scoreEquations -- score equations of the log-likelihood function of a Gaussian graphical model

Synopsis

Usage:

scoreEquations(R,U)
Inputs:
- R, a ring, defined as a gaussianRing of Graph, or Digraph, or Bigraph, or MixedGraph
- U, a matrix, or List of sample data. Alternatively, the input can be the sample covariance Matrix by setting the optional input SampleData to false
Optional inputs:
- CovarianceMatrix => ..., default value false, output covariance matrix
- RealPrecision => ..., default value 53, the number of bits of precision to use in the computation
- SampleData => ..., default value true, input sample covariance matrix instead of sample data
- Saturate => ..., default value true, whether to saturate
- SaturateOptions => ..., default value new OptionTable from {Strategy => null, BasisElementLimit => infinity, DegreeLimit => {}, MinimalGenerators => true, PairLimit => infinity}, use options from "saturate"
Outputs:
- a sequence, consisting either of (Ideal) or (Ideal,Matrix) where the ideal is generated by the score equations of the log-likelihood function of the Gaussian model and the matrix is the symbolic covariance matrix of the model

Description

This function computes the score equations that arise from taking partial derivatives of the log-likelihood function of the concentration matrix (the inverse of the covariance matrix) of a Gaussian graphical statistical model and returns the ideal generated by such equations.

The input of this function is a gaussianRing and statistical data. The latter can be given as a matrix or a list of observations. The rows of the matrix or the elements of the list are observation vectors given as lists. It is possible to input the sample covariance matrix directly by using the optional input SampleData.

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 : JU=scoreEquations(R,U)

o4 = ideal (192199680p    - 99333449, 267221621760p    - 849243924773,
                      3,4                          4,4                
     ------------------------------------------------------------------------
     1353974896462794079472640p    - 142165262245288892244817, 6898968p    -
                               3,3                                     2,2  
     ------------------------------------------------------------------------
     11533057, 19600p    - 95819, 20855l    + 90447,
                     1,1                3,4         
     ------------------------------------------------------------------------
     146915678869660815915l    - 4228634793402814499,
                           2,3                       
     ------------------------------------------------------------------------
     58766271547864326366l    + 4167005135395196717, 574914l    - 896035)
                          1,3                               1,2

o4 : Ideal of 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

i5 : V = sampleCovarianceMatrix U

o5 = | 95819/19600 25601/3360 -2129/4480 -1313/16800 |
     | 25601/3360  867/64     -2321/2304 -173/192    |
     | -2129/4480  -2321/2304 337/3072   473/11520   |
     | -1313/16800 -173/192   473/11520  3641/4800   |

              4       4
o5 : Matrix QQ  <-- QQ

i6 : JV=scoreEquations(R,V,SampleData=>false)

o6 = ideal (192199680p    - 99333449, 267221621760p    - 849243924773,
                      3,4                          4,4                
     ------------------------------------------------------------------------
     1353974896462794079472640p    - 142165262245288892244817, 6898968p    -
                               3,3                                     2,2  
     ------------------------------------------------------------------------
     11533057, 19600p    - 95819, 20855l    + 90447,
                     1,1                3,4         
     ------------------------------------------------------------------------
     146915678869660815915l    - 4228634793402814499,
                           2,3                       
     ------------------------------------------------------------------------
     58766271547864326366l    + 4167005135395196717, 574914l    - 896035)
                          1,3                               1,2

o6 : Ideal of 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

SaturateOptions allows to use all functionalities of saturate. Saturate determines whether to saturate. Note that the latter will not provide the score equations of the model.

i7 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph{{3,4}});

i8 : R = gaussianRing(G);

i9 : 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
o9 : Matrix QQ  <-- QQ

i10 : J=scoreEquations(R,U,SaturateOptions => {Strategy => Eliminate})

o10 = ideal (192199680p    - 99333449, 267221621760p    - 849243924773,
                       3,4                          4,4                
      -----------------------------------------------------------------------
      1353974896462794079472640p    - 142165262245288892244817, 6898968p    -
                                3,3                                     2,2  
      -----------------------------------------------------------------------
      11533057, 19600p    - 95819, 20855l    + 90447,
                      1,1                3,4         
      -----------------------------------------------------------------------
      146915678869660815915l    - 4228634793402814499,
                            2,3                       
      -----------------------------------------------------------------------
      58766271547864326366l    + 4167005135395196717, 574914l    - 896035)
                           1,3                               1,2

o10 : Ideal of 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

i11 : JnoSat=scoreEquations(R,U,Saturate=>false)

                                                                         
o11 = ideal (- 574914l    + 896035, - 2299656l   p    - 3584140l   p    -
                      1,2                     1,3 4,4           2,3 4,4  
      -----------------------------------------------------------------------
                                                                 
