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

Synopsis

Usage:

bettiStats(List)
Inputs:
- L, a list, of objects of type MonomialIdeal, or any objects to which betti resolution can be applied.
Optional inputs:
- CountPure => ..., default value false, optional input to show the number of objects in the list whose Betti tables are pure
- IncludeZeroIdeals => ..., default value true, optional input to choose whether zero ideals should be included
- SaveBettis => ..., default value "", optional input to store all Betti tables computed
- Verbose => ..., default value false, optional input to request verbose feedback
Outputs:
- a sequence, of objects of type BettiTally, representing the mean Betti table shape and the mean Betti table of the elements in the list L.

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 $beta_{ij}$ is not zero), and it also computes the average Betti table (that is, the table whose (ij) entry is the mean value of $beta_{ij}$ 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 
o2 = {monomialIdeal (a b, b*c), monomialIdeal (a*b, b*c ), monomialIdeal (a ,
     ------------------------------------------------------------------------
           3        3             3                  3
     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: $\sum_{all ideals} 1_{beta_{ij}>0} / N$; that is, the proportion of ideals with a nonzero $beta_{ij}$. 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 $\sum_{I\in ideals}beta_{ij}(R/I) / N$:

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:

bettiStats(List)

For the programmer

The object bettiStats is a method function with options.