# Mastery

Originally called 'mastership' before the devs got their act together, this is a property of the hero for which the formula is as follows:

**Where:**

**E**= The sum of equipment durability values ('none' = -11)

**S**= The sum of skill levels

Since the pantheon of mastery (where heroes are ranked by mastery) is only updated hourly, a hero may change their equipment in between pantheon updates rendering a slight disparity between the expected and observed mastership values.

## Explanation

The sum of equipment durabilities shall be denoted **E**, sum of skill levels denoted **S** and hero level denoted **L**. **Y** is the mastery rating.
Spode collated much data. Using various online calculators for multiple linear regression,a few formulas were found. The first one found was:

Since the lowest mastery rating in the pantheon was 30, the +300 term was taken to be true with the guess that maybe everything was multiplied by ten in the end. The coefficient of S was 10 to 2 d.p. so this was also taken to be true. As more data was added, more formulas were found. Here are three of them:

The last two were calculated here. Although the others vary, the 10S and +300 seem to be pretty constant. CRogers then had a go himself and found this formula:

After some testing, it became obvious that CRogers' model was very close in its predictions. He explained that the 77 part came from being the minimum sum of equipment durabilities. All of these approximations however seemed to break down for lower and higher mastery ratings. Spode now thought that maybe the hero level was not related, as CRogers had shown. Although the formula worked pretty well, Spode did not like the seemingly arbitrary 4.2855 so carried on with his investigation in order to find the actual value. Before CRogers had shown his formula though, he had plotted E values against [Mastery rating - (10S + 300)] and found a positive correlation with a coefficient of determination equal to 1 indicating that he was indeed on the right track; the unknown part of the formula was definitely a function of E only.

The equation of the line was given as:

Since the final rating is rounded to an integer, this precision is not needed anyway because it is not a nice fraction. In fact it seems that only 2 d.p. are needed to get the correct rating in most cases, but 3 d.p. is given in our current final approximation formula at the top of the page. Nothing over 3 d.p. gives a non-continued fraction so the mystery now was where this number comes from. Whatever, it was, he now tried to find the actual fraction that was in the real formula.

Spode now calculated [(Rating - 10S)/(E+77)] and found the mode of the results to be 4.28571428571429. Sure enough, substituting this into the formula in place of 4.2855 resulted in quite an increase in accuracy; the formula had been improved. It is worth noting that 4.28571428571429 is pretty much equal to 30/7 so this was what was used in the final formula. This was the closest approximation to date. It still didn't account for all the data though so it was thought that L might come into it after all: it could be (30/7 – kL )(E+77) as the remaining residuals tended to be around either +4 or -4. Through more multilinear regression analysis, the coefficient -1.074697987x10^-2 was found for L. Adding the term +(-0.01074697987*L) onto the formula at the top of the page increased the residuals that were already small and decreased the residuals that were large, so didn't seem to help at all and the idea was abandoned.

During discussion, Spode now posted some of the residuals on the forum:

-4.285714286

-4.142857143

-0.428571429

0

-0.428571429

0.142857143

-8.571428571

-12.85714286

0.142857143

0

0.285714286

-4.285714286

0.142857143

0.428571429

0

4.714285714

0

0

0.285714286

-8.428571429

0.142857143

0.428571429

-4.571428571

-4.142857143

0.142857143

0.142857143

-0.285714286

0.142857143

-0.428571429

0.428571429

-4.714285714

-0.428571429

0.142857143

0

0

0

-0.428571429

0

It was then noticed by mathematics that at least some of them were multiples of seven and he therefore suggested that the residuals came about from changes to equipment durability values after the pantheon had been updated. CRogers had flippantly wondered this before but it had been assumed that the pantheon was updated in real-time so the 'lag' idea was not taken too seriously. However, multiplying all residues by seven resulted in integers and therefore provided final proof that lag was responsible. Nexus then found that the pantheon updates hourly and that was this problem solved. It could now be said that the mastery formula was known, but it still remains unclear as to where the 30/7 value comes from.