Pearson correlation coefficient @ model

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

where

$\begin{array}{l}\{ (x, y) \} = \{ (x_1, y_1), \, (x_2, y_2), \, ... (x_n, y_n) \}\end{array}$	discrete array of variables $\begin{array}{l}x\end{array}$ and $\begin{array}{l}y\end{array}$
$\begin{array}{l}{\rm cov}(x,y)\end{array}$	covariance between variables $\begin{array}{l}x\end{array}$ and $\begin{array}{l}y\end{array}$
$\begin{array}{l}\sigma(x)\end{array}$ , $\begin{array}{l}\sigma(y)\end{array}$	standard deviation of property $\begin{array}{l}x\end{array}$ and $\begin{array}{l}y\end{array}$

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 $\begin{array}{l}b\end{array}$ (see Fig. 1 – Fig. 3 for examples)

Maximum value relates to perfect linear correlation and $\begin{array}{l}a>0\end{array}$
Zero value relates to random correlation between $\begin{array}{l}x\end{array}$ and $\begin{array}{l}y\end{array}$
Minimum value relates to perfect linear correlation and $\begin{array}{l}a< 0\end{array}$


Fig. 1. Highly correlated variables	Fig. 2. Poorly correlated variables	Fig. 3. Highly anti-correlated variables

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