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When employing non-linear methods to characterise complex systems, it isimportant to determine to what extent they are capturing genuine non-linearphenomena that could not be assessed by simpler spectral methods. Specifically,we are concerned with the problem of quantifying spectral and phasic effects onan observed difference in a non-linear feature between two systems (or twostates of the same system). Here we derive, from a sequence of null models, adecomposition of the difference in an observable into spectral, phasic, andspectrum-phase interaction components. Our approach makes no assumptions aboutthe structure of the data and adds nuance to a wide range of time seriesanalyses.

Decomposing spectral and phasic differences in non-linear features between datasets