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

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 (Formula presented.) 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 (Formula presented.) and the within-day resolution (Formula presented.). 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.