# Fitting of Gaussian Ancestral Graph Models Iterative conditional fitting of Gaussian Ancestral Graph Models. ```r fitAncestralGraph(amat, S, n, tol = 1e-06) ``` ## Arguments - `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. ## Details 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: $$ \hat\Sigma = \hat B^{-1}(\hat \Lambda+\hat\Omega)\hatB^{-T}.S = B^(-1) (L+O) B^(-t). $$ Note that in Richardson and Spirtes (2002), the matrices $L$ and $O$ are defined as submatrices. ## Returns - **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 `Bhat` contains ones on the diagonal and the off-diagonal entry $(i,j)$ equals the **negated** MLE of the regression coefficient for variable $j$ in the regression of variable $i$ on its parents. Note that this $(i,j)$ entry in `Bhat` corresponds to a directed edge $j -> i$, and thus to a one as $(j,i)$ entry in the adjacency matrix. - **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. ## References 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. ## Author(s) Mathias Drton ## See Also `fitCovGraph`, `icf`, `makeMG`, `fitDag` ## Examples ```r ## 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 ```

fitAncestralGraph function