TDPdensity: Semiparametric Bayesian density estimation using DP Mixtures...

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/TDPdensity.R

Description

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

Usage

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TDPdensity(y,support=3,transform=1,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.

transform

an integer number giving the type of transformation to be considered, 1=Uniform, 2=Normal,3=Logistic,4=Cauchy. The types 2-4 can be only used when the support is the real line.

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 process prior, alpha giving the value of the precision parameter (it must be specified if aa0 is missing, see details below), a0 and b0 giving the parameters of the beta centering distribution of the DP prior, and kmax giving the maximum value of the discrete uniform prior for number of components in the Mixture of Triangular distributions. Optionally, when the support of the data is the real line and the parametric transformation 2-4 are considered, the location mu and the scale parameter sigma2 can be included here. If not, they are taked as the mean and the variance of the data, respectively.

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.

data

data frame.

na.action

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

Details

This generic function fits a Triangular-Dirichlet model for density estimation:

yi | G ~ G, i=1,…,n

G | kmax, alpha, G0 ~ TDP(kmax, alpha G0)

where, yi is the transformed data to lie in [0,1], kmax is the upper limit of the discrete uniform prior for the number of components in the Mixture of Triangular distributions, alpha is the total mass parameter of the Dirichlet process component, and G0 is the centering distribution of the DP. The centering distribution corresponds to a G0=Beta(a0,b0) distribution.

Note that our representation is different to the Mixture of Triangular distributions proposed by Perron and Mengersen (2001). In this function we consider random weights following a Dirichlet prior and we exploit the underlying DP structure. By so doing, we avoid using Reversible-Jumps algorithms.

The precision or total mass parameter, α, of the DP prior can be considered as random, having a gamma distribution, Gamma(a0,b0), 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 a0 to NULL in the prior specification.

Value

An object of class TDPdensity representing the Triangular-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 number of components in the Mixture of Triangular distriutions.

Author(s)

Alejandro Jara <atjara@uc.cl>

References

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

Perron, F. and Mengersen, K. (2001) Bayesian Nonparametric Modeling Using Mixtures of Triangular Distributions. Biometrics, 57(2): 518-528.

See Also

DPdensity, PTdensity, BDPdensity

Examples

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## Not run: 
    # 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=50,
                  a0=1,
                  b0=1)

    # Fitting the model

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


## End(Not run)

DPpackage documentation built on May 1, 2019, 10:23 p.m.