CAST: Causal Anchored Simplex Transport for Distribution-Valued Time Series

TL;DR

This paper introduces CAST, a novel method for causal forecasting of distribution-valued time series on the probability simplex, demonstrating superior performance across diverse real-world benchmarks.

Contribution

CAST is a new successor-local operator that stabilizes and transports distributions on the simplex, addressing aliasing issues and improving forecasting accuracy.

Findings

CAST achieves the best average rank on 8/11 benchmarks for one-step KL and autoregressive JSD.

CAST outperforms various baselines including statistical, recurrent, convolutional, and Transformer models.

Component ablations confirm the theoretical advantages of CAST and its robustness against aliasing.

Abstract

Many decision-facing stochastic systems are observed through aggregate distributions rather than scalar trajectories: queue occupancies, mobility shares, public-health mixtures, generation-source shares, ecological compositions, and air-quality severity profiles all live on the probability simplex and evolve over time. We study causal (online) forecasting for these distribution-valued time series and argue that the transition operator itself should be structured around the simplex. We introduce CAST (Causal Anchored Simplex Transport), a successor-local operator that (i) retrieves empirical successors from causal context, (ii) stabilizes them with a persistence anchor, and (iii) applies a bounded local stochastic transport on ordered supports; every stage preserves the simplex by construction. We identify a structural failure mode, latent transition-kernel aliasing, where similar observed distributions evolve differently under different contextual regimes, and prove that any forecaster depending only on an aliased summary incurs an irreducible weighted Jensen-Shannon excess-risk lower bound, while the CAST hypothesis class contains the regime-aware Bayes successor; for ordered supports an additional Pinsker separation holds whenever the transported successor lies outside the no-transport anchor hull. On eleven public and simulated benchmarks spanning ecology, energy, diet, mortality, employment, air quality, severe weather, mobility, and G/G/1, G_t/G/1 queue occupancy, CAST attains the best average rank on both one-step KL (1.27) and autoregressive rollout JSD (1.91), winning 8/11 sections on each metric against a broad statistical, compositional, recurrent, convolutional, and Transformer baseline set, and top-2 on all 11 sections for offline KL. Component ablations and a controlled synthetic aliasing experiment corroborate the theory.