PaperlayerView exact revision on arXiv

Source-linked AI summary

Beyond Model Ranking: Regime Diagnosis for Distributional-Statistical Misspecification in Industrial Time-Series Forecasting

Pengyu Nie, Chenglang Xu, Yaoshi Chen, Chaogan Ren, Wei Hu, Chao Yang, Jiangong Zhang

arXiv:2609.40117v1cs.LG

TL;DR

Canonical losses impose fixed statistical priors that can misfit industrial demand regimes, while aggregate metrics hide regime-specific bias. The paper introduces an RBV-centered diagnostic and controlled attribution analysis; its study reports that regime-aware diagnosis separates failure types and regime-aware training resolves pooling-induced bias.

  • Problem

    Canonical losses encode fixed statistical priors that are violated by industrial demand regimes, and aggregate metrics can conceal the resulting systematic mismatch.

    Source: §1 · p. 2

  • Method

    The paper uses a regime-decomposed, metric-agnostic RBV diagnostic and controlled attribution to separate loss-estimand bias from excess attributable to training.

    Source: §4.1 · p. 5, §4.2 · p. 6

  • Results

    Across 13 losses, 3 seeds, and 60,000+ series, regime-aware diagnosis separates optimization-type from bias-type failure, and regime-aware training resolves the pooling-induced bias it exposes.

    Source: §1 · p. 2

  • Takeaways & Limitations

    RBV exposes cross-regime bias allocation that accuracy metrics cannot show, but should be read alongside bias levels and accuracy.

    Source: §4.2 · p. 6

    RBV diagnoses the aggregation trade-off but does not correct forecasts, and the framework targets non-negative series, requiring recalibration for mixed-sign data.

    Source: §7 · p. 9

Abstract

from arXiv · show

Time-series forecasting models achieve strong benchmark performance but exhibit severe systematic bias in industrial deployments. This train--deploy gap is conventionally attributed to temporal-structural errors or distribution shifts. We characterize a complementary source that these explanations overlook: canonical losses embed fixed statistical priors, while industrial demand mixes benign and pathological regimes---zero-inflation, skewness, high variability---in which these priors are systematically violated. The induced bias persists even under perfect temporal modeling, remains in a distributional-shape component that normalization cannot remove, and creates an aggregation trade-off invisible to aggregate metrics. We turn these observations into an evaluation toolkit centered on the Regime-wise Relative Bias Vector (RBV): a metric-agnostic, regime-decomposed diagnostic that audits how pooled training allocates systematic mismatch across pathological subpopulations. A controlled attribution analysis decomposes RBV into a model-independent intrinsic floor, set by each loss's estimand, and an excess component attributable to training, tracing observed bias to the loss rather than the model. A large-scale study---13 loss objectives, 3 seeds, 60,000+ series spanning RetailShiftBench and M5, with random-split controls---shows that regime-aware diagnosis separates optimization-type from bias-type failure, and that regime-aware training resolves the pooling-induced bias that capacity scaling cannot, for mean-type losses. A formal structural observation, that risk under evaluation-distribution contamination is affine in the pathology mixture weight, grounds these findings. Our work complements model ranking with mechanism-grounded, regime-oriented evaluation.

1 INTRODUCTION

Industrial forecasts can remain systematically biased because canonical losses impose fixed statistical priors that do not fit heterogeneous demand regimes, while aggregate metrics hide the resulting mismatch. The paper proposes regime-oriented diagnosis and a controlled study of how loss and training relate to regime-specific bias.

1 INTRODUCTION

  • Canonical losses misfit zero-inflated, skewed, and high-variability demand, producing regime-dependent bias that can persist despite perfect temporal modeling.Mean-targeting losses overshoot sparse regimes, while median-targeting losses suppress bursts; aggregate metrics can hide both directions.Source: §1 · p. 2
  • The paper shifts evaluation from model ranking to mechanism-grounded diagnosis of misspecification across pathological regimes.Its contributions include a pathological-regime taxonomy, RetailShiftBench, and an RBV-based audit with controlled attribution of intrinsic and training-related bias.Source: §1 · p. 2
  • The study covers 13 losses, 3 seeds, and 60,000+ series, and reports that regime-aware diagnosis separates optimization-type from bias-type failures.It also reports that regime-aware training resolves the pooling-induced bias it exposes.Source: §1 · p. 2
  • Global aggregate metrics hide regime-specific failures, while pattern-sliced evaluation retains uniform averaging over label-side pathologies; RBV and PCS assess regimes at model and dataset levels.The figure contrasts global metrics, pattern-sliced benchmarks, and the proposed model-diagnosis and dataset-audit approach.Source: §1 · p. 3, Figure 1 · p. 3

2 RELATED WORK

Prior forecasting evaluations and misspecification research largely emphasize temporal structure, feature-side properties, or domain-specific methods. This paper positions its approach as a complementary framework for diagnosing label-side distributional pathologies across regimes.

