  We propose the Wasserstein-Fourier (WF) distance to measure the
(dis)similarity between time series by quantifying the displacement of their
energy across frequencies. The WF distance operates by calculating the
Wasserstein distance between the (normalised) power spectral densities (NPSD)
of time series. Yet this rationale has been considered in the past, we fill a
gap in the open literature providing a formal introduction of this distance,
together with its main properties from the joint perspective of Fourier
analysis and optimal transport. As the main aim of this work is to validate WF
as a general-purpose metric for time series, we illustrate its applicability on
three broad contexts. First, we rely on WF to implement a PCA-like
dimensionality reduction for NPSDs which allows for meaningful visualisation
and pattern recognition applications. Second, we show that the geometry induced
by WF on the space of NPSDs admits a geodesic interpolant between time series,
thus enabling data augmentation on the spectral domain, by averaging the
dynamic content of two signals. Third, we implement WF for time series
classification using parametric/non-parametric classifiers and compare it to
other classical metrics. Supported on theoretical results, as well as synthetic
illustrations and experiments on real-world data, this work establishes WF as a
meaningful and capable resource pertinent to general distance-based
applications of time series.
