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

Synopsis

Function: gaussianRing
Usage:

gaussianRing G
Inputs:
- G, an instance of the type 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