Local Linear Dependence Measure for Functionally Correlated Variables

We propose a new correlation measure for functionally correlated variables based on local linear dependence. It is able to detect non-linear, non-monotonic and even implicit relationships. Applying the classical linear correlation in a local framework combined with tools from Principal Components Analysis the statistic is capable of detecting very complex dependences among the data. In a first part we prove that it meets the properties of independence, similarity invariance and dependence and the axiom of continuity. In a second part we run a numerical simulation over a variety of dependences and compare it to other dependence measures in the literature. The results indicate that we outperform existing coefficients. We also show better stability and robustness to noise.