Correlation Map Reference Information

Pearson (Linear) Correlations

The Pearson, or linear, correlation is calculated as follows for parameters X and Y:

r_{x y} = \frac{\sum_{k = 1}^{N} (X_{k} - \bar{X}) (Y_{k} - \bar{Y})}{\sqrt{\sum_{k = 1}^{N} {(X_{k} - \bar{X})}^{2}} \sqrt{\sum_{k = 1}^{N} {(Y_{k} - \bar{Y})}^{2}}}

The values of r will be between –1 (representing a perfect inverse linear correlation) and 1 (representing a perfect direct linear correlation). Values of 0 or close to zero signify a lack of correlation between the parameters.

Spearman (Rank) Correlations

For the Spearman (rank) correlation, slight nonlinearity is removed by ranking the parameter values from 1 to N for N values and replacing the parameter values with their rank value. The ranking can be based on highest to lowest parameter values or lowest to highest. The same correlation formulation is then applied to the ranked parameter values, R and S, substituted for X and Y:

s_{x y} = \frac{\sum_{k = 1}^{N} (R_{k} - \bar{R}) (S_{k} - \bar{S})}{\sqrt{\sum_{k = 1}^{N} {(R_{k} - \bar{R})}^{2}} \sqrt{\sum_{k = 1}^{N} {(S_{k} - \bar{S})}^{2}}}

The Spearman correlation values are again between –1 and 1.