D&D.Sci Hypersphere Analysis Part 3: Beat it with Linear Algebra

As before, this is composed entirely of spoilers for abstractapplic’s D&D.Sci scenario, don’t read if you want to solve on your own!

Last time, we left off with a solution that could reliably get us ~95% performance on all our generators. Sadly, in the Glorious and Righteous Interstellar Empire, 95% performance still means Just and Reasonable Beheading (praise our Eternal Empress for Her leniency)!

Quality Score

We plotted Performance in our first writeup, and saw this:

But we’ve seen a few things since then. In particular, looking at how different effects interacted makes them look multiplicative rather than additive. So now let’s try instead getting a frequency chart of Log(Performance):

That looks somewhat familiar!

So! Our new theory is as follows:

There are a variety of different things that can hurt performance.
All of these things are multiplicative. So one 50% reduction and one 20% reduction make a 60% reduction.
All or almost all of these things are independent, we don’t have to worry about interactions.

Given that, we’re going to try out linear regression:

We are going to try to predict Penalty (negative log of performance).
We’re going to use as predictors some existing variables:
- Four boolean columns for air tasting of Apples/Burning/Copper/Mint (all four being 0 means air tastes of nothing).
- Four boolean columns for Skittering/Buzzing/Squelching/Humming.
- Feng Shui (as two separate booleans for Exceptional/Disharmonious, with the default being Adequate)
- Strange Smell (as two separate booleans for None/Extremely, with the default being Somewhat)
And some derivatives of existing variables:
- Latitude: rather than including it directly, we include a 0 or 1 ‘Is In Death Zone?’
- Longitude: again rather than including it directly, we include a sine wave that seems to match the ‘sunlight effect’: Cosine(Longitude-60).
- Value of Pi: since High and Low values look different, and high values less than ~3.16 look like they might not do anything at all, we include ‘amount pi is below 3.14159’ and ‘amount pi is above 3.14159’ as separate cols so that we can derive e.g. ‘being too low is 4x worse than being too high’.
- Murphy’s Constant: based on trying to fit a curve to what we saw in part 2, I’m including all of MC, MC^2,(2 ^ MC), (3^MC), and (4^MC) as indicator cols.

This produces something that does a pretty good job of predicting score:

but still has more error in it than we’d like. There’s that one weird outlier above the 70% mark, but even leaving that aside there are a lot of differences—the average difference size is only 0.7%, which is not too bad, but the average difference size among sites where we predicted a score of 90%+ is 3.6%, and (much worse) is systematically in the wrong direction: we’re currently overconfident when we think we’ve found a good site.

Improving the Regression

Let’s plot the error in our Penalty term against various inputs.

Latitude:

Nothing to see here, there are fewer points near the poles and equator as we knew. It looks like we’ve successfully incorporated latitude.

Longitude:

Here it looks like our cosine function isn’t quite right. We’re treating longitudes around 0 through −50 as worse than we should, and those around 100 wrapping around to −150 as better than we should. Let’s try shifting the cosine fn a little next run and using 50 instead of 60 as our center?

Value of Pi:

Now that is interesting. It seems that the Pi effect is more complicated than we realized: while we wrote our logic assuming a transition at the true value of pi, we see another transition at 3.15 exactly. For the next run, as well as using ‘amount greater than 3.14159’ for the ‘too-high’ penalty, let’s use ‘amount greater than 3.15’ for a separate penalty field and see how that goes.

Regression #2

That looks like a much tighter correlation!

Our largest difference now is 4.74%: we have never been off by more than that on any site. The average error size is 0.29%, and (even better), when we look at entries where we predict scores >90% we don’t tend to be overestimating that much: on average our >90% scores are overestimated only by 0.2%!

We still don’t seem to be handling Longitude quite right—perhaps 60 down to 50 was too far and we should shift back up a few degrees?

But our regression seems to be doing well enough that we can at least get some answers that will probably not get us beheaded!

So using these horrible horrible regression coefficients:

Intercept: 0.3575762

In Death Zone : 0.2067350364928715
Sunlight : −0.052730543411768305
Hi Pi 1 : −4.114745626642051
Hi Pi 2 : 8.962689400353522
Low Pi : 5.756356131255769
Extreme Smell : −0.09552188338412596
No Smell : −0.1208155320309348
Excellent Shui : −0.02206536448091116
Bad Shui : 0.0750532882477745
Apple : −0.14396623550258927
Burning : −0.2982007933940471
Copper : −0.24478956406742985
Mint : −0.2797668454177703
Humming : 0.6335930965019667
Squelching : 0.30120208578549734
Buzzing : 0.12587463622379025
Skittering : 0.0173229850179329
Murphy-Linear : −0.020934759603605683
Murphy-Square : −0.02445781210524455
Murphy-2^N : 0.07794407327664456
Murphy-3^N : −0.01045445650523344
Murphy-4^N : 0.0011159301815777192

to predict the ‘Penalty’ (the negative log-base-10 of Performance, so negative coeffs mean something is good and positive coeffs mean it is bad), we go off and see how that evaluates our options for sites!

Potentially Adequate Answer

When we try our regression model on the sites we have available, there are with scores >100%.

We may still be slightly overestimating these sites:

But it doesn’t look like we’re overestimating them drastically—there’s a slight tendency in that chart for the very highest scores to be overestimates rather than underestimates, but not by much on average. And when we look at the top 12 available sites according to our prediction model, we have predicted scores for them well over 100%:

Site_ID	Est. Performance
23565	111.9%
96286	111.9%
9344	109.3%
68204	108.8%
107278	108.7%
905	108.6%
43792	106.4%
62718	105.8%
8415	105.6%
16423	105.6%
42742	105.4%
83512	104.8%

If our errors cap out at 4.7% (the largest error we saw our model have in the data), all of these are going to beat 100%. Even if our errors grow a little, our average estimated performance here is 107.7%, and I don’t think it’s likely we have a systematic error that large that didn’t show up in our test data.

In real life I still wouldn’t be willing to bet my life on this. Happily, my D&D.Sci character is allowed to have ~~no sense of self-preservation~~ a burning urge to prove himself in service to the Empress’s Glorious Reign! Hail Her Eternal Majesty!

Further Improvements?

There are some things I still want to look into:

I continue to suspect that something is up with Shortitude/Deltitude.
I want to look into Skittering more: it shows up as a very slight penalty, but an awful lot of the largest errors in our model in both directions have Skittering on them. It might be a penalty that only occurs some of the time (e.g. a 1-in-5 chance that the Skitterers eat 10% of your Performance), in which case it might be worth giving up a few percentage points to avoid Sites 68204 and 62718 (the only ones in our top-12 list that have Skitterers).
I want to fine-tune the model further: track down the best way to handle Longitude, see if we can make handling of Murphy’s Const less awful, etc.
It looks like pi going from 3.14159 up to 3.15 is beneficial to performance? If so, there might not be an effect at pi=3.14159 at all, and pi=3.15 might be the only inflection point? If true, we could simplify the model there.
Burning air didn’t look as good as Minty air at first, but our regression actually thinks it’s slightly better, and it’s on a lot of the best sites we’ve found here. I would like to look into why: did we underestimate sites with burning-tasting air earlier due to a correlation with something I missed?
I still haven’t figured out if ‘fewer equatorial points’ is the natural effect of a hypersphere or not.

I might or might not get around to those. If I don’t, my current submission is the following sites:

23565
96286
9344
68204
107278
905
43792
62718
8415
16423
42742
83512