Spectra in low‐rank localized layers (SpeLLL) for interpretable time–frequency analysis

The time‐varying frequency characteristics of many biomedical time series contain important scientific information. However, the high‐dimensional nature of the time‐varying power spectrum as a surface in time and frequency limits its direct use by applied researchers and clinicians for elucidating complex mechanisms. In this article, we introduce a new approach to time–frequency analysis that decomposes the time‐varying power spectrum in to orthogonal rank‐one layers in time and frequency to provide a parsimonious representation that illustrates relationships between power at different times and frequencies. The approach can be used in fully nonparametric analyses or in semiparametric analyses that account for exogenous information and time‐varying covariates. An estimation procedure is formulated within a penalized reduced‐rank regression framework that provides estimates of layers that are interpretable as power localized within time blocks and frequency bands. Empirical properties of the procedure are illustrated in simulation studies and its practical use is demonstrated through an analysis of heart rate variability during sleep.