\HeaderA{estimateSigmaMVbeta}{Zero mean multivariate t-dist. with covariate dependent scale.}{estimateSigmaMVbeta}
\keyword{htest}{estimateSigmaMVbeta}
\keyword{models}{estimateSigmaMVbeta}
\begin{Description}\relax
Estimate the parameters \eqn{\Sigma}{Sigma}, \eqn{m}{m} and \eqn{\nu}{v} of the multivariate
t-distribution with zero expectation, where \eqn{\nu}{v} is modeled as smooth function of a covariate.
\end{Description}
\begin{Usage}
\begin{verbatim}
estimateSigmaMVbeta(y, x, maxIter = 200, epsilon = 1e-06,
    verbose = FALSE, nknots = 10, nOut = 2000, nIn = 4000,
    iterInit = 3, br = NULL)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{y}] Data matrix
\item[\code{x}] Covariate vector
\item[\code{maxIter}] Maximum number of iterations
\item[\code{epsilon}] Convergence criteria
\item[\code{verbose}] Print computation info or not
\item[\code{nknots}] Number of knots of spline for \eqn{\nu}{v}
\item[\code{nOut}] Parameter for calculating knots, see getKnots
\item[\code{nIn}] Parameter for calculating knots, see getKnots
\item[\code{iterInit}] Number of iteration in when initiating \eqn{\Sigma}{Sigma}
\item[\code{br}] Knots, overrides nknots, n.out and n.in
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The multivariate t-distribution is parametrized as:
\deqn{y|c \sim N(\mu,c\Sigma)}{y|c ~ N(mu,c*Sigma)}
\deqn{c \sim \mbox{InvGamma}(m/2,m\nu/2)}{c ~ InvGamma(m/2,m*v/2)}
where \eqn{\nu}{v} is function of the covariate x: \eqn{\nu(x)}{v(x)} and
\eqn{N}{N} denotes a multivariate normal distribution, 
\eqn{\Sigma}{Sigma} is a covariance matrix and 
\eqn{\mbox{InvGamma}(a,b)}{InvGamma(a,b)} is the inverse-gamma distribution with density function
\deqn{f(x)=(b)^{a} \exp\{-b/x\} x^{-a-1}/\Gamma(a)}{f(x)=b^a exp\{-b/x\} x^{-a-1} /Gamma(a)}

A cubic spline is used to parameterize the smooth function \eqn{\nu(x)}{v(x)} 
\deqn{\nu(x) = \exp\{ H(x)^T \beta \}}{v(x)=exp{H(x)^T beta}}
where 
\eqn{H:R \to R^{2p-1}}{H:R->R^(2p-1)} 
is a set B-spline basis functions for a given set of p interior spline-knots, see chapter 5 of Hastie et al. (2001).
In this application \eqn{\mu}{mu} equals zero, and m is the degrees of freedom.

For details about the model see Kristiansson et al. (2005), \eqn{\mbox{\AA}}{A}strand et al. (2007a,2007b).
\end{Details}
\begin{Value}
\begin{ldescription}
\item[\code{Sigma}] Estimated covariance matrix for y
\item[\code{m}] Estimated shape parameter for inverse-gamma prior for gene variances
\item[\code{v}] Estimated scale parameter curve for inverse-gamma prior for gene variances
\item[\code{converged}] T if the EM algorithms converged
\item[\code{iter}] Number of iterations
\item[\code{modS2}] Moderated estimator of gene-specific variances
\item[\code{histLogS2}] Histogram of log(s2) where s2 is the ordinary variance estimator
\item[\code{fittedDensityLogS2}] The fitted density for log(s2)
\item[\code{logs2}] Variance estimators, logged with base 2.
\item[\code{beta}] Estimated parameter vector \eqn{\beta}{beta} of spline for \eqn{\nu(x)}{v(x)} 
\item[\code{knots}] The knots used in spline for \eqn{\nu(x)}{v(x)}
\item[\code{x}] The input vector covariate vector x
\end{ldescription}
\end{Value}
\begin{Author}\relax
Magnus \eqn{\mbox{\AA}}{A}strand
\end{Author}
\begin{References}\relax
Hastie, T., Tibshirani, R., and Friedman, J. (2001). The Elements of Statistical Learning, volume 1. Springer, first edition.

Kristiansson, E., Sj\eqn{\mbox{\"o}}{o}gren, A., Rudemo, M., Nerman, O. (2005). Weighted Analysis of Paired Microarray Experiments. Statistical Applications in Genetics and Molecular Biology 4(1)

\eqn{\mbox{\AA}}{A}strand, M. et al. (2007a). Improved covariance matrix estimators for weighted analysis of microarray data. Journal of Computational Biology, Accepted.

\eqn{\mbox{\AA}}{A}strand, M. et al. (2007b). Empirical Bayes models for multiple-probe type arrays at the probe level. Bioinformatics, Submitted 1 October 2007.
\end{References}
\begin{SeeAlso}\relax
plw, lmw
\end{SeeAlso}

