Quantification of Unidirectional Nonlinear Associations Between Multidimensional Signals

In this paper, we present a rigorous, general definition of the nonlinear association index, known as h². 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 h² can be generalized. We show that upon choosing suitable aperture functions the bin-related intuitive definition can be deduced. Special attention is dedicated to the direction of the association index that in general is taken in only one sense. We show that for linearly coupled signals high associations are always bidirectional. As a consequence, high asymmetric nonlinear associations are indicators of nonlinear relations, possibly critical, between the dynamic systems underlying the measured signals. We give a simple simulated example to illustrate this property. As a potential clinical application, we show that unidirectional associations between electroencephalogram (EEG) and electromyogram (EMG) recorded from patient with pharmacologically intractable epilepsy can be used to study the cortical involvement in the generation of motor seizures