The Wasserstein-Fourier Distance for Stationary Time Series

TL;DR

The paper introduces the Wasserstein-Fourier distance, a novel metric for measuring similarity between time series based on their spectral content, validated through theoretical analysis and practical applications like visualization, interpolation, and classification.

Contribution

It formally introduces the Wasserstein-Fourier distance, exploring its properties and demonstrating its utility in time series analysis tasks such as dimensionality reduction, data augmentation, and classification.

Findings

WF enables meaningful spectral visualization and pattern recognition.

WF supports spectral domain data augmentation via geodesic interpolation.

WF improves classification performance compared to classical metrics.

Abstract

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.