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

1. Problem Statement

Compositional Time Series and Structural Breaks:
Compositional data consists of vectors of non-negative components summing to unity (e.g., market shares, booking distributions). These data reside on a simplex, requiring specialized geometric handling (Aitchison geometry) to avoid spurious correlations.
The paper addresses a persistent challenge: structural breaks (abrupt or gradual shifts caused by shocks like the COVID-19 pandemic) in these series.

• Limitations of Existing Methods:
  • Fixed Effects: Can fit in-sample perfectly but cannot extrapolate to future periods because future dummy variables are undefined.
  • Step-Function Dummies: Can extrapolate but assume instantaneous transitions, which is often unrealistic for behavioral shifts.
  • Standard DARMA: Assumes stationarity and fails to capture regime changes, leading to poor forecasts during transitions.
• Goal: Develop a forecasting model that preserves compositional constraints, handles gradual structural breaks, and provides interpretable parameters for the direction, magnitude, and timing of the shift.

2. Methodology

The author proposes a Bayesian Directional-Shift Dirichlet ARMA (DARMA) model.

A. Core Framework

The model builds upon the Bayesian DARMA framework (Katz et al., 2024), where the composition $Y_t$ follows a Dirichlet distribution with a time-varying mean $\mu_t$ and concentration $\lambda_t$. The mean evolves in Isometric Log-Ratio (ILR) space ($Z_t = \text{ilr}(Y_t)$) using ARMA dynamics.

B. The Directional-Shift Intervention

The core innovation is the decomposition of a structural break into three interpretable parameters within the drift term ($d_t$) of the ILR space:

1. Direction ($v$): A unit vector ($\|v\|=1$) in $\mathbb{R}^{C-1}$ specifying the axis of change. It defines which components gain or lose share.
2. Amplitude ($\Delta$): A scalar controlling the magnitude of the shift along direction $v$.
3. Logistic Gate ($w_t$): A function governing the timing and speed of the transition.
   • Defined as a piecewise logistic function: $w_t(\tau, \kappa) = \frac{\sigma(\kappa(t-\tau)) - \sigma(\kappa(\ell-\tau))}{1 - \sigma(\kappa(\ell-\tau))}$.
   • $\tau$: Transition location.
   • $\kappa$: Transition speed (smoothness).
   • This allows for gradual rather than instantaneous shifts.

The drift term becomes:
$$d_t = b + BX_t + \Delta \cdot w_t(\tau, \kappa) \cdot v$$

C. Geometric Properties

• Geodesic Motion: The intervention trajectory corresponds to geodesic motion on the simplex under Aitchison geometry (the shortest path between pre- and post-break compositions).
• Basis Invariance: While the computation uses a specific ILR basis (e.g., Helmert), the induced trajectory on the simplex is invariant to the choice of basis.
• Sign Identification: The product $\Delta \cdot v$ is identified, but the pair $(\Delta, v)$ has a sign ambiguity. The model resolves this by constraining the first element of the raw direction vector to be positive ($v_1 > 0$).

D. Implementation

• Dynamics: Uses diagonal AR and MA matrices for parsimony and computational tractability, assuming ILR dimensions evolve independently given the drift.
• Computation: Implemented in Stan using Hamiltonian Monte Carlo (NUTS sampler).
• Priors: Weakly informative priors are used, with specific constraints on the direction vector to ensure identifiability.

3. Key Contributions

1. Model Development: A parsimonious Bayesian framework for structural breaks in compositional time series that respects simplex constraints by construction.
2. Interpretability: Decomposes breaks into direction, magnitude, and timing, facilitating domain-specific interpretation (e.g., "shift toward shorter booking windows").
3. Geometric Insight: Establishes that directional shifts correspond to geodesic motion on the simplex and are basis-invariant.
4. Comprehensive Evaluation:
   • Simulation: 400 fits across 8 scenarios (varying speed, amplitude, precision shift) plus supplementary studies on extreme speeds and partial reversibility.
   • Empirical Applications: Two distinct COVID-era Airbnb case studies (Lead-time distributions and Stay-length compositions).
5. Practical Benchmarking: A unique "full-matrix sensitivity" benchmark comparing the diagonal restriction against unrestricted AR/MA matrices.

4. Key Results

A. Simulation Study

• Parameter Recovery: When the direction is correctly identified (77.5% of cases), amplitude bias is near-zero, and timing bias is small.
• Calibration: The model achieves nominal 80% coverage for the mean composition credible intervals, even when direction recovery is imperfect.
• Robustness:
  • Transition Speed: Calibration remains nominal across extreme $\kappa$ values (slow to fast).
  • Reversibility: Under partial reversibility (non-monotone dynamics), coverage degrades gracefully (from ~79% to ~73% at 80% reversal) but remains usable.

B. Empirical Application 1: Fee Recognition Lead Times (Monotone Break)

• Context: Shift toward shorter booking lead times during COVID (monotone, ongoing transition).
• Findings:
  • Calibration: The intervention model achieved near-nominal coverage (79.6%), significantly outperforming the fixed-effect model (66.1%) and the baseline (53.6%).
  • Accuracy: Point accuracy was comparable to the fixed-effect model.
  • Full-Matrix Benchmark: Replacing the diagonal AR/MA operators with full matrices worsened both point accuracy and calibration (coverage dropped to ~45%), confirming the diagonal restriction acts as a necessary regularizer in rolling-window settings with limited post-break data.

C. Empirical Application 2: Stay-Length Compositions (Non-Monotone Break)

• Context: Shift in stay lengths with a "spike-and-reversion" pattern (non-monotone).
• Findings:
  • Calibration: Both break-aware models were acceptably calibrated (~83-86%). The fixed-effect model was slightly closer to the 80% target in the pooled sample.
  • Accuracy: The fixed-effect model outperformed the intervention model on point accuracy (Aitchison distance) in the pooled sample, particularly in regions with strong reversion dynamics.
  • Interpretation: The monotone gate cannot fully capture the reversion, leading to wider intervals (preserving calibration) but slightly higher point error compared to the fixed-effect model which settles on a flat post-break level.

5. Significance and Practical Implications

• Decision Rule for Practitioners:
  • Use the Directional-Shift Model when the primary goal is calibrated uncertainty quantification (risk management, scenario analysis) and the structural break is expected to be roughly monotone.
  • Use the Fixed-Effect Model when the primary goal is point accuracy or when post-break dynamics are known to be non-monotone (e.g., temporary spikes followed by reversion).
• Regularization Insight: The study demonstrates that in compositional time series with structural breaks, complex full-matrix dynamic specifications can lead to overfitting in rolling-window settings. The diagonal restriction serves as a crucial regularizing approximation.
• Handling Structural Breaks: The paper provides a principled, geometrically consistent method to model gradual regime changes in compositional data, filling a gap between simple fixed effects and complex, often intractable, change-point detection methods.

In summary, the paper introduces a robust, interpretable Bayesian model that successfully balances the trade-off between capturing complex structural dynamics and maintaining forecast reliability in compositional time series.