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
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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
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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
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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
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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
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The object bettiStats is a method function with options.