NMixMCMC: MCMC estimation of (multivariate) normal mixtures with possibly censored data.

Description

This function runs MCMC for a model in which unknown density is specified as a normal mixture with either known or unknown number of components. With a prespecified number of components, MCMC is implemented through Gibbs sampling (see Diebolt and Robert, 1994) and dimension of the data can be arbitrary. With unknown number of components, currently only univariate case is implemented using the reversible jump MCMC (Richardson and Green, 1997). Further, the data are allowed to be censored in which case additional Gibbs step is used within the MCMC algorithm

Usage

NMixMCMC(y0, y1, censor, scale, prior,
         init, init2, RJMCMC,
         nMCMC=c(burn=10, keep=10, thin=1, info=10),
         PED, keep.chains=TRUE, onlyInit=FALSE, dens.zero=1e-300,
         parallel=FALSE)
## S3 method for class 'NMixMCMC':
print(x, dic, \dots)
## S3 method for class 'NMixMCMClist':
print(x, ped, dic, \dots)

Arguments

numeric vector of length $n$ or $n\times p$ matrix with observed data. It contains exactly observed, right-censored, left-censored data and lower limits for interval-censored data.

numeric vector of length $n$ or $n\times p$ matrix with upper limits for interval-censored data. Elements corresponding to exactly observed, right-censored or left-censored data are ignored and can be filled arbitrarily (by NA

censor

numeric vector of length $n$ or $n\times p$ matrix with censoring indicators. The following values indicate:

[object Object],[object Object],[object Object],[object Object]

If it is not supplied then it is assumed that all values are exactly

scale

a list specifying how to scale the data before running MCMC. It should have two components:

[object Object],[object Object]

If there is no censoring, and argument scale is missing then the data are scaled to have zero mean

prior

a list with the parameters of the prior distribution. It should have the following components (for some of them, the program can assign default values and the user does not have to specify them if he/she wishes to use the defaults): [objec

init

a list with the initial values for the MCMC. All initials can be determined by the program if they are not specified. The list may have the following components: [object Object],[object Object],[object Object],[object Object],[object Object],[

init2

a list with the initial values for the second chain needed to estimate the penalized expected deviance of Plummer (2008). The list init2 has the same structure as the list init. All initials in init2 can

RJMCMC

a list with the parameters needed to run reversible jump MCMC for mixtures with varying number of components. It does not have to be specified if the number of components is fixed. Most of the parameters can be determined by the program if the

nMCMC

numeric vector of length 4 giving parameters of the MCMC simulation. Its components may be named (ordering is then unimportant) as: [object Object],[object Object],[object Object],[object Object] In total $(M_{burn} + M_{keep}) \cdot M_{thin}$

PED

a logical value which indicates whether the penalized expected deviance (see Plummer, 2008 for more details) is to be computed (which requires two parallel chains). If not specified, PED is set to TRUE for models

keep.chains

logical. If FALSE, only summary statistics are returned in the resulting object. This might be useful in the model searching step to save some memory.

onlyInit

logical. If TRUE then the function only determines parameters of the prior distribution, initial values, values of scale and parameters for the reversible jump MCMC.

dens.zero

a small value used instead of zero when computing deviance related quantities.

an object of class NMixMCMC or NMixMCMClist to be printed.

dic

logical which indicates whether DIC should be printed. By default, DIC is printed only for models with a fixed number of mixture components.

ped

logical which indicates whether PED should be printed. By default, PED is printed only for models with a fixed number of mixture components.

parallel

a logical value which indicates whether parallel computation (based on packages snow and snowfall-a-package) should be used when running two chains for the purpos

...

additional arguments passed to the default print method.

