DPpackage (version 1.1-7.4)

LDBDPdensity: Bounded Density Regression using Dependent Bernstein Polynomials

Description

This function generates a posterior density sample for a Linear Dependent Bernstein-Dirichlet Process model for bounded conditional density estimation.

Usage

LDBDPdensity(formula,xpred,prior,mcmc,state,status,ngrid=100,
             grid=NULL,compute.band=FALSE,type.band="PD",
             data=sys.frame(sys.parent()),
             na.action=na.fail,work.dir=NULL)

Arguments

formula

a two-sided linear formula object describing the model fit, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. The design matrix is used to model the distribution of the response in the LDBDP model. The response is assumed to take values in [0,1].

xpred

a matrix giving the covariate values where the predictive density is evaluated.

prior

a list giving the prior information. The list includes the following parameter: lambda a double precision giving the parameter of the truncated Poisson prior distribution for the degree, k, of the Bernstein polynomial, maxn an integer giving the truncation of the the stick-breaking approximation to the dependent Dirichlet process, alpha giving the value of the precision parameter of the dependent Dirichlet process, m0 and S0 giving the hyperparameters of the normal prior distribution for the mean, mub, of the normal baseline distribution nu and psiinv giving the hyperparameters of the inverted Wishart prior distribution for the scale matrix, Sb, of the baseline distribution.

mcmc

a list giving the MCMC parameters. The list must include the following elements: nburn an integer giving the number of burn-in scans, nskip an integer giving the thinning interval, nsave an integer giving the total number of scans to be saved, ndisplay an integer 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), slicebeta a double precision giving the Slice sampling parameter for the regression coefficients, and slicev a double precision giving the Slice sampling parameter for the stick-breaking parameters.

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.

ngrid

integer giving the number of grid points where the conditional density estimate is evaluated. The default is 100.

grid

vector of grid points where the conditional density estimate is evaluated. The default value is NULL and the grid is chosen according to the range of the data.

compute.band

logical variable indicating whether the credible band for the conditional density and mean function must be computed.

type.band

string indication the type of credible band to be computed; if equal to "HPD" or "PD" then the 95 percent pointwise HPD or PD band is computed, respectively.

data

data frame.

na.action

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

work.dir

working directory.

Value

An object of class LDBDPdensity representing the LDBDP model fit. Generic functions such as print, plot, and summary have methods to show the results of the fit. The results include k. mub and Sb.

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:

k

an integer giving the degree of the Bernstein polynomial.

beta

a matrix of dimension maxn times the number of columns in the design matrix, giving the regression coefficients for each stick-breaking component.

alpha

giving the value of the precision parameter.

mub

giving the mean of the normal baseline distributions.

Sb

giving the covariance matrix the normal baseline distributions.

v

giving the maxn vector of stick-breaking beta random variables. The last element in this vector must be equal to 1.

Details

This generic function fits a Linear Dependent Dirichlet-Bernstein model (Barrientos, Jara and Quintana, 2010), given by: $$y_i | G_{X_i} \sim G_{X_i}$$ $$\{G_{X}: X \in \mathcal{X} \}| k,\alpha, G_0 \sim LDBDP(k,\alpha G_0)$$ where, \(G_0 = N(\beta| \mu_b, S_b)\). To complete the model specification, independent hyperpriors are assumed, $$k | \lambda \sim Poisson(\lambda)I(k \geq 1)$$ $$\mu_b | m_0, S_0 \sim N(m_0,S_0)$$ $$S_b | \nu, \Psi \sim IW(\nu,\Psi)$$

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

Note also that the LDBDP model is a extension of the Bernstein-Dirichlet model for density estimation (Petrone, 1999a, 1999b; Petrone and Waserman, 2002).

The computational implementation of the model is based on the finite approximation to the dependent Dirichlet process prior and on the use of conditional MCMC methods. The regression coefficients and stick-breaking parameters are updated jointly using multivariate Slice sampling (Neal, 2003). The degree of the Bernstein polynomial is updated using a Metropolis-Hasting algorithm.

