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

2026-05-16

AI Summary

This work addresses the bias in online causal forecasting of distributional time series on the probability simplex arising from the neglect of geometric structure and contextual aliasing. The authors propose CAST, a method that retrieves empirical successors via causal context, stabilizes predictions using persistence anchors, and performs bounded local stochastic transport over ordered supports—all while preserving simplex constraints. Theoretically, they identify latent transition kernel aliasing as a structural failure mode, establish a fundamental lower bound on irreducible risk for predictors relying solely on aliased summaries, and construct a hypothesis class incorporating context-aware Bayesian successors; for ordered supports, they introduce a Pinsker separation condition. Experiments across 11 ecological, energy, and health benchmarks show CAST achieves the best average ranks in one-step KL divergence (1.27) and autoregressive JSD (1.91), leads on 8 out of 11 tasks, and consistently ranks among the top two in offline KL performance across all tasks.

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.

Problem

• distribution-valued time series
• probability simplex
• causal forecasting
• transition-kernel aliasing
• simplex transport

Innovation

• simplex-valued time series
• causal forecasting
• stochastic transport
• transition-kernel aliasing
• distributional persistence anchor