feat: matrix

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 $r=1$).
 
 $$