ggm (version 1.0.4)

makeAG: Ancestral Graphs

Description

Defines an ancestral graph from the directed, undirected and undirected components and checks if the components are compatible with an ancestral graph.

Usage

makeAG(dag = NULL, ug = NULL, bg = NULL)

Arguments

dag

the adjacency matrix of a directed acyclic graph specifying the arrows of the ancestral graph.

ug

the adjacency matrix of an undirected graph specifying the lines of the ancestral graph.

bg

the adjacency matrix of an undirected graph specifying the bidirected edges of the ancestral graph.

Value

• a square matrix obtained by combining the three graph components into an adjacency matrix of an ancestral graph. See the details for the coding of the adjacency matrix.

Details

An ancestral graph is a mixed graph with three types of edges: undirected, directed and bidirected edges. The following conditions must hold: (i) no undirected edge meets an arrowhead; (ii) no directed cycles; (iii) spouses cannot be ancestors. For details see Richardson & Spirtes (2002).

The function checks if, given the matrices of the undirected, directed and bidirected edges, the above three conditions are respected. If so, a resulting adjacency matrix $E=[e_{ij}$ is returned, with the following convention. If $(i,j)$ is a directed edge, then $e_{ij} = 1$ and $e_{ji}=0$. If $(i,j)$ is an undirected edge, then $e_{ij} = e_{ji} = 1$. Finally, if $(i,j)$ is a bidirected edge, then $e_{ij} = e_{ji} = 2$.

Note that the three adjacency matrices must have labels and may be defined using the functions DAG and UG.

References

Richardson, T. Spirtes, P. (2002). Ancestral Graph Markov Models. Annals of Statistics. 30, 4, 962--1030.

See Also

UG, DAG

Examples

## Examples from Richardson and Spirtes (2002)
a1 <- makeAG(dag=DAG(a~b, b~d, d~c), bg=UG(~a*c))  # Not an AG. (a2) p.969
a2 <- makeAG(dag=DAG(b ~ a, d~c), bg=UG(~a*c+c*b+b*d))           # Fig. 3 (b1) p.969
 a3 <- makeAG(ug = UG(~ a*c), dag=DAG(b ~ a, d~c), bg=UG(~ b*d)) # Fig. 3 (b2) p.969
 a5 <- makeAG(bg=UG(~alpha*beta+gamma*delta), dag=DAG(alpha~gamma,
delta~beta))  # Fig. 6 p. 973
## Another Example
 a4 <- makeAG(ug=UG(~y0*y1), dag=DAG(y4~y2, y2~y1), bg=UG(~y2*y3+y3*y4))