====== Pearson ======

Pearson product-moment correlation coefficient

The Correlation coefficients (ρ<sub>x,y</sub>) usually known as Pearson’s product moment correlation coefficients, provide a measure of the strength of the linear relationship between two variables. The correlation coefficient between two N-dimensional vectors x and y is defined by:

ρ<sub>x,y</sub> = ∑(x<sub>k</sub>-xbar)(y<sub>k</sub>-ybar) / ((∑((x<sub>k</sub>-xbar)<sup>2</sup>))<sup>1/2</sup>(∑((y<sub>k</sub>-ybar)<sup>2</sup>))<sup>1/2</sup>), k=1:N

where xbar and ybar are defined as the mean of x and y respectively.

The correlation coefficient could also be reformulated as:

ρ<sub>x,y</sub> = cov(x,y)/(σ(x)σ(y))

where cov(x,y) is the covariance between the data sets x and y and σ(x) and σ(y) are the sampled standard deviations.

The correlation coefficient is then the normalized covariance between the two data sets and (as [[SRC|SRC]]) produces a unitless index between -1 and +1. Correlation coefficient is equal in absolute value to the square root of the model coefficient of determination R<sup>2</sup> associated with the linear regression.

===== Reference =====

  * Francesca Campolongo, Andrea Saltelli, Tine Sørensen, and Stefano Tarantola. Hitchhiker’s guide to sensitivity analysis. In //Sensitivity analysis//, Wiley Ser. Probab. Stat., pages 15–47. Wiley, Chichester, 2000.