      223545l   p    - 223545p    + 36764p   , - 614424l   p    -
             3,4 3,4          4,4         3,4           1,3 4,4  
      -----------------------------------------------------------------------
                                                                
      1092420l   p    - 81235l   p    - 81235p    + 72660p   , -
              2,3 4,4         3,4 3,4         4,4         3,4   
      -----------------------------------------------------------------------
                                                                   
      35385l   p    - 153288l   p    - 324940l   p    + 13244p    -
            3,4 3,3          1,3 3,4          2,3 3,4         3,3  
      -----------------------------------------------------------------------
                                              2                             
      35385p   , - 19600p    + 95819, 1149828l    - 3584140l    - 235200p   
            3,4          1,1                  1,2           1,2          2,2
      -----------------------------------------------------------------------
                          2   2                       2               2   2  
      + 3186225, 55191744l   p    + 172038720l   l   p    + 152938800l   p   
                          1,3 4,4             1,3 2,3 4,4             2,3 4,4
      -----------------------------------------------------------------------
                                                                      2   2  
      + 10730160l   l   p   p    + 22745800l   l   p   p    + 1238475l   p   
                 1,3 3,4 4,4 3,4            2,3 3,4 4,4 3,4           3,4 3,4
      -----------------------------------------------------------------------
                     2                  2                  2    
      + 10730160l   p    + 22745800l   p    - 11289600p   p    -
                 1,3 4,4            2,3 4,4            3,3 4,4  
      -----------------------------------------------------------------------
                                                                        
      1764672l   p   p    - 20344800l   p   p    + 2476950l   p   p    -
              1,3 4,4 3,4            2,3 4,4 3,4           3,4 4,4 3,4  
      -----------------------------------------------------------------------
                 2                  2             2                     
      927080l   p    + 11289600p   p    + 1238475p    - 927080p   p    +
             3,4 3,4            4,4 3,4           4,4          4,4 3,4  
      -----------------------------------------------------------------------
              2            2   2                               
      8563632p   , 1238475l   p    + 10730160l   l   p   p    +
              3,4          3,4 3,3            1,3 3,4 3,3 3,4  
      -----------------------------------------------------------------------
                                          2   2                       2    
      22745800l   l   p   p    + 55191744l   p    + 172038720l   l   p    +
               2,3 3,4 3,3 3,4            1,3 3,4             1,3 2,3 3,4  
      -----------------------------------------------------------------------
                2   2                2              2        
      152938800l   p    - 927080l   p    - 11289600p   p    -
                2,3 3,4          3,4 3,3            3,3 4,4  
      -----------------------------------------------------------------------
                                                                        
      1764672l   p   p    - 20344800l   p   p    + 2476950l   p   p    +
              1,3 3,3 3,4            2,3 3,3 3,4           3,4 3,3 3,4  
      -----------------------------------------------------------------------
                   2                  2                  2             2    
      10730160l   p    + 22745800l   p    + 11289600p   p    + 8563632p    -
               1,3 3,4            2,3 3,4            3,3 3,4           3,3  
      -----------------------------------------------------------------------
                               2                               
      927080p   p    + 1238475p   , - 5365080l   l   p   p    -
             3,3 3,4           3,4            1,3 3,4 3,3 4,4  
      -----------------------------------------------------------------------
                                         2                      2            
      11372900l   l   p   p    - 1238475l   p   p    - 55191744l   p   p    -
               2,3 3,4 3,3 4,4           3,4 3,3 3,4            1,3 4,4 3,4  
      -----------------------------------------------------------------------
                                            2                             2  
      172038720l   l   p   p    - 152938800l   p   p    - 5365080l   l   p   
                1,3 2,3 4,4 3,4             2,3 4,4 3,4           1,3 3,4 3,4
      -----------------------------------------------------------------------
                         2                                                
      - 11372900l   l   p    + 882336l   p   p    + 10172400l   p   p    -
                 2,3 3,4 3,4          1,3 3,3 4,4            2,3 3,3 4,4  
      -----------------------------------------------------------------------
                                                                       
      1238475l   p   p    + 927080l   p   p    - 10730160l   p   p    -
              3,4 3,3 4,4          3,4 3,3 3,4            1,3 4,4 3,4  
      -----------------------------------------------------------------------
                                                               2    
      22745800l   p   p    + 11289600p   p   p    + 882336l   p    +
               2,3 4,4 3,4            3,3 4,4 3,4          1,3 3,4  
      -----------------------------------------------------------------------
                   2                 2              3                     
      10172400l   p    - 1238475l   p    - 11289600p    + 463540p   p    -
               2,3 3,4           3,4 3,4            3,4          3,3 4,4  
      -----------------------------------------------------------------------
                                                 2
      8563632p   p    - 1238475p   p    + 463540p   )
              3,3 3,4           4,4 3,4          3,4

o11 : Ideal of 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

The ML-degree of the model is the degree of the score equations ideal. The ML-degree of the running example is 1:

