# Wasserstein Exponential Smoothing for Distributional Time Series Forecasting

arXiv: https://arxiv.org/abs/2606.05560 · HTML: https://mlsanity.com/paper/2606.05560/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 31 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Wasserstein Autoregressive Models for Density Time Series](https://mlsanity.com/paper/2006.12640/) (2020)
- [Autoregressive Optimal Transport Models](https://mlsanity.com/paper/2105.05439/) (2021)
- [Dynamic Quantile Function Models](https://mlsanity.com/paper/1707.02587/) (2017)
- [On Distributional Autoregression and Iterated Transportation](https://mlsanity.com/paper/2303.09469/) (2023)

## Abstract (arXiv)

Distributional time series arise when each temporal observation is a probability distribution rather than a scalar. We propose Wasserstein exponential smoothing (WES), a one-parameter recursive forecasting method for distributional time series on ℝ. The method adapts the practical logic of classical exponential smoothing to probability distributions by updating forecast distributions along Wasserstein geodesics. This yields a simple filter that can be applied directly to empirical distributions without parametric density modeling. We estimate the smoothing parameter by minimizing an in-sample Wasserstein prediction loss and establish consistency under a distributional local-level data-generating process. In applications to high-frequency equity-index return distributions and household electricity-demand distributions, WES attains the lowest one-step-ahead Wasserstein prediction error among existing distributional autoregressive and regression-based benchmarks for all 20 series considered, and is retained in the 90% model confidence set in every case.