D&D.Sci(-Fi) June 2021: The Duel with Earwax

Afaict each resonance pilot strength result is a multiple of the result of a pilot-independent, resonance-specific rule times a pilot-specific power level with that resonance. (though, tbh, I haven't checked this that closely). 

With Maria out, the highest pilot power levels per resonance appear to be:

Alpha: Corazon. This resonance has apparently random variation about a constant value that jumped slightly somewhere around Floorday 500. It is too weak to save us.

Beta:  Janelle.  This resonance has apparently random variation about a constant value. It is too weak to be likely to save us.

Gamma: Janelle. Credit to GuySrinivasan for finding the specific formula of (1+k*amplitude) (times pilot gamma power level). The integer k is from 0 to 5 with 1 being slightly more common than 0 (could be random variation) but dropping off beyond that. Though that isn't the most expected random distribution and may hint at something non-random, I haven't found the pattern if there is one. Janelle needs k to be 1 or higher to save us or 2 or higher to overwhelm Earwax. Without knowing a pattern for the k value, this seems too risky.

Delta: Amir. This resonance has apparently random variation along with a moderate upward slope with heteropneum amplitude. It is too weak to save us.

Epsilon: Will. This resonance follows a cubic formula (credit to GuySrinivasan for reporting the cubic dependence first, though I hadn't read his comment when I reported it). Though GuySrinivasan expresses low confidence in Epsilon, it seems to me that, assuming the assumptions of the cubic formula plus the multiplicative relationship between power values for different pilots is correct, there is no way the coefficients could possibly off by enough for Will not to beat Earwax. And these assumptions seem to me more solid than for Zeta below, so I see this as the safe choice (but not my current choice, because Epsilon will not overwhelm).

Eta: Will. This resonance has a non-random constant value with several jumps over time. One of the jumps appears to coincide with Alpha's jump. Without any reason to expect a further jump since the last observed data, it is not strong enough to save us.

Zeta: Corazon. Zeta is either zero, or one of two non-zero values. Afaict whether it is zero is random, except that no zero values have been observed for heteropneum amplitudes above 2.27, so I weakly infer that there will not be a non-zero value against Earwax. It seems that which non-zero value occurs depends on which of two or more populations the heteropneum belongs to. The large majority of heteropneums belong to a population with amplitudes that are (before rounding) multiples of 0.142 or something very close to 0.142. These always get a low Zeta value if they get a non-zero result. The minority that are not in this population always get a high Zeta result if they get a non-zero result. Earwax's rounded 3.2 value cannot be obtained by rounding a multiple of 0.142, so we can expect a high Zeta result and for Earwax to be overwhelmed. Thus, I pick this choice, despite my uncertainty as to whether I have enough evidence against a zero result.

A potential wild card is that we don't know Flint's power levels except for alpha, since he never overwhelmed any heteropneums. If there is a way to predict power levels without seeing a strength result with that resonance, this could reveal further opportunities with Flint.

It's cubic not sine; I can fit Maria's Epsilon data so that the curve rounds to the exactly correct value for every data point, and also for Janelle's data (separately) to round to the exactly correct value; I still need to check if I can make a single curve and multiplier between Maria and Janelle to round exactly for both, but it does look like the curves are at least fairly close to exact multiples of each other. 

Interestingly, no x-value rounding needs to be assumed, at least to get the correctly rounding values for Maria and Janelle separately. So, perhaps the x (heteropneum amplitude) values are exact? No, see below

The cubic curve does take a big dive at high heteropneum amplitudes, but fortunately not until after Earwax's ~3.2 amplitude. Also, the fit for Maria's 0.57 amplitude result of 0.1 is actually around 0.096.  Will getting 0.21 suggests he is at least around 2.13 times stronger than Maria using Epsilon and is projected to get at least about 3.85 against a 3.2 amplitude heteropneum. So, Will using Epsilon still looks like a safe pick to survive if we can't find guaranteed survival another way.

edited to add: note that the speculation that the x-axis values might be exact should  only apply to the overwhelmed heteropneums - there are the these are the only ones we have epsilon data on and also the only ones we have data to 2 decimal places on. Irrelevant, see below

All overwhelmed heteropneums that are duplicates in power of another overwhelmed heteropneum is a multiple of an integer from 2 to 22 times 0.142 (the integers probably go higher than this, but Maria stops overwhelming them at that point). This value of 0.142 might not be the exact value, but it makes the numbers round correctly, whereas the rounded values are not exactly the right ratios, so presumably the amplitude values are rounded.

