Correlation Coefficient

  The correlation coefficient between two random variables X_i and X_j covariance divided by the square root of the product of the variances

It has the range and vanishes for independent variables. If and X_j are linearly dependent and the covariance matrix is singular.

The correlation coefficient can be regarded as a measure of the relation between the statistical distributions of the two random variables considered: if and are the variances along the uncorrelated major and minor axes in the plane defined by the two variables, the correlation coefficient after a rotation by the angle a ( Bivariate Normal Distribution) is given by

with

If no minor/major axes can be defined ( ), the variables are uncorrelated.

The global correlation coefficient is defined by

where C_ii and (C^-1)_ii are elements in the diagonal of the covariance matrix and of its inverse, respectively.