[seqfan] Re: The mean and variance of a sequence (new transforms)

[Chai Wah Wu]

Hi Neil,
  That is quite interesting, another empirical law like Benford's law. I
computed Pearson's sample correlation coefficient r of log(var) versus
log(mu) for various b-files (with at least 1000 terms) that I have lying
around on my hard drive. If we assume the exponent B in the formula to be
positive (I think in many of the ecology examples they found B > 0), we
expect the correlation coefficient to be close to +1 if Taylor's law holds.
For many sequences I looked at, this is indeed the case, but for some
others, the correlation coefficient is relatively small in absolute value
and some of them are negative, suggesting a negative correlation (implying
that the exponent B < 0). I put a plot of r for over 1000 sequences here:
https://drive.google.com/file/d/1P3dUHLbPX8EJ85ndfiPgd7LTUDO9T
t9q/view?usp=sharing where you see most of the points clustered around r =
1.
Here are some examples where |r| is small or r is negative:

OEIS sequence number                    r
10060 -0.671536970646
31286 -0.689168370738
50278 0.407158565862
71989 -0.306520534891
72449 0.354648169188
79585 0.357299556119
90550 -0.457252335521
99004 -0.000716729229114
99874 0.482546466714
99876 0.464402377777
99877 -0.308177311171
105817 -0.0787593379899
124687 0.410013635437
175499 -0.00233525102281
178816 -0.414182483865
187950 -0.43797952372
188068 -0.931511998455
188082 -0.979279705266
188090 -0.970419673585
188593 0.29994341336
210973 0.425157907992
243908 0.00588214535901
245555 0.416672935337
246709 -0.110447584087
247108 -0.138917091354
257341 -0.332537512628
257342 -0.364709775268
257893 0.407158565862
257899 0.147427419432
258403 -0.212595601425
268517 0.34761603859
269927 -0.446228527477
272695 -0.00209320783492
274087 -0.330418629967
274088 -0.345859698171
274090 -0.378384186111
274091 -0.370216475665
274092 -0.362565034455
274093 0.0397204294317
274094 0.0247421385104
274095 -0.222871852376
274096 -0.247962450854
274097 -0.218411422827




On Tue, Mar 20, 2018 at 4:56 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Dear Seq Fans,
> Given an infinite sequence s(m) (I assume m >= 1 for simplicity, but that's
> not important, it is trivial to change the formulas) of integers, or
> rationals, we can define the associated sequences of means and variances as
> follows:
>
> m(n) = (1/n) * Sum_{i=1..n} s(i), n>= 1,
>
> v(1) = 0, and for n >= 2,
> v(n) = (1/(n-1)) * Sum_{i=1..n} (s(i)-m(n))^2.
>
> Of course m(n) and v(n) are rationals, so we get 6 sequences,
> numerators of m(n), denominators of m(n),
> ditto for variances,
> nearest integer to m(n), ditto for v(n).
>
> There's a belief in certain circles that
> if the sequence is nonnegative then m(n) and v(n) are usually related by
> something
> called Taylor's Law, which says that for large n, v(n) approx-equals
> A*m(n)^B
> for constants A and B.  This paper
>
> Joel E. Cohen, Statistics of Primes (and Probably Twin Primes) Satisfy
> Taylor’s Law from Ecology, The American Statistician, 70 (2016), 399-404.
>
> shows that this is true for the primes, and this led me to create the
> following sequences:
>
> Mean and variance of primes: A301273/A301274, A301275/A301276, A301277,
> A273462.
> (The last one was already there)
>
> A sequel to that paper recently appeared, or is about to appear:
>
> Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and
> Binomial Coefficients, American Statistician, online: 19 Jan 2018;
> https://doi.org/10.1080/00031305.2017.1422439;
> https://www.tandfonline.com/doi/abs/10.1080/00031305.2017.1422439
>
> where he checked that Taylor's Law holds for 100 "nice" sequences in the
> OEIS.
> ("Finite" in his title refers to the fact that he looked at the first n
> terms. The sequences he looked at were all infinite).  He doesn't say much
> about the details.
>
> I looked at the mean and variance of the terms in the n-th row of Pascal's
> triangle,
> and this produced these sequences:
>
> Mean and variance of n-th row of Pascal's triangle: A084623/A000265,
> A301278/A301279, A054650, A301280.
>
> One could now do the same thing for any of the core sequences, and this
> will probably produce a lot of new sequences (or link together old
> sequences).
> I invite you all to join the fun and send in these sequences
>
> What about Catalans, Motzkins, squares, powers of 2, 3, etc, Fibonaccis,
> Bernoullis,
> harmonic numbers s(n) = Sum_{i=1..n} 1/i, etc ?
> For Bernoullis we should do two versions, one for all B_n, and one for
> B_{2n}
>
> I can send copies of the above papers to anyone interested.
>
> --
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>