Eta is simply a multiplication of a character-dependent Eta power level and a date-dependent Eta strength. The date-dependent Eta strength is constant except occasionally it jumps. The lowest strength was from floordays 2-253, then second lowest from 280-297, then next level from 316-395495, then it jumped to the highest level from 516-746, and then dropped to the second highest level from 749-804. (no relevant data in the time gaps). It has never jumped back to a level after going to a different level.

Will's sole eta value of 0.9 occurred on floorday 110 when Eta was at its lowest strength. This means Will is almost as strong as Maria at Eta (everyone else for whom we have Eta data is lower). Unfortunately, this is still not strong enough to beat Earwax if the Eta strength remains, now at floorday 814, at the same level it's been from 749-804. 

edited to add: Alpha also shows a jump between floorday 495 and floorday 516. (This is the reason for the bimodal appearance of its distribution). Since this jump occurred in both Alpha and Eta, but the others only occurred in Eta, this suggests that it might have a different cause than the other jumps. 

Though duplicate amplitude values are common, all verifiably high-tier Zeta values so far have been against heteropneums with unique amplitudes. Admittedly, this is only 5 datapoints.

The good news: Corazon got her Zeta results against duplicate-valued heteropneums, so if the pattern holds true for her, her results have been low-tier so far and she is strong enough to overwhelm Earwax if she gets a high-tier result.

The bad news: Earwax has a duplicate amplitude value (as long as the formatting including rounding if applicable is consistent between Earwax and the other entries) so if the pattern holds true for Earwax, there will be no high-tier Zeta result against Earwax. Wrong, see below

Edited to add: Earwax and the "duplicate" (Divisor, floorday 389) have not been overwhelmed and are likely rounded to 1 decimal place, but all of our zeta data is from overwhelmed heteropneums, reducing the likely relevance of the "duplicate".  More detailed info below.

Further addition: I failed to mention earlier that all the non-duplicated entries have either high-tier or zero Zeta results (whereas all the duplicated entries have zero or low-tier Zeta). So, this is very likely significant.

On reviewing the relationships between the duplicated entries for which we have overwhelm results, all are equal to 0.142 multiplied by an integer from 2 to 22 (when that is rounded to 2 decimal places). The 0.142 might not be the exact value but it makes them round correctly. The 22 is probably not the highest but is simply where Maria last overwhelms heteropneums (3.12 amplitude).

Importantly, Earwax's value of 3.2 cannot be rounded from a multiple of 0.142 (3.12 is too low, and the next value would be 3.266, which would round up to 3.3). If I try to lower the base value to 0.1419, this already prevents correct rounding of the known values (it would predict the 18x number would round to 2.55 but the 18x number needs to round to 2.56), and this too-low base value still predicts 3.2637 for the next multiple). Thus, Earwax is not from this population of heteropneums, which accounts for all the low tier Zeta results! 

However, we still need to find a way to predict if we will get a zero result. Of the 9 non-duplicated overwhelmed heteropneums, there was a zero pilot strength Zeta result in 4 of these cases.

Still further addition: For amplitudes above 2.27, we have no cases of zero zeta. Among the overwhelmed heteropneums (which is all we have Zeta data for) we have 27 total cases of zero Zeta among 150 data points, and there are 41 cases with more than 2.27 amplitude. So, if all are statistically independent, then the probability of this happening by chance is (123/150)x(122/149)x...x(83/110)=0.00006457.

There are a variety of reasons not to be too impressed by this probability number.

