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# gaussianRing(Digraph) -- ring of Gaussian correlations of a graphical model coming from a digraph

- Function: gaussianRing
- Usage:
    `gaussianRing G`
- Inputs:
  - G, a digraph,
- 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 $s_{(i,j)},l_{(i,j)}, p_{(i,j)}$ .

## Description

This function creates a polynomial ring in the indeterminates $s_{(i,j)}$ associated to the covariance matrix of the model plus two new lists of indeterminates:

- The $l_{(i,j)}$ indeterminates consist of regression coefficients associated to the directed edges in the graph.

- The $p_{(i,j)}$ indeterminates in the gaussianRing are the nonzero entries in the covariance matrix of the error terms in the graphical model associated to a mixed graph with bidirected edges.

Note that since version 2.0 of the package, gaussianRing of a directed graph is built as a gaussianRing of a mixed graph with only directed edges, see gaussianRing(MixedGraph).

    i1 : G = digraph {{a,{b,c}}, {b,{c,d}}, {c,{}}, {d,{}}};

    i2 : R = gaussianRing G;

    i3 : gens R

    o3 = {l   , l   , l   , l   , p   , p   , p   , p   , s   , s   , s   , s   ,
           a,c   a,b   b,c   b,d   a,a   b,b   c,c   d,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 : directedEdgesMatrix R

    o5 = | 0 l_(a,b) l_(a,c) 0       |
         | 0 0       l_(b,c) l_(b,d) |
         | 0 0       0       0       |
         | 0 0       0       0       |

                 4      4
    o5 : Matrix R  <-- R

    i6 : bidirectedEdgesMatrix R

    o6 = | p_(a,a) 0       0       0       |
         | 0       p_(b,b) 0       0       |
         | 0       0       p_(c,c) 0       |
         | 0       0       0       p_(d,d) |

                 4      4
    o6 : Matrix R  <-- R

## Ways to use this method:

- gaussianRing(Digraph) -- ring of Gaussian correlations of a graphical model coming from a digraph

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GraphicalModels.m2:2166:0.