Flow Matching in Hierarchical Residual Coordinates for Non-Stationary Probabilistic Forecasting

Probabilistic forecasting for non-stationary time series is difficult because the conditional target distribution entangles drifting location-scale statistics with residual stochastic dynamics. This entanglement forces generative models to learn both macroscopic distribution drift across contexts and microscopic stochastic dynamics within each context in a single transport map. We propose a novel extensible framework of hierarchical residual coordinates: an order-indexed composition of context-dependent invertible maps removes predictable distributional components from the full future trajectory before generation, a residual generator learns the remaining joint law while retaining temporal and cross-variate dependence, and the inverse coordinate map transports the learned residual flow back to reconstruct forecasts in the original space. We establish a conditional pushforward consistency guarantee for this construction. As a practical second-order instantiation, LS-Flow combines location-scale residualization with Conditional Flow Matching. Across eight real-world non-stationary multivariate benchmarks, LS-Flow consistently improves probabilistic forecasting over state-of-the-art generative baselines; order-wise ablations and a third-order pilot examine the framework's extensibility.