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NPflow (version 0.13.5)

DPMGibbsSkewN: Slice Sampling of Dirichlet Process Mixture of skew normal distributions

Description

Slice Sampling of Dirichlet Process Mixture of skew normal distributions

Usage

DPMGibbsSkewN(
  z,
  hyperG0,
  a = 1e-04,
  b = 1e-04,
  N,
  doPlot = TRUE,
  nbclust_init = 30,
  plotevery = N/10,
  diagVar = TRUE,
  use_variance_hyperprior = TRUE,
  verbose = 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:

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 iterations

clust_distrib:

the parametric distribution of the mixture component - "skewnorm"

hyperG0:

the prior on the cluster location

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. Default is 0.0001.

b

scale hyperparameter of the Gamma prior on the concentration parameter of the Dirichlet Process. Default is 0.0001. If 0, then the concentration is fixed set to a.

N

number of MCMC iterations.

doPlot

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

nbclust_init

number of clusters at initialization. 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 whether the variance of a cluster is a diagonal matrix. Default is FALSE (full matrix).

use_variance_hyperprior

logical flag indicating whether a hyperprior is added for the variance parameter. Default is TRUE which decrease the impact of the variance prior on the posterior. FALSE is useful for using an informative prior.

verbose

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

...

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

Author

Boris Hejblum

References

Hejblum BP, Alkhassim C, Gottardo R, Caron F and Thiebaut R (2019) Sequential Dirichlet Process Mixtures of Multivariate Skew t-distributions for Model-based Clustering of Flow Cytometry Data. The Annals of Applied Statistics, 13(1): 638-660. <doi: 10.1214/18-AOAS1209> <arXiv: 1702.04407> https://arxiv.org/abs/1702.04407 tools:::Rd_expr_doi("10.1214/18-AOAS1209")

Examples

Run this code
rm(list=ls())

#Number of data
n <- 1000
set.seed(123)

d <- 2
ncl <- 4

# Sample data

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

#xi <- matrix(nrow=d, ncol=ncl, c(-1.5, 1, 1.5, 1, 1.5, -2, -2, -2))
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(-0.3, 0, 0.5, 0.5, 0.5, -1.2, -1, 1))
psi <- matrix(nrow=d, ncol=4, c(0.4, -0.6, 0.8, 0, 0.3, -0.7, -0.3, -1.2))
p <- c(0.2, 0.1, 0.4, 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)
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] <- xi[, c[k]] + psi[, c[k]]*abs(rnorm(1)) + 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[["b_xi"]] <- rep(0,d)
hyperG0[["b_psi"]] <- rep(0,d)
hyperG0[["kappa"]] <- 0.0001
hyperG0[["D_xi"]] <- 100
hyperG0[["D_psi"]] <- 100
hyperG0[["nu"]] <- d + 1
hyperG0[["lambda"]] <- diag(d)

 # 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

 c2plot <- factor(c)
 levels(c2plot) <- c("3", "2", "4", "1")
 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


 ## 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


if(interactive()){
 # Gibbs sampler for Dirichlet Process Mixtures
 ##############################################

 MCMCsample_sn <- DPMGibbsSkewN(z, hyperG0, a, b, N=2500,
                                doPlot, nbclust_init, plotevery=200,
                                gg.add=list(theme_bw(),
                                 guides(shape=guide_legend(override.aes = list(fill="grey45")))),
                               diagVar=FALSE)

 s <- summary(MCMCsample_sn, burnin = 2000, thin=10)
 #cluster_est_binder(MCMCsample_sn$mcmc_partitions[1000:1500])
 print(s)
 plot(s)
 #plot_ConvDPM(MCMCsample_sn, from=2)






 # 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

 postalpha <- data.frame("alpha"=MCMCsample_sn$alpha[501:1000],
                         "distribution" = factor(rep("posterior",1000-500),
                         levels=c("prior", "posterior")))

 postK <- data.frame("K"=sapply(lapply(postalpha$alpha, "["),
                                function(x){sum(x/(x+0:(1000-1)))}))

 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=T)),
                    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=T)),
                    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()
       + xlim(0,100)
     )
 p

