D&D.Sci August 2021: The Oracle and the Monk

• Energy types are not created equal.  Ash, say, is almost invariably single-digit, and is basically useless.
• Several types of energy correlate: Solar and Lunar correlate very positively (around a ~30-day cycle that makes me suspect solar energy is a misclassified type of lunar energy).  Earth and Ocean correlate extraordinarily negatively.  Flame and Ash correlate very positively (though this is meaningless, since Ash is not useful to us).
• Solar+Lunar considered together spends just over half its time above 70.  We're currently above 70, but we are trending down, and probably won't be in 10 days.  Additionally, there was a recent anomaly in Solar energy (the supernova?) which makes me leery of using it.
• Earth+Ocean considered together varies very stably and predictably (on a 22-day cycle).  This cycle spends just under half of its time over 70...we are currently in one of its low points, but 10 days from now we should be in a high point.  If you plot (Earth+Ocean Energy) against (Day Number mod 22), 10 days from now (Day Number mod 22) will be 10, and there has never been such a day when Earth+Ocean was <70.
• Therefore, I plan to recommend the monk use Earth and Ocean energy.

Ignoring any time dependence, solar+lunar and solar+earth are the most successfuly combinations; they would have succeeded on 246 and 235 respectively of the existing 374 datapoints.

Solar seems to have a 27 day cycle with 3 peaks within it (so 9 day cycle?) but which peaks are stronger has been changing over time. The current cycle, cycle 14, has been weird with days 15-20 of the cycle (days 366-371) being higher than expected. The last 3 days (372-374) are low, but not far from expected. Other slightly weird cycles include cycle 8 (slightly higher than expected values from days 12-14 of the cycle, i.e. days 201-203) and cycle 10 (slightly higher than expected values from days 20 to 25 of the cycle, i.e. days 263-267). I'm counting "day 1" of a cycle to be the one that's a multiple of 27 from the day 1 of the overall data.

If solar is back to its normal pattern, on day 384 it should be on the way down from a high peak and approaching a high trough, so still doing pretty well (>40 expected), making solar a good candidate for one of the mana choices.

Lunar, like solar, shows a 27 day cycle (shouldn't it be 28 days?) with 3 peaks changing which is stronger over time; outliers include days 26 of cycle 8 to day 1 of cycle 9 (days 215-217) which are unexpectedly weak. Like solar, lunar should be declining to a high trough on day 384, I expect >35.  Assuming solar is back to normal, solar + lunar should succeed.

Ocean varies greatly from single digits to over 60. While some possible patterns appear (e.g. some short range autocorrelation, and a degree of autocorrelation at a 4-day displacement almost as high as at 1-day displacement) it does not seem to have a fixed period of variation. The possible high values make this a potentially interesting choice if it can be predicted, but more analysis needed to determine what ocean will be at at day 384.

Breeeze looks fairly random, distribution peaks at 13 and varies from 6 to 20.

Like Ocean, has short-range autocorrelation but no single period. Varies from 11 to 41.

Like Ocean and Flame, has short range autocorrelation but does not seem to have a single period. Varies from 2 to 10, so of scientific interest only.

Looks random except for some possible 1-displacement autocorrelation. Varies from 9 to 74, so definitely of interest if a pattern can be found.

Looks fairly random, varies between 17 and 31. This is the same size of range as for Breeze, but has a fatter and asymmetrical distribution.

8 days sawtooth pattern with some possible random variation (peak on day 2, bottom on day 3, then steady rise). Notable outliers from expected pattern: Day 9 of cycle 1 (i.e., day 9), day 5 of cycle 22 (i.e. day 181), day 3 of cycle 11 (i.e. day 91), and day 1 of cycle 31 (i.e. day 249).

On day 384, it will be day 8 of the cycle, and something in the range of 27-33 is likely. Not as good as solar and lunar, even discounting the risk of a miscalculation with one of the dangerous mana types.

28-day periodicity with lots of peaks and troughs within the period; the troughs are often (but not always) 0. A reliable spike on day 5 gives the period away. Seems to also prefer specific values instead of a smooth distribution. Day 384 will be day 20 of the period, and a low value can be expected (7 or 0).

So far Solar+Lunar seems the best choice.

This is also the choice that looked best before getting time dependence information for Solar, Lunar, Doom and Spite, so further research on the time dependence of other mana types (especially Ocean or Earth which can have high values) might find an alternative, better answer.

Aphyr found that Earth and Ocean are anticorrelated and their sum has a smooth 22-day pattern. As Aphyr reports, day 384 should be close to the peak. Expected value 75-80 or so. This looks like a good, safe solution.

