How's about this. I found one that gets really close to the smallest SoS we've seen, but using one submetric only (soapfar). I think by your definition this uses 6 parameters though.
sum-of-squares: 0.004643131 by a soapfar where:
ap = p → logα
(p + x) + w
ar = r → ry
k = 0.5920238095238095
α = 2.0107142857142857
w = -2.341928094887362
x = 3.069642857142857
y = 1.6476190476190475
t = 1.658452380952381
and the numinosity is determined post-soapfar.
What's disturbing, however, is how sensitive this situation is. You might think you could get away with substituting all these simple numbers:
k = 3/5
a = 2
w = log2
5 ≈ 2.32192809489
x = 3
y = 5/3
t = 5/3
but that knocks it to 0.0080 something.
It's interesting, I think, that y is > 1. When I was using the copfr function along with sopafar, y was < 1. I can see it going either way.
Interestingly, we can get really close to that, a SoS of 0.004715651, with a soapfar where
k = 0.5970238095238095
a = 2.0125
w = -2.334428094887362
x = 3.069642857142857
y = 1.6226190476190474
t = 1.618452380952381
In which y and t are unmistakably close to the golden ratio, φ. Which doesn't feel psychologically motivated. But interesting perhaps, especially when you realize that 2φ
= 3.06956450765 which is just absurdly close to that x. Also k is not far off from 1/φ. But again, you budge any of these just a wee bit and the SoS changes dramatically.
The sensitivity of these numbers makes me think that the formula is overspecified in a way that makes it really brittle. In other words, even though its summed over 80 data points, there might be a bit of a "luck" element in there.
I am able to get a 0.005680348 SoS with only 5 parameters, dropping the t:
but that doesn't quite make your 0.0055 cutoff. Man, it's distressing what a wide variety of parameters will get you something so close to the same level of ranking fit.
Again and again I find that prime limit does not help enough.
And I cannot find a situation where any other function besides soapifar (where the prime counting function pi is applied to the prime too) which ever moves the needle.
And I haven't found a situation where using the original numerator and denominator as the numinosity function (identity, as you called it; pre-soapfar, I guess, would be another way to call it) improved things for me.
I almost turned in for the day when I stumbled across this, almost by accident:
I can get 0.006689884017318771 SoS with a metric so simple it's almost unbelievable: w = -1. That's right: no k, no a, no y. Just subtract one from each prime as you sum them up. It's clear how this would help rate 25/7 better than 17/1. Of course it ignores the difference between 35/1 and 7/5 though... but apparently that doesn't matter? That's a reduction in SoS of about 40% from good ol' SoPF>3, which gave 0.011375524.
Awwww... well, I got super excited about that until I looked closer and realized the problem: lots of the values tie for the same rank, and the way my code works is that it then essentially gives the benefit of the doubt (i.e. if something goes wrong and our metric ends up assigning the exact same number of "antivotes" to each ratio, then my code gives that run a perfect score of 0 SoS, since the rank comes out 1,2,3,4,5... perfectly matching the real ranks which are of course already sorted). So I need to confront this issue of "fractional ranks" which I had noticed earlier but figured with all the complexity to our metric I wouldn't really have to deal with.
Okay, I've fixed that problem. I'm not super nervous that other results of mine would have suffered from this bug. The actual SoS for w=-1 is 0.012420586... slightly worse
I think I'm about ready to turn in and suggest that the one I found earlier, with only 4 parameters, is the way to go:
k = 0
a = 1.994
y = 0.455
w = -2.08
c = 0.577 (the weight on copfr)
It's "4" parameters, but involves 2 top-level submetrics: soapfar and copfr (and you might count k=0 as a parameter...) I wonder if you might prefer this 4-parameter soapfar-only one I found:
a = 3
w = -1
y = 0.8766666666666667
(all of my a's are now bases, by the way, not powers)
It's not terribly dissimilar from some of the others you've thrown out in the past couple pages ~0.006 but with the special behavior for 5 in the mcopfr. It may not be quite as good SoS but it's simpler I think.
Dave Keenan wrote: ↑
Fri Jul 03, 2020 2:02 pm
I may be about spent on this front. Sorry.