DPpackage (version 1.1-0)

BDPdensity: Semiparametric Bayesian density estimation using Bernstein Polynomials

Description

This function generates a posterior density sample for a Bernstein-Dirichlet model.

Usage

BDPdensity(y,support=3,ngrid=1000,prior,mcmc,state,status,
          data=sys.frame(sys.parent()),na.action=na.fail)

Arguments

y

a vector giving the data from which the density estimate is to be computed.

support

an integer number giving the support of the random density, 1=[0,1], 2=(0, +Inf], and 3=(-In,+Inf). Depending on this, the data is transformed to lie in the [0,1] interval.

ngrid

number of grid points where the density estimate is evaluated. This is only used if dimension of y is lower or equal than 2. The default value is 1000.

prior

a list giving the prior information. The list includes the following parameter: aa0 and ab0 giving the hyperparameters for prior distribution of the precision parameter of the Dirichlet

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

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 specifie

data

data frame.

na.action

a function that indicates what should happen when the data contain NAs. The default action (na.fail) causes BDPdensity to print an error message and terminate if there are

Value

• An object of class BDPdensity representing the Bernstein-Dirichlet model fit. Generic functions such as print, summary, and plot have methods to show the results of the fit. The results include the degree of the polynomial k, alpha, and the number of clusters. 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.
• yclus
  a real vector giving the y latent variables 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.
• k
  giving the degree of the Bernstein polynomial.

Details

This generic function fits a Bernstein-Dirichlet model for density estimation (Petrone, 1999a, 1999b; Petrone and Waserman, 2002): $$y_i | G \sim G, i=1,\ldots,n$$ $$G | kmax, \alpha, G_0 \sim BDP(kmax,\alpha G_0)$$ where, $y_i$ is the transformed data to lie in [0,1], kmax is the upper limit of the discrete uniform prior for the degree of the Bernstein polynomial, $\alpha$ is the total mass parameter of the Dirichlet process component, and $G_0$ is the centering distribution of the DP. The centering distribution corresponds to a $G_0=Beta(a_0,b_0)$ distribution. The precision or total mass parameter, $\alpha$, of the DP prior can be considered as random, having a gamma distribution, $Gamma(a_0,b_0)$, or fixed at some particular value. When $\alpha$ is random the method described by Escobar and West (1995) is used. To let $\alpha$ to be fixed at a particular value, set $a_0$ to NULL in the prior specification.

References

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588. Petrone, S. (1999a) Random Bernstein Polynomials. Scandinavian Journal of Statistics, 26: 373-393. Petrone, S. (1999b) Bayesian density estimation using Bernstein polynomials. The Canadian Journal of Statistics, 27: 105-126. Petrone, S. and Waserman, L. (2002) Consistency of Bernstein polynomial posterior. Journal of the Royal Statistical Society, Series B, 64: 79-100.

See Also

DPdensity, PTdensity

Examples

# Data
      data(galaxy)
      galaxy<-data.frame(galaxy,speeds=galaxy$speed/1000) 
      attach(galaxy)

    # Initial state
      state <- NULL

    # MCMC parameters

      nburn<-1000
      nsave<-10000
      nskip<-10
      ndisplay<-100
      mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)

    # Prior
      prior<-list(aa0=2.01,
                  ab0=0.01,
                  kmax=1000,
                  a0=1,
                  b0=1)

    # Fitting the model

      fit<-BDPdensity(y=speeds,prior=prior,mcmc=mcmc,state=state,status=TRUE)
      
      plot(fit)