DPdensity {DPpackage} | R Documentation |
This function generates a posterior density sample for a Dirichlet process mixture of normals model.
DPdensity(y,ngrid=1000,prior,mcmc,state,status,method="neal", data=sys.frame(sys.parent()),na.action=na.fail)
y |
a vector or matrix giving the data from which the density estimate is to be computed. |
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: a0 and b0 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 a0 is missing, see details
below), nu2 and psiinv2 giving the hyperparameters of the
inverted Wishart prior distribution for the scale matrix, Psi1 ,
of the inverted Wishart part of the baseline distribution,
tau1 and tau2 giving the hyperparameters for the
gamma prior distribution of the scale parameter k0 of the normal
part of the baseline distribution, m2 and s2 giving the mean
and the covariance of the normal prior for the mean, m1 , of the normal
component of the baseline distribution, respectively, nu1 and
psiinv1 (it must be specified if nu2 is missing, see details
below) giving the hyperparameters of the inverted Wishart part of the baseline
distribution and, m1 giving the mean of the normal part of the baseline
distribution (it must be specified if m2 is missing,
see details below) and, k0 giving the scale parameter of the normal part
of the baseline distribution (it must be specified if tau1 is missing,
see details below). |
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 . |
method |
the method to be used. See Details . |
data |
data frame. |
na.action |
a function that indicates what should happen when the data
contain NA s. The default action (na.fail ) causes
DPdensity to print an error message and terminate if there are any
incomplete observations. |
This generic function fits a Dirichlet process mixture of normal model for density estimation (Escobar and West, 1995):
yi | mui, Sigmai ~ N(mui,Sigmai), i=1,...,n
(mui,Sigmai) | G ~ G
G | alpha, G0 ~ DP(alpha G0)
where, the baseline distribution is the conjugate normal-inverted-Wishart,
G0 = N(mu| m1, (1/k0) Sigma) IW (Sigma | nu1, psi1)
To complete the model specification, independent hyperpriors are assumed (optional),
alpha | a0, b0 ~ Gamma(a0,b0)
m1 | m2, s2 ~ N(m2,s2)
k0 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
psi1 | nu2, psi2 ~ IW(nu2,psi2)
Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).
To let part of the baseline distribution fixed at a particular value, set the corresponding hyperparameters of the prior distributions to NULL in the hyperprior specification of the model.
Although the baseline distribution, G0, is a conjugate prior in
this model specification, the algorithms with auxiliary parameters
described in MacEachern and Muller (1998) and Neal (2000) are
adopted. Specifically, the no-gaps algorithm of
MacEachern and Muller (1998), "no-gaps"
and the algorithm 8 with m=1
of Neal (2000), "neal"
, are considered in the DPdensity
function.
The default method is the algorithm 8 of Neal.
An object of class DPdensity
representing the DP mixture of normals
model fit. Generic functions such as print
, summary
, and plot
have methods to
show the results of the fit. The results include the baseline parameters, alpha
, and the
number of clusters.
The function DPrandom
can be used to extract the posterior mean of the
subject-specific means and covariance matrices.
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. |
muclus |
a matrix of dimension (nobservations+2)*(nvariables) giving the means of the clusters
(only the first ncluster are considered to start the chain). |
sigmaclus |
a matrix of dimension (nobservations+2)*( (nvariables)*((nvariables)+1)/2) giving
the lower matrix of the covariance matrix 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. |
m1 |
giving the mean of the normal components of the baseline distribution. |
k0 |
giving the scale parameter of the normal part of the baseline distribution. |
psi1 |
giving the scale matrix of the inverted-Wishart part of the baseline distribution. |
Alejandro Jara <ajarav@udec.cl>
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338.
Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.
