The Wasserstein-Fourier Distance for Stationary Time Series
This work provides a formalized distance metric for time series analysis, addressing a gap in the literature, but it is incremental as it builds on existing ideas from optimal transport and Fourier analysis.
The authors tackled the problem of measuring similarity between stationary time series by proposing the Wasserstein-Fourier distance, which quantifies energy displacement across frequencies, and validated it as a general-purpose metric through applications like dimensionality reduction, data augmentation, and classification, showing competitive performance in experiments.
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.