The Wasserstein-Fourier Distance for Stationary Time Series

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