#Skew Normal
n=100000
xi <- 0
d <- 0.995
alpha <- d/sqrt(1-d^2)
z <- rtruncnorm(n,a=0, b=Inf)
e <- rnorm(n, mean = 0, sd = 1)
x <- d*z + sqrt(1-d^2)*e
o <- 1
y <- xi+o*x
nu=1.3
w <- rgamma(n,scale=nu/2, shape=nu/2)
yy <- xi+o*x/w
snd <- data.frame("Y"=y,"YY"=yy)
p <- (ggplot(snd)+geom_density(aes(x=Y), fill="blue", alpha=.2)
     + theme_bw()
     + ylab("Density")
     + ggtitle("Y~SN(0,1,10)\n")
     + xlim(-1,6)
     + ylim(0,0.8)
     )
p

p <- (ggplot(snd)+geom_density(aes(x=YY), fill="blue", alpha=.2)
     + theme_bw()
     + ylab("Density")
     + ggtitle("Y~ST(0,1,10,1.3)\n")
     + xlim(-2,40)
     + ylim(0,0.8)
     )
p

p <- (ggplot(snd)
     + geom_density(aes(x=Y, fill="blue"), alpha=.2)
     + geom_density(aes(x=YY, fill="red"), alpha=.2)
     + theme_bw()
     +theme(legend.text = element_text(size = 13), legend.position="bottom")
     + ylab("Density")
     + xlim(-2,40)
     + ylim(0,0.8)
     + scale_fill_manual(name="", labels=c("Y~SN(0,1,10)       ", "Y~ST(0,1,10,1.3)"),
     guide="legend", values=c("blue", "red"))
     )
p

#Variations
n=100000
xi <- -1
d <- 0.995
alpha <- d/sqrt(1-d^2)
z <- rtruncnorm(n,a=0, b=Inf)
e <- rnorm(n, mean = 0, sd = 1)
x <- d*z + sqrt(1-d^2)*e
snd <- data.frame("X"=x)
p <- (ggplot(snd)+geom_density(aes(x=X), fill="blue", alpha=.2)
     + theme_bw()
     + ylab("Density")
     + ggtitle("X~SN(10)\n")
     + xlim(-1.5,4)
     + ylim(0,1.6)
     )
p

o <- 0.5
y <- xi+o*x
snd <- data.frame("Y"=y)
p <- (ggplot(snd)+geom_density(aes(x=Y), fill="blue", alpha=.2)
     + theme_bw()
     + ylab("Density")
     + ggtitle("Y~SN(-1,1,10)\n")
     + xlim(-1.5,4)
     + ylim(0,1.6)
     )
p




#Simple toy example
###################

n <- 500
set.seed(12345)


d <- 2
ncl <- 4

# Sample data

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

xi <- matrix(nrow=d, ncol=ncl, c(-1.5, 1, 1.5, 1, 1.5, -2, -2, -2))
psi <- matrix(nrow=d, ncol=4, c(0.4, -0.6, 0.8, 0, 0.3, -0.7, -0.3, -1.2))
p <- c(0.2, 0.1, 0.4, 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)

#' # Set parameters of G0
hyperG0 <- list()
hyperG0[["b_xi"]] <- rep(0,d)
hyperG0[["b_psi"]] <- rep(0,d)
hyperG0[["kappa"]] <- 0.0001
hyperG0[["D_xi"]] <- 100
hyperG0[["D_psi"]] <- 100
hyperG0[["nu"]] <- d + 1
hyperG0[["lambda"]] <- 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] <- xi[, c[k]] + psi[, c[k]]*abs(rnorm(1)) + sdev[, , c[k]]%*%matrix(rnorm(d, mean = 0,
                                                                        sd = 1), nrow=d, ncol=1)
 cat(k, "/", n, " observations simulated\n", sep="")
}

 MCMCsample_sn_sep <- DPMGibbsSkewN(z, hyperG0, a, b, N=600,
                                    doPlot, nbclust_init, plotevery=100,
                                    gg.add=list(theme_bw(),
                               guides(shape=guide_legend(override.aes = list(fill="grey45")))),
                                    diagVar=TRUE)

 s <- summary(MCMCsample_sn, burnin = 400)

}

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