Functional Time Series Forecasting of Distributions: A Koopman-Wasserstein Approach

TL;DR

This paper introduces a novel Koopman-Wasserstein method for forecasting the evolution of probability distributions over time, combining optimal transport, spectral operator theory, and functional data analysis for improved accuracy and interpretability.

Contribution

It extends the Dynamic Probability Density Decomposition framework with a Wasserstein geometry embedding and importance-weighted EDMD, providing theoretical guarantees and superior empirical performance.

Findings

Achieves spectral convergence and optimal finite-sample Wasserstein error.

Demonstrates substantial improvements over Wasserstein Autoregression in simulations.

Applicable to diverse domains like finance, health, and neuroimaging.

Abstract

We propose a novel method for forecasting the temporal evolution of probability distributions observed at discrete time points. Extending the Dynamic Probability Density Decomposition (DPDD), we embed distributional dynamics into Wasserstein geometry via a Koopman operator framework. Our approach introduces an importance-weighted variant of Extended Dynamic Mode Decomposition (EDMD), enabling accurate, closed-form forecasts in 2-Wasserstein space. Theoretical guarantees are established: our estimator achieves spectral convergence and optimal finite-sample Wasserstein error. Simulation studies and a real-world application to U.S. housing price distributions show substantial improvements over existing methods such as Wasserstein Autoregression. By integrating optimal transport, functional time series modeling, and spectral operator theory, DPDD offers a scalable and interpretable solution for distributional forecasting. This work has broad implications for behavioral science, public health, finance, and neuroimaging--domains where evolving distributions arise naturally. Our framework contributes to functional data analysis on non-Euclidean spaces and provides a general tool for modeling and forecasting distributional time series.