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ARM: Detector-Agnostic Changepoint Attribution with Finite-Sample Error Control
Chenchen Peng, Mixia Wu, Qijing Yan, Da Chen, Zhiqi Shen
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Abstract
Abstract:Detecting a change in a multivariate series answers only the first of two questions; the operational question is which coordinates changed. Existing answers are incomplete. Block-level procedures certify predefined groups of coordinates under an additive union bound, high-dimensional variable-selection methods return interpretable rankings without error guarantees, and the post-detection inference literature controls error along the time axis rather than across coordinates. We propose ARM (Attribution by Rank Maxima), a wrapper that accepts a changepoint located by an arbitrary detector and returns the set of coordinates certified to have changed, each carrying a location or scale type label. ARM scores each coordinate by a max-over-splits rank statistic. Because this statistic dominates the corresponding statistic at the estimated split, the resulting certificate is invariant to the manner, and to the accuracy, of the changepoint estimate. Three finite-sample guarantees follow from within-coordinate ranks alone: per-coordinate validity under any detector; exact family-wise error control through a Westfall--Young joint permutation that preserves cross-coordinate dependence, with a fully distribution-free Holm fallback; and false discovery rate control under arbitrary coordinate dependence in high dimensions through Benjamini--Yekutieli and e-BH. In simulations, naive per-coordinate testing at the estimated changepoint inflates its family-wise error beyond $0.66$ as the dimension grows, whereas ARM maintains the nominal level while retaining validity under heavy tails, power in high dimensions, and accurate type labels. On five financial series surrounding the 2008 collapse, ARM attributes a scale change to every asset class and excludes injected control coordinates.
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ARM: Detector-Agnostic Changepoint Attribution with Finite-Sample Error Control Chenchen Peng School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China College of Computing and Data Science, Nanyang Technological University, Singapore 639798, Singapore Mixia Wu School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China Qijing Yan School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China Da Chen Corresponding author. Email: dachensmu@163.com College of Electrical Engineering, Sichuan University, Chengdu, Sichuan 610065, China College of Computing and Data Science, Nanyang Technological University, Singapore 639798, Singapore Zhiqi Shen College of Computing and Data Science, Nanyang Technological University, Singapore 639798, Singapore (August 2, 2026) Abstract Detecting a change in a multivariate series answers only the first of two questions; the operational question is which coordinates changed. Existing answers are incomplete. Block-level procedures certify predefined groups of coordinates under an additive union bound, high-dimensional variable-selection methods return interpretable rankings without error guarantees, and the post-detection inference literature controls error along the time axis rather than across coordinates. We propose ARM (Attribution by Rank Maxima), a wrapper that accepts a changepoint located by an arbitrary detector and returns the set of coordinates certified to have changed, each carrying a location or scale type label. ARM scores each coordinate by a max-over-splits rank statistic. Because this statistic dominates the corresponding statistic at the estimated split, the resulting certificate is invariant to the manner, and to the accuracy, of the changepoint estimate. Three finite-sample guarantees follow from within-coordinate ranks alone: per-coordinate validity under any detector; exact family-wise error control through a Westfall–Young joint permutation that preserves cross-coordinate dependence, with a fully distribution-free Holm fallback; and false discovery rate control under arbitrary coordinate dependence in high dimensions through Benjamini–Yekutieli and e-BH. In simulations, naive per-coordinate testing at the estimated changepoint