Minimal Gelbrich Distance to Uncorrelation

This letter reports new properties of the Wasserstein/Gelbrich distance and associated ambiguity sets to analyze the correlation between two scalar random variables. A simple closed expression is derived for the Gelbrich distance between two bidimensional random distributions. Moreover, the minimum disturbance in the Gelbrich metric required to reach uncorrelation between two random variables is obtained. This allows us to determine the robustness of the Pearson coefficient within an ambiguity set. A numerical example showcases the potential use of the obtained results in the field of variable selection.