I actually expect Solar+Lunar to be slightly higher, since the peaks and troughs have been shifting in height/depth over time, and while they will be near a trough at day 384, the trough is one that has been shifting upward. I expect ~43- 45 from Solar and ~36-40 from Lunar on day 384. However, this is less certain than the Earth+Ocean expectation and as Aphyr notes Solar has been weird lately. Solar+Lunar is definitely the riskier pick and probably is objectively not what one should pick based on the available info, so I'd switch to Aphyr's solution in real life but I'll stick with Solar+Lunar as what I want to get credit for (for now) since I have an excuse that it might be better and it's what my analysis was on.

Whoops so much for Lunar.

The following predicts solar with +-1 accuracy:

28 day cycle:  32,32,32,27,27,27,27,27,27,32,32,32,34,34,35,36,37,40,41,42,42,41,40,37,36,35,34,34

9 day cycle: 9,10,10,10,9,3,0,0,3

anomalies: +8 days 61 and 62, +12 dats 201-204, +9 days 263-268, +24 days 366-371

predicted result for day 384 is 45+-1, unless there's an anomaly.

Doom's 8 day cycle is 30,32,18,20,22,24,26,28 plus 0-5 with anomalies on days 34, 91, 181, 249. Unlike solar there are both positive and negative anomalies. Expected result (no anomaly) is 28-33 on day 384.

Solar+doom should give (with no anomaly) 72-81, does not look as good as Earth+Ocean's 74-80 (from GuySrinivasan) though obviously Doom is better than Lunar's known value of 16 at day 384.

With respect to Earth and Ocean, both include many values larger than the minimum of the sum of the two, and Ocean has a sharp minimum value at 4, while Earth's minimum value is not so sharp with 9 being the smallest but more of the bottom edge of the Earth-Ocean x-y plot being at 11 or so. This probably says something about how they are made but I have not thought of it. So far, nothing better than Earth+Ocean found.

Obviously, the best candidates for beating earth+ocean are solar+ocean or solar+earth (whichever we can find out will be bigger).

Spite correlates a bit with Ocean and anticorrelates a bit with Earth. Not a super large effect but relating ocean/earth (which we need to predict) to Spite (which we know deterministically) is very interesting)

solar seems to be (period 28) + (period 9) + (shortish upward transients) + (very small noise), and the periodic parts are predicted to be at 45 on day 384.

lunar appears to be 75-solar, shifted by 14 days. On day 384 we expect it to be experiencing the reverse of solar's most recent upward transient; it will not be a good candidate for Morgan's purposes.

ocean, breeze, flame, ash and earth all look rather unpredictable, except that ash is basically a scaled copy of the previous day's flame; none of them will be predictably at 25ish or more on day 384. I would be extremely unsurprised to find that some of these are more predictable than I think (e.g., breeze has a suspicious number of peaks at the same height and 6 or 7 days apart; maybe this is real, and maybe it's caused by something I could exploit with more work or ingenuity) but for now I'm leaving them alone. (Flame/ash also seems like it's not just uncorrelated random noise, but it might e.g. be low-pass-filtered noise.) Earth seems to have a pretty good chance of being big enough on any given day, but something more predictable would be better.

Spite is perfectly predictable, being a sum of periodic spikes with periods 4, 5, 7, and 14; on dat 384 it will be zero. Void seems rather unpredictable, though its overall level isn't far off what we need -- but if we have to rely on luck we have better chances with earth than with void.

doom has a strong periodic component of period 8 (and, alas, occasional single-day transients); its periodic component will be 30 on day 384. Unless we get unlucky with a sizeable negative transient, it seems like solar+doom is a likely winner.

It doesn't look like it. Solar is almost always bigger than doom; the largest doom-solar has ever been is 13, when our periodic estimates would have predicted an excess of 9; we are predicting solar=45 and doom=30 so we have plenty of slack to spare.

an abrupt change in the data that might correspond to the supernova.

Strong periodic components (either sinusoidal with any period, as found e.g. by FFT or periodogram calculations, or with integer period and arbitrary shape). Close correspondences between different kinds of mana, but I haven't looked for fancy things as opposed to the simple solar/lunar and flame/ash relationships. Short-term autocorrelation within a single type of mana. Long-term trends (no sign of any of these).

Matches between either the raw data or the locations of peaks or troughs or anomalies, and things in the OEIS. (Despite our character's name.) That sort of thing would be amusing but feels like it would be a cheat even if it were there. Also, most OEIS sequences don't look much like random-ish noise.

to go for solar + doom

almost certainly yield a total a little over 70, and almost certainly not cause a catastrophe; I think unexpectedly good outcomes are a bit more likely than unexpectedly bad ones.

that aphyer's suggestion looks strictly better than mine; annoyingly, I had noticed that there was some weak evidence of period-22 components in both earth and ocean but never bothered looking at their sum; in hindsight, eyeballing the plots of [EDITED to add for clarity: the sums of] all pairs of mana types with any chance of reaching 70 should have been a super-obvious thing to do, but I am a moron and didn't do that.

