diff --git a/docs/technical/bkpm.md b/docs/technical/bkpm.md index bd018283b..479f9628e 100644 --- a/docs/technical/bkpm.md +++ b/docs/technical/bkpm.md @@ -37,12 +37,24 @@ $$ \sigma=\sqrt{\frac{\sum_{ }^{ }\left(x_{i}-\mu\right)^{2}}{n}} $$ -An individual atomic statistic is provided by the equation below where $x$ is the player's statistic being evaluated, $\mu$ is the mean of the statistic across all players, and $\sigma$ is the standard deviation of the statistic across all players. During computation, one may pre-compute and reuse $\gamma=x-\mu$. +A normalized atomic statistic is provided by the equation below where $x$ is the player's statistic being evaluated, $\mu$ is the mean of the statistic across all players, and $\sigma$ is the standard deviation of the statistic across all players. During computation, one may pre-compute and reuse $\gamma=x-\mu$. $$ A=\frac{x-\mu}{\left|x-\mu\right|}\left(1-e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}\right)=\frac{\gamma}{\left|\gamma\right|}\left(1-e^{-\frac{1}{2}\left(\frac{\gamma}{\sigma}\right)^{2}}\right) $$ +$m$ is the slope of the least squares regression line. Below is a refresher of the formula where $x_i$ is the player's statistic being evaluated, $y_i$ is the player's win rate, and $n$ is the number of players (or the sample size). + +$$ +m=\frac{n\sum_{ }^{ }x_{i}y_{i}-\sum_{ }^{ }x_{i}\sum_{ }^{ }y_{i}}{n\sum_{ }^{ }x_{i}^{2}-\left(\sum_{ }^{ }x_{i}\right)^{2}} +$$ + +The weight of an atomic statistic is $w$ where $r$ is the correlation coefficient between the statistic and win rate, $\mu$ is the mean of the statistic across all players, and $m$ is the slope of the least squares regression line. + +$$ +w=\mu mr +$$ + The raw metric is the weighted average of all atomic statistics. $$ @@ -124,7 +136,7 @@ $$ \frac{\gamma}{\left|\gamma\right|}\left[1-e^{-\frac{1}{2}\left(\frac{\gamma}{\sigma}\right)^{2}}\right] $$ -Here is an example of HisRoyalFatness scoring a very high $0.988$ with the frags metric with his [XM551 Sheridan](https://blitzkit.app/tools/tankopedia/20257). $\mu=\frac{1081.164}{1405.2273}$, $\sigma=0.2492$, and $x=\frac{7310}{4837}$, HisRoyalFatness' average frags. At $x = 0$ (if the player manages to get no frags), the metric drops nearly to $-1$ and vice versa. And, at $x=\mu$, the metric is exactly $0$. +Here is an example of HisRoyalFatness scoring a very high $0.988$ with the frags metric with his [XM551 Sheridan](https://blitzkit.app/tools/tankopedia/20257). $\mu=0.7694$, $\sigma=0.2492$, and $x=1.5113$, HisRoyalFatness' average frags. At $x = 0$ (if the player manages to get no frags), the metric drops nearly to $-1$ and vice versa. And, at $x=\mu$, the metric is exactly $0$. ![](https://i.imgur.com/aIUQmGF.png) @@ -161,9 +173,3 @@ $$ $$ ![](https://i.imgur.com/O8stVlJ.png) - -For cases where increased $x$ negativey impacts gameplay, like in the case of `composite.damage_received`, the sign of the coefficient can simply be flipped. In fact, the sign of the function can be determined by the sign of the corelation coefficient. - -$$ -\frac{r\gamma}{\left|r\gamma\right|}\left[1-e^{-\frac{1}{2}\left(\frac{\gamma}{\sigma}\right)^{2}}\right] -$$