Value

An object of class NMixMCMC or class NMixMCMClist. Object of class NMixMCMC is returned if PED is FALSE. Object of class NMixMCMClist is returned if PED is TRUE.
Objects of class NMixMCMC have the following components:
iterindex of the last iteration performed.
nMCMCused value of the argument nMCMC.
dimdimension $p$ of the distribution of data
priora list containing the used value of the argument prior.
inita list containing the used initial values for the MCMC (the first iteration of the burn-in).
state.firsta list having the components labeled y, K, w, mu, Li, Q, Sigma, gammaInv, r containing the values of generic parameters at the first stored (after burn-in) iteration of the MCMC.
state.lasta list having the components labeled y, K, w, mu, Li, Q, Sigma, gammaInv, r containing the last sampled values of generic parameters.
RJMCMCa list containing the used value of the argument RJMCMC.
scalea list containing the used value of the argument scale.
freqKfrequency table of $K$ based on the sampled chain.
propKposterior distribution of $K$ based on the sampled chain.
DICa data.frame having columns labeled DIC, pD, D.bar, D.in.bar containing values used to compute deviance information criterion (DIC). Currently only $DIC_3$ of Celeux et al. (2006) is implemented.
movesa data.frame which summarizes the acceptance probabilities of different move types of the sampler.
Knumeric vector with a chain for $K$ (number of mixture components).
wnumeric vector or matrix with a chain for $w$ (mixture weights). It is a matrix with $K$ columns when $K$ is fixed. Otherwise, it is a vector with weights put sequentially after each other.
munumeric vector or matrix with a chain for $\mu$ (mixture means). It is a matrix with $p\cdot K$ columns when $K$ is fixed. Otherwise, it is a vector with means put sequentially after each other.
Qnumeric vector or matrix with a chain for lower triangles of $\boldsymbol{Q}$ (mixture inverse variances). It is a matrix with $\frac{p(p+1)}{2}\cdot K$ columns when $K$ is fixed. Otherwise, it is a vector with lower triangles of $\boldsymbol{Q}$ matrices put sequentially after each other.
Sigmanumeric vector or matrix with a chain for lower triangles of $\Sigma$ (mixture variances). It is a matrix with $\frac{p(p+1)}{2}\cdot K$ columns when $K$ is fixed. Otherwise, it is a vector with lower triangles of $\Sigma$ matrices put sequentially after each other.
Linumeric vector or matrix with a chain for lower triangles of Cholesky decompositions of $\boldsymbol{Q}$ matrices. It is a matrix with $\frac{p(p+1)}{2}\cdot K$ columns when $K$ is fixed. Otherwise, it is a vector with lower triangles put sequentially after each other.
gammaInvmatrix with $p$ columns with a chain for inverses of the hyperparameter $\boldsymbol{\gamma}$.
ordernumeric vector or matrix with order indeces of mixture components related to artificial identifiability constraint defined by a suitable re-labeling algorithm (by default, simple ordering of the first component of the mixture means is used).
It is a matrix with $K$ columns when $K$ is fixed. Otherwise it is a vector with orders put sequentially after each other.
ranknumeric vector or matrix with rank indeces of mixture components. related to artificial identifiability constraint defined by a suitable re-labeling algorithm (by default, simple ordering of the first component of the mixture means is used).
It is a matrix with $K$ columns when $K$ is fixed. Otherwise it is a vector with ranks put sequentially after each other.
mixturedata.frame with columns labeled y.Mean.*, y.SD.*, y.Corr.*.*, z.Mean.*, z.SD.*, z.Corr.*.* containing the chains for the means, standard deviations and correlations of the distribution of the original (y) and scaled (z) data based on a normal mixture at each iteration.
deviancedata.frame with columns labeles LogL0, LogL1, dev.complete, dev.observed containing the chains of quantities needed to compute DIC.
pm.ya data.frame with $p$ columns with posterior means for (latent) values of observed data (useful when there is censoring).
pm.za data.frame with $p$ columns with posterior means for (latent) values of scaled observed data (useful when there is censoring).
pm.indDeva data.frame with columns labeled LogL0, LogL1, dev.complete, dev.observed, pred.dens containing posterior means of individual contributions to the deviance.
pred.densa numeric vector with the predictive density of the data based on the MCMC sample evaluated at data points.
Note that when there is censoring, this is not exactly the predictive density as it is computed as the average of densities at each iteration evaluated at sampled values of latent observations at iterations.
poster.comp.prob1a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to $K$. In that case, poster.comp.prob1 is a matrix with $K$ columns and $n$ rows with estimated posterior component probabilities -- posterior means of the components of the underlying 0/1 allocation vector.
WARNING: By default, the labels of components are based on artificial identifiability constraints based on ordering of the mixture means in the first margin. Very often, such identifiability constraint is not satisfactory!
poster.comp.prob2a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to $K$. In that case, poster.comp.prob2 is a matrix with $K$ columns and $n$ rows with estimated posterior component probabilities -- posterior mean over model parameters.
WARNING: By default, the labels of components are based on artificial identifiability constraints based on ordering of the mixture means in the first margin. Very often, such identifiability constraint is not satisfactory!
summ.y.MeanPosterior summary statistics based on chains stored in y.Mean.* columns of the data.frame mixture.
summ.y.SDCorrPosterior summary statistics based on chains stored in y.SD.* and y.Corr.*.* columns of the data.frame mixture.
summ.z.MeanPosterior summary statistics based on chains stored in z.Mean.* columns of the data.frame mixture.
summ.z.SDCorrPosterior summary statistics based on chains stored in z.SD.* and z.Corr.*.* columns of the data.frame mixture.
poster.mean.wa numeric vector with posterior means of mixture weights after re-labeling. It is computed only if $K$ is fixed and even then I am not convinced that these are useful posterior summary statistics (see label switching problem mentioned above). In any case, they should be used with care.
poster.mean.mua matrix with posterior means of mixture means after re-labeling. It is computed only if $K$ is fixed and even then I am not convinced that these are useful posterior summary statistics (see label switching problem mentioned above). In any case, they should be used with care.
poster.mean.Qa list with posterior means of mixture inverse variances after re-labeling. It is computed only if $K$ is fixed and even then I am not convinced that these are useful posterior summary statistics (see label switching problem mentioned above). In any case, they should be used with care.
poster.mean.Sigmaa list with posterior means of mixture variances after re-labeling. It is computed only if $K$ is fixed and even then I am not convinced that these are useful posterior summary statistics (see label switching problem mentioned above). In any case, they should be used with care.
poster.mean.Lia list with posterior means of Cholesky decompositions of mixture inverse variances after re-labeling. It is computed only if $K$ is fixed and even then I am not convinced that these are useful posterior summary statistics (see label switching problem mentioned above). In any case, they should be used with care.
relabela list which specifies the algorithm used to re-label the MCMC output to compute order, rank, poster.comp.prob1, poster.comp.prob2, poster.mean.w, poster.mean.mu, poster.mean.Q, poster.mean.Sigma, poster.mean.Li.
Cpara list with components useful to call underlying C++ functions (not interesting for ordinary users).

