This method returns the $n \times{} n$ covariance matrix of the noise variables in the Gaussian graphical model. The diagonal in this matrix consists of the indeterminates $p_{(i,i)}$. Each off-diagonal entry is zero unless there is a bidirected edge between i and j in which case the corresponding entry in the matrix is the indeterminate $p_{(i,j)}$. The documentation of gaussianRing further describes the indeterminates $p_{(i,j)}$.
i1 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})
o1 = MixedGraph{Bigraph => Bigraph{a => {d}} }
d => {a}
Digraph => Digraph{b => {c, d}}
c => {d}
d => {}
Graph => Graph{}
o1 : MixedGraph
|
i2 : R = gaussianRing G o2 = R o2 : PolynomialRing |
i3 : compactMatrixForm =false; |
i4 : bidirectedEdgesMatrix R
o4 = | p 0 0 p |
| a,a a,d |
| |
| 0 p 0 0 |
| b,b |
| |
| 0 0 p 0 |
| c,c |
| |
| p 0 0 p |
| a,d d,d |
4 4
o4 : Matrix R <--- R
|
For mixed graphs that also have undirected edges, the size of the matrix coincides with the number of elements in compW, which depends on the vertex partition built in partitionLMG.
i5 : G = mixedGraph(digraph {{1,3},{2,4}},bigraph {{3,4}},graph {{1,2}});
|
i6 : R = gaussianRing G o6 = R o6 : PolynomialRing |
i7 : bidirectedEdgesMatrix R
o7 = | p p |
| 3,3 3,4 |
| |
| p p |
| 3,4 4,4 |
2 2
o7 : Matrix R <--- R
|
Bidirected graphs can also be considered:
i8 : G = bigraph {{a,d},{b},{c}}
o8 = Bigraph{a => {d}}
b => {}
c => {}
d => {a}
o8 : Bigraph
|
i9 : R = gaussianRing G o9 = R o9 : PolynomialRing |
i10 : bidirectedEdgesMatrix R
o10 = | p 0 0 p |
| a,a a,d |
| |
| 0 p 0 0 |
| b,b |
| |
| 0 0 p 0 |
| c,c |
| |
| p 0 0 p |
| a,d d,d |
4 4
o10 : Matrix R <--- R
|
The object bidirectedEdgesMatrix is a method function.