fitAncestralGraph | R Documentation |
Iterative conditional fitting of Gaussian Ancestral Graph Models.
fitAncestralGraph(amat, S, n, tol = 1e-06)
amat |
a square matrix, representing the adjacency matrix of an ancestral graph. |
S |
a symmetric positive definite matrix with row and col names, the sample covariance matrix. |
n |
the sample size, a positive integer. |
tol |
a small positive number indicating the tolerance used in convergence checks. |
In the Gaussian case, the models can be parameterized
using precision parameters, regression coefficients, and error
covariances (compare Richardson and Spirtes, 2002, Section 8). This
function finds the MLE \hat \Lambda
of the precision
parameters by fitting a concentration
graph model. The MLE \hat B
of the regression coefficients and
the MLE \hat\Omega
of the error covariances are obtained by
iterative conditional fitting (Drton and Richardson, 2003, 2004). The
three sets of parameters are
combined to the MLE \hat\Sigma
of the covariance matrix by
matrix multiplication:
\hat\Sigma = \hat B^{-1}(\hat \Lambda+\hat\Omega)\hat
B^{-T}.
Note that in Richardson and Spirtes (2002), the matrices \Lambda
and \Omega
are defined as submatrices.
Shat |
the fitted covariance matrix. |
Lhat |
matrix of the fitted precisions associated with undirected edges and vertices that do not have an arrowhead pointing at them. |
Bhat |
matrix of the fitted regression coefficients
associated to the directed edges. Precisely said |
Ohat |
matrix of the error covariances and variances of the residuals between regression equations associated with bi-directed edges and vertices with an arrowhead pointing at them. |
dev |
the ‘deviance’ of the model. |
df |
the degrees of freedom. |
it |
the iterations. |
Mathias Drton
Drton, M. and Richardson, T. S. (2003). A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence. Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, 184-191.
Drton, M. and Richardson, T. S. (2004). Iterative Conditional Fitting for Gaussian Ancestral Graph Models. Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, Department of Statistics, 130-137.
Richardson, T. S. and Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics. 30(4), 962-1030.
fitCovGraph
, icf
,
makeMG
, fitDag
## A covariance matrix
"S" <- structure(c(2.93, -1.7, 0.76, -0.06,
-1.7, 1.64, -0.78, 0.1,
0.76, -0.78, 1.66, -0.78,
-0.06, 0.1, -0.78, 0.81), .Dim = c(4,4),
.Dimnames = list(c("y", "x", "z", "u"), c("y", "x", "z", "u")))
## The following should give the same fit.
## Fit an ancestral graph y -> x <-> z <- u
fitAncestralGraph(ag1 <- makeMG(dg=DAG(x~y,z~u), bg = UG(~x*z)), S, n=100)
## Fit an ancestral graph y <-> x <-> z <-> u
fitAncestralGraph(ag2 <- makeMG(bg= UG(~y*x+x*z+z*u)), S, n=100)
## Fit the same graph with fitCovGraph
fitCovGraph(ag2, S, n=100)
## Another example for the mathematics marks data
data(marks)
S <- var(marks)
mag1 <- makeMG(bg=UG(~mechanics*vectors*algebra+algebra*analysis*statistics))
fitAncestralGraph(mag1, S, n=88)
mag2 <- makeMG(ug=UG(~mechanics*vectors+analysis*statistics),
dg=DAG(algebra~mechanics+vectors+analysis+statistics))
fitAncestralGraph(mag2, S, n=88) # Same fit as above
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