#'Slice Sampling of the Dirichlet Process Mixture Model with a prior on alpha
#'
#'
#'@param z data matrix \code{d x n} with \code{d} dimensions in rows and \code{n} observations in
#' columns.
#'
#'@param hyperG0 prior mixing distribution.
#'
#'@param a shape hyperparameter of the Gamma prior on the concentration parameter of the Dirichlet
#' Process. Default is \code{0.0001}.
#'
#'@param b scale hyperparameter of the Gamma prior on the concentration parameter of the Dirichlet
#' Process. Default is \code{0.0001}. If \code{0}, then the concentration is fixed set to \code{a}.
#'
#'@param N number of MCMC iterations.
#'
#'@param doPlot logical flag indicating whether to plot MCMC iteration or not. Default to
#' \code{TRUE}.
#'
#'@param plotevery an integer indicating the interval between plotted iterations when \code{doPlot}
#' is \code{TRUE}.
#'
#'@param nbclust_init number of clusters at initialization. Default to 30 (or less if there are less
#' than 30 observations).
#'
#'@param diagVar logical flag indicating whether the variance of each cluster is estimated as a
#' diagonal matrix, or as a full matrix. Default is \code{TRUE} (diagonal variance).
#'
#'@param use_variance_hyperprior logical flag indicating whether a hyperprior is added
#'for the variance parameter. Default is \code{TRUE} which decrease the impact of the variance prior
#'on the posterior. \code{FALSE} is useful for using an informative prior.
#'
#'@param verbose logical flag indicating whether partition info is written in the console at each
#' MCMC iteration.
#'
#'@param ... additional arguments to be passed to \code{\link{plot_DPM}}. Only used if \code{doPlot}
#' is \code{TRUE}.
#'
#'@return a object of class \code{DPMclust} with the following attributes:
#' \item{\code{mcmc_partitions}:}{ a list of length \code{N}. Each
#' element \code{mcmc_partitions[n]} is a vector of length
#' \code{n} giving the partition of the \code{n} observations.}
#' \item{\code{alpha}:}{ a vector of length \code{N}. \code{cost[j]} is the cost
#' associated to partition \code{c[[j]]}}
#' \item{\code{listU_mu}:}{a list of length \code{N} containing the matrices of
#' mean vectors for all the mixture components at each MCMC iteration}
#' \item{\code{listU_Sigma}:}{a list of length \code{N} containing the arrays of
#' covariances matrices for all the mixture components at each MCMC iteration}
#' \item{\code{U_SS_list}:}{a list of length \code{N} containing the lists of
#' sufficient statistics for all the mixture components at each MCMC iteration}
#' \item{\code{weights_list}:}{a list of length \code{N} containing the logposterior values
#' at each MCMC iterations}
#' \item{\code{logposterior_list}:}{a list of length \code{N} containing the logposterior values
#' at each MCMC iterations}
#' \item{\code{data}:}{ the data matrix \code{d x n} with \code{d} dimensions in rows
#'and \code{n} observations in columns.}
#' \item{\code{nb_mcmcit}:}{ the number of MCMC iterations}
#' \item{\code{clust_distrib}:}{the parametric distribution of the mixture component - \code{"gaussian"}}
#' \item{\code{hyperG0}:}{the prior on the cluster location}
#'
