D&D.Sci Hypersphere Analysis Part 2: Nonlinear Effects & Interactions

post by aphyer · 2024-01-14T19:59:37.911Z · LW · GW · 0 comments

Contents

  Nonlinear Effects
    Longitude
    Latitude
    Shortitude/Deltitude
    Strange Smell
    Feng Shui
    Value of Pi
    Murphy's Constant
  Interactions
  Playing With Maps
  Intermediate Still-Inadequate Solution
None
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Following on from Part 1 [LW · GW], again don't read this if you're trying to avoid spoilers!

Nonlinear Effects

The correlation analysis we did in Part 1 can find simple effects.  If the Evil Squelchers break our machines, we will be able to see a negative correlation between Evil Squelching and Performance.

It can't find complicated effects.  If a low Feng Shui is bad, a high Feng Shui is also bad, and a moderate Feng Shui is good, the correlation analysis won't tell us that.

While summary statistics are nice, there's a lot that they miss, and one of the best things to do is just get the data somewhere you can look at it.

Probably the best place for us to start is how stats vary with Latitude/Longitude/etc. 

Longitude

In our correlation analysis in Part 1, we saw a slight positive correlation between performance and Longitude.  Here we can see that does seem to appear in the data, in an interesting not-entirely-linear way:

I can't seem to find any other column that drives this.  Every other column's chart with Longitude looks something like this:

with no apparent trend.

If this planet is aligned similarly to Earth, a Longitude pattern like this would be 'things receiving sunlight do slightly better'.  However, if this were the case, it would also change as the planet rotates, and the sun might be elsewhere when our superiors check our performance.

I've asked the GM [LW · GW] whether we have information about this, and been told that we can assume a lack of time effects in the data.  In a real-world situation I think I'd be nervous about making assumptions on this until I actually had firm data on how the planet rotated: happily, Word Of GM can override that.

Latitude

In our last analysis, we saw a surprising frequency pattern for Latitude/Shortitude/Deltitude, where the polar values around +-90 were low (as expected from choosing a random location on the planet), but the equatorial values around 0 were also strangely low:


I still don't know why this is the case.

We have to be careful when looking at those buckets near the middle/ end, because they're very small.  If we plot the value of Pi against Latitude:

Those unusually-high outliers are not particularly meaningful.  The top left one is derived from a total of 4 datapoints.  Overall, I won't believe I've found an actual effect unless it shows an effect across the whole dataset...

...which it doesn't seem to for any of the existing datapoints.  Even things like the sounds, where I'd hope that the Evil HummingBuzzers live at certain latitudes, show no non-noise effect:

Despite that, there is a real effect of Latitude on performance:

(Part 1 missed this due to the limits of correlational analysis: this effect is symmetrical, and so causes no correlation).

It looks like within ~35 degrees of the equator there is an abrupt cutoff to some sort of zone where performance is much worse.  Looking more closely:

I think there's a sharp transition at exactly +-36 degrees.  Why?

Shortitude/Deltitude

These don't seem to show the same effect on performance as Latitude:

(which is kind of odd, and makes me wonder again about what landmarks there are in the system: something in the Latitudinal dimension is behaving differently from the Shortitudinal dimension?  The obvious effect of Latitude on an Earth-like planet is it being warmer near the equator and colder further away, but on the way I imagine a hypersphere being aligned that should also be true of the other dimensions.  Perhaps the sun extends into the Shortitudinal and Deltitudinal dimensions but not into the Latitudinal one?)

Strange Smell

When we see performance at various smell levels, 'Somewhat' does worse than 'No', but 'EXTREMELY' actually does fairly well:

This might just be randomness, because there aren't that many EXTREMELY smelly sites - or it might be confounded by something else.  Extreme smell doesn't show any obvious correlation with other parameters.  For now I'll probably assume that EXTREMELY smelly sites are also not good - it doesn't cost us many sites, and I don't feel confident gambling on their good performance here.

Feng Shui

Again we compressed three possible values into a linear scale.  This time that doesn't seem to have been a silly move:

Again this isn't super reliable, because the Exceptional bucket is small, but it certainly doesn't seem like the trend of better Feng Shui being good reverses.

