# covarianceMatrix -- covariance matrix of a Gaussian graphical model

## Synopsis

• Usage:
covarianceMatrix R
• Inputs:
• R, a ring, which should be a gaussianRing
• Outputs:
• , the $n \times{} n$ covariance matrix of the Gaussian graphical model.

## Description

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