1. We don't have a very good reason to believe that the results are statistically independent. Duplicates in amplitude values do vary in whether they get zero Zeta (if amplitude less than or equal to 2.27), so they might be statistically independent, though.
2. I came up with the hypothesis (that Zeta is never zero above some amplitude) after seeing the data, not before, and need to adjust for the prior with possible hindsight bias.
3. Even if there is a non-random pattern causing the results, it doesn't necessarily imply that it will hold for Earwax.

That being said, my expectation is that abstractapplic did leave us a way to overwhelm Earwax, so I'm confident enough (barely) to switch my proposed response to Corazon with Zeta. In real life, I'd stick with Will and Epsilon, which I am far more confident in.

What do you think has the best chance of getting in a strike that's double the amplitude of Earwax? (And otherwise ignoring chances of survival - I'm suggesting this partially because if Earwax has suddenly gotten stronger once, if it does so again, then a smaller chance of a larger magnitude seems like a better play.)

Summary: Send Janelle, using Gamma Resonance. Great chance of winning, maybe 60-70%, and half the time you win you double also. Honorable mention to Will's Epsilon Resonance, which if we had more data or a better theory we might be convinced could win 100% of the time, but we just don't have the data or theory to justify it yet.

Alpha: Maria does not show any real amplitude-dependent EFS, and no potential pilot shows 3.2+ EFS. Reject.

Beta: No amplitude-dependence. Maria and Janelle look pretty similar, but all the trainees do far less well, so this is person-dependent. As such probably Janelle's chances of winning are probably best estimated using only her data? Either way, I get somewhere between a 2%-3.5% chance of winning for Janelle and nothing for trainees. We can do better.

Gamma: Very clear dependence on amplitude A, which is good. Each person has some base value B; Maria's is 0.66, Janelle's is 0.89. The EFS generated is of the form B x (1 + k x A), for observed values of k from 0 to 4, and maybe one day we'll see higher. I don't see a way to predict k, but this is quite promising for Janelle. k=0 loses, k=1 wins, and k=2, 3, or 4 wins and doubles. That's a 64% chance of winning overall, with 39% (60% of the time given that we win) of doubling. Going off of observed frequency of Janelle hitting various k; the relative ratios are different enough from Maria's that it's not clear we can combine them in any nice way. They both show large k=0,1,2 and small k=3 and tiny k=4. (None of the trainees has a higher B.)

Delta: Maria's EFS shows a linear upward trend dependent on amplitude. Janelle's is too low to be interesting, and the trainees' are all very low and none suggest having a super-positive slope. Reject.

Epsilon: Ooookay this one's interesting. I didn't get sin to fit as well as a cubic, and I did get a cubic form that can be described with just one parameter varying per Maria vs Janelle, which I thought was likely given how Gamma worked, which means we can generate the entire cubic for Will and check how it does at A=3.2 and... it predicts 3.31 EFS. The Epsilon Resonance is entirely predictable given an amplitude. However, there are two big problems with simply sending Will to use Epsilon. First, even if our model is precisely right, the precision of our instruments is not perfect, and once we take that into account, Will's Epsilon EFS predictions vary a fair amount, leading to only about a 60% chance of winning. Second, our model is almost certainly wrong, because we haven't found a simple model. So Janelle's Gamma is better than Will's Epsilon in every way according to our current uncertainty.

Zeta: You might get 0, or your base Z, or rarely 3.5 x Z. The problem is that Janelle's 3.5 x Z only just barely beats 3.2 (and doesn't beat 3.25!), and she gets 3.5 x Z maybe like 5% of the time which is << 64%. While Corazon's 3.5 x Z would handily double, it loses otherwise. Again << 64%.

Eta: Janelle's too small here, but Flint has a 2.3, and it looks like the possible EFS's are a base E, or x1.5, or x1.5x1.5, or x1.5x1.5x1.125, or 1.5x1.5x1.125x1.25. If Flint's E=2.3, this might be promising. Unfortunately it seems much more likely that 2.3 is one of the multiples, and if there is an amplitude dependence, it's probably one of the high multiples.

