rDPGibbs: Density Estimation with Dirichlet Process Prior and Normal Base

Description

rDPGibbs implements a Gibbs Sampler to draw from the posterior for a normal mixture problem with a Dirichlet Process prior. A natural conjugate base prior is used along with priors on the hyper parameters of this distribution. One interpretation of this model is as a normal mixture with a random number of components that can grow with the sample size.

Usage

rDPGibbs(Prior, Data, Mcmc)

Value

A list containing:

nmix: a list containing: probdraw, zdraw, compdraw (see “nmix Details” section)
alphadraw: $R/keep x 1$ vector of alpha draws
nudraw: $R/keep x 1$ vector of nu draws
adraw: $R/keep x 1$ vector of a draws
vdraw: $R/keep x 1$ vector of v draws

Arguments

Data: list(y)
Prior: list(Prioralpha, lambda_hyper)
Mcmc: list(R, keep, nprint, maxuniq, SCALE, gridsize)

Author

Peter Rossi, Anderson School, UCLA, perossichi@gmail.com.

Details

Model and Priors

$y_i$ $\sim$ $N(\mu_i, \Sigma_i)$

$\theta_i=(\mu_i,\Sigma_i)$ $\sim$ $DP(G_0(\lambda),alpha)$

$G_0(\lambda):$
$\mu_i | \Sigma_i$ $\sim$ $N(0,\Sigma_i (x) a^{-1})$
$\Sigma_i$ $\sim$ $IW(nu,nu*v*I)$

$\lambda(a,nu,v):$
$a$ $\sim$ uniform on grid[alim[1], alimb[2]]
$nu$ $\sim$ uniform on grid[dim(data)-1 + exp(nulim[1]), dim(data)-1 + exp(nulim[2])]
$v$ $\sim$ uniform on grid[vlim[1], vlim[2]]

$alpha$ $\sim$ $(1-(\alpha-alphamin)/(alphamax-alphamin))^{power}$
$alpha$ = alphamin then expected number of components = Istarmin
$alpha$ = alphamax then expected number of components = Istarmax

We parameterize the prior on $\Sigma_i$ such that $mode(\Sigma)= nu/(nu+2) vI$. The support of nu enforces valid IW density; $nulim[1] > 0$

We use the structure for nmix that is compatible with the bayesm routines for finite mixtures of normals. This allows us to use the same summary and plotting methods.

The default choices of alim, nulim, and vlim determine the location and approximate size of candidate "atoms" or possible normal components. The defaults are sensible given that we scale the data. Without scaling, you want to insure that alim is set for a wide enough range of values (remember a is a precision parameter) and the v is big enough to propose Sigma matrices wide enough to cover the data range.

A careful analyst should look at the posterior distribution of a, nu, v to make sure that the support is set correctly in alim, nulim, vlim. In other words, if we see the posterior bunched up at one end of these support ranges, we should widen the range and rerun.

If you want to force the procedure to use many small atoms, then set nulim to consider only large values and set vlim to consider only small scaling constants. Set Istarmax to a large number. This will create a very "lumpy" density estimate somewhat like the classical Kernel density estimates. Of course, this is not advised if you have a prior belief that densities are relatively smooth.

Argument Details

Data = list(y)

`y:`	$n x k$ matrix of observations on $k$ dimensional data

Prior = list(Prioralpha, lambda_hyper) [optional]

`Prioralpha:`	`list(Istarmin, Istarmax, power)`
`$Istarmin:`	is expected number of components at lower bound of support of alpha (def: 1)
`$Istarmax:`	is expected number of components at upper bound of support of alpha (def: `min(50, 0.1*nrow(y))`)
`$power:`	is the power parameter for alpha prior (def: 0.8)
`lambda_hyper:`	`list(alim, nulim, vlim)`
`$alim:`	defines support of a distribution (def: `c(0.01, 10)`)
`$nulim:`	defines support of nu distribution (def: `c(0.01, 3)`)
`$vlim:`	defines support of v distribution (def: `c(0.1, 4)`)

Mcmc = list(R, keep, nprint, maxuniq, SCALE, gridsize) [only R required]

`R:`	number of MCMC draws
`keep:`	MCMC thinning parameter -- keep every `keep`th draw (def: 1)
`nprint:`	print the estimated time remaining for every `nprint`'th draw (def: 100, set to 0 for no print)
`maxuniq:`	storage constraint on the number of unique components (def: 200)
`SCALE:`	should data be scaled by mean,std deviation before posterior draws (def: `TRUE`)
`gridsize:`	number of discrete points for hyperparameter priors (def: 20)

`nmix` Details

nmix is a list with 3 components. Several functions in the bayesm package that involve a Dirichlet Process or mixture-of-normals return nmix. Across these functions, a common structure is used for nmix in order to utilize generic summary and plotting functions.

