Fitting of Gaussian Ancestral Graph Models
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 L of the precision parameters by fitting a concentration graph model. The MLE B of the regression coefficients and the MLE O 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 S of the covariance matrix by matrix multiplication:
S = B^(-1) (L+O) B^(-t).
Note that in Richardson and Spirtes (2002), the matrices L and O 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.
## 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 abovePlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.