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

Synopsis

• Usage:

  gaussianRing n

  gaussianRing G

• Inputs:
  • n, an integer, the number of random variables
  • G, a graph, or a digraph, or an instance of the type Bigraph, or an instance of the type MixedGraph
• 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 associated to the graphical model

Description

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

See also

• bidirectedEdgesMatrix -- matrix corresponding to the bidirected edges of a bigraph or a mixed graph
• conditionalIndependenceIdeal -- ideal of a list of conditional independent statements
• covarianceMatrix -- covariance matrix of a Gaussian graphical model
• directedEdgesMatrix -- matrix corresponding to the directed edges of a digraph or a mixed graph
• gaussianVanishingIdeal -- vanishing ideal of a Gaussian graphical model
• trekIdeal -- trek separation ideal of a mixed graph
• undirectedEdgesMatrix -- matrix corresponding to the edges of an undirected graph