DPcdensity: Bayesian Semiparametric Conditional Density Estimation using...
In DPpackage: Bayesian Nonparametric Modeling in R
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
This function generates a posterior density sample for a
Bayesian density regression model with continuous predictors using
a Dirichlet process mixture of normals model.
Usage
1
2
3
DPcdensity(y,x,xpred,ngrid=100,grid=NULL,compute.band=FALSE,
type.band="PD",prior,mcmc,state,status,
data=sys.frame(sys.parent()),work.dir=NULL)
Arguments
y
a vector giving the data from which the density estimate
is to be computed.
x
a vector or matrix giving the continuous predictors of
dimension nrec times nx.
xpred
a vector or matrix giving the values of the continuous
predictors used for prediction.
ngrid
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.
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). Notice that the dimension of the baseline measure
includes the predictor and the response, i.e., nx+1.
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.
work.dir
working directory.
Details
This generic function fits a Dirichlet process mixture of normal model
(Escobar and West, 1995) for the conditional density estimation
f(y|x) as proposed by Muller, Erkanli and West (1996). They proposed to
specify a Dirichlet process mixture of normals for the joint distribution of the response
and predictors. Although in the original paper these authors focussed on the mean regression
function, their method can be used to model the conditional density of the response giving the
predictors in a semiparametric way. Indeed, their method is essentially a locally weighted
mixture of normal regression models with weigths predictor-dependent.
Let yi and Xi be the response and the vector of predictors, respectively.
Further, let Zi=(yi,Xi). The model for the joint distribution of the response and predictors is as follows:
Zi | 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 Neal (2000) is adopted. Specifically, the algorithm 8 with m=1
of Neal (2000) is considered in the DPcdensity function.
Value
An object of class DPcdensity 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 MCMC samples of the parameters and the errors in the model are stored in the object
thetasave. The object is 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.
z
giving the matrix of response and predictors. This must be included if missing data
(response and/or predictors) are present. Those are imputed during the MCMC.
Author(s)
Alejandro Jara <atjara@uc.cl>
Tim Hanson <hansont@stat.sc.edu>
References
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference
Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
Muller, P., Erkanli, A. and West, M. (1996) Bayesian curve fitting using
multivariate normal mixtures. Biometrika, 83: 67-79.
Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models.
Journal of Computational and Graphical Statistics, 9: 249-265.
See Also
DPdensity, PTdensity, BDPdensity
Examples
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## Not run:
##########################################################
# Simulated data:
# Here we replicate the results reported with
# simulated data by Dunson, Pillai and Park (2007,
# JRSS Ser. B, 69: 163-183, pag 177) where a different
# approach is proposed.
##########################################################
dtrue <- function(grid,x)
{
exp(-2*x)*dnorm(grid,mean=x,sd=sqrt(0.01))+
(1-exp(-2*x))*dnorm(grid,mean=x^4,sd=sqrt(0.04))
}
nobs <- 500
x <- runif(nobs)
y1 <- x + rnorm(nobs, 0, sqrt(0.01))
y2 <- x^4 + rnorm(nobs, 0, sqrt(0.04))
u <- runif(nobs)
prob <- exp(-2*x)
y <- ifelse(u<prob,y1,y2)
# Prior information
w <- cbind(y,x)
wbar <- apply(w,2,mean)
wcov <- var(w)
prior <- list(a0=10,
b0=1,
nu1=4,
nu2=4,
s2=0.5*wcov,
m2=wbar,
psiinv2=2*solve(wcov),
tau1=6.01,
tau2=2.01)
# Initial state
state <- NULL
# MCMC parameters
mcmc <- list(nburn=5000,
nsave=5000,
nskip=3,
ndisplay=100)
# fitting the model
xpred <- c(0.00,0.05,0.10,0.15,0.20,0.25,
0.30,0.35,0.40,0.45,0.49,0.55,
0.60,0.65,0.70,0.75,0.80,0.85,
0.88,0.95,1.00)
fit <- DPcdensity(y=y,x=x,xpred=xpred,ngrid=100,
prior=prior,
mcmc=mcmc,
state=state,
status=TRUE,
compute.band=TRUE,type.band="PD")
# true model and estimates
par(mfrow=c(2,3))
plot(fit$grid,fit$densp.h[3,],lwd=1,type="l",lty=2,
main="x=0.10",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[3,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[3,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[3]),lwd=2,
type="l",lty=1,col="red")
plot(fit$grid,fit$densp.h[6,],lwd=1,type="l",lty=2,
main="x=0.25",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[6,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[6,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[6]),lwd=2,
type="l",lty=1,col="red")
plot(fit$grid,fit$densp.h[11,],lwd=1,type="l",lty=2,
main="x=0.49",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[11,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[11,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[11]),lwd=2,type="l",
lty=1,col="red")
plot(fit$grid,fit$densp.h[16,],lwd=1,type="l",lty=2,
main="x=0.75",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[16,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[16,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[16]),lwd=2,type="l",
lty=1,col="red")
plot(fit$grid,fit$densp.h[19,],lwd=1,type="l",lty=2,
main="x=0.75",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[19,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[19,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[19]),lwd=2,type="l",
lty=1,col="red")
# data and mean function
plot(x,y,xlab="x",ylab="y",main="")
lines(xpred,fit$meanfp.m,type="l",lwd=2,lty=1)
lines(xpred,fit$meanfp.l,type="l",lwd=2,lty=2)
lines(xpred,fit$meanfp.h,type="l",lwd=2,lty=2)
## End(Not run)
DPpackage documentation built on May 1, 2019, 10:23 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
This function generates a posterior density sample for a Bayesian density regression model with continuous predictors using a Dirichlet process mixture of normals model.
