This function generates a sample from the posterior distribution
of a linear regression model with Gaussian errors in which the
design matrix has been decomposed with singular value
decomposition.The sampling is done via the Gibbs sampling algorithm.
The user supplies data and priors, and
a sample from the posterior distribution is returned as an mcmc
object, which can be subsequently analyzed with functions
provided in the coda package.
Model formula. Predictions are returned for elements
of y that are coded as NA.
data
Data frame.
burnin
The number of burn-in iterations for the sampler.
mcmc
The number of MCMC iterations after burnin.
thin
The thinning interval used in the simulation. The number of
MCMC iterations must be divisible by this value.
verbose
A switch which determines whether or not the progress of
the sampler is printed to the screen. If verbose is greater
than 0 the iteration number, the
beta vector, and the error variance are printed to
the screen every verboseth iteration.
seed
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is
passed it is used to seed the Mersenne twister. The user can also
pass a list of length two to use the L'Ecuyer random number generator,
which is suitable for parallel computation. The first element of the
list is the L'Ecuyer seed, which is a vector of length six or NA (if NA
a default seed of rep(12345,6) is used). The second element of
list is a positive substream number. See the MCMCpack
specification for more details.
tau2.start
The starting values for the tau^2 vector. Can
be either a scalar or a vector. If a scalar is passed then that value
will be the starting value for all elements of tau^2.
g0
The prior mean of gamma. This can either be a
scalar or a
column vector with dimension equal to the number of gammas. If this
takes a scalar value, then that value will serve as the prior
mean for all of the betas.
a0
a0/2 is the shape parameter for the inverse
Gamma prior on sigma^2 (the variance of the
disturbances). The amount of information in the inverse Gamma prior
is something like that from a0 pseudo-observations.
b0
b0/2 is the scale parameter for the
inverse Gamma prior on sigma^2 (the variance of the
disturbances). In constructing the inverse Gamma prior,
b0 acts like the sum of squared errors from the
a0 pseudo-observations.
c0
c0/2 is the shape parameter for the inverse
Gamma prior on tau[i]^2.
d0
d0/2 is the scale parameter for the
inverse Gamma prior on tau[i]^2.
w0
The prior probability that gamma[i] = 0.
Can be either a scalar or an N vector where N is the
number of observations.
beta.samp
Logical indicating whether the sampled elements of
beta should be stored and returned.
intercept
Logical indicating whether the original design matrix
should include a constant term.
...
further arguments to be passed
Details
The model takes the following form:
y = X beta + epsilon
Where the errors are assumed to be iid Gaussian:
epsilon_i ~ N(0,
sigma^2)
Let N denote the number of rows of X and P
the number of columns of X. Unlike the standard regression
setup where N >> P here it is the case that P >> N.
To deal with this problem a singular value decomposition of
X' is performed: X' = ADF and the regression
model becomes
where delta0 is a unit point mass at 0 and
d[i] is the ith diagonal element of D.
Value
An mcmc object that contains the posterior sample. This
object can be summarized by functions provided by the coda package.
References
Mike West, Josheph Nevins, Jeffrey Marks, Rainer Spang, and Harry
Zuzan. 2000. “DNA Microarray Data Analysis and Regression Modeling
for Genetic Expression Profiling." Duke ISDS working paper.
Gottardo, Raphael, and Adrian Raftery. 2004. “Markov chain Monte
Carlo with mixtures of singular distributions.” Statistics
Department, University of Washington, Technical Report 470.
Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011.
“MCMCpack: Markov Chain Monte Carlo in R.”,
Journal of Statistical Software. 42(9): 1-21.
http://www.jstatsoft.org/v42/i09/.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007.
Scythe Statistical Library 1.0.http://scythe.wustl.edu.