gaussianRing(ZZ) -- ring of Gaussian correlations on n random variables

• Function: gaussianRing

• Usage:

  gaussianRing n

• Inputs:
  • n, an integer, the number of random variables
• 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 ring with indeterminates $s_{(i,j)}$ for $1 \leq i \leq j \leq n$

i1 : R = gaussianRing 5;

i2 : gens R

o2 = {s   , s   , s   , s   , s   , s   , s   , s   , s   , s   , s   , s   ,
       1,1   1,2   1,3   1,4   1,5   2,2   2,3   2,4   2,5   3,3   3,4   3,5 
     ------------------------------------------------------------------------
     s   , s   , s   }
      4,4   4,5   5,5

o2 : List

i3 : compactMatrixForm =false;

i4 : covarianceMatrix R

o4 = |  s     s     s     s     s     |
     |   1,1   1,2   1,3   1,4   1,5  |
     |                                |
     |  s     s     s     s     s     |
     |   1,2   2,2   2,3   2,4   2,5  |
     |                                |
     |  s     s     s     s     s     |
     |   1,3   2,3   3,3   3,4   3,5  |
     |                                |
     |  s     s     s     s     s     |
     |   1,4   2,4   3,4   4,4   4,5  |
     |                                |
     |  s     s     s     s     s     |
     |   1,5   2,5   3,5   4,5   5,5  |

             5      5
o4 : Matrix R  <-- R