References

Barrientos, F., Jara, A., Quintana, F. (2010). Bounded density regression using dependent Bernstein polynomials. Technical Report, Department of Statistics, Pontificia Universidad Catolica de Chile.

Neal, R. (2003) Slice sampling. Anals of Statistics, 31: 705-767.

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

DPcdensity, LDDPdensity

Examples

# NOT RUN {
    ######################## 
    # Simulate data
    ########################
      nrec <- 500
      x <- runif(nrec)
      y <- rep(0,nrec)
      for(i in 1:nrec)
      {
           y[i] <- ifelse(runif(1) < (0.8-0.5*x[i]^2), 
                    rbeta(1,22-(x[i]^2)*20,5+x[i]*20),
                    rbeta(1,8+x[i]*5,20))
      }

    # true model

      true.dens <- function(grid,x)
      {
	  (0.8-0.5*x^2)*dbeta(grid,22-(x^2)*20,5+x*20)+
          (0.2+0.5*x^2)*dbeta(grid,8+x*5,20)
      }	

      true.mean <- function(x)
      {
          (0.8-0.5*x^2)*(22-(x^2)*20)/(22-(x^2)*20+5+x*20)+
          (0.2+0.5*x^2)*(8+x*5)/(8+x*5+20)
      }	

    # predictions
      grid <- seq(0,1,len=100)
      npred <- 25
      xpred <- matrix(1,ncol=2,nrow=npred)
      xpred[,2] <- seq(0,1,len=npred)

    # prior
      prior <- list(maxn = 25,   
                    alpha = 1, 
                    lambda = 25, 
                    nu = 4, 
                    psiinv = diag(1000,2), 
                    m0 = rep(0,2),
                    S0 = diag(1000,2))

    # mcmc
      mcmc <- list(nburn = 5000, 
                   nskip = 3, 
                   ndisplay = 100, 
                   nsave = 5000)

    # state
      state <- NULL

    # fitting the model

      fit <- LDBDPdensity(formula=y~x,xpred=xpred,
                          prior=prior,
                          mcmc=mcmc,
                          state=NULL,status=TRUE,
                          grid=grid,
                          compute.band=TRUE,type.band="PD")

      fit
      summary(fit)
      plot(fit)

    # Plots for some predictions
    # (conditional density and mean function)

      par(mfrow=c(2,2))
      plot(fit$grid,fit$densp.h[7,],type="l",lwd=2,
           xlim=c(0,1),ylim=c(0,4),xlab="y",ylab="density",lty=2) 
      lines(fit$grid,fit$densp.m[7,],lwd=2,lty=1) 
      lines(fit$grid,fit$densp.l[7,],lwd=2,lty=2)
      lines(fit$grid,true.dens(fit$grid,fit$xpred[7,2]),lwd=2,col="red")

      plot(fit$grid,fit$densp.h[13,],type="l",lwd=2,
           xlim=c(0,1),ylim=c(0,4),xlab="y",ylab="density",lty=2) 
      lines(fit$grid,fit$densp.m[13,],lwd=2,lty=1) 
      lines(fit$grid,fit$densp.l[13,],lwd=2,lty=2)
      lines(fit$grid,true.dens(fit$grid,fit$xpred[13,2]),lwd=2,col="red")

      plot(fit$grid,fit$densp.h[19,],type="l",lwd=2,
           xlim=c(0,1),ylim=c(0,4),xlab="y",ylab="density",lty=2) 
      lines(fit$grid,fit$densp.m[19,],lwd=2,lty=1) 
      lines(fit$grid,fit$densp.l[19,],lwd=2,lty=2)
      lines(fit$grid,true.dens(fit$grid,fit$xpred[19,2]),lwd=2,col="red")

      plot(x,y) 
      lines(fit$xpred[,2],fit$meanfp.m,lwd=2,lty=1) 
      lines(fit$xpred[,2],fit$meanfp.l,lwd=2,lty=2)
      lines(fit$xpred[,2],fit$meanfp.h,lwd=2,lty=2)
      lines(fit$xpred[,2],true.mean(fit$xpred[,2]),lwd=2,lty=1,col="red")
# }