This function creates a ring whose indeterminates are the covariances of an n dimensional Gaussian random vector. Using a graph, digraph, or mixed graph $G$ as input gives a gaussianRing with extra indeterminates related to the parametrization of the graphical model associated to that graph. Check the details of the gaussianRing for each type of input:
The indeterminates of the ring - $s_{(i,j)},k_{(i,j)},l_{(i,j)},p_{(i,j)}$ - can be placed into an appropriate matrix format using the functions covarianceMatrix, undirectedEdgesMatrix, directedEdgesMatrix, and bidirectedEdgesMatrix respectively.
The variable names that appear can be changed using the options sVariableName, lVariableName, pVariableName, and kVariableName
i1 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}}) o1 = MixedGraph{Bigraph => Bigraph{a => {d}} } d => {a} Digraph => Digraph{b => {c, d}} c => {d} d => {} Graph => Graph{} o1 : MixedGraph |
i2 : R = gaussianRing (G,pVariableName => psi) o2 = R o2 : PolynomialRing |
i3 : gens R o3 = {l , l , l , psi , psi , psi , psi , psi , s , s , b,c b,d c,d a,a b,b c,c d,d a,d a,a a,b ------------------------------------------------------------------------ s , s , s , s , s , s , s , s } a,c a,d b,b b,c b,d c,c c,d d,d o3 : List |
The routines conditionalIndependenceIdeal, trekIdeal, covarianceMatrix, undirectedEdgesMatrix, directedEdgesMatrix, bidirectedEdgesMatrix, gaussianVanishingIdeal and gaussianParametrization require that the ring be created by this function.
The object gaussianRing is a function closure.