Trend and rhythm analysis of time-series data using complex demodulation

Biological time-series data collected over long intervals generally show combined systematic and periodic fluctuations. Comprehensive analysis of such data requires separation of the trend and rhythmic components. Most available time-series analytic techniques do not explicitly extract the trend, and do implicitly assume the underlying rhythms are simple symmetrical sinusoids, whose amplitude and phase values remain constant throughout the recorded interval. Neither assumption is very accurate when dealing with biological data, and the stationarity assumption in particular becomes harder to defend as experiments extend over days or even weeks. Complex demodulation (CD) is described here as a technique for separation of trend from cyclic components, and multiple complex demodulation (MCD) as a technique for extraction of all possible frequencies in the data set, along with their moment-by-moment amplitude and phase values.