Value of Pi

Our correlation matrix saw a high value of Pi as slightly correlated with good performance.  Looking closer:

The peak seems to be at the true value of Pi, which makes a lot of sense.  However, it looks like 'too-high' values of Pi have only a small effect on performance, while 'too-low' values are both further from the true value and more harmful.

Murphy's Constant

Our correlation matrix showed Murphy's Constant as being very bad.  Looking at performance:

we do indeed see a rapid (and smooth once we get to the areas that have lots of datapoints) falloff.  The falloff seems exponential in nature: if we have trouble finding enough sites, we're okay to accept Murphy-values up to 1 or perhaps even 2 without that having too large an impact on performance, but beyond there performance falls off increasingly rapidly.

Interactions

Another thing we'd miss with the correlation matrix is multivariate effects.  If the Squelchers will eat you unless rendered docile by the smell of mint, we will see that as  two independent effects 'Squelchers bad, Mint good' when actually the real effect is 'Squelchers without Mint bad'.

In the last part, I speculated:

In particular, we see what looks like a slight benefit from Skittering, but I suspect that this is a mirage.  Skittering is mutually exclusive with Buzzing, and Buzzing is quite bad.  It's plausible that Skitterers are bad but not as bad as Buzzers: that would suggest that we should avoid Skittering as well, and favor Silence.

We can test this by considering all possible combinations of Skittering and Buzzing:

 No SkitteringSkittering
No Buzzing26.0%25.0%
Buzzing19.5%N/A

We do in fact see what we suspected: Skittering is not in fact good in the absence of Buzzing (and might actually be slightly bad), it only appeared good because things with Skittering never had Buzzing.

We can look for whether there are interaction effects by doing something like this:

 No XX
No YAB
YCD

Looking at a chart like this:

We add in some boolean variables:

and consider those along with the various air-tastes and noises.  For each pair, we determine a 'Sum of Individual Effects' (i.e. (B-A) + (C-A)) and a 'Joint Effect' (i.e. (D-A)).

There are two interesting takeaways here:

Playing With Maps

The next thing we want to look at, which is sort of like interactions, is the effects of the coordinates (particularly of Shortitude/Deltitude, which haven't had any effect so far).  In a piece of cynical GM-psychoanalysis, I expect based on abstractapplic's past scenarios that most parameters will have exactly one effect: if this holds true, Latitude and Longitude (which we've found effects of) have no further secrets, but Shortitude and Deltitude (which we have not) have some effect we haven't found yet.

While my pitiful human eyes cannot see four dimensions at once, we can do 2D projections.  We'll start with a Latitude/Longitude map and see if this works out:

This looks as expected: we can see both the Death Zone at middling latitudes, and the trend for things to be nicer in the sunlight towards the middle-right of the graph.

(There are some extreme values, both good and bad, towards the top and bottom - these are the rarely-populated locations where we only have a few sites, and if those few sites do well/badly we can see extreme averages).

When we make the same map for Shortitude/Deltitude:

We see basically nothing.  Damn.

...there are a few places with near-zero or near-polar values where it looks like there might be some effects, but again this is because those values are rare.

...what if we try some other pair?  Longitude and Shortitude?

...Longitude and Deltitude?

...Latitude and Deltitude?

...no, just the Death Zone effect we already know about.  Latitude and Shortitude?

...damn.  Okay, this didn't go anywhere.

Intermediate Still-Inadequate Solution

Our current parameters for an ideal location are:

There are 2 such sites in the data, Site 2336 (performance 96.56%) and Site  9953 (performance 94.42%).  There are 20 such sites available for us now.  We're clearly closing in on a solution, but we also clearly aren't there yet.

Next time (hopefully), I'll try to get to the point of a solution that actually seems likely not to get me RIGHTEOUSLY BEHEADED FOR DISOBEYING ORDERS PASSED DOWN FROM THE GLORIOUS EMPRESS HERSELF (LONG MAY SHE REIGN).  Current plans are:

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