Properties of the Weighted and Robust Implicitly Weighted Correlation Coefficients

Pearson product-moment correlation coefficient represents a fundamental measure of similarity between two data vectors. In various applications, it is meaningful to consider its weighted version known as the weighted Pearson correlation coefficient. Its properties are studied in this theoretical paper; these include the robustness to rounding, as it is an important issue in approximate neurocomputing, or specific robustness properties for the context of template matching in image analysis. For a highly robust correlation coefficient inspired by the least weighted estimator, properties are derived and novel hypothesis tests are proposed. This robust measure is recommendable particularly for data contaminated by outliers (not only) in the context of image analysis.