DPMGibbsSkewT: Slice Sampling of Dirichlet Process Mixture of skew Students's t-distibutions

Description

Slice Sampling of Dirichlet Process Mixture of skew Students's t-distibutions

Usage

DPMGibbsSkewT(z, hyperG0, a, b, N, doPlot = TRUE, nbclust_init = 30,
  plotevery = N/10, diagVar = TRUE, verbose = TRUE, ...)

Arguments

data matrix d x n with d dimensions in rows and n observations in columns.

hyperG0

prior mixing distribution.

shape hyperparameter of the Gamma prior on the parameter of the Dirichlet Process.

scale hyperparameter of the Gamma prior on the parameter of the Dirichlet Process.

number of MCMC iterations.

doPlot

logical flag indicating wether to plot MCMC iteration or not. Default to TRUE.

nbclust_init

number of clusters at initialisation. Default to 30 (or less if there are less than 30 observations).

plotevery

an integer indicating the interval between plotted iterations when doPlot is TRUE.

diagVar

logical flag indicating wether the variance of each cluster is estimated as a diagonal matrix, or as a full matrix. Default is TRUE (diagonal variance).

verbose

logical flag indicating wether partition info is written in the console at each MCMC iteration.

...

additional arguments to be passed to plot_DPMst. Only used if doPlot is TRUE.

Value

a object of class DPMclust with the following attributes:
- mcmc_partitions:
{ a list of length N. Each element mcmc_partitions[n] is a vector of length n giving the partition of the n observations.}
alpha:a vector of length N. cost[j] is the cost associated to partition c[[j]]
U_SS_list:a list of length N containing the lists of sufficient statistics for all the mixture components at each MCMC iteration
weights_list:a list of length N containing the weights of each mixture component for each MCMC iterations
logposterior_list:a list of length N containing the logposterior values at each MCMC iterations
data:the data matrix d x n with d dimensions in rows and n observations in columns
nb_mcmcit:the number of MCMC itertations
clust_distrib:the parametric distribution of the mixture component - "skewt"
hyperG0:the prior on the cluster location

References

Hejblum BP, Alkhassim C, Gottardo R, Caron F, Thiebaut R, Sequential Dirichlet Process Mixtures of Multivariate Skew t-distributions for Model-based Clustering of Flow Cytometry Data, in preparation.

Examples

Run this code

rm(list=ls())

#Number of data
n <- 2000
set.seed(4321)


d <- 2
ncl <- 4

# Sample data
library(truncnorm)

sdev <- array(dim=c(d,d,ncl))

#xi <- matrix(nrow=d, ncol=ncl, c(-1.5, 1.5, 1.5, 1.5, 2, -2.5, -2.5, -3))
#xi <- matrix(nrow=d, ncol=ncl, c(-0.5, 0, 0.5, 0, 0.5, -1, -1, 1))
xi <- matrix(nrow=d, ncol=ncl, c(-0.2, 0.5, 2.4, 0.4, 0.6, -1.3, -0.9, -2.7))
psi <- matrix(nrow=d, ncol=4, c(0.3, -0.7, -0.8, 0, 0.3, -0.7, 0.2, 0.9))
nu <- c(100,25,8,5)
p <- c(0.15, 0.05, 0.5, 0.3) # frequence des clusters
sdev[, ,1] <- matrix(nrow=d, ncol=d, c(0.3, 0, 0, 0.3))
sdev[, ,2] <- matrix(nrow=d, ncol=d, c(0.1, 0, 0, 0.3))
sdev[, ,3] <- matrix(nrow=d, ncol=d, c(0.3, 0.15, 0.15, 0.3))
sdev[, ,4] <- .3*diag(2)


c <- rep(0,n)
w <- rep(1,n)
z <- matrix(0, nrow=d, ncol=n)
for(k in 1:n){
 c[k] = which(rmultinom(n=1, size=1, prob=p)!=0)
 w[k] <- rgamma(1, shape=nu[c[k]]/2, rate=nu[c[k]]/2)
 z[,k] <- xi[, c[k]] + psi[, c[k]]*rtruncnorm(n=1, a=0, b=Inf, mean=0, sd=1/sqrt(w[k])) +
                (sdev[, , c[k]]/sqrt(w[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[["b_xi"]] <- rowMeans(z)
hyperG0[["b_psi"]] <- rep(0,d)
hyperG0[["kappa"]] <- 0.001
hyperG0[["D_xi"]] <- 100
hyperG0[["D_psi"]] <- 100
hyperG0[["nu"]] <- d+1
hyperG0[["lambda"]] <- diag(apply(z,MARGIN=1, FUN=var))/3

