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Pearson product-moment correlation coefficient

The Correlation coefficients (ρ_{x,y}) 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_{x,y} = ∑(x_{k}-xbar)(y_{k}-ybar) / ((∑((x_{k}-xbar)^{2}))^{1/2}(∑((y_{k}-ybar)^{2}))^{1/2}), 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_{x,y} = 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) 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^{2} associated with the linear regression.

- 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.