Pearson Product-moment Correlation Coefficient - Reflective Correlation

Reflective Correlation

The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values. The population reflective correlation is

\text{Corr}_r(X,Y) = \frac{E}{\sqrt{EX^2\cdot EY^2}}.

The reflective correlation is symmetric, but it is not invariant under translation:

\text{Corr}_r(X, Y) = \text{Corr}_r(Y, X) = \text{Corr}_r(X, bY) \neq \text{Corr}_r(X, a + b Y), \quad a \neq 0, b > 0.

The sample reflective correlation is

rr_{xy} = \frac{\sum x_i y_i}{\sqrt{(\sum x_i^2)(\sum y_i^2)}}.

The weighted version of the sample reflective correlation is

rr_{xy, w} = \frac{\sum w_i x_i y_i}{\sqrt{(\sum w_i x_i^2)(\sum w_i y_i^2)}}.

Read more about this topic:  Pearson Product-moment Correlation Coefficient

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