Testing (Non-)Existence of Input–Output Relationships by Estimating Fractal Dimensions

Our aim is to propose tests for (non-)existence of nonlinear relationships between signals, which, after passing a test, can be interpreted as input and output signals of a certain system, if its characteristic is sufficiently smooth. The proposed tests are based on the theoretical results on equality of fractal dimensions of these signals as well as on estimation of fractal dimensions from observations. They are applicable when at least one of these signals has the fractal dimensions strictly larger than one, i.e., it is rough enough. The tests are then verified on simulated data. Their applicability is illustrated by two sets of real data, namely, observations of two financial time series and samples of displacement-force signals in a magneto-hydrological damper.