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gaussianRing -- ring of Gaussian correlations on n random variables or a graphical model



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:

* gaussianRing(ZZ)

* gaussianRing(Graph)

* gaussianRing(Digraph)

* gaussianRing(Bigraph)

* gaussianRing(MixedGraph)

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.

See also

Ways to use gaussianRing :

For the programmer

The object gaussianRing is a method function with a single argument.