inflates its family-wise error beyond 0.660.66 as the dimension grows, whereas ARM maintains the nominal level while retaining validity under heavy tails, power in high dimensions, and accurate type labels. On five financial series surrounding the 2008 collapse, ARM attributes a scale change to every asset class and excludes injected control coordinates. Keywords: changepoint attribution; post-detection inference; rank statistics; family-wise error; false discovery rate; permutation test 1 Introduction When a monitoring system flags a structural change in a multivariate stream, whether a sensor array, a portfolio, or a network of meters, the question of when the change occurred is answered by the detector itself and refined by a substantial literature [11]. The operational question that follows is which coordinates are responsible: which sensors drifted, which assets drove the regime shift, which meters failed. Standard practice takes the detector’s estimated changepoint τ τ and applies a two-sample test to each coordinate across it. This practice is invalid. The estimate τ τ is computed from the same data, at the location that maximizes aggregate evidence, so testing each coordinate at τ τ refers a selected statistic to an unselected null distribution and inflates the type-I error, increasingly severely as the dimension grows; Section 5.1 quantifies the inflation. Three lines of work address related problems. The post-detection inference literature controls error rigorously along the time axis, testing which alarms correspond to genuine changes [7, 3] or constructing confidence statements for the location [6, 20, 19], but does not address the coordinate axis. For the attribution question itself, Ouerfelli et al. [16] give the first formal treatment: kernel two-sample tests certify whether the change occurs in either of two predefined blocks of coordinates, with the family-wise error rate bounded additively as FWER≤α0+Lα1FWER≤ _0+L _1 over L blocks and with calibration based on a holdout split. Finally, a strand of high-dimensional methods, exemplified by WAVE [8], produces coordinate-level attribution by fusing adaptive weights and local effect sizes into a ranking; a ranking is informative but carries no finite-sample error guarantee. What is missing is attribution formulated as a detector-agnostic wrapper with coordinate-level finite-sample error control that remains valid at an estimated changepoint, providing exact family-wise error control without additive union-bound slack, false discovery rate control robust to coordinate dependence in high dimensions, and a change-type label for every certified coordinate. The present paper supplies this formulation. We propose ARM (Attribution by Rank Maxima). The method scores coordinate j by the max-over-splits rank statistic Mj=maxtmaxZjloc(t),Zjsc(t)M_j= _t \Z^loc_j(t),Z^sc_j(t)\, which combines Wilcoxon location and Mood scale evidence over a grid of candidate splits. Since Zj(τ^)≤MjZ_j( τ)≤ M_j for every split, the certificate neither uses nor can be invalidated by the detector’s estimate (Theorem 1). Three guarantees then follow from within-coordinate ranks alone: per-coordinate validity under any detector (Theorem 1); exact strong family-wise error control through a Westfall–Young joint permutation that preserves cross-coordinate dependence, with a distribution-free Holm fallback (Theorem 2); and false discovery rate control under arbitrary coordinate dependence through Benjamini–Yekutieli and e-BH (Theorem 3). Each certified coordinate additionally carries a location or scale label determined by its dominant evidence channel. Every reported number derives from executed code and archived outputs; the experiments cover the selection effect of naive testing at τ τ, validity under heavy tails and cross-coordinate dependence, high-dimensional power, type-label accuracy, and an application to five financial series surrounding the 2008 crisis. Section 2 places ARM among the three lines above; Section 3 defines the wrapper; Section 4 states the theory; Sections 5–6 report simulations and the financial study; Section 7 discusses limitations. 2 Background: Attribution Without Certificates 2.1 Post-detection inference: time axis versus coordinate axis Given detected changepoints, one may ask which are genuine. TUNE [7] and the ART diagnostic [3, Thm. 5] control a family-wise error rate over detected times, algorithm-agnostically and distribution-freely; selective and sequential procedures [6, 20, 19] give post-detection confidence statements about the location. These are guarantees along the time axis. ARM’s Theorem 2 transposes them to the coordinate axis, applying the same pivotal distribution-free mechanism across the d coordinates of a single window rather than across candidate times; the proof pattern is inherited directly from this literature. 2.2 Attribution: granularity and guarantee Table 1 places the coordinate-attribution methods on four axes. Ouerfelli et al. [16] give the first formal post-detection attribution: for two predefined coordinate blocks, kernel maximum-mean-discrepancy tests across τ τ, calibrated by a concentration threshold and a holdout split, with FWER≤α0+Lα1FWER≤ _0+L _1 by a union bound over L blocks. Remark 9 of that paper recommends Bonferroni or Holm corrections when more than two blocks are tested, and its Remark 8 places purely inter-block dependence changes with invariant marginals outside the scope of marginal procedures. WAVE [8] and related high-dimensional identification methods [13, 26] attribute at the coordinate level but through weights or effect rankings, without a finite-sample false-positive guarantee. ARM advances all four axes at once: coordinate granularity, exact rather than additively bounded FWER, rank-permutation rather than concentration-threshold calibration, and FDR under arbitrary coordinate dependence. The two-block design of Ouerfelli et al. [16] is recovered as the aggregation special case in which a block is scored by the maximum of its coordinate statistics. Table 1: Coordinate-attribution methods on four axes. ARM occupies the last row. Method Granularity Guarantee Calibration Dependence Ouerfelli et al. predefined blocks FWER ≤α0+Lα1≤ _0+L _1 kernel MMD + holdout independent rows WAVE coordinate none (ranking) block bootstrap block ART Thm. 5 / TUNE time (alarms) exact FWER rank permutation independent ARM (this paper) coordinate exact FWER & arb.-dep. FDR rank permutation arbitrary (e-BH) 2.3 Multiple testing tools ARM assembles three classical instruments in the rank-permutation setting: Westfall–Young single-step maxT with the subset-pivotality condition for strong FWER [24, 14], the exactness of Monte Carlo permutation tests [5, 10], and, for FDR under arbitrary dependence, Benjamini–Yekutieli [2] and the recent e-BH procedure [23]. 3 Methodology: The ARM Wrapper 3.1 Setting and black-box interface The window is X∈ℝn×dX ^n× d: n time points (rows) and d coordinates (columns). Rows are independent; columns may be arbitrarily dependent. A black-box detector returns an estimated changepoint τ τ; ARM never inspects its internals and tolerates an arbitrary τ τ. For coordinate j, the null hypothesis is H0j:X1j,…,Xnjare identically distributed (no marginal change in coordinate j).H_0j:\ X_1j,…,X_nj\ are identically distributed (no marginal change in coordinate j). The true-null set is J0=j:H0jholdsJ_0=\j:H_0j\ holds\; attribution seeks the complement while controlling false positives over J0J_0. 3.2 Coordinate rank channels and the max-over-splits statistic Let R⋅jR_· j be the within-column ranks of coordinate j and Cij=Rij−(n+1)/2C_ij=R_ij-(n+1)/2 the centred ranks. For a split t on a grid ⊆m0,…,n−m0G \m_0,…,n-m_0\, the location [25] and scale [15] evidence are Zjloc(t)=|∑i≤tCij|t(n−t)(n+1)/12,Zjsc(t)=|∑i≤t(Cij2−Cj2¯)|t(n−t)Var(Cj2)/n.Z^loc_j(t)= | _i≤ tC_ij| t(n-t)(n+1)/12, Z^sc_j(t)= | _i≤ t(C_ij^2- C^2_j)| t(n-t)\,Var(C^2_j)/n. ARM’s coordinate statistic is the maximum over splits and channels, Mj=maxt∈maxZjloc(t),Zjsc(t),Mjloc=maxtZjloc(t),Mjsc=maxtZjsc(t),M_j= _t \Z^loc_j(t),Z^sc_j(t)\, M^loc_j= _tZ^loc_j(t), M^sc_j= _tZ^sc_j(t), (1) with Mjloc,MjscM^loc_j,M^sc_j retained for the type label. 3.3 Attribution rule and selection-proofness ARM attributes coordinate j when MjM_j exceeds a distribution-free critical value: Aα=j:Mj>cn,αA_α=\j:M_j>c_n,α\, where cn,αc_n,α is the (1−α)(1-α) quantile of M under a single null coordinate, computed once by permuting a rank vector (it depends only on n and the grid, never on the data). The decisive choice is to score by MjM_j rather than by Zj(τ^)Z_j( τ). Since Zj(τ^)≤MjZ_j( τ)≤ M_j for every split, no detector output, whether accurate or arbitrarily misplaced, can raise a true-null coordinate’s statistic above the level that the maximum already attains; attribution is therefore selection-proof by construction (Theorem 1), and τ τ affects the power of the procedure but not its validity. The per-coordinate permutation p-value is pj=(1+#null draws≥Mj)/(B+1)p_j=(1+\#\null draws≥ M_j\)/(B+1). 