Wasserstein Exponential Smoothing

A new study extends exponential smoothing, a widely used forecasting technique, to distributional time series where each observation is a full probability distribution rather than a single number, proposing a method grounded in Wasserstein space geometry [1]. The paper, posted to arXiv on June 4, 2026, introduces Wasserstein exponential smoothing (WES), a generalization that retains the single-parameter parsimony of classical exponential smoothing while operating on probability distributions on the real line [1][3]. Classical exponential smoothing has long been valued for its simplicity and strong empirical performance across diverse data-generating processes, but its application has been limited to scalar-valued time series [3]. The authors note that the methodological toolkit for distributional time series remains substantially underdeveloped compared to its scalar counterpart, with no existing analogue of exponential smoothing [3]. Distributional data analysis, a branch of nonparametric statistics related to functional data analysis, deals with random objects that are themselves probability distributions, a setting where the space of distributions is convex but not a vector space [6]. The WES framework lifts the defining recursion of classical exponential smoothing from Euclidean space to Wasserstein space by replacing ordinary linear interpolation with geometric interpolation under the Wasserstein distance [3]. The focus on one-dimensional distributions is crucial because the one-dimensional Wasserstein geometry admits a tractable representation through quantile functions, making both the smoothing recursion and loss-based estimation computationally feasible [3]. The authors demonstrate that the smoothing parameter can be consistently estimated by minimizing a Wasserstein distance based on the observed series [1][3]. They also establish theoretical properties including extended notions of mean stationarity and autocovariance decay for the smoothing dynamics [3]. The method was applied to distributional time series of high-frequency financial returns and household electricity demands, confirming its practical effectiveness [1][2]. Related work has explored autoregressive models for distributional time series in Wasserstein space, with recent contributions extending vector autoregressive models to multivariate distributional time series for studying both within-series and cross-series dependencies [5]. A separate line of research has examined how exponential smoothing emerges as a special case in data-driven decision problems where historical observations come from a time-evolving distribution with bounded Wasserstein-distance shifts, providing principled choices for the smoothing parameter [4].