Are any of the other monks doing anything like this?

How much good will it do, channeling good, if another monk is 'possessed by darkness and transformed into an immortal demon, wreaking horror and bloodshed upon the world' and interrupts him amidst channeling? (Probably by killing him.)

Is the temple close? Even if it is far away, might any of the channeling help banish or counter such demons?

(What happens if someone channels two of Doom/Void/Spite? If bad, might there be an evil order of monks that might do such a thing?)

First condition can be met by...not channeling Void, Doom, or Spite at all.

Second, over the past 10 days, Solar and Lunar look like together they meet the second condition.

I haven't balanced the third condition - I don't have a model yet.

Eyeballing it, Earth got high once. Higher than 70, on day 253. (Solar also happened to be somewhat high on that day as well. Not sure how likely, or if it's worth it - do we have any idea what this utility function looks like?

Solar + Lunar seems 70-80, versus this moonshot at 116.

aphyer seems to have it figured out.

Solar and Lunar Mana, which I think has about 66% chance of working. The  only mana type that can be predicted 10 days in advance (beyond the prediction of a random sample from the previous data) is Doom. And that still doesn't work out with as high probability. (Doom mana will be on a high at the time, but using it with solar gives a 69% chance of reaching 70 and an 11% chance of demons. So a 58% chance of doing good.) If the utility is 0 for any amount of mana <70, and the amount of mana if its >=70, then solar + earth does slightly better in expected utility. 54.6 vs the 54.2 for solar+lunar. It has slightly lower success probability 63%, but slightly more mana if it does succeed. 

Solar - Appears to operate on approximately a 27 day cycle which suggests it's value should be around 50 on day 384. However this has risen to anomalously high values recently (The supernova being a star may be the cause), so there is a risk it may not behave as predicted in the immediate future.
 
 Lunar - Seems to go through a pattern of relative highs and lows that repeat every 27-28 days, probably not coincentally the same as the orbital period of the moon. On day 356-357 lunar was at a relatively high value of 46 - 39, so this looks like a good prospect. 
 
 Ocean - Can be very strong, but also very weak at times with no obvious pattern.
 
 Breeze -  Has never been more than 20 or less than 6. This is unlikely to be good enough.
 
 Flame - Can be anywhere between 11 and 41 with no obvious pattern
 
 Ash - Is consistently <= 10 , so should probably be avoided
 
 Earth - Another one that can easily be very high or low with no obvious pattern.
 
 Void - Highly erratic osscillations between 17 and 30,
 
 Doom - The qualitative pattern seems to repeat itself once every 8 days, suggesting it's value on day 384
        will be about 30.
 
 Spite - Is 0 about half the time. It takes on a high value regularly once every 28 days though. The last time
            this happened was on day 369 which is no good for us. On other days it is either 0 or a relatively low
            value, so this is one to avoid.
        
  Looking at the ones that seem reasonably predictable Solar + Lunar should yield over 90 so this should be a good prospect, providing the supernova doesn't unexpectedly disrupt things.
  
  Lunar + Doom is another possibility, however unless the supernove disrupts things it has a lower expected value.
  

 On balance I think Solar + Lunar looks like the best bet.

• Solar is on a 28-day cycle, on each day there are a few possible options based on the day number mod 28
• Sometimes solar is randomly higher but this is rare
• Solar has recently gone up in power by (at least?) 15
• On day 384, solar will be very strong, I think minimum power would normally be 42, so I suspect minimum will now be 57
• Lunar is directly related to Solar, a 100% negative correlation with the solar of 14 days previously, I think the linear transformation is 75 - Solar
• On day 384, Lunar will be 16 (unless Lunar isn't affected by the increase in solar, in which case it will be more, which would be sufficient, and better than doom)
• Earth and Ocean are strongly negatively correlated
• Flame is correlated with itself on a day-to-day basis but seems to be a random walk, difficult to predict into the future
• Ash is entirely determined by the previous day's flame mana
• Doom is on an 8-day cycle, with high variance
• Minimum Doom on day 384 will be 19, maximum will be 31
• Spite is entirely deterministic on a 140 cycle
• On day 384 spite will be 0

Therefore Doom + Solar is my strategy

Spite is 100% deterministic. Add these four components to get its value on any day. No supernova effects. Will be 0 on T=384.
spite1 = cycle([0,7,0,0,0])
spite2 = cycle([0,0,6,0,0,0,0,0,0,0,0,0,0,0])
spite3 = cycle([13,0,0,0])
spite4 = cycle([0,0,0,0,18,0,0])

Solar and Lunar are highly related. Solar is always more than Lunar. Solar - Luna has a strong 9-day cycle; subtracting out its mean leaves a fairly strong pattern of outliers where Solar is even stronger, predicting that Solar will be "even stronger" on T=384. Needs more scrutiny.