Towards a Localized Version of Pearson’s Correlation Coefficient

Pearson’s correlation coefficient is used to describe dependence between random variables X and Y . In some practical situations, however, we have strong correlation for some values X and/or Y and no correlation for other values ofX and Y . To describe such a local dependence, we come up with a natural localized version of Pearson’s correlation coefficient. We also study the properties of the newly defined localized coefficient. 1 Formulation of the Problem Pearson’s correlation coefficient: reminder. To describe relation between two random variables X and Y , Pearson’s correlation coefficient r is often used. This coefficient is defined as r[X,Y ] def = C[X,Y ] σ[X] · σ(Y ) , (1)