PTlm {DPpackage} | R Documentation |
This function generates a posterior density sample from a semiparametric linear regression model using a Mixture of Polya Trees prior for the distribution of the errors.
PTlm(formula,ngrid=200,grid=NULL,prior,mcmc,state,status, data=sys.frame(sys.parent()),na.action=na.fail)
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. |
ngrid |
number of grid points where the error density estimate is evaluated. The default value is 200. |
grid |
grid points where the density estimate is evaluated. The default is NULL. |
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 Polya Tree
prior, alpha giving the value of the precision parameter (it
must be specified if a0 and b0 are missing, see details
below), tau1 and tau2 giving the
hyperparameters for the prior distribution of the variance
of the normal baseline distribution, beta0 and Sbeta0
giving the hyperparameters of the normal prior distribution for the regression
coefficients, optionally M giving the finite level
to be considered, and frstlprob a logical variable
indicating whether the first level probabilities of the PT are fixed
defining a median regression model (the default is TRUE). Note that
if M is specified, a Partially Specified
Mixture of Polya trees is fitted. |
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 the 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
PTlm to print an error message and terminate if there are any
incomplete observations. |
By default, this generic function fits a median regression model using a Scale Mixture of Polya Trees prior for the distribution of the errors (see, e.g., Lavine, 1992 and 1994, Hanson and Johnson, 2004):
yi = Xi beta + Vi, i=1,...,n
Vi | G ~ G
G | alpha,sigma2 ~ PT(Pi^{sigma2},textit{A})
where, the PT is centered around a N(0,sigma2) distribution, by taking each m level of the partition Pi^{sigma2} to coincide with the k/2^m, k=0,...,2^m quantile of the N(0,sigma2) distribution. The family textit{A}={alphae: e in E^{*}}, where E^{*}=bigcup_{m=1}^{+infty} E^m and E^m is the m-fold product of E={0,1}, was specified as alpha{e1 ... em}=α m^2. To complete the model specification, independent hyperpriors are assumed,
alpha | a0, b0 ~ Gamma(a0,b0)
sigma^-2 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
Optionally, if frstlprob=FALSE (the default value is TRUE) is specified, a mean regression model is considered. In this case, the following PT prior is considered:
G | alpha,mu,sigma2 ~ PT(Pi^{mu,sigma2},textit{A})
where, the PT is centered around a N(mu,sigma2) distribution. In this case, the intercept term is automatically excluded from the model and the hyperparameters for the normal prior for mu must be specified. The normal prior is given by,
mu | mub, Sb ~ N(mub,Sb)
The precision parameter, alpha, of the PT
prior
can be considered as random, having a gamma
distribution, Gamma(a0,b0),
or fixed at some particular value. To let alpha to be fixed at a particular
value, set a0 to NULL in the prior specification.
In the computational implementation of the model, Metropolis-Hastings steps are used to sample the posterior distribution of the regression coefficients and hyperparameters.
An object of class PTlm
representing the semiparametric median regression
model fit. Generic functions such as print
, plot
,
summary
, and anova
have methods to show the results of the fit.
The results include beta
, mu
, sigma2
, and the precision
parameter \alpha
.
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:
alpha |
giving the value of the precision parameter. |
beta |
giving the value of the regression coefficients. |
mu |
giving the mean of the normal baseline distribution (If needed). |
sigma2 |
giving the variance of the normal baseline distribution. |
v |
giving the value of the errors (it must be consistent with the data. |
Alejandro Jara <ajarav@udec.cl>
Hanson, T., and Johnson, W. (2002) Modeling regression error with a Mixture of Polya Trees. Journal of the American Statistical Association, 97: 1020 - 1033.
Lavine, M. (1992) Some aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 20: 1222-11235.
Lavine, M. (1994) More aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 22: 1161-1176.
## Not run: #################################### # A simulated Data Set # (Mixture of Normals) #################################### ind<-rbinom(100,1,0.5) vsim<-ind*rnorm(100,1,0.15)+(1-ind)*rnorm(100,3,0.15) x1<-rep(c(0,1),50) x2<-rnorm(100,0,1) etasim<-x1+-1*x2 y<-etasim+vsim # Initial state state <- NULL # MCMC parameters nburn<-5000 nsave<-10000 nskip<-20 ndisplay<-100 mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip, ndisplay=ndisplay) # Prior information prior <- list(alpha=1,beta0=rep(0,3),Sbeta0=diag(1000,3), tau1=0.01,tau2=0.01,M=6) # Fit the model fit1 <- PTlm(formula=y~x1+x2,prior=prior,mcmc=mcmc,state=state, status=TRUE) # Summary with HPD and Credibility intervals summary(fit1) summary(fit1,hpd=FALSE) # Plot model parameters (to see the plots gradually set ask=TRUE) plot(fit1) plot(fit1,nfigr=2,nfigc=2) # Table of Pseudo Contour Probabilities anova(fit1) ############################################ # The Australian Institute of Sport's data # (Skew data example) ############################################ data(sports) attach(sports) # Initial state state <- NULL # MCMC parameters nburn<-5000 nsave<-10000 nskip<-20 ndisplay<-100 mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip, ndisplay=ndisplay) # Prior information prior <- list(alpha=1,beta0=rep(0,3),Sbeta0=diag(1000,3), tau1=0.01,tau2=0.01,M=8) # Fit the model fit2 <- PTlm(formula=bmi~lbm+gender,prior=prior,mcmc=mcmc, state=state,status=TRUE) # Summary with HPD and Credibility intervals summary(fit2) summary(fit2,hpd=FALSE) # Plot model parameters (to see the plots gradually set ask=TRUE) plot(fit2) plot(fit2,nfigr=2,nfigc=2) # Table of Pseudo Contour Probabilities anova(fit2) ## End(Not run)