\name{fitCovGraph}
\alias{fitCovGraph}
\title{Fitting of Gaussian covariance graph models}
\description{
Fits a Gaussian covariance graph model by maximum likelihood.
}
\usage{
fitCovGraph(amat, S,n ,alg = "icf", dual.alg = 2, start.icf = NULL, tol = 1e-06)
}
\arguments{
  \item{amat}{A symmetric Booloean matrix with dimnames representing
    the adjacency matrix of an UG.}
  \item{S}{A symmetric positive definite matrix with dimnames, the
	  sample covariance matrix.}
	  \item{n}{A positive integer, the sample size.}
  \item{alg}{A character string, the algorithm used.
    If  \code{alg="icf"} (the default) the algorithm is based on iterative
    conditional fitting (see Drton and Richardson, 2003). In this case
    the ML estimates are returned.
    If  \code{alg="dual"} the algorithm is based on the dual
    likelihood (see Kauermann, 1996). The fitted values are an approximation
    of the ML estimates.
    }
    \item{dual.alg}{And integer equal to 1 or 2. It is used if
      \code{alg="dual"}. In this case a concentration graph model
      is fitted to the inverse of the sample covariance matrix, and
      \code{dual.alg} is passed to \code{fitConGraph} to
      specify the algorithm used in \code{fitConGraph}.
    }
  \item{start.icf}{A symmetric matrix used as starting value
    of the algorithm. If \code{start=NULL} the starting value
    is a diagonal matrix with diagonal entries equal to sample
    variances.}
  \item{tol}{A small positive number indicating the tolerance
    used in convergence tests.}
}
\details{
  A covariance graph is an undirected graph  in which
  the variables associated to two non-adjacent nodes
  are marginally independent. The edges of these
  models are represented by bi-directed edges
  (Drton and Richardson, 2003) or by dashed lines
  (Cox and Wermuth, 1996).

  By default, this function gives the ML estimates in the covariance
  graph model, by iterative conditional fitting (Drton and
  Richardson, 2003).  Otherwise, the estimates from a ``dual
  likelihood'' estimator can be obtained (Kauermann, 1996; Edwards, 2000,
  section 7.4).
}
\value{
\item{Shat}{the fitted covariance matrix.}
\item{dev}{the `deviance' of the model.}
\item{df}{the degrees of freedom.}
\item{it}{the iterations.}
}
\references{
  Cox, D. R. & Wermuth, N. (1996). \emph{Multivariate
    dependencies}. London: Chapman & Hall.

  Drton, M. and Richardson, T. S. (2003). A new algorithm for
  maximum likelihood estimation in Gaussian graphical models for
  marginal independence. \emph{Proceedings
    of the Nineteenth Conference on Uncertainty in Artificial
    Intelligence}, 184--191.

  Kauermann, G. (1996). On a dualization of graphical
  Gaussian models. \emph{Scandinavian Journal of Statistics}.
  23, 105--116.
}

\author{Mathias Drton}
\seealso{\code{\link{fitConGraph}}, \code{\link{icf}}}
\examples{
## Correlations among four strategies to cope with stress for
## 72 students. Cox & Wermuth (1996), p. 73.

data(stress)

## A chordless 4-cycle covariance graph
G <- UG(~ Y*X + X*U + U*V + V*Y)

fitCovGraph(G, S = stress, n=72)
fitCovGraph(G, S = stress, n=72, alg="dual")
}
\keyword{graphs}
\keyword{models}
\keyword{multivariate}