Using Python in a Jupyter notebook. Seaborn has a fantastic little function to quickly see pairwise graphs, it's great to begin with. Here's what Maria looks like after sns.pairplot(df[df.name=='Maria N.']): https://drive.google.com/file/d/12Q_11ZTPnyak87EXO89Vbg4am3TXc4px/view?usp=sharing

check whether resonances are affected by days of the week, you're going to have to create days of the week

as well as come up with your own mapping between days of the week and 7 resonances. Is this number a coincidence?

First of all, I'm sorting by Pilot, via right clicking the column. This seems to scramble the floornight column, so everything seems to be working right, to figure out who has what resonances down. (Sorting a resonance column to bring the days to the top that maximize it is isn't super simple. Sorting by maximum values, however, is.)

Glancing at Will on non-double events, makes it seem as though it isn't likely the best resonance. Except on double events it looks...best one time, second best the other. Alpha came out higher the other time though. The time it didn't it was narrowly third best. There's an argument to be made for having Will use Alpha,

Nevermind, I was reading an entry from the next pilot. Will, beta resonance seems the way to go (not a lot of data). Best (out of six): 1.7. Best out of overloads: 1.6.

Maria. Does a lot more missions. Seems strong with delta, harder to confirm across all data points considering there's so many. Best out of first 7 is 6.01. Highest probably around 7.

Janelle is tricky. Beta? Gamma? Those 2 seem higher than the others, either way. Highest seems to be 4.66, with gamma.

The rest are short, like Will.

Corazon, alpha. Highest: 2.3, and is an order of magnitude worse with everything else.

Amir, as likely alpha as beta. Overall slightly better with alpha, though the difference is like 0.0x, where x is a number. Highest 2.something.

assuming you can use multiple people at once, but each can only have one resonance

Alpha, corazon

Beta, Amir

Gamma, Janelle

Maria, delta

If you can only send one, in order:

Maria

Janelle

Corazon
Amir

I could write a function to tell if, say, delta is max, and then paste it in, and count how many that's true for versus not...or I could move on. What if floordays are the secret? Chances of survival due seem higher than usual - ish - given the circumstances.

Maybe today is a supercharge day, so everyone is stronger, heterpneums and pilots alike! Maybe.

Are some resonances better on some days? Maybe. A formula to check for the best resonance (for days where all resonances are available can work like this: (F2>E2)+(F2>G2)+(F2>H2)+(F2>I2)+(F2>J2) + (F2>K2)

Name the column appropriately, [resonance name] is max, and if it is it will say 6 if it is, and the heteropneum gets overloaded in that row. Repeat as necessary, then move the floor day row over to see if there's a (simple) matchup. If it's a function also involving the pilot, then things are going to get complicated.

Beta resonance is best if Janelle, or (unclear subset of even numbered days which just happens to coincide...). Also happens with Will, once...

And it's also happened on odd numbered days. Doesn't seem to be about days.

Floor day doesn't seem to matter (for beta resonance). Though perhaps a plot of when one resonance is best will yield a different equation, more complicated than day of week?

There's also the possibility of trying to:

- figure out conditional probability of overload (independent of heteropneum strength)
- try plotting the distributions
- try calculating the distributions for the pilots with only a few datapoints, and figure out who to send out on the small chance they will succeed today like they've never succeeded before.

--figure out relationships between resonances, and heteropneum strength, in case that helps

--relationships between days and pilots and resonances

None of these people are likely to beat this thing. Here's a shot in the dark: you go. We have no idea what your resonance is, or how powerful it will be. This suggests 'figure out the most likely/most powerful resonance.' There's also 'don't evacuate everyone just throw who you can in the sub, and hope they can warn ??? about anomalous heteropneums. Maybe a team can be dispatched, that will be able to take it out (and then some) and build your new theory.

Or maybe the submarine has some sort of charge, based off of its last use, that you can make use of. What to look for: effects based on last use.

The other possible analysis change, would be: don't just focus on overload days to try to figure out the distribution, for the best combination.