2.1 TIME-SERIES DIAGNOSTIC FRAMEWORK ECOSYSTEMS

  • Existing diagnostic benchmarks slice temporal patterns or compare architectures, leaving label-side distributional pathologies outside their evaluation axes.Prior work also addresses temporal-structural misspecification, which temporal alignment methods cannot resolve when distributional mismatches persist under stationary inputs.Source: §2.1 · p. 3, §2.2 · p. 3
  • Related subgroup and dataset-auditing efforts generally target feature-side properties or temporal patterns rather than label-side zero-inflation, skewness, and scale variability.The paper presents regime-decomposed evaluation and RetailShiftBench as addressing this gap at model and dataset levels.Source: §2.1 · p. 3

2.2 DUAL DIMENSIONS OF LOSS MISSPECIFICATION

  • Intermittent-demand research recognizes demand pathologies, but prior domain treatments lack systematic regime-decomposed risk quantification for deep forecasting.The paper distinguishes its evaluation taxonomy from ADI-CV2 groupings and describes canonical losses as eliciting fixed statistical functionals that cannot adapt to regime-varying distributions.Source: §2.2 · pp. 3–4
  • Decision-focused learning and inventory research address mismatch between statistical losses and a decision target, whereas the paper focuses on aggregation trade-offs from pooled training across heterogeneous regimes.It also distinguishes its stationary mismatch focus from work refining temporal fitting objectives and from per-model likelihood–distribution matching.Source: §2.1 · p. 3, §2.2 · p. 4

3 INDUSTRIAL PATHOLOGY TAXONOMY

The taxonomy describes industrial demand through three statistical pathologies—skewness, within-series scale variability, and zero-inflation—and their coupled regimes. It treats temporal-structure effects as amplifiers rather than core distributional pathologies.

3 INDUSTRIAL PATHOLOGY TAXONOMY

  • The taxonomy defines three foundational, conceptually orthogonal distributional traits and frames loss failure in statistical rather than business-label terms.The same pathologies can recur across domains under different business guises; kurtosis and multimodality are treated as secondary factors.Source: §3 · p. 4, §3.1 · p. 4

3.1.1 DISTRIBUTION SHAPE SKEWNESS

  • Skewness measures distributional asymmetry, with high skewness defined by |γ1| ≥4.Promotional and seasonal demand bursts are described as a business source of asymmetric, long-tailed distributions.Source: §3.1.1 · p. 4, §3 · p. 4

3.1.2 WITHIN-SERIES SCALE VARIABILITY

  • Within-series scale variability is measured by CV; high CV is defined as CV ≥3.0.The paper distinguishes this sequence-level variability from differences in absolute magnitude across SKUs.Source: §3 · p. 4, §3.1.2 · p. 5

3.1.3 TEMPORAL SPARSITY (ZERO-INFLATION RATE)

  • Zero-inflation rate (ZIR) measures intermittent zero observations; high ZIR is defined as ZIR ≥0.4.The paper relates inventory shortage and cold-start scenarios to sparse zeros and notes that thresholds are fixed before evaluation and robust to ±20% perturbations.Source: §3.1.3 · p. 5

3.2 COUPLED DISTRIBUTIONAL PATHOLOGICAL REGIMES

  • Coupled regimes require joint satisfaction of high-threshold conditions, and overlapping pathologies can yield saturation or super-additive mismatch.The taxonomy includes ZIR–Skew, ZIR–Vol, Skew–Vol, and ZIR–Skew–Vol; the disjoint-support ZIR×CV coupling can induce super-additive mismatch at intermediate intensity.Source: §3.2 · p. 5
  • Temporal regime shifts and cyclic patterns can amplify mismatch on pathological data, but are excluded from the core taxonomy to preserve its theoretical rigor.Examples include non-stationarity increasing inference complexity and superimposed cycles creating intermittent-burst transitions.Source: §3.2 · p. 5

4 REGIME-DECOMPOSED BIAS DIAGNOSTIC

The diagnostic section addresses how global averaging can conceal regime-specific bias, especially when pathological long-tail series contribute little to pooled training. Its protocol partitions sequences into regimes, estimates their bias separately, and summarizes cross-regime imbalance with RBV.

