Quantification of Unidirectional Nonlinear Associations Between Multidimensional Signals

In this paper, we present a rigorous, general definition of the nonlinear association index, known as $h ^{2}$ . Proving equivalence between different definitions we show that the index measures the best dynamic range of any nonlinear map between signals. We present also a construction for removing the influence of one signal from another, providing, thus, the basis of an independent component analysis. Our definition applies to arbitrary multidimensional vector-valued signals and depends on an aperture function. In this way, the bin-related classic definition of $h ^{2}$ can be generalized. We show that upon choosing suitable aperture functions the bin-related intuitive definition can be deduced.