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

Harrison Katz

TL;DR

The paper develops a directional-shift extension to Bayesian Dirichlet ARMA (B-DARMA) models for compositional time series experiencing structural breaks. It decomposes breaks into a direction, amplitude, and timing via a logistic gate, yielding geodesic trajectories on the simplex and preserving compositional coherence. Simulation shows accurate parameter recovery when the break direction is identified and nominal calibration, while an empirical COVID-19 application demonstrates superior point accuracy and interval calibration relative to baseline and fixed-effect approaches. The method provides interpretable, extrapolatable forecasts through and after breaks, offering practical guidance for forecasting under regime changes in compositional data.

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.

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

TL;DR

The paper develops a directional-shift extension to Bayesian Dirichlet ARMA (B-DARMA) models for compositional time series experiencing structural breaks. It decomposes breaks into a direction, amplitude, and timing via a logistic gate, yielding geodesic trajectories on the simplex and preserving compositional coherence. Simulation shows accurate parameter recovery when the break direction is identified and nominal calibration, while an empirical COVID-19 application demonstrates superior point accuracy and interval calibration relative to baseline and fixed-effect approaches. The method provides interpretable, extrapolatable forecasts through and after breaks, offering practical guidance for forecasting under regime changes in compositional data.

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.
Paper Structure (53 sections, 2 theorems, 7 equations, 4 figures, 4 tables)

This paper contains 53 sections, 2 theorems, 7 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background: Compositional Time Series
  3. Directional-Shift Intervention Model
  4. Motivation and Overview
  5. The Directional Shift Parameterization
  6. The Logistic Gate Function
  7. Concentration Parameter Dynamics
  8. Full Model Specification
  9. Prior Specification and Computation
  10. Sign identification.
  11. Priors.
  12. Computation.
  13. Simulation Study
  14. Simulation Design
  15. Data generating process.
  16. ...and 38 more sections

Key Result

Proposition 1

Let $\bm{\eta}_0 \in \mathbb{R}^{C-1}$ be an ILR coordinate and $\bm{v} \in \mathbb{R}^{C-1}$ a unit direction. The curve $\bm{\mu}(w) = \operatorname{ilr}^{-1}(\bm{\eta}_0 + w\bm{v})$ for $w \in \mathbb{R}$ is a geodesic on $(\mathcal{S}^{C}, d_A)$.

Figures (4)

  • Figure 1: Lead-time distribution over time (heatmap). Each column represents a month; each row represents a lead-time category (0 months at bottom, 9+ months at top). Darker shading indicates higher share. The COVID-19 structural break (March 2020) is clearly visible as a shift toward shorter lead times.
  • Figure 2: Rolling 1-step-ahead Aitchison distance by forecast origin. The baseline model (no break mechanism) performs poorly throughout. The intervention model consistently outperforms the fixed effect model on point accuracy; its additional advantage in uncertainty quantification emerges in coverage statistics (Table \ref{['tab:empirical_results']}).
  • Figure 3: In-sample model fit: posterior mean of $\bm{\mu}_t$ (blue) versus observed lead-time shares (black) for all categories. Shaded bands show 80% credible intervals. The intervention model captures both the pre-COVID seasonal patterns and the gradual structural shift beginning March 2020. Out-of-sample forecast performance is evaluated separately in Table \ref{['tab:empirical_results']}.
  • Figure 4: Mean Aitchison distance by forecast horizon. The intervention model outperforms alternatives at all horizons, with all models improving as forecasts extend into the post-break steady state.

Theorems & Definitions (4)

  • Proposition 1: Geodesic motion
  • proof
  • Proposition 2: Basis invariance
  • proof