4 REGIME-DECOMPOSED BIAS DIAGNOSTIC

  • Global averaging can conceal regime-specific distortion, while low-volume long-tail SKUs can have their gradient contributions drowned out in global multi-SKU training.The section frames regime-decomposed diagnosis as a way to make these failures measurable.Source: §4 · p. 5

4.1 METRIC-AGNOSTIC REGIME-DECOMPOSED PROTOCOL

  • The protocol partitions test sequences by skewness, ZIR, and CV, estimates bias independently by regime, then applies a forecasting metric to quantify RBV.The same partition supports a dataset-level pathological-coverage audit; metric-agnostic means any base functional can be used, not that RBV magnitudes are invariant across functionals.Source: §4.1 · pp. 5–6

4.2 THE RBV DIAGNOSTIC

  • RBV is the ℓ2 norm of a four-component vector of regime-level bias deviations from pooled bias, exposing cross-regime imbalance rather than predictive accuracy.The regimes are Benign, ZIR, Skew, and CV; regime flags can overlap, though partition cells are mutually exclusive.Source: §4.2 · p. 6
  • RBV must be interpreted alongside bias levels and accuracy: uniformly poor but constant bias can produce low RBV.For non-mean estimands, the training-attributable signal is excess RBV above the intrinsic floor set by the loss estimand.Source: §4.2 · p. 6

5 MECHANISTIC ACCOUNT

The section distinguishes loss-implied estimand mismatch from pooling-induced aggregation bias, and formalizes how pathology in the evaluation mixture affects risk.

5.1 DUAL BIAS MODES

  • Canonical losses target distinct estimands—MSE the mean, MAE the median, and pinball fixed quantiles—so prior–objective mismatch can arise independently of pathology.Table 1 maps loss priors to industrial departures and mismatch mechanisms.Source: §5.1 · p. 6, §5 · p. 6, Table 1 · p. 7
  • Under pathological marginals, global pooling shifts prior-dependent optima toward a cross-regime compromise that may be optimal for no single regime.This aggregation bias is independent of temporal modeling and cannot be eliminated by capacity or architectural improvements under a fixed loss prior.Source: §5.1 · p. 7

5.2 ERROR SEPARATION AND INTENSITY TREND

  • With perfect temporal fitting, distributional-statistical deviation remains independent of temporal modeling, separating it from temporal-structural bias.The separation is presented as a measurement-theoretic basis for isolating error channels, not as a claim about joint optimization.Source: §5.2 · p. 7
  • For a fixed predictor, evaluation risk is affine in pathology mixture weight and non-decreasing when pathological data raises risk above its benign-data level.Source: §5.2 · p. 7
  • Decomposition mitigates the aggregation trade-off for mean-type losses, while capacity scaling cannot remove it and decomposition exposes rather than repairs median-type estimand mismatch.The intervention separates the pooling trade-off from the estimand gap by holding the loss-implied target fixed.Source: §5.2 · p. 7

6 EXPERIMENTAL EVALUATION

The evaluation audits pooled training on RetailShiftBench and tests whether regime-aware training and other interventions separate loss-driven bias from fitting limitations.

6 EXPERIMENTAL EVALUATION

  • The study audits pooled bias, contrasts global with regime-aware training, and probes robustness across capacity, quantile, and architecture axes.Source: §6 · p. 7

6.1 EXPERIMENTAL SETUP

  • RetailShiftBench contains 31,213 desensitized retail series of 89 time steps, with stable weekly periodicity and no trend breaks or long-range drift.The design removes temporal non-stationarity as a confound so residual errors can be attributed to distributional misspecification rather than temporal distribution shift.Source: §6.1 · p. 8

6.2 REGIME-DEPENDENT LOSS MISALIGNMENT

  • MSE misaligns under ZIR, MAE under Skew, and static-weight QL, Huber, and Tweedie under high-variability CV, while all losses perform well on Benign data.The RBV diagnostic treats lower values as more stable cross-regime fitting and higher values as more severe heterogeneous systematic misalignment.Source: §6.2 · p. 8, Table 2 · p. 8
  • Because every configuration receives identical pathology-aware prior information, persistent cross-regime imbalance is attributed to loss-function misspecification rather than capacity, architecture, or missing statistical cues.The ablation varies plug-in loss objectives while fixing the backbones.Source: §6.2 · p. 8

7 DISCUSSION

The discussion connects pooled evaluation and training to hidden regime misalignment, and frames the problem as one of aggregation structure rather than model capacity.