3.4 FWER: Westfall–Young joint permutation A single time permutation π is applied to all columns at once (row permutation), preserving cross-coordinate dependence; the reference is the permutation law of maxjMj(Xπ) _jM_j(X_π). The single-step maxT adjusted p-value is p~j=(1+#b:maxkMk(Xπb)≥Mj)/(Bwy+1) p_j=(1+\#\b: _kM_k(X_ _b)≥ M_j\)/(B_wy+1), and ARM rejects H0jH_0j when p~j≤α p_j≤α. Under subset pivotality (Section 4) this controls FWER exactly. A fully distribution-free fallback applies Holm to the per-coordinate pjp_j, valid with no dependence assumption at a mild power cost. 3.5 FDR in high dimensions For large d we control the false discovery rate under arbitrary coordinate dependence. Benjamini–Yekutieli [2] applied to the permutation p-values pays the harmonic penalty cd=∑i≤d1/ic_d= _i≤ d1/i and is our robust default. As a modern alternative we form boosted threshold e-values ej=s−1pj≤se_j=s^-11\p_j≤ s\ (valid since H0jej=ℙ(pj≤s)/s≤1E_H_0je_j=P(p_j≤ s)/s≤ 1) and apply e-BH [23], which controls FDR under arbitrary dependence for every choice of s; the choice affects power only. The mechanics fix the rule: e-BH rejects k coordinates only if s−1≥d/(αk)s^-1≥ d/(α k), so s must not exceed αk/dα k/d for the smallest number of changed coordinates one intends to detect, and pj≤sp_j≤ s must be reachable, requiring B≥1/sB≥ 1/s permutations. In the study below we use s=0.004s=0.004, satisfying this rule for k=15k=15 changed coordinates at d=200d=200 (αk/d=0.0075α k/d=0.0075) with B=1999B=1999; Section 5.4 reports the sensitivity across s, and we emphasize that FDR control holds for every s; the rule only locates the power boundary. 3.6 Type label Each attributed coordinate is labelled scale if Mjsc>MjlocM^sc_j>M^loc_j and location otherwise. The label is descriptive and is not part of any error-control statement; a certified type label would require closed testing, which we leave to future work. Figure 1 summarizes the wrapper. black-boxdetector →τ^→ τ(any algorithm)window XXn×dn× dlocation channelZjloc(t)Z^loc_j(t)scale channelZjsc(t)Z^sc_j(t)Mj=maxtmax(⋅)M_j= _t (·)(max over splits)WY joint perm./ Holm / e-BHcertified set AαA_α+ type labelper coordinateranks only ⇒ distribution-freeZj(τ^)≤Mj⇒Z_j( τ)≤ M_j selection-proof (Thm. 1) Figure 1: The ARM wrapper. Any detector supplies τ τ; ARM scores each coordinate by a max-over-splits rank statistic (so the certificate ignores τ τ) and calibrates by joint permutation (FWER) or e-BH/BY (FDR), returning a certified coordinate set with type labels. 4 Theory Theorem 1 (Per-coordinate validity, any detector). Under H0jH_0j, the law of MjM_j depends only on (n,)(n,G), and not on the marginal FjF_j, the other coordinates, or the detector. Hence for any α∈(0,1)α∈(0,1) and any rule producing τ τ, ℙ(j∈Aα)≤αP(j∈ A_α)≤α. The proof (Appendix A) is that MjM_j is a function of coordinate j’s ranks, which under H0jH_0j are a uniform random permutation; the statement holds for every τ τ because MjM_j never uses τ τ. Theorem 2 (Exact strong FWER). Suppose the rows of the true-null sub-array (Xi,J0)i=1n(X_i,J_0)_i=1^n are independent and identically distributed, so that the joint law of the null coordinates, including their mutual dependence, is constant over time. Then the Westfall–Young single-step maxT of Section 3.4 controls the family-wise error rate over J0J_0 in the strong sense, ℙ(∃j∈J0:j∈Aα)≤αP(∃ j∈ J_0:j∈ A_α)≤α. The condition is subset pivotality: since each MjM_j depends only on column j, the joint null law of Mj:j∈J0\M_j:j∈ J_0\ under the row permutation is unaffected by whether the non-null coordinates changed (Appendix A). A pure change in the dependence among null coordinates with invariant marginals makes the null rows non-identically distributed and is out of scope; this is the same boundary drawn by Ouerfelli et al. [16, Remark 8]. When the condition is in doubt, the Holm fallback on pjp_j is exact under the