diff --git a/docs/technical/kitscore.md b/docs/technical/kitscore.md index ab7730e01..ec49b2c17 100644 --- a/docs/technical/kitscore.md +++ b/docs/technical/kitscore.md @@ -49,6 +49,48 @@ $$ A=\frac{\sum w_{i}a_{i}}{\sum w_{i}} $$ +The "atomic" score $a$ is the difference between the predicted target statistic $P_{i \leftrightarrow j}(x_i)$ acquired using a polynomial regression of degree $m$ and the actual newly observed $x_j$. + +$$ +a_i=x_j-P_{i \leftrightarrow j}(x_i) +$$ + +The polynomial regression of $m$th degree can be written as follows: + +$$ +P_i(x)=\beta_0x^0+\beta_1x^1+\beta_2x^2+\dots+\beta_mx^m +$$ + +Polynomial regression coefficients $\beta$ can be acquired through basic linear algebra where an $m \times m$ of constants is multiplied with $m \times 1$ of unknowns which equals another $m \times 1$ of constants. + +$$ +\begin{bmatrix} +\sum_{k=1}^n x_i^0 & \sum_{k=1}^n x_i^1 & \sum_{k=1}^n x_i^2 & \dots & \sum_{k=1}^n x_i^m \\ +\sum_{k=1}^n x_i^1 & \sum_{k=1}^n x_i^2 & \sum_{k=1}^n x_i^3 & \dots & \sum_{k=1}^n x_i^{m+1} \\ +\sum_{k=1}^n x_i^2 & \sum_{k=1}^n x_i^3 & \sum_{k=1}^n x_i^4 & \dots & \sum_{k=1}^n x_i^{m+2} \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +\sum_{k=1}^n x_i^m & \sum_{k=1}^n x_i^{m+1} & \sum_{k=1}^n x_i^{m+1} & \dots & \sum_{k=1}^n x_i^{2m} \\ +\end{bmatrix} + +\begin{bmatrix} +\beta_0 \\ +\beta_1 \\ +\beta_2 \\ +\vdots \\ +\beta_m +\end{bmatrix} + += + +\begin{bmatrix} +\sum_{k=1}^n x_jx_i^0 \\ +\sum_{k=1}^n x_jx_i^1 \\ +\sum_{k=1}^n x_jx_i^2 \\ +\vdots \\ +\sum_{k=1}^n x_jx_i^m +\end{bmatrix} +$$ + The weight $w$ is a slice of the Pearson corelation matrix. The numerator is the corelation between the $i$th statistic and the target statistic $j$ while the denominator the sum of the corelations between the $i$th statistic and all statistics but the target statistic $j$ (including itself which will always result in $r=1$). $$