This method returns the $n \times{} n$ covariance matrix of the Gaussian graphical model where $n$ is the number of random variables in the model. If the gaussianRing was created using a graph, $n$ will be the number of vertices of the graph. If this function is called without a graph $G$, it is assumed that $R$ is the gaussianRing of a directed acyclic graph.
i1 : compactMatrixForm =false; |
i2 : covarianceMatrix gaussianRing 4
o2 = | s s s s |
| 1,1 1,2 1,3 1,4 |
| |
| s s s s |
| 1,2 2,2 2,3 2,4 |
| |
| s s s s |
| 1,3 2,3 3,3 3,4 |
| |
| s s s s |
| 1,4 2,4 3,4 4,4 |
4 4
o2 : Matrix (QQ[s ..s , s ..s , s ..s , s ]) <--- (QQ[s ..s , s ..s , s ..s , s ])
1,1 1,4 2,2 2,4 3,3 3,4 4,4 1,1 1,4 2,2 2,4 3,3 3,4 4,4
|
i3 : G = digraph {{a,{b,c}}, {b,{c,d}}, {c,{}}, {d,{}}}
o3 = Digraph{a => {c, b}}
b => {c, d}
c => {}
d => {}
o3 : Digraph
|
i4 : R = gaussianRing G o4 = R o4 : PolynomialRing |
i5 : S = covarianceMatrix R
o5 = | s s s s |
| a,a a,b a,c a,d |
| |
| s s s s |
| a,b b,b b,c b,d |
| |
| s s s s |
| a,c b,c c,c c,d |
| |
| s s s s |
| a,d b,d c,d d,d |
4 4
o5 : Matrix R <--- R
|
This function also works for gaussianRings created with a graph or mixedGraph.
i6 : G = graph({{a,b},{b,c},{c,d},{a,d}})
o6 = Graph{a => {b, d}}
b => {a, c}
c => {b, d}
d => {a, c}
o6 : Graph
|
i7 : R = gaussianRing G o7 = R o7 : PolynomialRing |
i8 : S = covarianceMatrix R
o8 = | s s s s |
| a,a a,b a,c a,d |
| |
| s s s s |
| a,b b,b b,c b,d |
| |
| s s s s |
| a,c b,c c,c c,d |
| |
| s s s s |
| a,d b,d c,d d,d |
4 4
o8 : Matrix R <--- R
|
i9 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})
o9 = MixedGraph{Bigraph => Bigraph{a => {d}} }
d => {a}
Digraph => Digraph{b => {c, d}}
c => {d}
d => {}
Graph => Graph{}
o9 : MixedGraph
|
i10 : R = gaussianRing G o10 = R o10 : PolynomialRing |
i11 : S = covarianceMatrix R
o11 = | s s s s |
| a,a a,b a,c a,d |
| |
| s s s s |
| a,b b,b b,c b,d |
| |
| s s s s |
| a,c b,c c,c c,d |
| |
| s s s s |
| a,d b,d c,d d,d |
4 4
o11 : Matrix R <--- R
|
The object covarianceMatrix is a method function.