#'@author Boris Hejblum
#'
#'@export
#'
#'@examples
#' rm(list=ls())
#' #Number of data
#' n <- 500
#' d <- 4
#' #n <- 2000
#' set.seed(1234)
#' #set.seed(123)
#' #set.seed(4321)
#'
#' # Sample data
#' m <- matrix(nrow=d, ncol=4, c(-1, 1, 1.5, 2, 2, -2, -1.5, -2))
#' p <- c(0.2, 0.1, 0.4, 0.3) # frequence des clusters
#'
#' sdev <- array(dim=c(d,d,4))
#' sdev[, ,1] <- 0.3*diag(d)
#' sdev[, ,2] <- c(0.1, 0.3)*diag(d)
#' sdev[, ,3] <- matrix(nrow=d, ncol=d, 0.15)
#' diag(sdev[, ,3]) <- 0.3
#' sdev[, ,4] <- 0.3*diag(d)
#' c <- rep(0,n)
#' z <- matrix(0, nrow=d, ncol=n)
#' for(k in 1:n){
#' c[k] = which(rmultinom(n=1, size=1, prob=p)!=0)
#' z[,k] <- m[, c[k]] + sdev[, , c[k]]%*%matrix(rnorm(d, mean = 0, sd = 1), nrow=d, ncol=1)
#' #cat(k, "/", n, " observations simulated\n", sep="")
#' }
#'
#' # Set parameters of G0
#' hyperG0 <- list()
#' hyperG0[["mu"]] <- rep(0,d)
#' hyperG0[["kappa"]] <- 0.001
#' hyperG0[["nu"]] <- d+2
#' hyperG0[["lambda"]] <- diag(d)/10
#'
#' # hyperprior on the Scale parameter of DPM
#' a <- 0.0001
#' b <- 0.0001
#'
#' # Number of iterations
#' N <- 30
#'
#' # do some plots
#' doPlot <- TRUE
#' nbclust_init <- 30
#'
#'
#'
#' ## Data
#' ########
#' library(ggplot2)
#' p <- (ggplot(data.frame("X"=z[1,], "Y"=z[2,]), aes(x=X, y=Y))
#' + geom_point()
#' + ggtitle("Toy example Data"))
#' p
#'
#'
#' ## alpha priors plots
#' #####################
#' prioralpha <- data.frame("alpha"=rgamma(n=5000, shape=a, scale=1/b),
#' "distribution" =factor(rep("prior",5000),
#' levels=c("prior", "posterior")))
#' p <- (ggplot(prioralpha, aes(x=alpha))
#' + geom_histogram(aes(y=..density..),
#' colour="black", fill="white", bins=30)
#' + geom_density(alpha=.6, fill="red", color=NA)
#' + ggtitle(paste("Prior distribution on alpha: Gamma(", a,
#' ",", b, ")\n", sep=""))
#' + theme_bw()
#' )
#' p
#'
#'
#'if(interactive()){
#' # Gibbs sampler for Dirichlet Process Mixtures
#' ##############################################
#'
#' MCMCsample <- DPMGibbsN(z, hyperG0, a, b, N=500, doPlot, nbclust_init, plotevery=100,
#' gg.add=list(theme_bw(),
#' guides(shape=guide_legend(override.aes = list(fill="grey45")))),
#' diagVar=FALSE)
#'
#' plot_ConvDPM(MCMCsample, from=2)
#'
#' s <- summary(MCMCsample, burnin = 200, thin=2, posterior_approx=FALSE,
#' lossFn = "MBinderN")
#'
#' F <- FmeasureC(pred=s$point_estim$c_est, ref=c)
#'
#' postalpha <- data.frame("alpha"=MCMCsample$alpha[50:500],
#' "distribution" = factor(rep("posterior",500-49),
#' levels=c("prior", "posterior")))
#' p <- (ggplot(postalpha, aes(x=alpha))
#' + geom_histogram(aes(y=..density..), binwidth=.1,
#' colour="black", fill="white")
#' + geom_density(alpha=.2, fill="blue")
#' + ggtitle("Posterior distribution of alpha\n")
#' # Ignore NA values for mean
#' # Overlay with transparent density plot
#' + geom_vline(aes(xintercept=mean(alpha, na.rm=TRUE)),
#' color="red", linetype="dashed", size=1)
#' )
#' p
#'
#' p <- (ggplot(drop=FALSE, alpha=.6)
#' + geom_density(aes(x=alpha, fill=distribution),
#' color=NA, alpha=.6,
#' data=prioralpha)
#' #+ geom_density(aes(x=alpha, fill=distribution),
#' # color=NA, alpha=.6,
#' # data=postalpha)
