What Moved the Alarm Score? Causal Attribution with Exact Computation and Scarce-Incident Estimation

Masaaki Takada; Ryusei Shingaki

stat.MLOct 20, 2026

What Moved the Alarm Score? Causal Attribution with Exact Computation and Scarce-Incident Estimation

Authors: Masaaki Takada, Ryusei Shingaki

Organizations: Toshiba Corporation

Abstract

Alarms in industrial systems and scientific instruments often demand immediate diagnosis despite scarce incident data. To guide intervention, an explanation must trace how changes in the data-generating process propagate through the causal structure to alter the alarm score, rather than merely flagging shifts that may be downstream effects. Yet exact attribution across interacting causal components appears combinatorial, while existing methods often either assume an unchanged causal model or require enough incident data to estimate the changed system reliably. We address this setting by formulating the alarm-score change as a Shapley game over hybrid structural causal model (SCM) states. Treating structural functions and exogenous laws as distinct players yields a finer-grained, function/law-resolved report of how each type of change propagates to the alarm score. The same framework defines realized-event and population reports, with an event--population coherence identity linking their attributions. For ordered linear SCMs and quadratic scores, a causal first- and second-moment recursion combined with quadrature computes the full Shapley vector exactly in polynomial time, eliminating combinatorial computation. To estimate this target from scarce incident data, we adapt a well-observed reference SCM to the incident regime using cross-fitted estimates of mechanism changes and residual moments, and derive attribution-error bounds and top-k recovery guarantees. Experiments on controlled and measured shifts illustrate the diagnoses and assess scaling to hundreds of variables and estimation from small incident samples.

Figures & tables

Report	Causal question	Source	Target	Contrast	Players	Causal estimand
Population shift	Which regime changes moved the mean score?	$M(F^{(0)},Q^{(0)})$	$M(F^{(1)},Q^{(1)})$	$T(M^{(1)})-T(M^{(0)})$	$\mathcal{U}_{\mathtt{F}\cup\mathtt{Q}}$	$\Phi_{u}$ : hybrid expected-score game
Shifted event	Which target-state components explain this event?	$M(F^{(0)},Q^{(0)})$	$M(F^{(1)},\delta_{e^{(1)}})$	$\tau(x)-T(M^{(0)})$	$\mathcal{U}_{\mathtt{F}\cup\mathtt{Q}}$	$\Phi_{u}^{\mathrm{shift}}(e^{(1)})$ : abducted-event game
Fixed-SCM event	Which disturbances explain the event under $M^{(0)}$ ?	$M(F^{(0)},Q^{(0)})$	$M(F^{(0)},\delta_{e^{(0)}})$	$\tau(x)-T(M^{(0)})$	$\mathcal{U}_{\mathtt{Q}}$	$\Phi_{u}^{\mathrm{fix}}(e^{(0)})$ : fixed-function event game

Table 1: Three representative causal score-attribution questions in the source–target–player–score framework.

Figure 1: Same-oracle Shapley aggregation on the full

K=2d

function/law game (mean

\pm

SE over three non-timeout problems; crosses: first 1000-second timeout). Only exact S 4 -RCA reaches

K=1000

. Panel (b) reports relative

\ell_{2}

error; DW-Fast’s diagnosed

K=20

failure is omitted (DoWhy 0.14; Supplement Sec. D.3).

Figure 2: Scarce-incident recovery over 30 ordered

d=100

SCM pairs with balanced function/law oracle mass (mean

\pm

SE). Panels report

F

and

Q

Recall@5 and

F+Q

Recall@10; node rankings are lifted for

F+Q

. Beyond the smallest regime, S 4 -RCA has the highest mean, most clearly for

F

(broader benchmark: Supplement Sec. D.4).

