N, function(i){
Sbuf.1 <- NULL
if(K != 1){
if(logit.PI){ # partial wrt the multivariate logit parameter
Sbuf.1 <- t[i, -K] - PI[-K]
} else{
tmp <- t[i, ] / PI
Sbuf.1 <- tmp[1:(K-1)] - tmp[K]
}
}
Sbuf.2 <- list()
Sbuf.3 <- list()
for(j in 1:K){
tmp <- x[i,] - MU[j,]
dim(tmp) <- c(p, 1)
tmp.1 <- t[i,j] * invS[,,j]
Sbuf.2[[j]] <- tmp.1 %*% tmp
Smat <- tmp.1 %*% (tmp %*% t(tmp) %*% invS[,,j] - Id) / 2
Sbuf.3[[j]] <- as.vector(Smat) %*% G
}
c(Sbuf.1, do.call("c", Sbuf.2), do.call("c", Sbuf.3))
})
do.call("cbind", SS)
} # End of partial.q().
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