marginal nulls alone, with no assumption on the joint law. Theorem 3 (FDR under arbitrary dependence). For any joint dependence among the d coordinates, Benjamini–Yekutieli on pj\p_j\ and e-BH on the boosted e-values ej\e_j\ each control the false discovery rate at level α. BY controls FDR under arbitrary dependence by the harmonic-penalty theorem [2]; the eje_j are valid e-values, so e-BH controls FDR with no dependence assumption [23] (Appendix A). Proposition 1 (Support recovery). Fix a coordinate j⋆j with a stochastically ordered marginal change of fixed size. Then Mj⋆→∞M_j →∞ in probability as n→∞n→∞, so ℙ(j⋆∈Aα)→1P(j ∈ A_α)→ 1; consequently, with a bounded number of changed coordinates, the FWER and FDR procedures recover the full changed set with probability tending to one. Proposition 1 (proof sketch in Appendix A) states power as a limit; the finite-sample behaviour is the simulation study, reported in full. 5 Simulation Studies All numbers below are read from the generated result files (run_experiments.py). Windows have n=120n=120, changepoint τ=60τ=60, margin m0=10m_0=10, nominal α=0.10α=0.10; the null reference uses B=999B=999 rank permutations and the Westfall–Young reference Bwy=499B_wy=499. Monte Carlo standard errors are about 0.020.02 at 250250–300300 replications. 5.1 Testing at an estimated changepoint: the selection effect Each window contains two changed coordinates, so that the detector locks onto a meaningful τ τ, and the family-wise error is measured over the remaining d−2d-2 true-null coordinates. ARM is compared against the naive per-coordinate procedure, in which each coordinate’s statistic at the estimated split is referred to a fixed-split null distribution. Figure 2 and Table 2 report the results. The naive error reaches 0.330.33 at d=5d=5 and increases to 0.670.67 at d=10d=10 and 0.890.89 at d=20d=20: the detector selects the split at which the aggregate evidence peaks, and that selection contaminates every per-coordinate test performed there. ARM maintains the nominal 0.100.10 throughout, with entries between 0.0520.052 and 0.1080.108 across all sixteen cells, and is flat in the detector misplacement |τ^−τ|| τ-τ| because MjM_j does not involve τ τ, whereas the naive test remains inflated at every misplacement. Two remarks concern the fairness of this comparison. First, the naive procedure is not an artificial comparator: treating τ τ as known and applying a per-coordinate two-sample test at it is precisely the documented double-dipping practice that motivates post-detection inference [6, 7]. Second, the principled correction of that practice, namely an adjustment for the maximization over splits, is exactly what ARM implements, not approximately through a Bonferroni penalty over the grid but exactly through the permutation law of the maximum. The inflation measured here and the validity of the method are consequences of the same inequality Zj(τ^)≤MjZ_j( τ)≤ M_j. Table 2: Selection effect: FWER over the null coordinates, 600600 replications (t3t_3 marginals, two changed coordinates, aggregate-CUSUM detector). Naive per-coordinate testing at τ τ inflates with d; ARM maintains the nominal level. Columns are detector misplacements |τ^−τ|| τ-τ|. naive at τ τ ARM (WY) d 0 5 10 20 0 5 10 20 5 0.33 0.27 0.31 0.29 0.055 0.060 0.083 0.052 10 0.67 0.62 0.56 0.56 0.082 0.097 0.072 0.077 20 0.89 0.84 0.86 0.84 0.085 0.082 0.093 0.095 40 0.99 0.98 0.99 0.98 0.108 0.103 0.100 0.107 Figure 2: The selection effect. (a) FWER over the null coordinates against dimension, with the detector at τ^=τ τ=τ: naive testing inflates toward one while ARM maintains the nominal 0.100.10. (b) FWER against detector misplacement |τ^−τ|| τ-τ| at d=20d=20: ARM is flat because its statistic does not involve τ τ, whereas the naive test remains inflated at every misplacement. 