Pearson correlation coefficient @ model

(1)	$\begin{array}{l}\displaystyle \rho_{xy} = \frac{{\rm cov}(x,y)}{\sigma(x) \sigma(y)}\end{array}$

where

$\begin{array}{l}\{ (x, y) \} = \{ (x_1, y_1), \, (x_2, y_2), \, x_3 ... (x_n, y_n) \}\end{array}$

dynamic properties on the same discrete argument ( $\begin{array}{l}i=1..N\end{array}$ )

(2)	$\begin{array}{l}\displaystyle {\rm cov}(x,y) = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x) (y_i - \bar y)\end{array}$

cross-property covariance

(3)	$\begin{array}{l}\displaystyle \sigma(x) = \sqrt { \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 }\end{array}$

standard deviation of property $\begin{array}{l}x\end{array}$

(4)	$\begin{array}{l}\displaystyle \sigma(y) = \sqrt { \frac{1}{n} \sum_{i=1}^n (y_i - \bar y)^2 }\end{array}$

standard deviation of property $\begin{array}{l}y\end{array}$

Pearson correlation coefficient ranges between -1 and 1 and indicates how close the two properties can be related by a linear correlation:

$\begin{array}{l}\displaystyle y_i = a x_i + b, \quad \forall \, i=1..N\end{array}$

with a certain pick on $\begin{array}{l}a\end{array}$ and $\begin{array}{l}b\end{array}$ .

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