# Pearson correlation coefficient¶

This method deals with the parametric modelling of a probability distribution for a random vector . It aims to measure a type of dependence (here a linear correlation) which may exist between two components and .

The Pearson’s correlation coefficient aims to measure the strength of a linear relationship between two random variables and . It is defined as follows:

where , , , and . If we have a sample made up of a set of pairs , Pearson’s correlation coefficient can be estimated using the formula:

where and represent the empirical means of the samples and .

Pearson’s correlation coefficient takes values between -1 and 1. The closer its absolute value is to 1, the stronger the indication is that a linear relationship exists between variables and . The sign of Pearson’s coefficient indicates if the two variables increase or decrease in the same direction (positive coefficient) or in opposite directions (negative coefficient). We note that a correlation coefficient equal to 0 does not necessarily imply the independence of variables and : this property is in fact theoretically guaranteed only if and both follow a Normal distribution. In all other cases, there are two possible situations in the event of a zero Pearson’s correlation coefficient:

• the variables and are in fact independent,

• or a non-linear relationship exists between and .

The estimate of Pearson’s correlation coefficient is sometimes denoted by .

Examples: