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GraphicalModels :: covarianceMatrix

covarianceMatrix -- the covariance matrix of a Gaussian graphical model

Synopsis

Description

This method returns the n × n covariance matrix of the Gaussian graphical model where n is the number of random variables in the model. If the gaussianRing was created using a graph, n will be the number of vertices of the graph. If this function is called without a graph G, it is assumed that R is the gaussianRing of a directed acyclic graph.

i1 : compactMatrixForm =false;
i2 : covarianceMatrix gaussianRing 4

o2 = | s     s     s     s    |
     |  1,1   1,2   1,3   1,4 |
     |                        |
     | s     s     s     s    |
     |  1,2   2,2   2,3   2,4 |
     |                        |
     | s     s     s     s    |
     |  1,3   2,3   3,3   3,4 |
     |                        |
     | s     s     s     s    |
     |  1,4   2,4   3,4   4,4 |

                                                                            4                                                                      4
o2 : Matrix (QQ[s   , s   , s   , s   , s   , s   , s   , s   , s   , s   ])  <--- (QQ[s   , s   , s   , s   , s   , s   , s   , s   , s   , s   ])
                 1,1   1,2   1,3   1,4   2,2   2,3   2,4   3,3   3,4   4,4              1,1   1,2   1,3   1,4   2,2   2,3   2,4   3,3   3,4   4,4
i3 : G = digraph {{a,{b,c}}, {b,{c,d}}, {c,{}}, {d,{}}}

o3 = Digraph{a => set {b, c}}
             b => set {c, d}
             c => set {}
             d => set {}

o3 : Digraph
i4 : R = gaussianRing G

o4 = R

o4 : PolynomialRing
i5 : S = covarianceMatrix R

o5 = | s     s     s     s    |
     |  a,a   a,b   a,c   a,d |
     |                        |
     | s     s     s     s    |
     |  a,b   b,b   b,c   b,d |
     |                        |
     | s     s     s     s    |
     |  a,c   b,c   c,c   c,d |
     |                        |
     | s     s     s     s    |
     |  a,d   b,d   c,d   d,d |

             4       4
o5 : Matrix R  <--- R

This function also works for gaussianRings created with a graph or mixedGraph.

i6 : G = graph({{a,b},{b,c},{c,d},{a,d}})

o6 = Graph{a => set {b, d}}
           b => set {a, c}
           c => set {b, d}
           d => set {a, c}

o6 : Graph
i7 : R = gaussianRing G

o7 = R

o7 : PolynomialRing
i8 : S = covarianceMatrix R

o8 = | s     s     s     s    |
     |  a,a   a,b   a,c   a,d |
     |                        |
     | s     s     s     s    |
     |  a,b   b,b   b,c   b,d |
     |                        |
     | s     s     s     s    |
     |  a,c   b,c   c,c   c,d |
     |                        |
     | s     s     s     s    |
     |  a,d   b,d   c,d   d,d |

             4       4
o8 : Matrix R  <--- R
i9 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})

o9 = MixedGraph{Bigraph => Bigraph{a => set {d}}   }
                                   d => set {a}
                Digraph => Digraph{b => set {c, d}}
                                   c => set {d}
                                   d => set {}
                Graph => Graph{}

o9 : MixedGraph
i10 : R = gaussianRing G

o10 = R

o10 : PolynomialRing
i11 : S = covarianceMatrix R

o11 = | s     s     s     s    |
      |  a,a   a,b   a,c   a,d |
      |                        |
      | s     s     s     s    |
      |  a,b   b,b   b,c   b,d |
      |                        |
      | s     s     s     s    |
      |  a,c   b,c   c,c   c,d |
      |                        |
      | s     s     s     s    |
      |  a,d   b,d   c,d   d,d |

              4       4
o11 : Matrix R  <--- R

See also

Ways to use covarianceMatrix :

  • covarianceMatrix(Ring)