Machine Learning Encyclopedia

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Correlation coefficient

The most well-known measure of similarity between two random variables is the correlation coefficient. Correlation coefficient \(\rho$ between two random variable\)X_1\(and\)X_2$$ is defined as

\begin{equation} \rho(X_1,X_2) = \frac{\text{cov}(X_1,X_2)}{\sqrt{\text{var}(X_1), \text{var}(X_2)}}.\nonumber \end{equation}

If \(X_1\) and \(X_2\) are completely correlated, i.e., exact linear dependency exists \(\rho(X_1,X_2)\) is \(1\) or \(-1\). If \(X_1\) and \(X_2\) are totally uncorrelated, \(\rho(X_1,X_2)\) is \(0\).