Functional time series forecasting of distributions: A Koopman-Wasserstein approach

Abstract We present a novel method for forecasting the temporal evolution of probability distributions observed at discrete time points. Building on the Dynamic Probability Density Decomposition (DPDD), we incorporate distributional dynamics into Wasserstein geometry using a Koopman operator framework. Our method introduces an importance-weighted variant of Extended Dynamic Mode Decomposition (EDMD), allowing for accurate, closed-form forecasts in 2-Wasserstein space. We establish theoretical guarantees, demonstrating that our estimator achieves spectral convergence and an optimal Wasserstein error. Simulation studies and a real-world application to U.S. housing price distributions reveal significant improvements over existing methods, such as Wasserstein Autoregression. By integrating optimal transport, functional time series modeling, and spectral operator theory, DPDD provides a scalable and interpretable solution for distributional forecasting. This work has broad implications for behavioral science, public health, finance, and neuroimaging–fields where evolving distributions are commonplace. Our framework contributes to functional data analysis on non-Euclidean spaces and serves as a general tool for modeling and forecasting distributional time series.