R: Slice Sampling of Dirichlet Process Mixture of skew Student's...

DPMGibbsSkewT {NPflow} | R Documentation

## Slice Sampling of Dirichlet Process Mixture of skew Student's t-distributions

### Description

Slice Sampling of Dirichlet Process Mixture of skew Student's t-distributions

### Usage

    DPMGibbsSkewT(
      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,
      ...
    )

### Arguments

`z` | data matrix `d x n` with `d` dimensions in rows and `n` observations in columns.
`hyperG0` | parameters of the prior mixing distribution in a `list` with the following named components: `"b_xi"`: a vector of length `d` with the mean location prior parameter. Can be set as the empirical mean of the data in an Empirical Bayes fashion. `"b_psi"`: a vector of length `d` with the skewness location prior parameter. Can be set as 0 a priori. `"kappa"`: a strictly positive number part of the inverse-Wishart component of the prior on the variance matrix. Can be set as very small (e.g. 0.001) a priori. `"D_xi"`: hyperprior controlling the information in ξ (the larger the less information is carried). 100 is a reasonable value, based on Fruhwirth-Schnatter et al., Biostatistics, 2010. `"D_psi"`: hyperprior controlling the information in ψ (the larger the less information is carried). 100 is a reasonable value, based on Fruhwirth-Schnatter et al., Biostatistics, 2010 `"nu"`: a prior number on the degrees of freedom of the t component that must be strictly greater than `d`. Can be set as `d + 1` for instance. `"lambda"`: a `d x d` symmetric definitive positive matrix part of the inverse-Wishart component of the prior on the variance matrix. Can be set as the diagonal of empirical variance of the data in an Empircal Bayes fashion divided by a factor 3 according to Fruhwirth-Schnatter et al., Biostatistics, 2010.
`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 each cluster is estimated as a diagonal matrix, or as a full matrix. Default is `TRUE` (diagonal variance).
`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_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 iterations

- `clust_distrib`: the parametric distribution of the mixture component - `"skewt"`

- `hyperG0`: the prior on the cluster location

### Author(s)

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 https://doi.org/10.1214/18-AOAS1209

Fruhwirth-Schnatter S, Pyne S, Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions, Biostatistics, 2010.

### Examples

    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

     ## 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

    if(interactive()){
     # Gibbs sampler for Dirichlet Process Mixtures
     ##############################################
     MCMCsample_st <- DPMGibbsSkewT(z, hyperG0, a, b, N=1500,
                                    doPlot=TRUE, nbclust_init=30, plotevery=100,
                                   diagVar=FALSE)
     s <- summary(MCMCsample_st, burnin = 1000, thin=10, 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])
    }

[Package NPflow version 0.13.3 Index]