Mining loot analysis

[falkao]

This thread will give basic insights into mining loot and here is a summary of what we were able to identify so far.

I have first to say thanks to Steffel and Noodles for providing their data and to make it public. This is not a matter of course. I have seen many Avas hiding their data or not willing to share.


The set consists first of 7360 drops with MF105 + MA104, 1998 of them had a find. This corresponds to a find rate of 27.1%. If you have followed the loot analysis thread, you'll know that the inverse of the find rate gives the mean waiting time for the next drop. This leads to 3.68 drops until a find. This can be depicted as follows.

fig.1: Wait in drops until find.


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Using an exponential distribution I can estimated the waiting time which leads again to 3.68. So the set fits to perfectly random waiting times.


Adding also Noodles data we have a total of n = 3925 finds, 78.9% of them are enmatter and 21.1% ore finds. Since taxation is evident in lower loot classes, the amp effect is multiplicative and the difference between enamtter and ore is known, all finds have been standardized using the following equation:

Loot Std = Loot /(1-Tax)/Amp/c,
where c is a correction factor, c=1 for enmatter, c=2 ore

For the find rate there are several possibilities. The estimated 95% confidence interval ranges from 26.1% to 28.2% and hence values between 26% and 28% do look reasonable. However, we do have no data about a skill or equipment related find rate and hence we're not sure if this find rate can be considered as universally valid.


There is a small difference in median loot (about 10 PEC) between Steffel and Noodles data, but not different between ore and enmatter (p = 0.027 and p = .152 respectively). Since Steffel was mining only 4 different enmatter types, the difference might also be related to rounding and having a large dataset we're able to detect small differences that might not be relevant.

tab. 1: Finds according to miners. n = 3925, values in PED, p=.027 Mann-Whitney U-test.


There is however an exception between enmatters. Melchi Water seems to have a lower mean loot, whereas Alicenies Liquid has a higher one.

Fig. 2: Mean loot according to miner and resource type


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Loots distribution is depicted in the next fig 3.


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Please note the log scale on mining loot (log base 3 was used as explained later on). As one can clearly see, loot is split up in so called loot classes, i.e. loot is more frequent in those classes and between classes there are gaps.

As you may have noticed, there are sometimes observations between classes in the gaps. Those observations are most probably related to rounding/truncation as a consequence of res TT or taxation. Therefore I do consider them as noise.

To get a more or less noise free signal I used a kernel estimator to derive class limits. Here is an example

Fig. 4: Noise identification using a kernel density estimator, showing class C1 and C2

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Every datapoint below the visually identified noise level (.117 in the figure) gets eliminated. Here C1 in a higher resolution:


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Loot class limits are derived from the denoised signal and means per class can be calculated. The denoising has the further advantage, that we get a more general representation not so bound on the observed data.

The most interesting find about those class means is, that they are closely related to the class number as shown in the next fig.

Fig. 5: Log Loot class means per class number


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I used a log scale (ln) so that the correlation is better visible. The real correlation has an exponential form. Fitting such a model to the observed data results in the following equation

Mean class loot = .215*exp(1.099*class)

This formula can be reexpressed as
Mean class loot = .215*exp(1.099)^class = .215*3^class


There is nothing natural behind this formula and since a human being invented it, I’m rather sure that the real one might look like this

Loot model

Mean class loot = .22*3^class (PED) or .21*3^class (PED).

All this would lead to the following loot table assuming loot = .22*3^class PED:

Table 2: Loot classes, for ore use a multiplier of 2


All values are expressed in PED
obs.mean = observed mean
w.mean = weighted mean
cum w.mean = cum. weighted mean
rr = return rate, assuming a find rate of 27% and the below explained costs.

Class 0 and classes above 6 are only given for completeness and are not observed in the data.

My first assumption about class 1.66 was that it comes from rounding, but that didn't prove to be correct. Class 2.66 looks also artificial but the gap from about 1 PED can't be explained in another way, therefore I kept it in the model.

Overall return rate till class 6 should be close to 95% assuming a cost of .528 PED per drop including finder and driller decay. Till class 5 (globals) you should get 88% back, till C4 (minis) 81%. Below that (normal loot) only about 71%.

Limitations:

Results are not validated using an independent dataset and hence the model might overfit the data.

