bmonomvn {monomvn}R Documentation

Bayesian Estimation for Multivariate Normal Data with Monotone Missingness

Description

Bayesian estimation via sampling from the posterior distribution of the of the mean and covariance matrix of multivariate normal (MVN) distributed data with a monotone missingness pattern via Gibbs Sampling. Through the use of parsimonious/shrinkage regressions (currently only lasso is supported), where standard regressions fail, this function can handle an (almost) arbitrary amount of missing data.

Usage

bmonomvn(y, pre = TRUE, p = 0.9, B = 100, T = 200, thin = 10,
         method = c("default", "rjlasso", "rjlsr", "lasso"),
         capm = method!="lasso", start = NULL, r = 2, delta = 0.1,
         rao.s2 = TRUE, verb = 1, trace = FALSE)

Arguments

y data matrix were each row is interpreted as a random sample from a MVN distribution with missing values indicated by NA
pre logical indicating whether pre-processing of the y is to be performed. This sorts the columns so that the number of NAs is non-decreasing with the column index
p when performing regressions, p is the proportion of the number of columns to rows in the design matrix before an alternative regression (lasso) is performed as if least-squares regression “failed”. Least-squares regression is known to fail when the number of columns equals the number of rows, hence a default of p = 0.9 close to 1. Alternatively, setting p = 0 forces lasso to be used for every regression. Intermediate settings of p allow the user to control when least-squares regressions stop and the lasso ones start
B number of Burn-In MCMC sampling rounds, during which samples are discarded
T total number of MCMC sampling rounds to take place after burn-in, during which samples are saved
thin level of thinning in the MCMC, i.e., the number of MCMC rounds must be collected before a sample is saved
method indicates the lasso regression specification to be used. The "default" uses the Trevor & Park formulation until p >= n at which point a Reversible Jump (RJ) is turned on in order to keep the number of non-zero regression coefficients beta at most n (when capm = TRUE. The "rjlasso" method uses RJ at all times that the lasso is used, whereas "rjlsr" uses RJ without the lasso prior. Finally, the "lasso" method never uses RJ. See blasso and note below for more details
capm when TRUE this argument indicates that the number of components of beta should not exceed n, the number of response variables in a particular regression. This argument is ignored when using method = "lasso"
start a list depicting starting values for the parameters that are use to initaliza the Markov chain. Usually this will be a "monomvn"-class object depsicting maximum likelihood estimates output from the monomvn function, but the relevant fields are the mean vector $mu, covariance matrix $S, monotone ordering $o (for sanity checking with intput y), component vector $ncomp and penalty parameter vector $lambda. See note below
r alpha (shape) parameter to the gamma distribution prior for the lasso parameter lambda
delta beta (rate) parameter to the gamma distribution prior for the lasso parameter lambda
rao.s2 indicates whether to use Rao-Blackwellized samples for s^2 should be used (default TRUE), see the details section of blasso for more information
verb verbosity level; currently only verb = 0 and verb = 1 are supported
trace if TRUE then samples from the regressions are saved to files in the CWD

Details

If pre = TRUE then bmonomvn first re-arranges the columns of y into nondecreasing order with respect to the number of missing (NA) entries. Then (at least) the first column should be completely observed.

Samples from the posterior distribution of the over a Multivariate Normal mean vector and covariance matrix are obtained by obtaining samples from the posterior distribution of Bayesian regression models. The method by which these samples from the regression posterior(s) are used to obtain samples from the mean vector and covariance matrix is outlined in the monomvn documentation, detailing a similarly structured maximum likelihood approach. See also the references below.

Whenever the regression model is ill–posed (i.e., when there are more covariates than responses, or a “big p small n” problem) then Bayesian Lasso regressions – possibly augmented with Reversible Jump (RJ) for model selection – are used instead. See the Park & Casella reference below, and the blasso documentation. The p argument can be used to turn on Lasso regressions at other times.

One difference between the Bayesian and MLE approach is that the MLE approach treats the complete (fully observed) columns jointly, without performing regressions. In contrast, the Bayesian only treats the first complete column without regressions. The remaining complete columns are processed via regression.

