Pearson correlation coefficient¶

The Pearson correlation coefficient $\rho_P(X,Y)$ measures the strength of a linear relationship between two random variables $X$ and $Y$ with finite variance. It is defined as follows:

$\rho_P(X,Y)= \dfrac{\Cov{X,Y}}{\sqrt{\Var{X}\Var{Y}}}$

where $\Cov{X,Y} = \Expect{ \left( X - \Expect{X} \right) \left( Y - \Expect{Y} \right) }$ .

Let $((x_1, y_1), \dots, (x_\sampleSize, y_\sampleSize))$ be a sample generated by the bivariate random vector $(X,Y)$ . The Pearson correlation coefficient is estimated:

(1)¶ $\hat{\rho}_P(X,Y) = \dfrac{\sum_{k=1}^\sampleSize (x_k- \bar{x})(y_k- \bar{y})} {\sqrt{\sum_{k=1}^\sampleSize(x_k- \bar{x})^2\sum_{k=1}^\sampleSize(y_k- \bar{y})^2}}$

where $\bar{x} = \dfrac{1}{\sampleSize} \sum_{k=1}^\sampleSize x_k$ and $\bar{y} = \dfrac{1}{\sampleSize} \sum_{k=1}^\sampleSize y_k$ are the empirical mean of each sample.

The estimate $\hat{\rho}_P(X,Y)$ of the Pearson correlation coefficient is sometimes denoted by $r$ .

We sum up some interesting features of the coefficient:

The Pearson’s correlation coefficient takes values between -1 and 1.
If $|\rho_P(X,Y)|=1$ then there exists a linear relationship between $X$ and $Y$ .
The closer $|\rho_P(X,Y)|$ is to 1, the stronger the indication is that a linear relationship exists between $X$ and $Y$ . The sign of the Pearson’s coefficient indicates if the two variables increase or decrease in the same direction (positive coefficient) or in opposite directions (negative coefficient).
If $X$ and $Y$ are independent, then $\rho_P(X,Y)=0$ .
If $\rho_P(X,Y)=0$ , it does not imply the independence of the variables $X$ and $Y$ . It may only means that the relation between both variables is not linear.