Figure 3: Causal Chamber all-channel score. (a) Target rank (1 best; worst of five splits; m / s : severity). (b) Target-

Q

and downstream- ir_3 mass;

F

denotes the fitted linear mechanism.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Framework	$M^{(0)}$ : source	$M^{(1)}$ : target	Modules $\mathcal{U}$	Scalar $T$	Report in words
S 4 -RCA (ours)	$M(F^{(0)},Q^{(0)})$	$M(F^{(1)},Q^{(1)})$	$\mathcal{U}_{\mathtt{F}\cup\mathtt{Q}}$ : separate $\mathtt{F}_{j},\mathtt{Q}_{j}$	$T(M)=\mathbb{E}_{M}[\tau(X)]$	Allocates a population alarm-score change to structural functions and disturbance laws separately.
Population CCA/MRCCA ( Budhathoki et al. 2021 ; Quintas-Martinez et al. 2024 )	$M(F^{(0)},Q^{(0)})$	$M(F^{(1)},Q^{(1)})$	$\mathcal{U}_{(\mathtt{F},\mathtt{Q})}$ : joint node switches	$T(M)=\psi(P_{Y}^{M})$	Switches each node conditional jointly. MRCCA estimates $T(M)=\mathbb{E}_{M}[h(Y)]$ multiply robustly; with $h(Y)=\tau(X)$ , the game is paired-node S 4 -RCA.
Unit-level change attribution ( Budhathoki et al. 2022a )	$M(F_{Y}^{(0)},\delta_{x^{(0)}})$	$M(F_{Y}^{(1)},\delta_{x^{(1)}})$	$\{\mathtt{F}_{Y},\mathtt{X}_{1},\ldots,\mathtt{X}_{d}\}$ : function/value switches	$T(M)=\mathbb{E}_{M}[Y]=F_{Y}(x)$	Explains one unit’s output change by switching the prediction function and observed input coordinates (fine game), or all inputs jointly (coarse game).
Fixed-SCM outlier RCA ( Budhathoki et al. 2022b )	$M(F^{(0)},Q^{(0)})$	$M(F^{(0)},\delta_{e^{(0)}(x)})$	$\mathcal{U}_{\mathtt{Q}}$ : factual-noise fixes	$T_{x}(M)=\log P_{M}(A_{x})$	Fixes reference noises to their factual values and allocates the information-theoretic score $S^{\tau}_{M^{(0)}}(x)$ .
FANS ( Seo et al. 2026 )	$M(F^{(0)},Q^{(0)})$	$M(F^{(1)},Q^{(1)})$	—; returns $\mathtt{F}_{j}/\mathtt{Q}_{j}$ labels	—	From samples in two environments, detects changed nodes and labels function or noise shifts; it does not allocate a score contrast.
Exact do-Shapley ( Witter et al. 2026 )	$M^{(0)}=M$	$M^{(1)}=M_{\operatorname{do}(X=x)}$	$\{\mathtt{X}_{j}:j\in[d]\}$ : variable interventions	$T(M)=\mathbb{E}_{M}[Y]$	Allocates an interventional outcome to variable interventions, not function/law shifts.

Appendix

Table 1: Closest frameworks translated into this paper’s notation. CCA/MRCCA neither identifies nor separately switches the function/law representation of an induced node conditional. For outlier RCA,

A_{x}=\{\tau(X)\geq\tau(x)\}

and

S^{\tau}_{M^{(0)}}(x)=-\log P_{M^{(0)}}(A_{x})

. Unit-level attribution observes paired input vectors rather than estimating input laws. Its deterministic one-output SCM embedding uses

\delta_{x^{(r)}}=\bigotimes_{j=1}^{d}\delta_{x_{j}^{(r)}}

, so

\mathtt{X}_{j}

switches an observed value, not a population disturbance law. FANS has no attribution modules or scalar game.