As depicted in the next figure, using bootstrapping, the observed mean loot is 1.94 PED ranging from 1.79 to 2.18 PED (95% confidence interval using bootstrapping with 100k samples). This would lead to a return rate from 90.5% to 110%. Hence the estimated 95% from the model lies within those ranges.

fig.6: Kernel density of mean loot using bootstrapping

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Some applications of the loot model.

Loot simulation

We can use the mining loot model to simulate loot. With this we'll get a feeling about the distribution of return or return rate. For simplicity we've used the return rate.

The following setup was used:
1) run length 1k drops, 10k runs
2) run length 10k drops, 10k runs
3) run length 100k drops, 10k runs

fig.7: Simulated return rate of mining runs


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The different runs can come from different avas or the same ava, counting every run separately.

What we can observe is, that all run types do lead to the same expected mean but do have a different variance. Runs with a short run length (less drops) do spread more. This is a consequence of having a right tailed loot distribution.

Furthermore, when runs do come from different avas, then there will be avas that do profit, about 40% with 1k drops and 20% with 10k drops. If all avas do 100k drops, then there are still differences between them but they are smaller and all close to expected mean return rate.

So what does all that imply? Is it possible to profit when doing short runs? Yes and no. Due to the loot distribution that MA is using, those that play only for limited time and hence doing a lower number of drops, do have a 40% chance to profit from loot only.
(Hence we will have 40% noobs that are telling how able miners they are.) If those go on, their return rate will become lower and converge to the expected mean return. The contrary is also true. If those that were unlucky continue mining, then they will higher their return rate.



Why are sizes missing when using amps?
Loot is based on loot classes with respective weights. Hence when looting, first a class is drawn, amp is applied and tax detracted. This loot value is then expressed as size. Since sizes have fixed limits, it might happen when amped that some sizes are missing.

Here an example of loot class limits with amp 4, values in PED and using a res with TT = .01 as reference:

Class	lower	upper
1	2.0	3.3
1.66	5.0	5.9
2	5.9	9.9
2.66	15.3	17.4
3	17.8	29.7
4	53.5	89.1
...

Size Tiny II goes from .32 to .99 PED (according to wiki). Since the first loot class starts at 2 PED size II can't be observed. Similarly, size III ranging from 1 to 1.99 PED can only be observed when taxed. However, the frequency of a size 3 will increase for higher TT res, due to rounding.

Here the same tab as before but with a .96 TT res.

Class	lower	upper
1	1.9	2.9
1.66	4.8	5.8
2	5.8	9.6
2.66	15.4	17.3
3	18.2	29.8
4	53.8	89.3
5	160.3	266.9
6	481.0	801.6
...

As one can see, limits are slightly different. This is a consequence of rounding.

If we go on we will discover that also size 12 (35-49.99) and size 17 (303-449) will be missing.

[Grrrrr]

The drop rate is fairly consistent to my analysis...

1,120 bombs dropped using OF-105 with OA-103 gave me a 29.2% claim rate.

588 bombs dropped using OF-105 with OA-102 is currently tracking at 27.2% claim rate (analysis 30% complete at this stage).

[steffel]

my pleasure

[Noodles]

Subscribing, with a quick contribution of what seems to be turning into an alternate (complementary) way of analyzing the single loot values.

The following is unamped enmatter data, showing the first three loot windows.

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X-axis is the loot value (in pecs)
Y-axis is the survivor function (chance of a larger loot)

The R^2 values for the three linear fits are 0.966, 0.984, and 0.973 (blue, red, yellow). I suspect that the blue R^2 is slightly low because sometimes rounding causes an artificially low claim (e.g. 38 pecs for two lytairian dust). I also tried an exponential fit for the red data, and the fit was worse.

The slopes of the fits can be controlled by MA in two ways: change the chance of getting a loot in that particular window, or change the width of the window.

The idea to explore here is that each loot window is a rectangular distribution. I think that this idea (which seems like it might be easier to program than some of the other functions we have discussed), is worth exploring further.

Incidentally, this phenomenon was also observed (but not explored further) when we were discussing kobolok's formidon data here. The graph is copied below.


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And I do seem to be seeing similar effects in my basic-filters-on-condition tests.

[Noodles]

Also note the similarity in the survivor functions (the % chance of each loot window) of Steffel's and Noodles' data. Factoring out the difference caused by amping on the size of the loots . . .

[falkao]

First of all, Noodles what you did is not completely wrong. It all depends from which side you look at the model. I’ll explain that later.
Here some further analysis:

The sample consisted of 4 different enmatters with different TT.
There frequency is not equally distributed and shows the following distribution
A = 11%, B = 18%, C=29% and D= 42% (p<.001).

