  When employing non-linear methods to characterise complex systems, it is
important to determine to what extent they are capturing genuine non-linear
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 non-linear 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.
