DPpackage (version 1.1-7.4)

DPMdencens: Bayesian density estimation for interval-censored data using a DPM of normals

Description

This function generates a posterior density sample for a Dirichlet process mixture of normals model for interval-censored data.

Usage

DPMdencens(left,right,ngrid=100,grid=NULL,prior,mcmc,state,status)

Arguments

left

a vector or matrix giving the lower limit for each response variable. Note that the responses are defined on the entire real line and that unknown limits should be indicated by NA.

right

a vector or matrix giving the upper limit for each response variable. Note that the responses are defined on the entire real line and that unknown limits should be indicated by NA.

ngrid

number of grid points where the density estimate is evaluated. The default value is 100.

grid

matrix of dimension ngrid*nvar of grid points where the density estimate is evaluated. The default value is NULL and the grid is chosen according to the range of the interval limits.

prior

a list giving the prior information. The list includes the following parameter: a0 and b0 giving the hyperparameters for prior distribution of the precision parameter of the Dirichlet process prior, alpha giving the value of the precision parameter (it must be specified if a0 is missing, see details below), nu2 and psiinv2 giving the hyperparameters of the inverted Wishart prior distribution for the scale matrix, Psi1, of the inverted Wishart part of the baseline distribution, tau1 and tau2 giving the hyperparameters for the gamma prior distribution of the scale parameter k0 of the normal part of the baseline distribution, m2 and s2 giving the mean and the covariance of the normal prior for the mean, m1, of the normal component of the baseline distribution, respectively, nu1 and psiinv1 (it must be specified if nu2 is missing, see details below) giving the hyperparameters of the inverted Wishart part of the baseline distribution and, m1 giving the mean of the normal part of the baseline distribution (it must be specified if m2 is missing, see details below) and, k0 giving the scale parameter of the normal part of the baseline distribution (it must be specified if tau1 is missing, see details below).

mcmc

a list giving the MCMC parameters. The list must include the following integers: nburn giving the number of burn-in scans, nskip giving the thinning interval, nsave giving the total number of scans to be saved, and ndisplay giving the number of saved scans to be displayed on screen (the function reports on the screen when every ndisplay iterations have been carried out).

state

a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis.

status

a logical variable indicating whether this run is new (TRUE) or the continuation of a previous analysis (FALSE). In the latter case the current value of the parameters must be specified in the object state.

Value

An object of class DPMdencens representing the DP mixture of normals model fit. Generic functions such as print, summary, and plot have methods to show the results of the fit. The results include the baseline parameters, alpha, and the number of clusters.

The function DPrandom can be used to extract the posterior mean of the subject-specific means and covariance matrices.

The MCMC samples of the parameters and the errors in the model are stored in the object thetasave and randsave, respectively. Both objects are included in the list save.state and are matrices which can be analyzed directly by functions provided by the coda package.

The list state in the output object contains the current value of the parameters necessary to restart the analysis. If you want to specify different starting values to run multiple chains set status=TRUE and create the list state based on this starting values. In this case the list state must include the following objects:

ncluster

an integer giving the number of clusters.

muclus

a matrix of dimension (nobservations+100)*(nvariables) giving the means of the clusters (only the first ncluster are considered to start the chain).

sigmaclus

a matrix of dimension (nobservations+100)*( (nvariables)*((nvariables)+1)/2) giving the lower matrix of the covariance matrix of the clusters (only the first ncluster are considered to start the chain).

ss

an interger vector defining to which of the ncluster clusters each observation belongs.

alpha

giving the value of the precision parameter.

m1

giving the mean of the normal components of the baseline distribution.

k0

giving the scale parameter of the normal part of the baseline distribution.

psi1

giving the scale matrix of the inverted-Wishart part of the baseline distribution.

y

giving the matrix of imputed data points.

Details

This generic function fits a Dirichlet process mixture of normal model for density estimation (Escobar and West, 1995) based on interval-censored data: $$y_{ij} \in [l_{ij},u_{ij}), i=1,\ldots,n, j=1,\ldots,m$$ $$y_i | \mu_i, \Sigma_i \sim N(\mu_i,\Sigma_i), i=1,\ldots,n,$$ $$(\mu_i,\Sigma_i) | G \sim G,$$ $$G | \alpha, G_0 \sim DP(\alpha G_0),$$ where, \(y_i=(y_{i1},\ldots,y_{im})\), and the baseline distribution is the conjugate normal-inverted-Wishart distribution, $$G_0 = N(\mu| m_1, (1/k_0) \Sigma) IW (\Sigma | \nu_1, \psi_1)$$

To complete the model specification, independent hyperpriors are assumed (optional), $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$m_1 | m_2, s_2 \sim N(m_2,s_2)$$ $$k_0 | \tau_1, \tau_2 \sim Gamma(\tau_1/2,\tau_2/2)$$ $$\psi_1 | \nu_2, \psi_2 \sim IW(\nu_2,\psi_2)$$

Note that the inverted-Wishart prior is parametrized such that if \(A \sim IW_q(\nu, \psi)\) then \(E(A)= \psi^{-1}/(\nu-q-1)\).