Usage
1 2 3 | DPcdensity(y,x,xpred,ngrid=100,grid=NULL,compute.band=FALSE,
type.band="PD",prior,mcmc,state,status,
data=sys.frame(sys.parent()),work.dir=NULL)
|
Arguments
y |
a vector giving the data from which the density estimate is to be computed. |
x |
a vector or matrix giving the continuous predictors of
dimension |
xpred |
a vector or matrix giving the values of the continuous predictors used for prediction. |
ngrid |
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. |
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 ( |
data |
data frame. |
work.dir |
working directory. |
Details
This generic function fits a Dirichlet process mixture of normal model (Escobar and West, 1995) for the conditional density estimation f(y|x) as proposed by Muller, Erkanli and West (1996). They proposed to specify a Dirichlet process mixture of normals for the joint distribution of the response and predictors. Although in the original paper these authors focussed on the mean regression function, their method can be used to model the conditional density of the response giving the predictors in a semiparametric way. Indeed, their method is essentially a locally weighted mixture of normal regression models with weigths predictor-dependent.
Let yi and Xi be the response and the vector of predictors, respectively. Further, let Zi=(yi,Xi). The model for the joint distribution of the response and predictors is as follows:
Zi | 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 Neal (2000) is adopted. Specifically, the algorithm 8 with m=1
of Neal (2000) is considered in the DPcdensity function.
Value
An object of class DPcdensity 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 MCMC samples of the parameters and the errors in the model are stored in the object
thetasave. The object is 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 |
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 |
ss |
an interger vector defining to which of the |
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. |
z |
giving the matrix of response and predictors. This must be included if missing data (response and/or predictors) are present. Those are imputed during the MCMC. |
Author(s)
Alejandro Jara <atjara@uc.cl>
Tim Hanson <hansont@stat.sc.edu>
References
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
Muller, P., Erkanli, A. and West, M. (1996) Bayesian curve fitting using multivariate normal mixtures. Biometrika, 83: 67-79.
Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.
See Also
DPdensity, PTdensity, BDPdensity
Examples
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 | ## Not run:
##########################################################
# Simulated data:
# Here we replicate the results reported with
# simulated data by Dunson, Pillai and Park (2007,
# JRSS Ser. B, 69: 163-183, pag 177) where a different
# approach is proposed.
##########################################################
dtrue <- function(grid,x)
{
exp(-2*x)*dnorm(grid,mean=x,sd=sqrt(0.01))+
(1-exp(-2*x))*dnorm(grid,mean=x^4,sd=sqrt(0.04))
}
nobs <- 500
x <- runif(nobs)
y1 <- x + rnorm(nobs, 0, sqrt(0.01))
y2 <- x^4 + rnorm(nobs, 0, sqrt(0.04))
u <- runif(nobs)
prob <- exp(-2*x)
y <- ifelse(u<prob,y1,y2)
# Prior information
w <- cbind(y,x)
wbar <- apply(w,2,mean)
wcov <- var(w)
prior <- list(a0=10,
b0=1,
nu1=4,
nu2=4,
s2=0.5*wcov,
m2=wbar,
psiinv2=2*solve(wcov),
tau1=6.01,
tau2=2.01)
# Initial state
state <- NULL
# MCMC parameters
mcmc <- list(nburn=5000,
nsave=5000,
nskip=3,
ndisplay=100)
# fitting the model
xpred <- c(0.00,0.05,0.10,0.15,0.20,0.25,
0.30,0.35,0.40,0.45,0.49,0.55,
0.60,0.65,0.70,0.75,0.80,0.85,
0.88,0.95,1.00)
fit <- DPcdensity(y=y,x=x,xpred=xpred,ngrid=100,
prior=prior,
mcmc=mcmc,
state=state,
status=TRUE,
compute.band=TRUE,type.band="PD")
# true model and estimates
par(mfrow=c(2,3))
plot(fit$grid,fit$densp.h[3,],lwd=1,type="l",lty=2,
main="x=0.10",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[3,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[3,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[3]),lwd=2,
type="l",lty=1,col="red")
plot(fit$grid,fit$densp.h[6,],lwd=1,type="l",lty=2,
main="x=0.25",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[6,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[6,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[6]),lwd=2,
type="l",lty=1,col="red")
plot(fit$grid,fit$densp.h[11,],lwd=1,type="l",lty=2,
main="x=0.49",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[11,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[11,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[11]),lwd=2,type="l",
lty=1,col="red")
plot(fit$grid,fit$densp.h[16,],lwd=1,type="l",lty=2,
main="x=0.75",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[16,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[16,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[16]),lwd=2,type="l",
lty=1,col="red")
plot(fit$grid,fit$densp.h[19,],lwd=1,type="l",lty=2,
main="x=0.75",xlab="values",ylab="density",ylim=c(0,4))
lines(fit$grid,fit$densp.l[19,],lwd=1,type="l",lty=2)
lines(fit$grid,fit$densp.m[19,],lwd=2,type="l",lty=1)
lines(fit$grid,dtrue(fit$grid,xpred[19]),lwd=2,type="l",
lty=1,col="red")
# data and mean function
plot(x,y,xlab="x",ylab="y",main="")
lines(xpred,fit$meanfp.m,type="l",lwd=2,lty=1)
lines(xpred,fit$meanfp.l,type="l",lwd=2,lty=2)
lines(xpred,fit$meanfp.h,type="l",lwd=2,lty=2)
## End(Not run)
|
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