Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a Linear Dependent Dirichlet Process Mixture of Normals model. Support provided by the NIH/NCI R01CA75981 grant.
1 2 3 4 |
formula |
a two-sided linear formula object describing the
model fit, with the response on the
left of a |
zpred |
a matrix giving the covariate values where the predictive density is evaluated. |
prior |
a list giving the prior information. The list includes the following
parameter: |
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. |
status |
a logical variable indicating whether this run is new ( |
ngrid |
integer giving the number of grid points where the conditional density estimate is evaluated. The default is 100. |
grid |
vector of grid points where the conditional density estimate is evaluated. The default value is NULL and the grid is chosen according to the range of the data. |
compute.band |
logical variable indicating whether the credible band for the conditional density and mean function must be computed. |
type.band |
string indication the type of credible band to be computed; if equal to "HPD" or "PD" then the 95 percent pointwise HPD or PD band is computed, respectively. |
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 Linear Dependent Dirichlet Process Mixture of Normals model,
yi | fXi ~ fXi
fXi = \int N(Xi beta, sigma2) G(d beta d sigma2)
G | alpha, G0 ~ DP(alpha G0)
where, G0 = N(beta| mub, sb)Gamma(sigma2|tau1/2,tau2/2). To complete the model specification, independent hyperpriors are assumed,
alpha | a0, b0 ~ Gamma(a0,b0)
mub | m0, Sbeta0 ~ N(m0,Sbeta0)
sb | nu, psi ~ IW(nu,psi)
tau2 ~ Gamma(tau2 | taus1, taus2 ~ Gamma(taus1/2,taus2/2)
Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).
Note also that the LDDP model is a natural and simple extension of the the ANOVA DDP model discussed in in De Iorio et al. (2004). The same model is used in Mueller et al.(2005) as the random effects distribution in a repeated measurements model.
The precision or total mass parameter, alpha, 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.
The computational implementation of the model is based on the marginalization of
the DP
and on the use of MCMC methods for non-conjugate DPM models (see, e.g,
MacEachern and Muller, 1998; Neal, 2000).
An object of class LDDPdensity
representing the LDDP mixture of normals model fit.
Generic functions such as print
, plot
,
and summary
have methods to show the results of the fit. The results include
mub
, sb
, tau2
, the precision parameter
alpha
, and the number of clusters.
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:
betaclus |
a matrix of dimension (number of subject + 100) times the
number of columns in the design matrix, giving the
regression coefficients for each cluster (only the first |
sigmaclus |
a vector of dimension (number of subjects + 100) giving the variance of the normal kernel for
each cluster (only the first |
alpha |
giving the value of the precision parameter. |
mub |
giving the mean of the normal baseline distributions. |
sb |
giving the covariance matrix the normal baseline distributions. |
ncluster |
an integer giving the number of clusters. |
ss |
an interger vector defining to which of the |
tau2 |
giving the value of the tau2 parameter. |
Alejandro Jara <atjara@uc.cl>
Peter Mueller <pmueller@math.utexas.edu>
Gary L. Rosner <grosner@jhmi.edu>
De Iorio, M., Muller, P., Rosner, G., and MacEachern, S. (2004), An ANOVA model for dependent random measures. Journal of the American Statistical Association, 99(465): 205-215.
De Iorio, M., Muller, P., Rosner, G.L., and MacEachern, S (2004) An ANOVA Model for Dependent Random Measures. Journal of the American Statistical Association, 99: 205-215
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.
Mueller, P., Rosner, G., De Iorio, M., and MacEachern, S. (2005). A Nonparametric Bayesian Model for Inference in Related Studies. Applied Statistics, 54 (3), 611-626.
Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | ## Not run:
##########################################################
# Simulate data from a mixture of two normal densities
##########################################################
nobs <- 500
y1 <-rnorm(nobs, 3,.8)
## y2 = 0.6
y21 <- rnorm(nobs,1.5, 0.8)
y22 <- rnorm(nobs,4.0, 0.6)
u <- runif(nobs)
y2 <- ifelse(u<0.6,y21,y22)
y <- c(y1,y2)
## design matrix including a single factor
trt <- c(rep(0,nobs),rep(1,nobs))
## design matrix for posterior predictive
zpred <- rbind(c(1,0),c(1,1))
# Prior information
S0 <- diag(100,2)
m0 <- rep(0,2)
psiinv <- diag(1,2)
prior <- list(a0=10,
b0=1,
nu=4,
m0=m0,
S0=S0,
psiinv=psiinv,
tau1=6.01,
taus1=6.01,
taus2=2.01)
# Initial state
state <- NULL
# MCMC parameters
nburn <- 5000
nsave <- 5000
nskip <- 3
ndisplay <- 100
mcmc <- list(nburn=nburn,
nsave=nsave,
nskip=nskip,
ndisplay=ndisplay)
# Fit the model
fit1 <- LDDPdensity(y~trt,prior=prior,mcmc=mcmc,
state=state,status=TRUE,
ngrid=200,zpred=zpred,
compute.band=TRUE,type.band="PD")
# Plot posterior density estimate
# with design vector x0=(1,0)
plot(fit1$grid,fit1$densp.h[1,],type="l",xlab="Y",
ylab="density",lty=2,lwd=2)
lines(fit1$grid,fit1$densp.l[1,],lty=2,lwd=2)
lines(fit1$grid,fit1$densp.m[1,],lty=1,lwd=3)
# add true density to the plot
p1 <- dnorm(fit1$grid, 3.0, 0.8)
lines(fit1$grid,p1,lwd=2,lty=1, col="red")
# Plot posterior density estimate
# with design vector x0=(1,1)
plot(fit1$grid,fit1$densp.h[2,],type="l",xlab="Y",
ylab="density",lty=2,lwd=2)
lines(fit1$grid,fit1$densp.l[2,],lty=2,lwd=2)
lines(fit1$grid,fit1$densp.m[2,],lty=1,lwd=3)
# add true density to the plot
p2 <- 0.6*dnorm(fit1$grid, 1.5, 0.8) +
0.4*dnorm(fit1$grid, 4.0, 0.6)
lines(fit1$grid,p2,lwd=2,lty=1, col="red")
# Plot posterior CDF estimate
# with design vector x0=(1,0)
plot(fit1$grid,fit1$cdfp.h[1,],type="l",xlab="Y",
ylab="density",lty=2,lwd=2)
lines(fit1$grid,fit1$cdfp.l[1,],lty=2,lwd=2)
lines(fit1$grid,fit1$cdfp.m[1,],lty=1,lwd=3)
# add true CDF to the plot
p1 <- pnorm(fit1$grid, 3.0, 0.8)
lines(fit1$grid,p1,lwd=2,lty=1, col="red")
# Plot posterior CDF estimate
# with design vector x0=(1,1)
plot(fit1$grid,fit1$cdfp.h[2,],type="l",xlab="Y",
ylab="density",lty=2,lwd=2)
lines(fit1$grid,fit1$cdfp.l[2,],lty=2,lwd=2)
lines(fit1$grid,fit1$cdfp.m[2,],lty=1,lwd=3)
# add true density to the plot
p2 <- 0.6*pnorm(fit1$grid, 1.5, 0.8) +
0.4*pnorm(fit1$grid, 4.0, 0.6)
lines(fit1$grid,p2,lwd=2,lty=1, col="red")
## End(Not run)
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