i12 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph{{3,4}});

i13 : R = gaussianRing(G);

i14 : 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
o14 : Matrix QQ  <-- QQ

i15 : J = scoreEquations(R,U)

o15 = ideal (192199680p    - 99333449, 267221621760p    - 849243924773,
                       3,4                          4,4                
      -----------------------------------------------------------------------
      1353974896462794079472640p    - 142165262245288892244817, 6898968p    -
                                3,3                                     2,2  
      -----------------------------------------------------------------------
      11533057, 19600p    - 95819, 20855l    + 90447,
                      1,1                3,4         
      -----------------------------------------------------------------------
      146915678869660815915l    - 4228634793402814499,
                            2,3                       
      -----------------------------------------------------------------------
      58766271547864326366l    + 4167005135395196717, 574914l    - 896035)
                           1,3                               1,2

o15 : Ideal of 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

i16 : dim J, degree J

o16 = (0, 1)

o16 : Sequence

Ways to use scoreEquations :

scoreEquations(Bigraph,Digraph,Graph,List)
scoreEquations(Bigraph,Digraph,Graph,Matrix)
scoreEquations(Bigraph,Digraph,List)
scoreEquations(Bigraph,Digraph,Matrix)
scoreEquations(Bigraph,Graph,Digraph,List)
scoreEquations(Bigraph,Graph,Digraph,Matrix)
scoreEquations(Bigraph,Graph,List)
scoreEquations(Bigraph,Graph,Matrix)
scoreEquations(Bigraph,List)
scoreEquations(Bigraph,Matrix)
scoreEquations(Digraph,Bigraph,Graph,List)
scoreEquations(Digraph,Bigraph,Graph,Matrix)
scoreEquations(Digraph,Bigraph,List)
scoreEquations(Digraph,Bigraph,Matrix)
scoreEquations(Digraph,Graph,Bigraph,List)
scoreEquations(Digraph,Graph,Bigraph,Matrix)
scoreEquations(Digraph,Graph,List)
scoreEquations(Digraph,Graph,Matrix)
scoreEquations(Digraph,List)
scoreEquations(Digraph,Matrix)
scoreEquations(Graph,Bigraph,Digraph,List)
scoreEquations(Graph,Bigraph,Digraph,Matrix)
scoreEquations(Graph,Bigraph,List)
scoreEquations(Graph,Bigraph,Matrix)
scoreEquations(Graph,Digraph,Bigraph,List)
scoreEquations(Graph,Digraph,Bigraph,Matrix)
scoreEquations(Graph,Digraph,List)
scoreEquations(Graph,Digraph,Matrix)
scoreEquations(Graph,List)
scoreEquations(Graph,Matrix)
scoreEquations(List,Bigraph)
scoreEquations(List,Bigraph,Digraph)
scoreEquations(List,Bigraph,Digraph,Graph)
scoreEquations(List,Bigraph,Graph)
scoreEquations(List,Bigraph,Graph,Digraph)
scoreEquations(List,Digraph)
scoreEquations(List,Digraph,Bigraph)
scoreEquations(List,Digraph,Bigraph,Graph)
scoreEquations(List,Digraph,Graph)
scoreEquations(List,Digraph,Graph,Bigraph)
scoreEquations(List,Graph)
scoreEquations(List,Graph,Bigraph)
scoreEquations(List,Graph,Bigraph,Digraph)
scoreEquations(List,Graph,Digraph)
scoreEquations(List,Graph,Digraph,Bigraph)
scoreEquations(List,MixedGraph)
scoreEquations(List,Ring)
scoreEquations(Matrix,Bigraph)
scoreEquations(Matrix,Bigraph,Digraph)
scoreEquations(Matrix,Bigraph,Digraph,Graph)
scoreEquations(Matrix,Bigraph,Graph)
scoreEquations(Matrix,Bigraph,Graph,Digraph)
scoreEquations(Matrix,Digraph)
scoreEquations(Matrix,Digraph,Bigraph)
scoreEquations(Matrix,Digraph,Bigraph,Graph)
scoreEquations(Matrix,Digraph,Graph)
scoreEquations(Matrix,Digraph,Graph,Bigraph)
scoreEquations(Matrix,Graph)
scoreEquations(Matrix,Graph,Bigraph)
scoreEquations(Matrix,Graph,Bigraph,Digraph)
scoreEquations(Matrix,Graph,Digraph)
scoreEquations(Matrix,Graph,Digraph,Bigraph)
scoreEquations(Matrix,MixedGraph)
scoreEquations(Matrix,Ring)
scoreEquations(MixedGraph,List)
scoreEquations(MixedGraph,Matrix)
scoreEquations(Ring,List)
scoreEquations(Ring,Matrix)

For the programmer

The object scoreEquations is a method function with options.