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GraphicalModels :: covarianceMatrix

covarianceMatrix -- the covariance matrix of a gaussian graphical model

Synopsis

Usage:
S = covarianceMatrix R or S = covarianceMatrix(R,G)
Inputs:
- R, a ring, which should be a gaussianRing
- G, an object of class Digraph, a directed acyclic graph, or an object of class MixedGraph with directed and bidirected edges
Outputs:
- S, a matrix, the n x n covariance matrix of symbols where n is the number of vertices in G

Description

If this function is called without a graph G, it is assumed that R is the gauss ring of a directed acyclic graph.

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

o1 = Digraph{a => set {b, c}}
             b => set {c, d}
             c => set {}
             d => set {}

o1 : Digraph

i2 : R = gaussianRing G

o2 = R

o2 : PolynomialRing

i3 : S = covarianceMatrix R

o3 = | 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
o3 : Matrix R  <--- R

Note that the covariance matrix is symmetric in the symbols.

i4 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})

o4 = MixedGraph{Bigraph => Bigraph{a => set {d}}   }
                                   d => set {a}
                Digraph => Digraph{b => set {c, d}}
                                   c => set {d}
                                   d => set {}
                Graph => Graph{}

o4 : MixedGraph

i5 : R = gaussianRing G

o5 = R

o5 : PolynomialRing

i6 : S = covarianceMatrix(R,G)

o6 = | 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
o6 : Matrix R  <--- R

See also

gaussianRing -- ring of gaussian correlations on n random variables
gaussianParametrization -- the parametrization of the covariance matrix in terms of treks
bidirectedEdgesMatrix -- the matrix corresponding to the bidirected edges of a mixed graph
directedEdgesMatrix -- the matrix corresponding to the directed edges of a mixed graph

Ways to use `covarianceMatrix` :

covarianceMatrix(Ring)
covarianceMatrix(Ring,Digraph)
covarianceMatrix(Ring,MixedGraph)