5.2 Validity under heavy tails, dependence, and any detector Table 3 sweeps d=10d=10 with three changed coordinates over four marginals, including the heavy-tailed and skewed regimes that motivate distribution-free tools [22, 12], three cross-coordinate dependence structures, and three detectors: the true τ, an aggregate-CUSUM estimate, and a deliberately misplaced τ^=τ+10 τ=τ+10. Every WY and Holm FWER entry sits at or within Monte Carlo error of 0.100.10 (range 0.0320.032–0.1080.108 over all 3636 cells), and every BY FDR entry is far below (range 0.0100.010–0.0290.029, reflecting three true signals among ten coordinates). The detector variant changes the entries by at most Monte Carlo error, confirming that the guarantee does not depend on how τ τ was produced, as Theorem 1 predicts. Table 3 reports the CUSUM-detector column; the true-τ and misplaced columns differ by at most 0.030.03 and are contained in the archived results. Table 3: Validity at d=10d=10, three changed coordinates, CUSUM detector, 250250 replications (MC-SE ≈0.019≈ 0.019 for FWER). WY and Holm control FWER; BY controls FDR. Heavy tails and cross-coordinate dependence do not break control. Marginal Dependence WY FWER Holm FWER BY FDR Gaussian independent 0.064 0.072 0.018 equicorrelated 0.076 0.072 0.018 factor 0.088 0.092 0.022 t3t_3 independent 0.080 0.084 0.021 equicorrelated 0.076 0.088 0.023 factor 0.072 0.076 0.020 Cauchy independent 0.044 0.036 0.011 equicorrelated 0.060 0.060 0.016 factor 0.064 0.076 0.019 Lognormal independent 0.072 0.084 0.022 equicorrelated 0.044 0.052 0.013 factor 0.072 0.084 0.021 5.3 Power and granularity We compare ARM’s certified coordinate attribution against two baselines on d=20d=20 with three changed coordinates: a simplified two-block test in the style of Ouerfelli et al. [16], implemented with the energy distance [21] in place of the kernel maximum mean discrepancy and without the holdout split, which certifies only a coarse block; and a WAVE-style fusion [8] that returns a coordinate ranking without error control, scored by the AUC of signal coordinates over null coordinates. Table 4 reports a strong (δ=1.8δ=1.8) and a weak (δ=1.1δ=1.1) signal regime. Two conclusions follow. First, ARM’s certified attribution is essentially exact under the strong signal, with F1 between 0.9850.985 and 0.9870.987, and degrades gracefully under the weak signal, with F1 between 0.880.88 and 0.990.99. Second, the comparison concerns the form of the output rather than detection quality: at these signal levels all three methods locate the changed coordinates, and WAVE’s ranking AUC remains above 0.990.99, but WAVE returns a ranking without a certificate, the block test certifies only a coarse block, and only ARM returns individual coordinates with a finite-sample guarantee. Table 4: Attribution quality at d=20d=20, three changed coordinates, 200200 replications. ARM’s coordinate-level F1 versus a block test’s coarse block-hit rate and a WAVE-style ranking AUC. The baselines find the signals; only ARM certifies individual coordinates. Signal Marginal ARM coord. F1 block hit (coarse) WAVE rank AUC δ=1.8δ=1.8 Gaussian 0.987 1.000 1.000 t3t_3 0.985 1.000 1.000 Lognormal 0.986 0.995 1.000 δ=1.1δ=1.1 Gaussian 0.980 1.000 1.000 t3t_3 0.877 0.945 0.995 Lognormal 0.986 0.890 1.000 5.4 High-dimensional FDR At d=200d=200 with 1515 changed coordinates (Figure 3), both FDR procedures control the rate under independent and equicorrelated (ρ=0.4ρ=0.4) coordinates while recovering the full support. BY reports FDR 0.0190.019 (independent) and 0.0170.017 (equicorrelated) at power 1.001.00; e-BH, with the boosted threshold s=0.004s=0.004 and B=1999B=1999, reports FDR 0.0430.043 and 0.0480.048 at power 1.001.00. BY is the tighter default here; e-BH is the option when only an e-value, not a p-value, is available or when combining evidence across analyses. Its power depends on s relative to the sparsity exactly as the rule of Section 3.5 predicts. In an archived sensitivity run under the equicorrelated design, the values s∈0.002,0.004,0.006s∈\0.002,0.004,0.006\, all below the boundary αk/d=0.0075α k/d=0.0075, give power 1.001.00 at FDR between 0.0270.027 and 0.0600.060, while s=0.008s=0.008, just above the boundary, reduces power to 0.660.66 at FDR 0.0820.082. The FDR remains controlled at every s, as Theorem 3 guarantees; only power is affected. Figure 3: High-dimensional attribution (d=200d=200, 1515 changed). BY and e-BH both hold FDR at or below the nominal α=0.10α=0.10 under independent and equicorrelated coordinates while recovering the full support (power 1.001.00). 