#' + ggtitle("Prior and posterior distributions of alpha\n")
#' + scale_fill_discrete(drop=FALSE)
#' + theme_bw()
#' +xlim(0,10)
#' +ylim(0, 1.3)
#' )
#' p
#'
#'}
#'
#'# k-means comparison
#'####################
#'
#' plot(x=z[1,], y=z[2,], col=kmeans(t(z), centers=4)$cluster,
#' xlab = "d = 1", ylab= "d = 2", main="k-means with K=4 clusters")
#'
#' KM <- kmeans(t(z), centers=4)
#' dataKM <- data.frame("X"=z[1,], "Y"=z[2,],
#' "Cluster"=as.character(KM$cluster))
#' dataCenters <- data.frame("X"=KM$centers[,1],
#' "Y"=KM$centers[,2],
#' "Cluster"=rownames(KM$centers))
#'
#' p <- (ggplot(dataKM)
#' + geom_point(aes(x=X, y=Y, col=Cluster))
#' + geom_point(aes(x=X, y=Y, fill=Cluster, order=Cluster),
#' data=dataCenters, shape=22, size=5)
#' + scale_colour_discrete(name="Cluster")
#' + ggtitle("K-means with K=4 clusters\n"))
#' p
#'
#'
#'
#'
DPMGibbsN <- function (z, hyperG0, a=0.0001, b=0.0001, N, doPlot=TRUE,
nbclust_init=30, plotevery=N/10,
diagVar=TRUE, use_variance_hyperprior=TRUE, verbose=TRUE,
...){
p <- nrow(z)
n <- ncol(z)
U_mu <- matrix(0, nrow=p, ncol=n)
U_Sigma = array(0, dim=c(p, p, n))
listU_mu<-list()
listU_Sigma<-list()
# U_SS is a list where each U_SS[[k]] contains the sufficient
# statistics associated to cluster k
U_SS <- list()
#store U_SS :
U_SS_list <- list()
#store clustering :
c_list <- list()
#store sliced weights
weights_list <- list()
#store concentration parameter
#alpha <- list()
#store log posterior probability
logposterior_list <- list()
m <- numeric(n) # number of obs in each clusters
c <-numeric(n)
# initial number of clusters
# Initialization----
# each observation is assigned to a different cluster
# or to 1 of the 50 initial clusters if there are more than
# 50 observations
i <- 1
if(n<nbclust_init){
for (k in 1:n){
c[k] <- k
#cat("cluster ", k, ":\n")
U_SS[[k]] <- update_SS(z=z[, k, drop=FALSE], S=hyperG0, hyperprior = NULL)
NiW <- rNiW(U_SS[[k]], diagVar=diagVar)
U_mu[, k] <- NiW[["mu"]]
U_SS[[k]][["mu"]] <- NiW[["mu"]]
U_Sigma[, , k] <- NiW[["S"]]
U_SS[[k]][["S"]] <- NiW[["S"]]
m[k] <- m[k]+1
U_SS[[k]][["weight"]] <- 1/n
}
} else{
c <- sample(x=1:nbclust_init, size=n, replace=TRUE)
for (k in unique(c)){
obs_k <- which(c==k)
#cat("cluster ", k, ":\n")
U_SS[[k]] <- update_SS(z=z[, obs_k,drop=FALSE], S=hyperG0)
NiW <- rNiW(U_SS[[k]], diagVar=diagVar)
U_mu[, k] <- NiW[["mu"]]
U_SS[[k]][["mu"]] <- NiW[["mu"]]
U_Sigma[, , k] <- NiW[["S"]]
U_SS[[k]][["S"]] <- NiW[["S"]]
m[k] <- length(obs_k)
U_SS[[k]][["weight"]] <- m[k]/n
}
}
listU_mu[[i]]<-U_mu
listU_Sigma[[i]]<-U_Sigma
alpha <- c(log(n))
U_SS_list[[i]] <- U_SS
c_list[[i]] <- c
weights_list[[i]] <- numeric(length(m))
weights_list[[i]][unique(c)] <- table(c)/length(c)
logposterior_list[[i]] <- logposterior_DPMG(z, mu=U_mu, Sigma=U_Sigma,
hyper=hyperG0, c=c, m=m, alpha=alpha[i], n=n, a=a, b=b, diagVar)
if(verbose){
cat(i, "/", N, " samplings:\n", sep="")
cat(" logposterior = ", sum(logposterior_list[[i]]), "\n", sep="")
cl2print <- unique(c)
cat(length(cl2print), "clusters:", cl2print[order(cl2print)], "\n\n")
}
if(doPlot){
plot_DPM(z=z, U_mu=U_mu, U_Sigma=U_Sigma,
c=c, i=i, alpha=alpha[[i]], U_SS=U_SS, ...)
