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

TDPdensity {DPpackage} | R Documentation

## Semiparametric Bayesian density estimation using DP Mixtures of Triangular Distributions

### Description

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

### Usage

    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 `NA`s. 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

    ## 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)

[Package DPpackage version 1.1-7.4 Index]