Directional-Shift Dirichlet ARMA Models for Compositional Time Series with Structural Break Intervention

2026-01-23

AI Summary

This study addresses the challenge of modeling compositional time series when structural breaks occur due to external shocks, a scenario where conventional methods often fail. The authors propose a Bayesian Dirichlet ARMA model incorporating a directional shift intervention mechanism based on geodesic motion over the simplex. This mechanism characterizes abrupt changes through a direction vector, magnitude, and a logistic gating function, rigorously preserving the compositional constraint while enabling coherent probabilistic forecasts across the break. The intervention parameters are interpretable, invariant to the choice of isometric log-ratio (ILR) basis, and resolve sign indeterminacy via a hemisphere constraint. Simulations show that the method correctly identifies the shift direction in 77.5% of scenarios, with nearly unbiased estimates of magnitude and timing, and achieves nominal coverage for 80% credible intervals. In empirical analysis, it yields a lower Aitchison distance (0.83 vs. 0.90) and improves prediction interval coverage to 87%.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Bayesian Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Compositional time series, vectors of proportions summing to unity observed over time, frequently exhibit structural breaks due to external shocks, policy changes, or market disruptions. Standard methods either ignore such breaks or handle them through ad-hoc dummy variables that cannot extrapolate beyond the estimation sample. We develop a Bayesian Dirichlet ARMA model augmented with a directional-shift intervention mechanism that captures structural breaks through three interpretable parameters: a unit direction vector specifying which components gain or lose share, an amplitude controlling the magnitude of redistribution, and a logistic gate governing the timing and speed of transition. The model preserves compositional constraints by construction, maintains innovation-form DARMA dynamics for short-run dependence, and produces coherent probabilistic forecasts during and after structural breaks. We establish that the directional shift corresponds to geodesic motion on the simplex and is invariant to the choice of ILR basis. A comprehensive simulation study with 400 fits across 8 scenarios demonstrates that when the shift direction is correctly identified (77.5% of cases), amplitude and timing parameters are recovered with near-zero bias, and credible intervals for the mean composition achieve nominal 80% coverage; we address the sign identification challenge through a hemisphere constraint. An empirical application to fee recognition lead-time distributions during COVID-19 compares baseline, fixed-effects, and intervention specifications in rolling forecast evaluation, demonstrating the intervention model's superior point accuracy (Aitchison distance 0.83 vs. 0.90) and calibration (87% vs. 71% coverage) during structural transitions.

Problem

• compositional time series
• structural breaks
• Dirichlet ARMA
• Bayesian modeling
• forecasting

Innovation

• Directional-Shift Intervention
• Compositional Time Series
• Dirichlet ARMA
• Structural Break
• Bayesian Forecasting