}
for(i in 2:N){
nbClust <- length(unique(c))
alpha <- c(alpha,sample_alpha(alpha_old=alpha[i-1], n=n,
K=nbClust, a=a, b=b))
slice <- sliceSampler_N(c=c, m=m, alpha=alpha[i],
z=z, hyperG0=hyperG0,
U_mu=U_mu, U_Sigma=U_Sigma,
diagVar=diagVar)
m <- slice[["m"]]
c <- slice[["c"]]
weights_list[[i]] <- slice[["weights"]]
U_mu<-slice[["U_mu"]]
U_Sigma<-slice[["U_Sigma"]]
# Update cluster locations
fullCl <- which(m!=0)
for(j in fullCl){
obs_j <- which(c==j)
#cat("cluster ", j, ":\n")
if(use_variance_hyperprior){
U_SS[[j]] <- update_SS(z=z[, obs_j, drop=FALSE], S=hyperG0, hyperprior = list("Sigma"=U_Sigma[,,j]))
}else{
U_SS[[j]] <- update_SS(z=z[, obs_j, drop=FALSE], S=hyperG0)
}
NiW <- rNiW(U_SS[[j]], diagVar=diagVar)
U_mu[, j] <- NiW[["mu"]]
U_SS[[j]][["mu"]] <- NiW[["mu"]]
U_Sigma[, , j] <- NiW[["S"]]
U_SS[[j]][["S"]] <- NiW[["S"]]
U_SS[[j]][["weight"]] <- weights_list[[i]][j]
#cat("sampled S =", NiW[["S"]], "\n\n\n")
}
listU_mu[[i]]<-U_mu
listU_Sigma[[i]]<-U_Sigma
U_SS_list[[i]] <- U_SS[which(m!=0)]
c_list[[i]] <- c
logposterior_list[[i]] <- logposterior_DPMG(z, mu=U_mu, Sigma=U_Sigma,
hyper=hyperG0, c=c, m=m, alpha=alpha[i], n=n, a=a, b=b, diagVar)
if(verbose){
cat(i, "/", N, " samplings:\n", sep="")
cat(" logposterior = ", sum(logposterior_list[[i]]), "\n", sep="")
cl2print <- unique(c)
cat(length(cl2print), "clusters:", cl2print[order(cl2print)], "\n\n")
}
if(doPlot && i/plotevery==floor(i/plotevery)){
plot_DPM(z=z, U_mu=U_mu, U_Sigma=U_Sigma, c=c, i=i,
alpha=alpha[i], U_SS=U_SS, ...)
}
}
# return(list("clusters" = c, "U_mu" = U_mu, "U_Sigma" = U_Sigma,
# "partition" = m, "alpha"=alpha, "U_SS_list"=U_SS_list,
# "c_list" = c_list, "weights_list"=weights_list,
# "logposterior_list"=logposterior_list,
# "nb_mcmcit"=N,
# "clust_distrib"="Normal",
# "hyperG0"=hyperG0))
dpmclus <- list("mcmc_partitions" = c_list,
"alpha"=alpha,
# "U_mu" = U_mu,
# "U_Sigma" = U_Sigma,
"listU_mu"=listU_mu,
"listU_Sigma"=listU_Sigma,
"U_SS_list"=U_SS_list,
"weights_list"=weights_list,
"logposterior_list"=logposterior_list,
"data"=z,
"nb_mcmcit"=N,
"clust_distrib"="gaussian",
"hyperG0"=hyperG0)
class(dpmclus) <- "DPMMclust"
return(dpmclus)
}
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