Metric	Function/law-resolved	Paired node
Target-Q selected (%)	100.0	0.0
Selected score reduction	1.934 $\pm$ 0.029	0.838 $\pm$ 0.016
Post-repair score shift	-0.543 $\pm$ 0.018	0.552 $\pm$ 0.011
Alarm cleared (%)	100.0	0.0

Appendix

Table 2: Q-actionable repair localization over 100 independent source problems (mean

\pm

standard error). Score reduction is

T(M^{(1)})-T(M^{(1),a})

; post-repair shift is

T(M^{(1),a})-T(M^{(0)})

, so a nonpositive value clears the alarm.

Figure 1: Broad comparison on the same 30 scarce-incident SCMs as Main Figure 2 (mean

\pm

SE). The main figure emphasizes transfer, resolution, and propagation ablations; this figure adds fixed-SCM event attribution, CCA/MRCCA, DCI, iSCAN, and feature-shift SHAP. Single-ranking outputs are copied to both

F

and

Q

labels only for a common localization diagnostic and do not thereby identify change type. Fixed-

F

is the expected-score event mean, not the native information-theoretic tail game. DCI has no

n_{1}=100

result because every such cell reached the 300-second cap.

$n_{1}$	$F$ R@5	$Q$ R@5	$F{+}Q$ R@10
20	0.21 (0.03)	0.18 (0.03)	0.20 (0.02)
40	0.37 (0.03)	0.33 (0.03)	0.30 (0.02)
60	0.46 (0.04)	0.39 (0.05)	0.38 (0.03)
80	0.59 (0.02)	0.45 (0.04)	0.43 (0.02)
100	0.69 (0.03)	0.52 (0.04)	0.47 (0.02)

Appendix

Table 3: Calibrated

d=100

cross-fitted diagnostics over 30 SCMs; entries are mean (SE).

F

and

Q

R@5 are family-specific changed-support recalls, and

F+Q

R@10 ranks the 200 typed players jointly. Relative

\ell_{1}

errors compare the family attribution vectors with their exact oracles. Endpoint errors first average the five folds within each SCM.

Reference-only order estimation
$n_{0}$	Oracle $\ell_{1}$	E2E $\ell_{1}$	E2E R@2
100	0.14 (0.09)	2.11 (0.19)	0.40 (0.12)
200	0.11 (0.11)	1.63 (0.22)	0.55 (0.09)
300	0.16 (0.16)	1.67 (0.20)	0.60 (0.07)
400	0.00 (0.00)	1.12 (0.15)	0.75 (0.08)

Appendix

Table 4: Core-assumption diagnostics over ten

d=5

SCMs (mean (SE)). The three blocks vary reference size, nonlinear-score quartic ratio, and nonlinear-mechanism strength. Oracle

\ell_{1}

isolates population distortion from the learned order or linear approximation; E2E

\ell_{1}

additionally includes endpoint estimation and cross-fitting. R@

k

is changed-player recall.

Polarizer coefficients
	$\to$ ir_3		$\to$ vis_3
Regime	P1	P2	P1	P2
Baseline	$-0.05$	$+0.05$	$-0.05$	$+0.04$
P1-mid	$-0.38$	$+0.35$	$-0.36$	$+0.34$
P1-strong	$-0.10$	$+0.10$	$-0.08$	$+0.09$
P2-mid	$+0.37$	$-0.38$	$+0.35$	$-0.36$

Appendix

Table 5: Reference-standardized polarization projections on the fixed official subgraph. P1/P2 are pol_1 / pol_2 ; coefficients and

R^{2}

use all rows and are descriptive rather than held-out evidence.

(a) Wind Tunnel
Method	H@1	H@3	Med.
S 4 -RCA (function/law)	0.808	1.000	1.0
Fixed- $F$ control	0.769	1.000	1.0
Delta shift	1.000	1.000	1.0
Observed PCA	0.731	0.923	1.0
Residual local KL	1.000	1.000	1.0

Appendix

Table 6: All-channel external interventions. H@

k

requires a target to rank within the top

k

in all five splits; Med. is the median worst rank across 26 Wind or seven Sachs conditions. Primary rows use official-edge support, and order-only S 4 -RCA is a sensitivity. Bold marks the within-dataset best, including ties.

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