Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a DP mixture of normals model for related random probability measures. Support provided by the NIH/NCI R01CA75981 grant.
1 2 3 | HDPMcdensity(formula,study,xpred,ngrid=100,prior,mcmc,
state,status,data=sys.frame(sys.parent()),
na.action=na.fail,work.dir=NULL)
|
formula |
a two-sided linear formula object describing the
model fit, with the response on the
left of a |
study |
a (1 by |
ngrid |
integer giving the number of grid points where the density estimate is evaluated. The default is 100. |
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
parameters: |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: |
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 (not available yet). |
status |
a logical variable indicating whether this run is new ( |
data |
data frame. |
na.action |
a function that indicates what should happen when the data
contain |
work.dir |
working directory. |
This generic function fits a hierarchical mixture of DPM of normals model for conditional density
estimation (Mueller, Quintana and Rosner, 2004). Let di=(yi,xi) be the vector of
full data, including the responses yi and predictors xi. The HDPMcdensity
function
fits the hierarchical mixture of DPM of normals model to the full data and then look at the implied conditional
distribution of the responses given the predictors. The model is given by:
d_{ij} | F_i \sim F_i
where, yij denote the j-th observation in the i-th study, i=1,...,I, and F_i is assumed to arise as a mixture F_i = ε H_0 + (1-ε) H_i of one common distribution H_0 and a distribution H_i that is specific or idiosyncratic to the i-th study.
The random probability measures H_i in turn are given a Dirichlet process mixture of normal prior. We assume
H_i(d) = \int N(μ,Σ) d G_i(μ),~ i=0,1,…,I
with
G_i | α_i, G_0 \sim DP(α G_0)
where, the baseline distribution is
G_0 = N(μ| μ_b,Σ_b)
.
To complete the model specification, independent hyperpriors are assumed (optional),
Σ | ν, T \sim IW(ν,T)
α_i | a_{0i}, b_{0i} \sim Gamma(a_{0i},b_{0i})
μ_b | m_0, S_0 \sim N(m_0,S_0)
Σ_b | ν_b, Tb \sim IW(ν_b,Tb)
and
p(ε) = π_0 δ_0+ π_1 δ_1+(1- π_0 -\ pi_1) Be(a_ε,b_ε)
Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).
An object of class HDPMcdensity
representing the hierarchical DPM of normals model.
Generic functions such as print
, plot
,
and summary
have methods to show the results of the fit. The results include
sigma
, eps
, the vector of precision parameters
alpha
, mub
and sigmab
.
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. |
ss |
an interger vector defining to which of the |
sc |
an integer vector defining to which DP each cluster belongs. Note that length(sc)=nrec (only
the first |
alpha |
giving the vector of dimension nsuties+1 of precision parameters. |
muclus |
a matrix of dimension (number of subject + 100) times the
total number of variables (responses + predictors), giving the
means for each cluster (only the first |
mub |
giving the mean of the normal baseline distributions. |
sigmab |
giving the covariance matrix the normal baseline distributions. |
sigma |
giving the normal kernel covariance matrix. |
eps |
giving the value of |
Alejandro Jara <atjara@uc.cl>
Peter Mueller <pmueller@mdanderson.org>
Mueller, P., Quintana, F. and Rosner, G. (2004). A Method for Combining Inference over Related Nonparametric Bayesian Models. Journal of the Royal Statistical Society, Series B, 66: 735-749.
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# Data
data(calgb)
attach(calgb)
y <- cbind(Z1,Z2,Z3,T1,T2,B0,B1)
x <- cbind(CTX,GM,AMOF)
z <- cbind(y,x)
# Data for prediction
data(calgb.pred)
xpred <- as.matrix(calgb.pred[,8:10])
# Prior information
prior <- list(pe1=0.1,
pe0=0.1,
ae=1,
be=1,
a0=rep(1,3),
b0=rep(1,3),
nu=12,
tinv=0.25*var(z),
m0=apply(z,2,mean),
S0=var(z),
nub=12,
tbinv=var(z))
# Initial state
state <- NULL
# MCMC parameters
mcmc <- list(nburn=5000,
nsave=5000,
nskip=3,
ndisplay=100)
# Fitting the model
fit1 <- HDPMcdensity(formula=y~x,
study=~study,
xpred=xpred,
prior=prior,
mcmc=mcmc,
state=state,
status=TRUE)
# Posterior inference
fit1
summary(fit1)
# Plot the parameters
# (to see the plots gradually set ask=TRUE)
plot(fit1,ask=FALSE)
# Plot the a specific parameters
# (to see the plots gradually set ask=TRUE)
plot(fit1,ask=FALSE,param="eps",nfigr=1,nfigc=2)
# Plot the measure for each study
# under first values for the predictors, xpred[1,]
predict(fit1,pred=1,i=1,r=1) # pred1, study 1
predict(fit1,pred=1,i=2,r=1) # pred1, study 2
# Plot the measure for each study
# under second values for the predictors, xpred[2,]
predict(fit1,pred=2,i=1,r=1) # pred2, study 1
predict(fit1,pred=2,i=2,r=1) # pred2, study 2
# Plot the idiosyncratic measure for each study
# under first values for the predictors, xpred[1,]
predict(fit1,pred=1,i=1,r=0) # study 1
predict(fit1,pred=1,i=2,r=0) # study 2
# Plot the common measure
# under first values for the predictors, xpred[1,]
predict(fit1,pred=1,i=0)
## End(Not run)
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