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RandomMonomialIdeals :: bettiStats

bettiStats -- statistics on Betti tables of a sample of monomial ideals or list of objects

Synopsis

Description

For a sample of ideals stored as a list, this function computes some basic Betti table statistics of the sample. Namely, it computes the average shape of the Betti tables (where 1 is recorded in entry (ij) for each element if betaij is not zero), and it also computes the average Betti table (that is, the table whose (ij) entry is the mean value of betaij for all ideals in the sample).

i1 : R = ZZ/101[a..e];
i2 : L={monomialIdeal"a2b,bc", monomialIdeal"ab,bc3",monomialIdeal"ab,ac,bd,be,ae,cd,ce,a3,b3,c3,d3,e3"}

                      2                                3                   3        3        3             3                  3
o2 = {monomialIdeal (a b, b*c), monomialIdeal (a*b, b*c ), monomialIdeal (a , a*b, b , a*c, c , b*d, c*d, d , a*e, b*e, c*e, e )}

o2 : List
i3 : (meanBettiShape,meanBetti,stdDevBetti) = bettiStats L;
i4 : meanBettiShape

            0       1       2       3       4       5
o4 = total: 1       2       2 1.33333 1.33333 .333333
         0: 1       .       .       .       .       .
         1: .       1 .333333 .333333 .333333       .
         2: . .666667 .666667 .333333 .333333       .
         3: . .333333 .666667 .333333 .333333       .
         4: .       . .333333 .333333 .333333 .333333

o4 : BettiTally
i5 : meanBetti

            0       1       2       3       4 5
o5 = total: 1 5.33333 10.3333      10       5 1
         0: 1       .       .       .       . .
         1: .       3 3.66667       2 .333333 .
         2: .       2       5 4.33333 1.33333 .
         3: . .333333 .666667 .666667 .333333 .
         4: .       .       1       3       3 1

o5 : BettiTally
i6 : stdDevBetti

                  1       2       3       4       5
o6 = total: 5.46008 13.4481 14.1421 7.07107 1.41421
         1: 2.82843 5.18545 2.82843 .471405       .
         2: 2.16025 6.37704 6.12826 1.88562       .
         3: .471405 .471405 .942809 .471405       .
         4:       . 1.41421 4.24264 4.24264 1.41421

o6 : BettiTally

For sample size N, the average Betti table shape considers nonzero Betti numbers. It is to be interpreted as follows: entry (i,j) encodes the following sum of indicators: all ideals 1betaij>0 / N; that is, the proportion of ideals with a nonzero betaij. Thus an entry of 0.33 means 33% of ideals have a non-zero Betti number there.

i7 : apply(L,i->betti res i)

             0 1 2         0 1 2         0  1  2  3  4 5
o7 = {total: 1 2 1, total: 1 2 1, total: 1 12 29 30 15 3}
          0: 1 . .      0: 1 . .      0: 1  .  .  .  . .
          1: . 1 .      1: . 1 .      1: .  7 11  6  1 .
          2: . 1 1      2: . . .      2: .  5 14 13  4 .
                        3: . 1 1      3: .  .  1  2  1 .
                                      4: .  .  3  9  9 3

o7 : List
i8 : meanBettiShape

            0       1       2       3       4       5
o8 = total: 1       2       2 1.33333 1.33333 .333333
         0: 1       .       .       .       .       .
         1: .       1 .333333 .333333 .333333       .
         2: . .666667 .666667 .333333 .333333       .
         3: . .333333 .666667 .333333 .333333       .
         4: .       . .333333 .333333 .333333 .333333

o8 : BettiTally

For sample size N, the average Betti table is to be interpreted as follows: entry (i,j) encodes I∈idealsbetaij(R/I) / N:

i9 : apply(L,i->betti res i)

             0 1 2         0 1 2         0  1  2  3  4 5
o9 = {total: 1 2 1, total: 1 2 1, total: 1 12 29 30 15 3}
          0: 1 . .      0: 1 . .      0: 1  .  .  .  . .
          1: . 1 .      1: . 1 .      1: .  7 11  6  1 .
          2: . 1 1      2: . . .      2: .  5 14 13  4 .
                        3: . 1 1      3: .  .  1  2  1 .
                                      4: .  .  3  9  9 3

o9 : List
i10 : meanBetti

             0       1       2       3       4 5
o10 = total: 1 5.33333 10.3333      10       5 1
          0: 1       .       .       .       . .
          1: .       3 3.66667       2 .333333 .
          2: .       2       5 4.33333 1.33333 .
          3: . .333333 .666667 .666667 .333333 .
          4: .       .       1       3       3 1

o10 : BettiTally

Ways to use bettiStats :