 # hyperprior on the Scale parameter of DPM
 a <- 0.0001
 b <- 0.0001

 # 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("Simple example in 2d data")
       +xlab("D1")
       +ylab("D2")
       +theme_bw())
 p #pdf(height=8.5, width=8.5)

 c2plot <- factor(c)
 levels(c2plot) <- c("4", "1", "3", "2")
 pp <- (ggplot(data.frame("X"=z[1,], "Y"=z[2,], "Cluster"=as.character(c2plot)))
       + geom_point(aes(x=X, y=Y, colour=Cluster, fill=Cluster))
       #+ ggtitle("Slightly overlapping skew-normal simulation\\n")
       + xlab("D1")
       + ylab("D2")
       + theme_bw()
       + scale_colour_discrete(guide=guide_legend(override.aes = list(size = 6, shape=22))))
 pp #pdf(height=7, width=7.5)


 ## 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")
       + geom_density(alpha=.2, fill="red")
       + ggtitle(paste("Prior distribution on alpha: Gamma(", a,
                 ",", b, ")\n", sep=""))
      )
 p



# Gibbs sampler for Dirichlet Process Mixtures
 ##############################################
 MCMCsample_st <- DPMGibbsSkewT(z, hyperG0, a, b, N=2000,
                                doPlot, nbclust_init, plotevery=100,
                                gg.add=list(theme_bw(),
                                 guides(shape=guide_legend(override.aes = list(fill="grey45")))),
                               diagVar=FALSE)
 s <- summary(MCMCsample_st, burnin = 900, thin=2, lossFn = "Binder")
 print(s)
 plot(s, hm=TRUE) #pdf(height=8.5, width=10.5) #png(height=700, width=720)
 plot_ConvDPM(MCMCsample_st, from=2)
 #cluster_est_binder(MCMCsample_st$mcmc_partitions[900:1000])

 postalpha <- data.frame("alpha"=s$alpha,
                         "distribution" = factor(rep("posterior",length(s$alpha)),
                         levels=c("prior", "posterior")))

 postK <- data.frame("K"=sapply(lapply(postalpha$alpha, "["),
                                function(x){sum(x/(x+0:(n-1)))}))

 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(postK, aes(x=K))
       + geom_histogram(aes(y=..density..),
                        colour="black", fill="white")
       + geom_density(alpha=.2, fill="blue")
       + ggtitle("Posterior distribution of predicted K\n")
       # Ignore NA values for mean
       # Overlay with transparent density plot
       + geom_vline(aes(xintercept=mean(K, na.rm=TRUE)),
                    color="red", linetype="dashed", size=1)
       #+ scale_x_continuous(breaks=c(0:6)*2, minor_breaks=c(0:6)*2+1)
       + scale_x_continuous(breaks=c(1:12))
     )
 p

 p <- (ggplot(drop=FALSE, alpha=.6)
       + geom_density(aes(x=alpha, fill=distribution),
                      color=NA, alpha=.6,
                      data=postalpha)
       + geom_density(aes(x=alpha, fill=distribution),
                      color=NA, alpha=.6,
                      data=prioralpha)
       + ggtitle("Prior and posterior distributions of alpha\n")
       + scale_fill_discrete(drop=FALSE)
       + theme_bw()
     )
 p

 postK <- data.frame("K"=sapply(lapply(s$mcmc_partitions, unique), length))
 (ggplot(postK, aes(x=K))
       + geom_histogram(aes(y=..density..),
                        colour="black", fill="grey45", binwidth=1)
       + ggtitle("Posterior distribution of K\n")
       # Ignore NA values for mean
       # Overlay with transparent density plot
       + geom_vline(aes(xintercept=mean(K, na.rm=TRUE)),
                    color="red", linetype="dashed", size=1)
       + theme_bw()
       + scale_x_continuous(breaks=c(3:11))
       + xlim(3,11)
     )




 # k-means

 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)
 KMclust <- factor(KM$cluster)
 levels(KMclust) <- c("2", "4", "1", "3")
 dataKM <- data.frame("X"=z[1,], "Y"=z[2,],
                    "Cluster"=as.character(KMclust))
 dataCenters <- data.frame("X"=KM$centers[,1],
                           "Y"=KM$centers[,2],
                           "Cluster"=c("2", "4", "1", "3"))

 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", guide=guide_legend(override.aes=list(size=6,
                                                                                    shape=22)))
       + ggtitle("K-means with K=4 clusters\n")
       + theme_bw()
 )
 p

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