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## functions to calculate partial derivatives of loglikelihood function of the
## multivariate Gaussian mixture model but using the multivariate logit
## transform for the PIs.
### partial.logL. Return a matrix with dimension M * N.
partial.logL <- function(x, PI, MU, S, mixdensvals = NULL, logit.PI = TRUE){
##
## PI is the vector of 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)
LTS <- variance2LTSigma(S)
if(is.null(mixdensvals)){
mixdensvals <- dmixmvn(x = x, pi = PI, Mu = MU, LTSigma = LTS)
}
SS <- lapply(1:N, function(i){
mixdens <- mixdensvals[i]
gauscomps <- unlist(lapply(1:K, function(k){
dmvn(x = x[i, ], mu = MU[k, ],
LTsigma = LTS[k, ]) }))
Sbuf.1 <- NULL
if(K != 1){
if(logit.PI){ # partial wrt the multivariate logit parameter
Sbuf.1 <- PI[-K] * gauscomps[-K] / mixdens - PI[-K]
} else{
Sbuf.1 <- (gauscomps[-K] - gauscomps[K]) / mixdens
}
}
Sbuf.2 <- list()
Sbuf.3 <- list()
for(j in 1:K){
tmp <- x[i,] - MU[j,]
dim(tmp) <- c(p, 1)
Sbuf.2[[j]] <- -1 * invS[,,j] %*% tmp * PI[j] * gauscomps[j] / mixdens
Smat <- 0.5 * invS[,,j] %*% (tmp %*% t(tmp) %*% invS[,,j] - Id) *
gauscomps[j] / mixdens
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.logL().
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