To let part of the baseline distribution fixed at a particular value, set the corresponding hyperparameters of the prior distributions to NULL in the hyperprior specification of the model.

Although the baseline distribution, \(G_0\), is a conjugate prior in this model specification, an algorithm based on auxiliary parameters is adopted. Specifically, the algorithm 8 with \(m=1\) of Neal (2000) is considered in the DPMdencens function.

Finally, note that this function can be used to fit the DPM of normals model for ordinal data proposed by Kottas, Mueller and Quintana (2005). In this case, the arbitrary cut-off points must be specified in left and right. Samples from the predictive distribution contained in the (last columns) of the object randsave (please see below) can be used to obtain an estimate of the cell probabilities.

References

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.

Kottas, A., Mueller, P., Quintana, F. (2005). Nonparametric Bayesian modeling for multivariate ordinal data. Journal of Computational and Graphical Statistics, 14: 610-625.

Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.

See Also

DPrandom, DPdensity

Examples

# NOT RUN {
    ####################################
    # Bivariate example:
    # Censored data is artificially
    # created 
    ####################################

      data(airquality)      
      attach(airquality)

      ozone <- Ozone**(1/3)
      radiation <- Solar.R
      y <- na.omit(cbind(radiation,ozone))

    # create censored-data 
      xxlim <- seq(0,300,50)
      yylim <- seq(1.5,5.5,1)

      left <- matrix(0,nrow=nrow(y),ncol=2)
      right <- matrix(0,nrow=nrow(y),ncol=2)

      for(i in 1:nrow(y))
      {
          left[i,1] <- NA
          right[i,1] <- NA
          if(y[i,1] < xxlim[1]) right[i,1] <- xxlim[1] 
          for(j in 1:length(xxlim))
          { 
              if(y[i,1] >= xxlim[j]) left[i,1] <- xxlim[j]	 
              if(y[i,1] >= xxlim[j]) right[i,1] <- xxlim[j+1]	 
          }	 
          left[i,2] <- NA
          right[i,2] <- NA
          if(y[i,2] < yylim[1]) right[i,2] <- yylim[1] 

          for(j in 1:length(yylim))
          { 
              if(y[i,2] >= yylim[j]) left[i,2] <- yylim[j]	 
              if(y[i,2] >= yylim[j]) right[i,2] <- yylim[j+1]	 
          }	 
      }

    # Prior information
      s2 <- matrix(c(10000,0,0,1),ncol=2)
      m2 <- c(180,3)
      psiinv2 <- diag(c(1/10000,1),2)

      prior <- list(alpha=1,nu1=4,nu2=4,s2=s2,
                    m2=m2,psiinv2=psiinv2,tau1=0.01,tau2=0.01)

    # Initial state
      state <- NULL

    # MCMC parameters
      nburn <- 5000
      nsave <- 5000
      nskip <- 3
      ndisplay <- 1000
      mcmc <- list(nburn=nburn, 
                   nsave=nsave,
                   nskip=nskip,
                   ndisplay=ndisplay)

    # Fitting the model
      fit1 <- DPMdencens(left=left,right=right,ngrid=100,
                         prior=prior,mcmc=mcmc,
                         state=state,status=TRUE)
      fit1
      summary(fit1)

    # Plot the estimated density
      plot(fit1)

    # Extracting the univariate density estimates
      cbind(fit1$grid[,1],fit1$funi[[1]])
      cbind(fit1$grid[,2],fit1$funi[[2]])
       
    # Extracting the bivariate density estimates
      fit1$grid[,1]
      fit1$grid[,2]
      fit1$fbiv[[1]]

    # Plot of the estimated density along with the 
    # true data points and censoring limits
      contour(fit1$grid[,1],fit1$grid[,2],fit1$fbiv[[1]])
      points(y)
      abline(v=xxlim)
      abline(h=yylim)
# }