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

Paper: arXiv 2601.16821

Authors: Harrison Katz

Abstract

Compositional time series frequently exhibit structural breaks due to external shocks, policy changes, or market disruptions. Standard methods either ignore such breaks or handle them through fixed effects that cannot extrapolate beyond the sample, or step-function dummies that impose instantaneous adjustment. We develop a Bayesian Dirichlet ARMA model augmented with a directional-shift intervention mechanism that captures structural breaks through three interpretable parameters: a direction vector specifying which components gain or lose share, an amplitude controlling redistribution magnitude, and a logistic gate governing transition timing and speed. The model preserves compositional constraints by construction, maintains DARMA dynamics for short-run dependence, and produces coherent probabilistic forecasts through and after structural breaks. The intervention trajectory corresponds to geodesic motion on the simplex and is invariant to the choice of ILR basis. A simulation study with 400 fits across 8 scenarios shows near-zero amplitude bias and nominal 80% credible interval coverage when the shift direction is correctly identified (77.5% of cases); supplementary studies confirm robustness across extreme transition speeds and non-monotone DGPs. Two empirical applications to COVID-era Airbnb data characterize performance relative to simpler alternatives. Where the break is monotone and ongoing, the intervention model achieves near-nominal calibration (79.6%) while the fixed effect substantially under-covers (66.1%). Where post-break dynamics are non-monotone, both models are acceptably calibrated and the fixed effect outperforms on point accuracy. The intervention model’s advantages are thus specific to settings with roughly monotone structural transitions.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework for structural break interventions in compositional time series using Bayesian Dirichlet ARMA models and Aitchison geometry. The empirical evaluation is extensive, featuring large simulation studies, full-matrix sensitivity benchmarks, and realistic applications to COVID-era Airbnb data that clearly delineate where the model excels and where simpler alternatives suffice.

Research Flowchart

  flowchart TD
    A[Research Goal: Model structural breaks<br/>in compositional time series] --> B[Data Inputs: Synthetic scenarios &<br/>COVID-era Airbnb data]
    B --> C[Methodology: Bayesian Dirichlet ARMA +<br/>Directional-Shift Intervention]
    C --> D[Model Properties: Geodesic motion on simplex,<br/>3-parameter shift, ILR basis invariant]
    D --> E[Computational Processes:<br/>MCMC Sampling & Simulation Study]
    E --> F[Outcomes: Near-zero amplitude bias,<br/>79.6% calibration in monotone breaks]
    F --> G[Key Finding: Superior probabilistic forecasting<br/>specifically for monotone structural shifts]

Cited References (57)

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8. Isometric logratio transformations for compositional data (2003)

0 of 57 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.