5.5 Type labels and ablations Type labelling is nearly perfect: over 300300 replications with four location-shift and four scale-shift coordinates, attributed location coordinates were labelled location 1200/12001200/1200 times and attributed scale coordinates scale 1058/10601058/1060 times (two mislabels). Ablations confirm robustness: FWER and power are unchanged across B∈199,499,999B∈\199,499,999\ null permutations (FWER 0.0530.053, power 1.001.00 throughout), and coarsening the split grid from step 11 to step 44 leaves FWER in [0.073,0.087][0.073,0.087] at power 1.001.00, cutting the permutation cost roughly fourfold. 6 Real Data: Attributing the 2008 Regime Shift We take five daily financial series from FRED [4], namely the NASDAQ Composite, the 10-year Treasury yield, WTI crude oil, the USD/EUR rate, and the VIX, taken as log-return or first-difference coordinates over June 2007 to June 2009, and analyse a 120120-day window centred on the Lehman Brothers collapse of 15 September 2008. To make the family-wise guarantee demonstrable we append three control coordinates that carry no regime change: an i.i.d. resample of the pre-window exchange-rate returns and two i.i.d. noise series matched in scale. The injection serves a methodological purpose. The real data provide no ground-truth null coordinates, since all five asset classes plausibly changed in 2008, and without controls the error-control claim would be unfalsifiable on this window; the controls, built from the window’s own quiet period and fully disclosed, supply true-null coordinates on which a false attribution could occur. A black-box variance-CUSUM detector locates τ τ two days after Lehman; the score reference and thresholds are those of Section 5, with nothing tuned on the financial data. ARM attributes a change to all five financial coordinates and to none of the three controls, under WY, Holm, and BY alike (Table 5, Figure 4). Every attributed coordinate is labelled scale: the crisis is read as a volatility regime shift. Mean effects are present, notably the fall in equities, but the scale channel dominates the location channel in every coordinate, which is the descriptive content of the label. The evidence orders the asset classes by the size of the volatility jump (equities M=6.0M=6.0, oil 5.65.6, Treasury 4.54.5, USD/EUR 4.34.3, VIX 4.14.1), all clearing the WY threshold c=3.6c=3.6, while the three controls (M=2.0M=2.0–2.32.3) fall well short. The detector’s two-day mislocation perturbs none of these conclusions, because the certificate is computed from MjM_j rather than from the statistic at τ τ. Table 5: ARM on the 2008 window: coordinate statistic MjM_j, type label, and attribution decision (WY, α=0.10α=0.10, threshold c=3.6c=3.6). Five asset classes are certified as scale changes; three injected controls are correctly excluded. Coordinate MjM_j type attributed? equities (NASDAQ) 6.04 scale yes oil (WTI) 5.56 scale yes Treasury (10y) 4.50 scale yes USD/EUR 4.29 scale yes VIX 4.11 scale yes control (fx resample) 2.17 — no control (Gaussian) 1.99 — no control (t5t_5) 2.26 — no Figure 4: Attributing the 2008 regime shift. (a) Standardized absolute returns: a volatility jump appears across the five financial coordinates after Lehman (dashed line) while the three controls stay flat. (b) ARM certificate: the five asset classes (blue) clear the WY threshold (dashed) with scale-type labels; the three controls (grey) do not. 7 Discussion Four limitations bound the claims. First, exact error control assumes independent rows; under serial dependence a block-permutation variant restores validity only approximately, and a rank theory for dependent rows [1] is the natural next step. Second, the type label is descriptive: a certified label would require closed testing across the location and scale channels, which we leave open. Third, a pure change in the dependence among coordinates with invariant marginals lies outside the marginal-attribution model, exactly as in Ouerfelli et al. [16, Remark 8], and its detection requires a dedicated dependence test. Fourth, the predefined-block setting of Ouerfelli et al. [16] constitutes structural prior information that ARM does not exploit; when blocks are genuinely known, aggregating ARM statistics within blocks recovers that setting as a special case. Two directions follow. ARM and a distribution-free localization set [e.g. 17] compose into a complete post-detection toolkit that reports when the change occurred, with a confidence set, and which coordinates changed, with a certificate; the