Wasserstein Regression, Forecasting, and Change‐Point Detection for Daily Traffic Flow Distributions

ABSTRACT We develop a distribution‐valued framework for modeling, forecasting, and monitoring traffic flow counts by treating each day as a probability distribution summarized by jittered empirical quantile signatures. Inference is conducted under the 2‐Wasserstein geometry, which in one dimension is isometric to the metric on quantile functions. This representation preserves the empirical distribution of within‐day traffic intensities beyond mean aggregation while deliberately abstracting away from the chronological ordering of the intraday curve. We introduce Wasserstein‐based distributional regression, one‐step‐ahead forecasting, and a Wasserstein CUSUM statistic for change‐point detection and localization. Our theory provides finite‐sample and asymptotic guarantees under the two‐stage sampling structure of traffic data, with error bounds that separate the roles of the number of days and the within‐day resolution . Simulations show competitive performance under location shifts and substantial gains under dispersion or shape changes. An analysis of publicly available interstate traffic volumes illustrates quantile‐dependent covariate effects and interpretable regime changes via quantile‐shift diagnostics.