Decomposing spectral and phasic differences in nonlinear features between datasets

When employing nonlinear methods to characterize complex systems, it is important to determine to what extent they are capturing genuine nonlinear phenomena that could not be assessed by simpler spectral methods. Specifically, we are concerned with the problem of quantifying spectral and phasic effects on an observed difference in a nonlinear feature between two systems (or two states of the same system). Here we derive, from a sequence of null models, a decomposition of the difference in an observable into spectral, phasic, and spectrum-phase interaction components. Our approach makes no assumptions about the structure of the data and adds nuance to a wide range of time series analyses.