Furthermore, they don’t show any differences in their cum. dist. Functions (Log Rank p = .177).

Fig. 1: Survival function by enmatters


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Moreover, there is no significant difference in mean loot and mean depth between enmatters (Kruskal-Wallis p = .146 and p=.4 respectively).

Fig. 2. Loot by enmatters


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Fig. 3. Depth by enmatters


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Sure one can discuss the fact that the p-values of .17 and .14 are quite low and maybe with a larger sample we might get some significant difference. Nevertheless, atm we can only conclude that if such a difference exists, then it might be quite low.

The only significant difference I find is a difference in size (p < .001). There is one enmatter that shows a lower mean find size. It’s the enmatter with the highest TT and there seems to be a limit somewhere around 1 PED, i.e. enmatters with a high TT tend to have lower size but in mean the same mean loot.

Fig. 4. Size by enmatter


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The most interesting about mining data is the knowledge of an MA given find size. So let’s have first a look within sizes.

Fig. 5. Within Size histograms


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The figures starts with size 3 and depicts then every size in ascending order. The higher a size the less frequent it is, but this we will analyze in more detail later on. As one can easily see, there is no clear picture within a size. The distribution can be similar to a uniform one, an exponential or a normal one. If I’ll take together size 3 to 8 and 9 to 11 I’ll get the following:

Fig. 6. Histograms of sizes 3-8 and 9-11

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This Noodles is what you were regressing. The first regression line given by you is an approximation of the distribution within the first classes and so on.

To understand what happens we need two further steps. So let’s have first a look to the distribution of the size classes itself.

Fig. 7. Survival function of Size


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The find size seems to follow a Weibull distribution. The first classes are rather frequent, then drop like in a normal distribution but with a heavy right tail.

Within each size class we have a distribution of loot itself. The next figure depicts the natural log of size (ln_size) versus the double ln of loot (lnln_loot)

Fig. 8. Size vs Loot

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As a result we do get a nearly linear relationship. There is more variability in low loot and less in higher ones. So the lower size classes do spread more.

If I undo the logarithms we do get the following form.

Loot = exp( exp(a) * size ^b), where a and b are constants, a typically less than 0.

This form is a combination of a power and exponential function, and similar to a Weibull distribution. The combination of 2 Weibull like distribution can be approximated by a GPD that I’ve depicted in my first post. We have now the basic ingredients to model loot itself.

There is one further interesting find. There seems to be no difference in loot between taxed and untaxed areas (p =.253).

Fig. 9: Loot vs tax


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That’s enough for the moment.

[Butch]

Lots of numbers...

Can somebody translate this into simple English? I mean, I'm interested, but the number and all of the math make my head swim.

[Immortal]

Originally Posted by Butch

Lots of numbers...

Can somebody translate this into simple English? I mean, I'm interested, but the number and all of the math make my head swim.

From the data:
Amps make hitrates slightly higher, and finds at the lowend bigger approximated by the TT cost of amp+bomb.
The loot looks the same as in hunting.

So nothing we didn't know before.

[falkao]

Originally Posted by Noodles

Also note the similarity in the survivor functions (the % chance of each loot window) of Steffel's and Noodles' data. Factoring out the difference caused by amping on the size of the loots . . .

can you provide some numbers on overall hit rate?

Furthermore I would like to know if you find similar things with your data. I have to further study my above mentioned findings. As it seems now we have a Weibull random variable to which an exp. function is applied. This however does only model the means within size classes and not their distribution. So I'm interested to see if you have something similar.

With hunting data I approximated base loot by an exponential distribution and the rest by GPD's. This works here similarly but we have now the advantage of given sizes and no influence of an additional factor like hp dmg done.

[steffel]

Originally Posted by falkao

There seems to be no difference in loot between taxed and untaxed areas (p =.253).

that's bugging me.

first, i did not find a single IX or XII. it's often been observed that specific combos cannot find specific sizes and is probably indicating loot classes/multipliers.

but, one should think a low (when found untaxed) X should transform into a IX when found taxed (4.3% in my sample). but it did not happen so how is tax generated if not taken from the claims...two ideas: 1) mining aint taxed (although i remember one landowner once was ableto confirm a tower being on his land due toimmediate tax receipt) or 2) tax does not apply to the find but is taken from your expenditures, ala "expenses - tax => lootpool"...

or could it be that 4.3% tax is causing non-significant variance?