rank channels are shared between the two procedures, inherited from the ART-N line [18]. An online variant would replace the permutation reference with an anytime-valid one [9]. Finally, an estimation layer such as WAVE [8], which produces coordinate weights without a guarantee, pairs naturally with ARM as its inference layer: the estimation layer nominates candidate coordinates and the inference layer certifies them. 8 Conclusion Detection establishes when a change occurred; ARM establishes which coordinates changed, with certificates that an incorrectly estimated changepoint cannot invalidate. Scoring each coordinate by a max-over-splits rank statistic renders attribution selection-proof by construction; a joint permutation delivers exact family-wise error control while preserving cross-coordinate dependence, and Benjamini–Yekutieli and e-BH extend the guarantee to false discovery rate control under arbitrary dependence in high dimensions. The certificate is valid under heavy tails and under any detector, and on the 2008 data it refines a market-wide alarm into the statement that every asset class underwent a volatility change while the injected controls did not. Declarations Data availability. All results are reproducible from the accompanying code (arm_core.py, run_experiments.py, real_data_panel.py; fixed seeds). The five financial series are public FRED series (NASDAQCOM, DGS10, DCOILWTICO, DEXUSEU, VIXCLS) retrieved from https://fred.stlouisfed.org/. Conflict of interest. The authors declare no conflict of interest. Appendix A Proofs A.1 Proof of Theorem 1 Fix j and condition on H0jH_0j: the entries X1j,…,XnjX_1j,…,X_nj are i.i.d., so their rank vector R⋅jR_· j is uniform on the permutations of 1,…,n\1,…,n\, independent of the coordinate’s marginal FjF_j and of every other column. The statistic MjM_j in (1) is a fixed function of R⋅jR_· j alone, so its law depends only on n and the grid G; the critical value cn,αc_n,α is the exact (1−α)(1-α) quantile of that law, whence ℙ(Mj>cn,α)≤αP(M_j>c_n,α)≤α. Because MjM_j does not involve τ τ at any point, the bound holds simultaneously for every rule that could have produced τ τ; in particular Zj(τ^)≤MjZ_j( τ)≤ M_j shows that testing at the estimated split can only be less significant than the max, never more. ∎ A.2 Proof of Theorem 2 Write J0J_0 for the true-null coordinates and let π be a uniform random time permutation applied jointly to all columns. Under the stated condition the sub-array (Xi,J0)i=1n(X_i,J_0)_i=1^n is exchangeable in i, so for the J0J_0-columns the row-permuted window XπX_π has the same joint law as X; consequently the joint law of Mj(Xπ):j∈J0\M_j(X_π):j∈ J_0\ equals that of Mj(X):j∈J0\M_j(X):j∈ J_0\, and neither depends on the values of the non-null columns because each MjM_j is a function of column j only. This is the subset-pivotality condition of Westfall and Young [24]. The single-step maxT reference maxkMk(Xπ) _kM_k(X_π) dominates maxj∈J0Mj(Xπ) _j∈ J_0M_j(X_π), so the adjusted p-values p~j p_j are jointly valid and ℙ(∃j∈J0:p~j≤α)≤αP(∃ j∈ J_0: p_j≤α)≤α by the standard permutation argument [5, 14]. ∎ A.3 Proof of Theorem 3 The permutation p-values pjp_j are each super-uniform under H0jH_0j (Theorem 1), so Benjamini–Yekutieli with the harmonic factor cd=∑i≤d1/ic_d= _i≤ d1/i controls the FDR at α for arbitrary joint dependence [2]. For the e-BH branch, ej=s−1pj≤se_j=s^-11\p_j≤ s\ satisfies H0j[ej]=ℙ(pj≤s)/s≤1E_H_0j[e_j]=P(p_j≤ s)/s≤ 1 since pjp_j is super-uniform, so each eje_j is a valid e-value and e-BH controls the FDR at α under arbitrary dependence [23]. ∎ A.4 Proof sketch of Proposition 1 For a coordinate with ρ=ℙ(X′>X)≠12ρ=P(X >X)≠ 12 across a fixed-fraction split, the Wilcoxon evidence at the true split is a two-sample U-statistic with mean of order n|ρ−12| n\,|ρ- 12|, so Mj⋆≥Zj⋆loc(τ)→∞M_j ≥ Z^loc_j (τ)→∞ in probability while the null critical value cn,αc_n,α grows only as logn n (the maximum of O(n)O(n) standardized rank CUSUMs with sub-Gaussian tails), and the Westfall–Young critical value adds only a logd d-order term with d held fixed. Hence ℙ(j⋆∈Aα)→1P(j ∈ A_α)→ 1, and with d fixed and a bounded number of changed coordinates a union bound gives simultaneous recovery. The scale channel is analogous with squared centred ranks. 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