ARM: Detector-Agnostic Changepoint Attribution with Finite-Sample Error Control

Technical Summary: ARM (Attribution by Rank Maxima)

Problem Statement
In multivariate change-point detection, identifying when a structural change occurs is distinct from the operational question of which coordinates (variables) are responsible. Standard practice involves taking a detector's estimated changepoint $\hat{\tau}$ and applying a two-sample test to each coordinate at that specific split. The paper argues this approach is statistically invalid. Because $\hat{\tau}$ is selected from the data to maximize aggregate evidence, testing coordinates at this selected point refers a selected statistic to an unselected null distribution. This "double-dipping" inflates the Type-I error rate, a phenomenon that worsens as the dimension $d$ increases. Existing literature offers incomplete solutions: block-level procedures rely on additive union bounds (introducing slack), high-dimensional variable selection methods provide rankings without finite-sample error guarantees, and post-detection inference literature typically controls error along the time axis rather than the coordinate axis.

Methodology: The ARM Wrapper
The authors propose ARM (Attribution by Rank Maxima), a detector-agnostic wrapper that accepts an estimated changepoint $\hat{\tau}$ from any black-box detector but does not rely on its accuracy for validity.

1. Max-Over-Splits Statistic: Instead of testing at $\hat{\tau}$, ARM scores each coordinate $j$ using a statistic $M_j$ defined as the maximum over a grid of candidate splits $t$ and two evidence channels (location and scale):
   $$M_j = \max_{t \in G} \max \{ Z^{\text{loc}}_j(t), Z^{\text{sc}}_j(t) \}$$
   where $Z^{\text{loc}}$ and $Z^{\text{sc}}$ are Wilcoxon location and Mood scale statistics based on within-column ranks. Since $Z_j(\hat{\tau}) \leq M_j$ for any split, the statistic $M_j$ is invariant to the detector's choice of $\hat{\tau}$.

2. Selection-Proofness: Because $M_j$ is computed independently of $\hat{\tau}$, the resulting attribution certificate is "selection-proof." The validity of the test does not depend on whether $\hat{\tau}$ is accurate or misplaced.

3. Error Control Mechanisms:

   • Per-Coordinate Validity: Under the null hypothesis of no marginal change, the distribution of $M_j$ depends only on the sample size and grid, not on the detector or other coordinates.
   • Family-Wise Error Rate (FWER): ARM employs a Westfall–Young joint permutation test. By permuting rows (time points) across all columns simultaneously, the method preserves cross-coordinate dependence. This provides exact strong FWER control. A fully distribution-free Holm fallback is available if subset pivotality conditions are not met.
   • False Discovery Rate (FDR): For high-dimensional settings, ARM controls FDR under arbitrary coordinate dependence using Benjamini–Yekutieli (BY) applied to permutation $p$-values, or the e-BH procedure applied to boosted $e$-values.

4. Type Labeling: Each certified coordinate is assigned a descriptive label ("location" or "scale") based on which channel ($Z^{\text{loc}}$ or $Z^{\text{sc}}$) contributed the maximum value to $M_j$. This label is descriptive and not part of the formal error control statement.

Key Contributions

• Formulation: The paper formulates attribution as a wrapper that provides coordinate-level finite-sample error control valid at an estimated changepoint, eliminating the need for additive union-bound slack.
• Theoretical Guarantees:
  • Theorem 1: Establishes per-coordinate validity under any detector.
  • Theorem 2: Proves exact strong FWER control via joint permutation, preserving dependence structures.
  • Theorem 3: Demonstrates FDR control under arbitrary dependence using BY and e-BH.
• Robustness: The method is distribution-free (relying on ranks) and robust to heavy tails and cross-coordinate dependence.

Results

• Selection Effect: Simulations show that standard per-coordinate testing at $\hat{\tau}$ inflates FWER to over 0.66 as dimension grows (e.g., $d=20$), whereas ARM maintains the nominal level (0.10) regardless of detector misplacement.
• Validity: ARM maintains control under heavy-tailed distributions (Cauchy, $t_3$), various dependence structures (equicorrelated, factor models), and different detector outputs.
• Power and Granularity: In scenarios with strong signals, ARM achieves high F1 scores (0.98+) for individual coordinate attribution, outperforming coarse block tests and providing certified results where ranking-based methods (like WAVE) offer only unguaranteed rankings.
• High Dimensions: At $d=200$, both BY and e-BH procedures successfully control FDR while recovering the full set of changed coordinates.
• Real Data Application: Applied to five financial series surrounding the 2008 Lehman Brothers collapse, ARM attributed a scale change (volatility shift) to all five asset classes while correctly excluding three injected control coordinates. The detector's two-day mislocation of the changepoint did not affect the attribution conclusions.

Significance and Claims
The paper claims that ARM provides the first formulation for changepoint attribution that is simultaneously:

1. Detector-agnostic: Valid regardless of the accuracy of the estimated changepoint.
2. Exact: Offering finite-sample error control without additive union-bound slack.
3. Robust: Valid under heavy tails and arbitrary coordinate dependence.
4. Granular: Providing certificates for individual coordinates rather than predefined blocks.

The authors position ARM as a complete inference layer that can be paired with any estimation layer (e.g., WAVE) or detector. The method refines a "market-wide alarm" into a specific, certified statement about which assets changed and the nature of that change, a capability previously unavailable with rigorous error guarantees. The paper acknowledges limitations, noting that exact control assumes independent rows (serial dependence requires approximation) and that type labels are currently descriptive rather than formally certified.