DPrandom
, PTdensity
, BDPdensity
## Not run: #################################### # Univariate example #################################### # Data data(galaxy) galaxy<-data.frame(galaxy,speeds=galaxy$speed/1000) attach(galaxy) # Initial state state <- NULL # MCMC parameters nburn<-1000 nsave<-10000 nskip<-10 ndisplay<-100 mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay) # Example of Prior information 1 # Fixing alpha, m1, and Psi1 prior1<-list(alpha=1,m1=rep(0,1),psiinv1=diag(0.5,1),nu1=4,tau1=1,tau2=100) # Example of Prior information 2 # Fixing alpha and m1 prior2<-list(alpha=1,m1=rep(0,1),psiinv2=solve(diag(0.5,1)),nu1=4,nu2=4, tau1=1,tau2=100) # Example of Prior information 3 # Fixing only alpha prior3<-list(alpha=1,m2=rep(0,1),s2=diag(100000,1), psiinv2=solve(diag(0.5,1)), nu1=4,nu2=4,tau1=1,tau2=100) # Example of Prior information 4 # Everything is random prior4<-list(a0=2,b0=1,m2=rep(0,1),s2=diag(100000,1), psiinv2=solve(diag(0.5,1)), nu1=4,nu2=4,tau1=1,tau2=100) # Fit the models fit1.1<-DPdensity(y=speeds,prior=prior1,mcmc=mcmc,state=state,status=TRUE) fit1.2<-DPdensity(y=speeds,prior=prior2,mcmc=mcmc,state=state,status=TRUE) fit1.3<-DPdensity(y=speeds,prior=prior3,mcmc=mcmc,state=state,status=TRUE) fit1.4<-DPdensity(y=speeds,prior=prior4,mcmc=mcmc,state=state,status=TRUE) # Posterior means fit1.1 fit1.2 fit1.3 fit1.4 # Plot the estimated density plot(fit1.1,ask=FALSE) plot(fit1.2,ask=FALSE) plot(fit1.3,ask=FALSE) plot(fit1.4,ask=FALSE) # Extracting the density estimate cbind(fit1.1$x1,fit1.1$dens) cbind(fit1.2$x1,fit1.2$dens) cbind(fit1.3$x1,fit1.3$dens) cbind(fit1.4$x1,fit1.4$dens) # Plot the parameters (only prior 2 for illustration) # (to see the plots gradually set ask=TRUE) plot(fit1.2,ask=FALSE,output="param") # Plot the a specific parameters # (to see the plots gradually set ask=TRUE) plot(fit1.2,ask=FALSE,output="param",param="psi1-speeds",nfigr=1,nfigc=2) # Extracting the posterior mean of the specific means and covariance matrices # (only prior 2 for illustration) DPrandom(fit1.2) # Ploting predictive information about the specific means and covariance matrices # with HPD and Credibility intervals # (only prior 2 for illustration) # (to see the plots gradually set ask=TRUE) plot(DPrandom(fit1.2,predictive=TRUE),ask=FALSE) plot(DPrandom(fit1.2,predictive=TRUE),ask=FALSE,hpd=FALSE) # Ploting information about all the specific means and covariance matrices # with HPD and Credibility intervals # (only prior 2 for illustration) # (to see the plots gradually set ask=TRUE) plot(DPrandom(fit1.2),ask=FALSE,hpd=FALSE) #################################### # Bivariate example #################################### # Data data(airquality) attach(airquality) ozone<-Ozone**(1/3) radiation<-Solar.R # Prior information s2<-matrix(c(10000,0,0,1),ncol=2) m2<-c(180,3) psiinv2<-solve(matrix(c(10000,0,0,1),ncol=2)) prior<-list(a0=1,b0=1/5,nu1=4,nu2=4,s2=s2, m2=m2,psiinv2=psiinv2,tau1=0.01,tau2=0.01) # Initial state state <- NULL # MCMC parameters nburn<-5000 nsave<-10000 nskip<-10 ndisplay<-1000 mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay) # Fit the model fit1<-DPdensity(y=cbind(radiation,ozone),prior=prior,mcmc=mcmc, state=state,status=TRUE,na.action=na.omit) # Plot the estimated density plot(fit1) # Extracting the density estimate fit1$x1 fit1$x2 fit1$dens ## End(Not run)