MCMCSVDreg  R Documentation 
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.
MCMCSVDreg(
formula,
data = NULL,
burnin = 1000,
mcmc = 10000,
thin = 1,
verbose = 0,
seed = NA,
tau2.start = 1,
g0 = 0,
a0 = 0.001,
b0 = 0.001,
c0 = 2,
d0 = 2,
w0 = 1,
beta.samp = FALSE,
intercept = TRUE,
...
)
formula 
Model formula. Predictions are returned for elements of y that are coded as NA. 
data 
Data frame. 
burnin 
The number of burnin 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

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

tau2.start 
The starting values for the 
g0 
The prior mean of 
a0 

b0 

c0 

d0 

w0 
The prior probability that 
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 
The model takes the following form:
y = X \beta +
\varepsilon
Where the errors are assumed to be iid Gaussian:
\varepsilon_{i} \sim \mathcal{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
y = F'D \gamma + \varepsilon
where \gamma = A' \beta
We assume the following priors:
\sigma^{2} \sim \mathcal{G}amma(a_0/2, b_0/2)
\tau^{2} \sim \mathcal{G}amma(c_0/2, d_0/2)
\gamma_i \sim w0_i \delta_0 + (1w0_i) \mathcal{N}(g0_i,
\sigma^2 \tau_i^2/ d_i^2)
where \delta_0
is a unit point mass at 0 and d_i
is the
i
th diagonal element of D
.
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
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): 121. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v042.i09")}.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.s3websiteuseast1.amazonaws.com/.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. “Output Analysis and Diagnostics for MCMC (CODA)”, R News. 6(1): 711. https://CRAN.Rproject.org/doc/Rnews/Rnews_20061.pdf.
plot.mcmc
,
summary.mcmc
, lm
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