Power spectral geodesics and tracking

It is often reasonable to assume that a power spectral density can be used to reflect, locally in time, properties of a non-stationary time series; for instance, the short-time Fourier transform is based on this hypothesis and provides a succession of estimated power spectra over successive time intervals. In this paper we consider such families of power spectral densities, indexed by time, to represent changes in the spectral content of non-stationary processes. We propose as a model for the drift in spectral power over time, the model of mass transport; that is, at least locally, spectral power shifts along geodesics of a suitable mass-transport metric. We show that fitting spectral geodesics in the Wasserstein metric to data is a convex quadratic program. Finally, we highlight the effectiveness of this proposition in tracking features (e.g., spectral lines, peaks) of non-stationary processes.