Details

See accompanying paper (Komárek, 2009). In the rest of the helpfile, the same notation is used as in the paper, namely, $n$ denotes the number of observations, $p$ is dimension of the data, $K$ is the number of mixture components, $w_1,\dots,w_K$ are mixture weights, $\boldsymbol{\mu}_1,\dots,\boldsymbol{\mu}_K$ are mixture means, $\boldsymbol{\Sigma}_1,\dots,\boldsymbol{\Sigma}_K$ are mixture variance-covariance matrices, $\boldsymbol{Q}_1,\dots,\boldsymbol{Q}_K$ are their inverses.

For the data $\boldsymbol{y}_1,\dots,\boldsymbol{y}_n$ the following $g_y(\boldsymbol{y})$ density is assumed $$g_y(\boldsymbol{y}) = |\boldsymbol{S}|^{-1} \sum_{j=1}^K w_j \varphi\bigl(\boldsymbol{S}^{-1}(\boldsymbol{y} - \boldsymbol{m}\,|\,\boldsymbol{\mu}_j,\,\boldsymbol{\Sigma}_j)\bigr),$$ where $\varphi(\cdot\,|\,\boldsymbol{\mu},\,\boldsymbol{\Sigma})$ denotes a density of the (multivariate) normal distribution with mean $\boldsymbol{\mu}$ and a~variance-covariance matrix $\boldsymbol{\Sigma}$. Finally, $\boldsymbol{S}$ is a pre-specified diagonal scale matrix and $\boldsymbol{m}$ is a pre-specified shift vector. Sometimes, by setting $\boldsymbol{m}$ to sample means of components of $\boldsymbol{y}$ and diagonal of $\boldsymbol{S}$ to sample standard deviations of $\boldsymbol{y}$ (considerable) improvement of the MCMC algorithm is achieved.

References

Celeux, G., Forbes, F., Robert, C. P., and Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651-674. Cappé, Robert and Rydén (2003). Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers. Journal of the Royal Statistical Society, Series B, 65, 679-700. Diebolt, J. and Robert, C. P. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56, 363--375.

Jasra, A., Holmes, C. C., and Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modelling. Statistical Science, 20, 50--67. Komárek, A. (2009). A new R package for Bayesian estimation of multivariate normal mixtures allowing for selection of the number of components and interval-censored data. Computational Statistics and Data Analysis, 53, 3932--3947.

Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523-539. Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with unknown number of components (with Discussion). Journal of the Royal Statistical Society, Series B, 59, 731-792.

Spiegelhalter, D. J.,Best, N. G., Carlin, B. P., and van der Linde, A. (2002). Bayesian measures of model complexity and fit (with Discussion). Journal of the Royal Statistical Society, Series B, 64, 583-639.

Examples

Run this code

## See also additional material available in 
## YOUR_R_DIR/library/mixAK/doc/
## or YOUR_R_DIR/site-library/mixAK/doc/
## - files Galaxy.pdf, Faithful.pdf, Tandmob.pdf
##         Galaxy.R, Faithful.R, Tandmob.R
## ==============================================

## Simple analysis of Anderson's iris data
## ==============================================
library("colorspace")

data(iris, package="datasets")
summary(iris)
VARS <- names(iris)[1:4]
#COLS <- rainbow_hcl(3, start = 60, end = 240)
COLS <- c("red", "darkblue", "darkgreen")
names(COLS) <- levels(iris[, "Species"])

### Prior distribution and the length of MCMC
Prior <- list(priorK = "fixed", Kmax = 3)
nMCMC <- c(burn=5000, keep=10000, thin=5, info=1000)

### Run MCMC
set.seed(20091230)
fit <- NMixMCMC(y0 = iris[, VARS], prior = Prior, nMCMC = nMCMC)

### Basic posterior summary
print(fit)

### Univariate marginal posterior predictive densities
### based on chain #1
pdens1 <- NMixPredDensMarg(fit[[1]], lgrid=150)
plot(pdens1)
plot(pdens1, main=VARS, xlab=VARS)

### Bivariate (for each pair of margins) predictive densities
### based on chain #1
pdens2a <- NMixPredDensJoint2(fit[[1]])
plot(pdens2a)

plot(pdens2a, xylab=VARS)
plot(pdens2a, xylab=VARS, contour=TRUE)

### Determine the grid to compute bivariate densities
grid <- list(Sepal.Length=seq(3.5, 8.5, length=75),
             Sepal.Width=seq(1.8, 4.5, length=75),
             Petal.Length=seq(0, 7, length=75),
             Petal.Width=seq(-0.2, 3, length=75))
pdens2b <- NMixPredDensJoint2(fit[[1]], grid=grid)
plot(pdens2b, xylab=VARS)

### Plot with contours
ICOL <- rev(heat_hcl(20, c=c(80, 30), l=c(30, 90), power=c(1/5, 2)))
oldPar <- par(mfrow=c(2, 3), bty="n")
for (i in 1:3){
  for (j in (i+1):4){
    NAME <- paste(i, "-", j, sep="")
    MAIN <- paste(VARS[i], "x", VARS[j])
    image(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], col=ICOL,
          xlab=VARS[i], ylab=VARS[j], main=MAIN)
    contour(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], add=TRUE, col="brown4")
  }  
}  

### Plot with data
for (i in 1:3){
  for (j in (i+1):4){
    NAME <- paste(i, "-", j, sep="")
    MAIN <- paste(VARS[i], "x", VARS[j])
    image(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], col=ICOL,
          xlab=VARS[i], ylab=VARS[j], main=MAIN)
    for (spec in levels(iris[, "Species"])){
      Data <- subset(iris, Species==spec)
      points(Data[,i], Data[,j], pch=16, col=COLS[spec])
    }  
  }  
}  

### Set the graphical parameters back to their original values
par(oldPar)

### Clustering based on posterior summary statistics of component allocations
### or on the posterior distribution of component allocations
### (these are two equivalent estimators of probabilities of belonging
###  to each mixture components for each observation)
p1 <- fit[[1]]$poster.comp.prob1
p2 <- fit[[1]]$poster.comp.prob2

### Clustering based on posterior summary statistics of mixture weight, means, variances
p3 <- NMixPlugDA(fit[[1]], iris[, VARS])
p3 <- p3[, paste("prob", 1:3, sep="")]

  ### Observations from "setosa" species (all would be allocated in component 1)
apply(p1[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p2[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p3[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))

  ### Observations from "versicolor" species (almost all would be allocated in component 2)
apply(p1[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p2[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p3[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))

  ### Observations from "virginica" species (all would be allocated in component 3)
apply(p1[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p2[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))
apply(p3[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))

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