# gaussianRing(Graph) -- ring of Gaussian correlations of a graphical model coming from an undirected graph

## Synopsis

• Function: gaussianRing
• Usage:
gaussianRing G
• Inputs:
• G, an instance of the type Graph
• Optional inputs:
• Coefficients => ..., default value QQ, optional input to choose the base field in markovRing or gaussianRing
• kVariableName => ..., default value k, symbol used for indeterminates in a ring of Gaussian joint probability distributions
• lVariableName => ..., default value l, symbol used for indeterminates in a ring of Gaussian joint probability distributions
• pVariableName => ..., default value p, symbol used for indeterminates in a ring of Gaussian joint probability distributions
• sVariableName => ..., default value s, symbol used for indeterminates in a ring of Gaussian joint probability distributions
• Outputs:
• a ring, a polynomial ring with indeterminates $s_{(i,j)}$ and $k_{(i,j)}$

## Description

This function creates a polynomial ring with indeterminates $s_{(i,j)}$ for $1 \leq i \leq j \leq n$, where $n$ is the number of vertices in $G$, and $k_{(i,j)}$.

The $s_{(i,j)}$ indeterminates in the gaussianRing are the entries in the covariance matrix of the jointly normal random variables.

The $k_{(i,j)}$ indeterminates in the gaussianRing are the nonzero entries in the concentration matrix in the graphical model associated to the undirected graph.

 i1 : G = graph({{a,b},{b,c},{c,d},{a,d}}) o1 = Graph{a => {b, d}} b => {a, c} c => {b, d} d => {a, c} o1 : Graph i2 : R = gaussianRing G o2 = R o2 : PolynomialRing i3 : gens R o3 = {k , k , k , k , k , k , k , k , s , s , s , s , a,a b,b c,c d,d a,b a,d b,c c,d a,a a,b a,c a,d ------------------------------------------------------------------------ s , s , s , s , s , s } b,b b,c b,d c,c c,d d,d o3 : List i4 : covarianceMatrix R o4 = | s_(a,a) s_(a,b) s_(a,c) s_(a,d) | | s_(a,b) s_(b,b) s_(b,c) s_(b,d) | | s_(a,c) s_(b,c) s_(c,c) s_(c,d) | | s_(a,d) s_(b,d) s_(c,d) s_(d,d) | 4 4 o4 : Matrix R <--- R i5 : undirectedEdgesMatrix R o5 = | k_(a,a) k_(a,b) 0 k_(a,d) | | k_(a,b) k_(b,b) k_(b,c) 0 | | 0 k_(b,c) k_(c,c) k_(c,d) | | k_(a,d) 0 k_(c,d) k_(d,d) | 4 4 o5 : Matrix R <--- R