Value

bmonomvn returns an object of class "monomvn", which is a list containing a subset of the components below.

call a copy of the function call as used
mu estimated mean vector with columns corresponding to the columns of y
S estimated covariance matrix with rows and columns corresponding to the columns of y
mu.var estimated variance of mean vector with columns corresponding to the columns of y
S.var estimated variance of the individual components of the covariance matrix with columns corresponding to the columns of y
na when pre = TRUE this is a vector containing number of NA entries in each column of y
o when pre = TRUE this is a vector containing the index of each column in the sorting of the columns of y obtained by o <- order(na)
method method of regression used on each column, or "bcomplete" indicating that no regression was used
lambda2 records the mean lambda^2 value found in the trace of the Bayesian Lasso regressions. This value will be zero if the corresponding column corresponds to a complete case or a ordinary least squares regression (these would be NA entries maximum likelihood estimates are sought via monomvn
ncomp records the mean number of components (columns of the design matrix) used in the regression model for each column of y. If input RJ = FALSE then this simply corresponds to the monotone ordering (these would correspond to the NA entries when maximum likelihood estimates are sought via monomvn. When RJ = TRUE the monotone ordering is an upper bound (on each entry)
trace if input trace = TRUE then this field contains traces of the samples of mu in the field $mu and of S in the field $S, and of all regression parameters for each of the m = length(mu) columns in the field $reg
B from inputs: number of Burn-In MCMC sampling rounds, during which samples are discarded
T from inputs: total number of MCMC sampling rounds to take place after burn-in, during which samples are saved
r from inputs: alpha (shape) parameter to the gamma distribution prior for the lasso parameter lambda
delta from inputs: beta (rate) parameter to the gamma distribution prior for the lasso parameter lambda

Note

Whenever the bmonomvn algorithm requires a regression where p >= n, i.e., if any of the columns in the Y matrix have fewer non–NA elements than the number of columns with more non–NA elements. then it is helpful to employ both lasso and RJ method. Therefore, should this case be detected and one of the "rjlsr" or "lasso" methods is specified, a warning will be printed

It is important than any starting values provided in the start input list be compatible with the regression model specified by inputs RJ and method. Any incompatibilites will result with a warning that (alternative) default action was taken and may result in an undesired (possibly inferior) model being fit

Author(s)

Robert B. Gramacy bobby@statslab.cam.ac.uk

References

Robert B. Gramacy and Joo Hee Lee (2007). On estimating covariances between many assets with histories of highly variable length. Preprint available on arXiv:0710.5837:
http://arxiv.org/abs/0710.5837

Roderick J.A. Little and Donald B. Rubin (2002). Statistical Analysis with Missing Data, Second Edition. Wilely.

Park, T., Casella, G. (2008). The Bayesian Lasso, (unpublished)
http://www.stat.ufl.edu/~casella/Papers/bayeslasso.pdf

http://www.statslab.cam.ac.uk/~bobby/monomvn.html

See Also

blasso, monomvn, em.norm in the norm package, and mlest in the mvnmle package

Examples

## standard usage, duplicating the results in
## Little and Rubin, section 7.4.3
data(cement.miss)
out <- bmonomvn(cement.miss)
out
out$mu
out$S

##
## A bigger example, comparing the various methods
##

## generate N=1000 samples from a random MVN
xmuS <- randmvn(1000, 100)

## randomly impose monotone missingness
xmiss <- rmono(xmuS$x)

## using least squares only when necessary
out.b <- bmonomvn(xmiss)
out.b
kl.norm(out.b$mu, out.b$S, xmuS$mu, xmuS$S, symm=TRUE)
out.mle <- monomvn(xmiss, method="lasso")
kl.norm(out.mle$mu, out.mle$S, xmuS$mu, xmuS$S, symm=TRUE)

## using least squares sparingly
out.b.s <- bmonomvn(xmiss, p=0.25)
kl.norm(out.b.s$mu, out.b.s$S, xmuS$mu, xmuS$S, symm=TRUE)
out.mle.s <- monomvn(xmiss, p=0.25, method="lasso")
kl.norm(out.mle.s$mu, out.mle.s$S, xmuS$mu, xmuS$S, symm=TRUE)

## using the maximum likelihood solution to initialize
## the Markov chain and avoid burn-in.  
out.b2.s <- bmonomvn(xmiss, p=0.25, B=0, start=out.mle.s, method="rjlasso")
kl.norm(out.b2.s$mu, out.b2.s$S, xmuS$mu, xmuS$S, symm=TRUE)

[Package monomvn version 1.3-1 Index]