Wasserstein Exponential Smoothing

2026-06-03

AI Summary

This study extends classical exponential smoothing to distributional time series—where observations are probability distributions on the real line—by introducing a principled and intuitive exponential smoothing framework in Wasserstein space. The smoothing parameter is consistently estimated by minimizing the Wasserstein distance, enabling effective capture and forecasting of the underlying distributional dynamics. Theoretical analysis establishes the consistency of this parameter estimator, while empirical evaluations on high-frequency financial returns and household electricity demand data demonstrate the model’s predictive accuracy and practical utility.

Abstract

Exponential smoothing (ES) often outperforms other techniques in time series forecasting across a wide range of data-generating processes. While ES has traditionally been applied to time series in $\mathbb{R}$, this paper extends the methodology to distributional time series, where each observation is a probability distribution on $\mathbb{R}$. The primary contribution of this work is twofold. First, we propose a principled and intuitive generalization of ES within the Wasserstein space, which retains the exceptional parsimony of classical ES. Second, we theoretically and empirically demonstrate that the smoothing parameter can be consistently estimated by minimizing a Wasserstein distance. Applications to distributional time series of high-frequency financial returns and household electricity demands confirm the practical effectiveness of our Wasserstein ES model.

Problem

• distributional time series
• exponential smoothing
• Wasserstein distance
• forecasting
• probability distributions

Innovation

• Wasserstein distance
• exponential smoothing
• distributional time series
• parameter estimation
• Wasserstein space