NPflow (version 0.9.0)

DPMGibbsN: Slice Sampling of the Dirichlet Process Mixture Model with a prior on alpha

Description

Slice Sampling of the Dirichlet Process Mixture Model with a prior on alpha

Usage

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

Arguments

z

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

hyperG0

prior mixing distribution.

a

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

b

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

N

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_DPM. 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]]
• listU_mu:
  a list of length N containing the matrices of mean vectors for all the mixture components at each MCMC iteration
• listU_Sigma:
  a list of length N containing the arrays of covariances matrices for all the mixture components at each MCMC iteration
• 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 logposterior values at 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 - "gaussian"
• hyperG0:
  the prior on the cluster location

Examples

rm(list=ls())
#Number of data
n <- 500
#n <- 2000
set.seed(1234)
#set.seed(123)
#set.seed(4321)

# Sample data
m <- matrix(nrow=2, 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(2,2,4))
sdev[, ,1] <- matrix(nrow=2, ncol=2, c(0.3, 0, 0, 0.3))
sdev[, ,2] <- matrix(nrow=2, ncol=2, c(0.1, 0, 0, 0.3))
sdev[, ,3] <- matrix(nrow=2, ncol=2, c(0.3, 0.15, 0.15, 0.3))
sdev[, ,4] <- .3*diag(2)
c <- rep(0,n)
z <- matrix(0, nrow=2, 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(2, mean = 0, sd = 1), nrow=2, ncol=1)
 #cat(k, "/", n, " observations simulated\\n", sep="")
}

 d<-2
 # 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



 # Gibbs sampler for Dirichlet Process Mixtures
 ##############################################
MCMCsample <- DPMGibbsN(z, hyperG0, a, b, N=500, doPlot, nbclust_init, plotevery=50,
                         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