`probdraw:`	$ncomp x R/keep$ matrix that reports the probability that each draw came from a particular component
`zdraw:`	$R/keep x nobs$ matrix that indicates which component each draw is assigned to
`compdraw:`	A list of $R/keep$ lists of $ncomp$ lists. Each of the inner-most lists has 2 elemens: a vector of draws for `mu` and a matrix of draws for the Cholesky root of `Sigma`.

Examples

Run this code

if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
set.seed(66)

## simulate univariate data from Chi-Sq

N = 200
chisqdf = 8
y1 = as.matrix(rchisq(N,df=chisqdf))

## set arguments for rDPGibbs

Data1 = list(y=y1)
Prioralpha = list(Istarmin=1, Istarmax=10, power=0.8)
Prior1 = list(Prioralpha=Prioralpha)
Mcmc = list(R=R, keep=1, maxuniq=200)

out1 = rDPGibbs(Prior=Prior1, Data=Data1, Mcmc=Mcmc)

if(0){
  ## plotting examples
  rgi = c(0,20)
  grid = matrix(seq(from=rgi[1],to=rgi[2],length.out=50), ncol=1)
  deltax = (rgi[2]-rgi[1]) / nrow(grid)
  plot(out1$nmix, Grid=grid, Data=y1)
  
  ## plot true density with historgram
  plot(range(grid[,1]), 1.5*range(dchisq(grid[,1],df=chisqdf)),
       type="n", xlab=paste("Chisq ; ",N," obs",sep=""), ylab="")
  hist(y1, xlim=rgi, freq=FALSE, col="yellow", breaks=20, add=TRUE)
  lines(grid[,1], dchisq(grid[,1],df=chisqdf) / (sum(dchisq(grid[,1],df=chisqdf))*deltax),
        col="blue", lwd=2)
}

## simulate bivariate data from the  "Banana" distribution (Meng and Barnard) 

banana = function(A, B, C1, C2, N, keep=10, init=10) { 
  R = init*keep + N*keep
  x1 = x2 = 0
  bimat = matrix(double(2*N), ncol=2)
  for (r in 1:R) { 
    x1 = rnorm(1,mean=(B*x2+C1) / (A*(x2^2)+1), sd=sqrt(1/(A*(x2^2)+1)))
    x2 = rnorm(1,mean=(B*x2+C2) / (A*(x1^2)+1), sd=sqrt(1/(A*(x1^2)+1)))
    if (r>init*keep && r%%keep==0) {
      mkeep = r/keep
      bimat[mkeep-init,] = c(x1,x2)
    } 
  }
  return(bimat)
}

set.seed(66)
nvar2 = 2
A = 0.5
B = 0
C1 = C2 = 3
y2 = banana(A=A, B=B, C1=C1, C2=C2, 1000)

Data2 = list(y=y2)
Prioralpha = list(Istarmin=1, Istarmax=10, power=0.8)
Prior2 = list(Prioralpha=Prioralpha)
Mcmc = list(R=R, keep=1, maxuniq=200)

out2 = rDPGibbs(Prior=Prior2, Data=Data2, Mcmc=Mcmc)


if(0){
  ## plotting examples
  
  rx1 = range(y2[,1])
  rx2 = range(y2[,2])
  x1 = seq(from=rx1[1], to=rx1[2], length.out=50)
  x2 = seq(from=rx2[1], to=rx2[2], length.out=50)
  grid = cbind(x1,x2)
  plot(out2$nmix, Grid=grid, Data=y2)
  
  ## plot true bivariate density
  tden = matrix(double(50*50), ncol=50)
  for (i in 1:50) { 
  for (j in 1:50) {
        tden[i,j] = exp(-0.5*(A*(x1[i]^2)*(x2[j]^2) + 
                    (x1[i]^2) + (x2[j]^2) - 2*B*x1[i]*x2[j] - 
                    2*C1*x1[i] - 2*C2*x2[j]))
  }}
  tden = tden / sum(tden)
  image(x1, x2, tden, col=terrain.colors(100), xlab="", ylab="")
  contour(x1, x2, tden, add=TRUE, drawlabels=FALSE)
  title("True Density")
}

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`probdraw:`	\(ncomp x R/keep\) matrix that reports the probability that each draw came from a particular component
`zdraw:`	\(R/keep x nobs\) matrix that indicates which component each draw is assigned to
`compdraw:`	A list of \(R/keep\) lists of \(ncomp\) lists. Each of the inner-most lists has 2 elemens: a vector of draws for `mu` and a matrix of draws for the Cholesky root of `Sigma`.