scoreEquations(...,CovarianceMatrix=>...) -- output covariance matrix

Synopsis

Usage:

scoreEquations(R,U,CovarianceMatrix=>b)
Inputs:
- b, a Boolean value, default is false

Description

scoreEquations(...,CovarianceMatrix=>...) is set to false by default. If b is true, scoreEquations gives an additional output: the covariance matrix with rational entries in the same variables as the ideal of score equations.

i1 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph {{3,4}});

i2 : R=gaussianRing(G);

i3 : U = matrix{{6, 10, 1/3, 1}, {3/5, 3, 1/2, 1}, {4/5, 3/2, 9/8, 3/10}, {10/7, 2/3,1, 8/3}};

              4       4
o3 : Matrix QQ  <-- QQ

i4 : (J,Sigma)=scoreEquations(R,U,CovarianceMatrix=>true);

i5 : Sigma

o5 = | p_(1,1)                                           
     | l_(1,2)p_(1,1)                                    
     | l_(1,2)l_(2,3)p_(1,1)+l_(1,3)p_(1,1)              
     | l_(1,2)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)l_(3,4)p_(1,1)
     ------------------------------------------------------------------------
     l_(1,2)p_(1,1)                                                          
     l_(1,2)^2p_(1,1)+p_(2,2)                                                
     l_(1,2)^2l_(2,3)p_(1,1)+l_(1,2)l_(1,3)p_(1,1)+l_(2,3)p_(2,2)            
     l_(1,2)^2l_(2,3)l_(3,4)p_(1,1)+l_(1,2)l_(1,3)l_(3,4)p_(1,1)+l_(2,3)l_(3,
     ------------------------------------------------------------------------
               l_(1,2)l_(2,3)p_(1,1)+l_(1,3)p_(1,1)                          
               l_(1,2)^2l_(2,3)p_(1,1)+l_(1,2)l_(1,3)p_(1,1)+l_(2,3)p_(2,2)  
               l_(1,2)^2l_(2,3)^2p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)p_(1,1)+l_(1,3
     4)p_(2,2) l_(1,2)^2l_(2,3)^2l_(3,4)p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)
     ------------------------------------------------------------------------
                                                                             
                                                                             
     )^2p_(1,1)+l_(2,3)^2p_(2,2)+p_(3,3)                                     
     p_(1,1)+l_(1,3)^2l_(3,4)p_(1,1)+l_(2,3)^2l_(3,4)p_(2,2)+l_(3,4)p_(3,3)+p
     ------------------------------------------------------------------------
            l_(1,2)l_(2,3)l_(3,4)p_(1,1)+l_(1,3)l_(3,4)p_(1,1)               
            l_(1,2)^2l_(2,3)l_(3,4)p_(1,1)+l_(1,2)l_(1,3)l_(3,4)p_(1,1)+l_(2,
            l_(1,2)^2l_(2,3)^2l_(3,4)p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)p_(
     _(3,4) l_(1,2)^2l_(2,3)^2l_(3,4)^2p_(1,1)+2l_(1,2)l_(1,3)l_(2,3)l_(3,4)^
     ------------------------------------------------------------------------
                                                                            
     3)l_(3,4)p_(2,2)                                                       
     1,1)+l_(1,3)^2l_(3,4)p_(1,1)+l_(2,3)^2l_(3,4)p_(2,2)+l_(3,4)p_(3,3)+p_(
     2p_(1,1)+l_(1,3)^2l_(3,4)^2p_(1,1)+l_(2,3)^2l_(3,4)^2p_(2,2)+l_(3,4)^2p
     ------------------------------------------------------------------------
                                    |
                                    |
     3,4)                           |
     _(3,3)+2l_(3,4)p_(3,4)+p_(4,4) |

                                                                            4                                                                     4
o5 : Matrix (frac(QQ[l   ..l   , l   , l   , p   , p   , p   , p   , p   ]))  <-- (frac(QQ[l   ..l   , l   , l   , p   , p   , p   , p   , p   ]))
                      1,2   1,3   2,3   3,4   1,1   2,2   3,3   4,4   3,4                   1,2   1,3   2,3   3,4   1,1   2,2   3,3   4,4   3,4

Further information

Default value: false
Function: scoreEquations -- score equations of the log-likelihood function of a Gaussian graphical model
Option key: CovarianceMatrix -- optional input to output covariance matrix

Functions with optional argument named CovarianceMatrix: