GRAIL: From Saliency to Semantics in Time-Series Explanation

1 Introduction

• Predictive patterns span multiple samples, making per-timestep saliency high-dimensional, unstable, and difficult for domain experts to interpret.A sample-level score may localize evidence without expressing its regional pattern or clinical meaning.Source: §1 · p. 1
• GRAIL uses contiguous, named regions as explanation units, such as ECG waves, injected pulses, seasonal stretches, or trends.These few human-readable regions carry domain knowledge about which patterns distinguish outcomes.Source: §1 · p. 1
• GRAIL places temporal regions and non-temporal covariates such as age and sex on one scale so their contributions can be compared.This makes model reliance on demographics alongside waveform morphology directly readable.Source: §1 · p. 2
• GRAIL represents regions and covariates as cooperative-game players and generates absent regions on-manifold instead of using fixed fillers.Its interpretable codec encodes regions as named features and conditionally regenerates them, keeping coalitions plausible real signals.Source: §1 · p. 2
• GRAIL estimates exactly complete Owen values with a common-random-number sweep and amortizes the game into a fast single-pass surrogate.The framework also reports the residual and supports deployment through the amortized model.Source: §1 · p. 2

2 Related Work

• Most time-series explainers report per-timestep saliency, while coarser methods group timesteps into windows or segments and use fixed off-manifold fillers for absent states.Examples include WindowSHAP [Nayebi et al. 2023], TimeSHAP [Bento et al. 2021], TS-MULE [Schlegel et al. 2021], LIMESegment, and Dynamask [Crabbé and Van Der Schaar 2021].Source: §2 · p. 2
• GRAIL replaces fixed removal with conditional on-manifold generation and lifts the flat Shapley game to two-level Owen values over features nested within regions.It evaluates methods under matched on- and off-manifold operators because perturbation faithfulness is confounded by the removal operator (Hooker et al. 2019; Jethani et al. 2021).Source: §2 · pp. 2–3
• C-SHAP [Jutte et al. 2025] and GroupSegment-SHAP [Kim and Park 2026] use time-series players but retain mean-replacement removal and do not admit non-temporal covariates.CausalConceptTS [Lopez Alcaraz and Strodthoff 2025] uses a generative premise but imputes over full recordings; ManifoldShap [Taufiq, Blöbaum, and Minorics 2023] uses an on-manifold interventional background rather than generating it.Source: §2 · pp. 2–3

3.1 Defining the Cooperative Game

• GRAIL defines a cooperative game whose players are human-readable features nested within semantic regions, with static covariates appended as singleton players.The framework evaluates temporal and tabular evidence on one scale.Source: §3 · p. 3, §3.1 · p. 3
• For each coalition, present features retain observed values while absent features are sampled from a reference and used to regenerate their region conditionally.The reference may be marginal or class-conditional; the default class-conditional reference uses the normal cohort, so feature attributions read as deviations from normal.Source: §3.1 · p. 3

3.2 Playing the Game On-manifold

• The default on-manifold generator is a Gaussian-process posterior, and Owen values attribute feature contributions consistently across their region blocks.A common-random-number sweep reuses randomness across coalitions; a morphology-residual player restores exact completeness when the codec only approximately reconstructs the input.Source: §3.2 · pp. 3–4
• When region priors are unavailable, GRAIL-auto combines PELT change points with a model-aware selector and snaps boundaries to clean-cut points without retraining.The selector uses a thresholded mask and moves boundaries to the nearest point minimizing a cleanliness cost.Source: §3.2 · p. 4

3.3 Amortizing for Deployment with GRAIL-fast

• GRAIL-fast trains a single-pass network against the exact game and projects its attributions to enforce exact completeness at inference.It runs approximately 103× faster, but its on-manifold faithfulness holds only where signals carry sufficient structure; the authors recommend it for real data and retain the exact GP game as the reference.Source: §3.3 · p. 4

4.1 Explanation Faithfulness and Completeness

• GRAIL is most faithful on-manifold across all six families, outperforming all twelve baselines on each dataset with joint statistical significance.Provided-region GRAIL leads on five families; GRAIL-auto leads on Wafer.Source: §4.1 · p. 5, Table 2 · p. 5
• The exact game has zero residual on every family, while its Owen estimates converge at the Monte-Carlo-optimal rate.At P=128, the Synth region ranking matches the largest-budget run at Spearman correlation 0.98.Source: §4.1 · p. 5, Figure 3 · p. 6

4.3 Temporal Signal and Covariates on One Scale

• Three LUDB leads improve every quality metric over one lead in both region regimes, at approximately 39× runtime.Within-setting localization is 0.745 versus 0.535 for provided versus discovered regions; the cross-column localization comparison is not comparable.Source: Table 3 · p. 5, §4.3 · p. 6

4.2 Comparison against Other Methods

• On Synth, the exact game beats both closest comparators on-manifold but loses to both off-manifold; on ECG, it wins on-manifold and performs approximately comparably off-manifold.The two-operator evaluation exposes how removal choice can favor methods evaluated with matching removal operators.Source: §4.2 · pp. 5–6

4.3 Temporal Signal and Covariates on One Scale

• On the LVH task, age’s mean |ϕ| is 1.663, exceeding the top ECG region’s 1.542, while adding static covariates lowers expected-region hit rate from 0.575 to 0.325.On-manifold faithfulness remains 0.598 to 0.597 (p=0.89); age ranks highest on 62.5% of instances.Source: §4.3 · p. 6

4.4 Region Discovery and Ablations

• For unaided region discovery, boundary-snapping raises the LUDB selector’s localization from 0.34 to 0.73, while annotated regions remain the strongest option when available.The model-aware selector requires no retraining; the ground-truth prior dominates unaided arms.Source: §4.4 · p. 6, Table 4 · p. 7
• Kernel choice trades ECG faithfulness against robustness, while the reference p0 barely changes faithfulness across its five arms.RBF is most faithful on ECG (0.872); rougher Matérn kernels have max-sensitivity 0.72/0.74 versus RBF’s 0.91.Source: §4.4 · pp. 6–7
• Conditional semantics reproduces the marginal arm to machine precision on region-level metrics, including real ECG, because absent features have no retained sibling within whole-block coalitions.Across six datasets, faithfulness is unchanged; localization precision and recall trade off.Source: §4.4 · p. 7

5 Conclusion

• GRAIL combines domain-aligned regions, covariates on a shared attribution scale, on-manifold counterfactuals, exactly-complete Owen values, and an amortized surrogate.The conclusion also describes a faithfulness-aware segmenter for region discovery and multi-lead support that recovers clinically expected leads.Source: §5 · p. 7

A Related Work

• Prior explainers use timestep, window, or segment players, while GRAIL changes the removal operator, supports non-temporal covariates, and generates absent regions individually.Owen values formalize nested credit assignment; GRAIL nests features in regions and, for multivariate signals, regions in channels.Source: A Related Work · p. 10

B Implementation details

• Implementation uses closed-form codec features, a boundary-conditioned GP by default, a reference corpus for feature sampling, and common-random-number sweeps for attribution.The experiments use shape features with an RBF residual kernel; the stated budget includes P=128 restricted permutations and Q=3 base-randomness draws.Source: B Implementation details · pp. 10–11

C A worked clinical example

• On one PVC beat, QRS was the top region with Σϕ=+5.70 and ρ=+0.0000; across 40 beats it led on 31/40 and completeness had max |ρ|<10^-4.The QRS complex is the clinical ground truth; the example also attributes a strong feature-level driver to elevated baseline/ST level (4.9σ; ϕ=+4.16±0.09 log-odds).Source: C A worked clinical example · p. 11

D Kernel functions

• On FordA, RBF has the highest on-manifold faithfulness (0.864), while Matérn-5/2 reaches 0.790; the filterbank codec lowers max-sensitivity from 0.39 to 0.21.FordA has no region ground truth, so localization is undefined; completeness remains exact.Source: D Kernel functions · p. 12
• On FordA, only 18% of attributed mass sits on named features under RBF, versus roughly half under Matérn-5/2.The passage relates matching generator roughness to signal properties with keeping explanations interpretable.Source: D Kernel functions · p. 12
• On ECG, exact-game kernels score 0.72–0.84 on-manifold faithfulness versus ≤0.66 for every baseline, and every exact-game arm is exactly complete.The rougher Matérn kernels are most robust; localization varies with saliency settings, while runtime can be one to three orders of magnitude slower.Source: D Kernel functions · pp. 11–12, Table 5 · p. 12
• Use Matérn-5/2 for non-stationary settings, RBF when on-manifold faithfulness is the priority, and the filterbank codec when jitter-stable component identity matters.RBF is infinitely smooth and is expected to perform poorly on non-smooth signals; it performed best in the non-stationary setting reported here.Source: D Kernel functions · pp. 11–12

E Experimental Setup

• Synth uses 1,500 univariate 100-step series with injected feature combinations and ground-truth region masks; Mixed adds static shortcut, informative, or distractor covariates.Mixed has known signal and covariate ground truth.Source: E Experimental Setup · p. 12
• ECG comprises 4,000 R-aligned beats with cardiologist P/QRS/T regions across five binary tasks; LUDB adds age, sex, three leads, and LVH labels.The listed dataset sizes are 4000×1×288 for ECG and 1799×3×400 for LUDB.Source: E Experimental Setup · pp. 12–13
• FordA and Wafer are UCR sensor series without region annotation, so localization is undefined for FordA.The reported sizes are 4921×1×100 for FordA and 7164×1×100 for Wafer.Source: E Experimental Setup · p. 13
• GRAIL is tested with provided regions, its model-aware GRAIL-auto segmenter, and GRAIL-fast; faithfulness uses matched on/off-manifold operators alongside localization, robustness, cost, completeness, and segmentation metrics.Methods share region support and the evaluation harness, while GRAIL-auto discovers its own regions; comparisons are paired within a family and gated by corrected Wilcoxon tests.Source: E Experimental Setup · p. 13

F “Absent” features

• On-manifold faithfulness_corr_on spans 0.620–0.638 across five Synth reference arms with provided regions and 0.533–0.582 with discovered regions; real-data ranges are tighter.The reported ranges are 0.669–0.696 and 0.462–0.478 on ECG, and 0.607–0.628 and 0.461–0.476 on LUDB, respectively.Source: F “Absent” features · p. 14
• Conditional sampling reproduces marginal region-level metrics to machine precision (max |∆| ≈10^-15), because whole-block coalitions have no retained sibling feature to condition on.Conditional draws can move feature-level Owen values, but region roll-ups remain identical; class-conditional cohorts can change the fully absent region’s reference pool.Source: F “Absent” features · p. 14
• All six Synth profile arms leave faithfulness bit-identical at 0.620 provided and 0.582 discovered; only rank-alignment perturbs it downward.The caption attributes the invariance to the roll-up property.Source: Table 7 · p. 14

G Where a region’s credit goes

• Six redistribution arms preserve faithfulness to three decimals; rank-align alone changes it, worsening scores from 0.620 to 0.561 and 0.582 to 0.534.Rank-align reorders regions while leaving localization untouched, so the authors leave it off by default.Source: G Where a region’s credit goes · p. 14
• With ground-truth regions, flat broadcast leads on AUPRC at 0.634 but trails on pointing at 0.525; discovered regions favor smoothed occlusion on both metrics.For discovered regions, smoothed occlusion reaches 0.543 AUPRC and 0.600 pointing; imprecise boundaries leave room for the profile to recover localization.Source: G Where a region’s credit goes · pp. 14–15
• Across all four real-data cells, occlusion profiles win the precision family, while flat broadcast wins the recall family.The synthetic sweep did not separate this precision–recall split; ground-truth support leaves less room for a profile to recover coverage than approximate boundaries do.Source: G Where a region’s credit goes · p. 15
• The default occ-6 +f is never worst, is best or tied-best at pointing in both discovered-region cells, and ranks second on AUPRC in all four.LUDB results are directional only: at n=16, between-seed SEMs on AUPRC reach ±0.10.Source: G Where a region’s credit goes · p. 15

H Swapping the generator, and asking whether it fits

• As generators become more learned and less analytic, on-manifold faithfulness falls from 0.662 to 0.610 to 0.430, while off-manifold faithfulness rises from −0.092 to 0.335 to 0.439.The DDPM’s on-manifold result is not treated as converged because it had an 8× smaller permutation budget; the off-manifold ordering is budget-insensitive in direction.Source: H Swapping the generator, and asking… · p. 15
• All three generators preserve exact completeness, while the analytic GP has the strongest on-manifold faithfulness among the compared generators.The learned generators drift toward counterfactuals recognized by a splice-based operator, raising off-manifold faithfulness.Source: H Swapping the generator, and asking… · p. 15
• On ECG with annotated regions, GP log-density is −12.4 versus −17.8 to −21.5 on synthetic data, and regeneration distances are 0.36 versus 2.24.The authors attribute this reversal to synthetic informative regions on i.i.d. background versus correlated QRS excursions that the smooth boundary-conditioned GP models well.Source: H Swapping the generator, and asking… · p. 15
• On ECG, discovered rather than annotated regions lower GP log-density by 54 nats per sample, from −12.4 to −67.0.The authors propose log-density as a segmentation diagnostic when ground-truth masks—and therefore boundary F1—are unavailable.Source: H Swapping the generator, and asking… · p. 15
• PIT KS distances of 0.23–0.40 indicate miscalibrated GP uncertainty, so counterfactual confidence intervals are not supported by this evidence.The authors report that the GP mean still drives faithful attributions, while downstream faithfulness metrics are insensitive to error-bar calibration.Source: H Swapping the generator, and asking… · p. 15

I Static covariates on one scale

• On LUDB, age’s mean |ϕ| is 1.663 versus 1.542 for the top temporal region; on mixed data, signed covariate means recover the injected roles.Age ranks above every ECG region on 62.5% of LUDB instances; the mixed control’s shortcut and informative covariates are 80σ and 8σ from zero, while the distractor is 1σ.Source: I Static covariates on one scale · p. 16, Table 11 · p. 16
• Adding covariates leaves faithfulness and localization statistically unmoved, but age out-ranks the annotated QRS on 25% of instances and lowers the joint expected-region hit rate to 0.325.The comparable re-run localization value is 0.473, not the frozen CSV’s 0.353, which predates the occ-w=6 default.Source: I Static covariates on one scale · pp. 16–17, Table 12 · p. 17

J Amortized deployment for fast attribution

• The exact GP game is slow, so the paper distills its region attributions into a network that reproduces them in one forward pass.The section frames amortization as a trade-off between deployment speed and faithfulness.Source: J Amortized deployment for fast… · p. 17, §J.1 · p. 17

J.1 Amortizing the game for deployment

• The deployed region-token head regresses exact on-manifold Owen values using region-aware tokens and a transformer, unlike the segmentation-agnostic FastSHAP density head.Both heads preserve completeness by construction; the attribution heads are smooth functions of input and stable under time shifts.Source: §J.1 · pp. 17–18

J.2 Runtime and robustness

• GRAIL-fast runs in 9–35 ms versus 2.7–12.5 s for GRAIL, a 290–710× speed-up, while remaining exactly complete.The variant with a model-evidence channel is 3–5× slower because it first runs an occlusion sweep over the signal.Source: Table 13 · p. 18, §J.2 · p. 18
• Both GRAIL and GRAIL-fast are jitter-stable (≥0.87), while max-sensitivity improves with amortization on real families but worsens on synthetic data.For max-sensitivity, ECG changes from 0.76 to 0.37 and LUDB from 0.22 to 0.05; synthetic results change from 0.68 to 1.00 and 0.85 to 1.07.Source: §J.2 · p. 18

J.3 On-manifold faithfulness tradeoff

• GRAIL-fast beats the best baseline on all four datasets, while the exact GRAIL game remains the on-manifold faithfulness leader.The GRAIL-fast gap to GRAIL on LUDB is not significant (q=0.12); the other gaps have q < 10−4.Source: Table 14 · p. 18, §J.3 · p. 18
• On real data, GRAIL-fast beats GRAIL on loc-AUPRC for ECG (0.648 vs. 0.587) and LUDB (0.633 vs. 0.502), but pointing is worse on ECG and mixed.It still trails GRAIL on mixed loc-AUPRC (0.518 vs. 0.648); the ECG result clears the best baseline in the full eight-method run (0.607).Source: §J.3 · p. 18

J.4 Is the surrogate’s negative a training-operator mismatch?

• Matching training and evaluation operators changes faithfulness by −0.012 on synthetic data and +0.011 on mixed data, both within one between-seed SEM.The synthetic change is in the wrong direction, so the negative result survives this confound; localization on synthetic data improves from 0.415 to 0.452.Source: §J.4 · pp. 18–19
• The authors retract operator mismatch as the explanation: low-signal synthetic responses to regenerated regions leave little stable structure for a one-pass surrogate to capture.On real data, where regenerating a QRS changes the decision repeatably, the same architecture and training budget reach 0.556 on ECG and 0.381 on LUDB.Source: §J.4 · p. 19

• GRAIL is the most faithful on-manifold method on every family and is exactly complete; GRAIL-fast is modestly less faithful on real data but disqualifying on low-signal synthetic data.GRAIL-fast recovers the result at a small fraction of the cost on real, structured data.Source: K Extended Discussion · p. 19
• Amortizing the exact game against the same value function reduces runtime without changing what is explained.Source: K Extended Discussion · p. 19
• Faithfulness is operator-relative, so evaluating every method with both on- and off-manifold operators exposes GRAIL-fast’s on-manifold limit.Source: K Extended Discussion · p. 19
• On LUDB LVH cases, adding age as a player takes top rank from annotated QRS in a quarter of instances while faithfulness remains unchanged.The example shows that a faithful explanation can point away from an annotated region when the model relies on static or unannotated evidence.Source: K Extended Discussion · p. 19
• The generator’s mean supports the most faithful attributions here, but its error bars are uncalibrated, so faithfulness metrics cannot certify the generative model.Source: K Extended Discussion · p. 19