Theoretical and Paradigm Insights

  • Global L1 and L2 evaluation metrics certify median- and mean-adequacy, respectively, repeating training priors instead of auditing them.The discussion says globally optimized forecasters can trade pathological-regime alignment for better aggregate accuracy.Source: §7 · p. 9

Structural barriers to scale-up

  • Capacity scaling cannot eliminate heterogeneity across zero-inflation, skewness, and CV, while hard regime cells remain a coarse diagnostic instrument rather than natural kinds.The reported cells are stable to ±20% threshold perturbations, but real-world pathologies form a continuous spectrum.Source: §7 · p. 9
  • Addressing both pooling and intrinsic estimand mismatch requires decomposition plus a curvature-appropriate estimand; data-volume-adaptive partial pooling is proposed when slow-converging estimands are unavoidable.Source: §7 · p. 9

Limitations & Future Work

  • The framework targets non-negative series and requires recalibration for mixed-sign data; combined design costs are justified only when RBV confirms both failure modes.Regime thresholds may also shift at extreme boundaries despite robustness to ±20% perturbation.Source: §7 · p. 9

8 CONCLUSION

The conclusion presents RBV and RetailShiftBench as tools for regime-oriented diagnosis, while the appendix schematic limits its trade-off claim to heterogeneous pooled forecasting settings.

8 CONCLUSION

  • Across 13 loss objectives, 3 seeds, and 60,000+ series spanning RetailShiftBench and M5, regime-aware diagnosis separates optimization-type from bias-type failure, and regime-aware training resolves pooling-induced bias for mean-type losses.The study includes random-split controls; the affine-risk observation is supporting rather than foundational.Source: §8 · p. 10
  • RBV decomposes observed bias into a model-independent intrinsic floor and a training-attributable excess component, alongside a benchmark with full pathological coverage.The work positions this toolkit as a complement to global accuracy ranking and an audit of dataset pathological validity.Source: §8 · p. 10

A AGGREGATION-DRIVEN BIAS-VARIANCE TRADE-OFF SCHEMATIC

  • The schematic has no derived closed-form curve; only global-pooled, random-cell R2, and pathology-cell operating points are measured.The qualitative trade-off concerns pooled multi-series forecasting under strong cross-series distributional heterogeneity.Source: A AGGREGATION-DRIVEN BIAS-VARIANCE… · p. 13, Figure 2 · p. 13
  • Near-homogeneous collections have a flatter trade-off, while the proposed partial-pooling regime targets an intermediate operating point where global scores can mask systematic deviations.Source: A AGGREGATION-DRIVEN BIAS-VARIANCE… · p. 13

B SUPPLEMENTARY FORMAL STATEMENTS, ANALYTICAL DERIVATIONS AND ROBUSTNESS VALIDATIONS

The appendix formalizes RBV and PCS, validates regime-specific findings across metrics and threshold choices, and uses synthetic experiments to probe loss–pathology interactions and their boundaries.

B.3 RBV: TEMPLATE, INTERPRETATION, AND UNCERTAINTY QUANTIFICATION

  • RBV variants share centering and ℓ2 aggregation but differ in their regime-level functional; pathology-bias RBV centers using absolute-demand mass rather than count share.The protocol allows different base functionals, so magnitudes depend on the chosen functional and cross-variant comparisons are ranking-level only.Source: §B.3 · pp. 15–16, Table 4 · p. 16
  • RBV localizes which regimes the pooled model sacrifices, but it is diagnostic rather than corrective; regime-localized intervention depends on whether bias persists after per-regime retraining.Capacity increases reduce RBV only to a loss-dependent floor, while quantile RBV must be interpreted per τ because quantiles are not additive under aggregation.Source: §B.3 · p. 16
  • SKU-level bootstrap intervals are tight, and all cell-level biases are significant at the 1% level; the loss ranking by RBV is stable.For example, the RBV 95% interval is 0.409 [0.391, 0.427] for MAE and 0.101 [0.080, 0.123] for the τ=0.6 quantile compromise.Source: §B.3 · p. 17, Table 5 · p. 17
  • Across threshold settings, RBV rankings and the U-shaped RBV–τ structure remain invariant; sensitivity is dominated by CV-cell extremity rather than the threshold definition.This analysis uses continuous pathology statistics rather than thresholded regime labels, so its RBV magnitudes are not directly comparable to Table 2.Source: Figure 3 · p. 18

B.5 METRIC-AGNOSTIC VALIDATION

  • Across MSE, MAE, and pinball evaluation, relative failure rankings remain invariant: MSE is most degraded by high ZIR, MAE by skewness, and Quantile, Huber, and Tweedie by high CV.Absolute values vary with metric sensitivity to tail risk, but the structural misspecification signatures persist.Source: §B.5 · p. 18

B.6 PATHOLOGICAL COVERAGE SCORE (PCS): DATASET AUDITING METRIC

  • PCS measures pathological-regime coverage on a 0–1 scale, with higher scores indicating a richer pathological spectrum; alternative dimension weights do not change mainstream-dataset ordering.It complements regime-decomposed evaluation, which isolates severe high-pathology misalignments.Source: §B.6 · pp. 18–19

B.7 SYNTHETIC ILLUSTRATION FOR PROP. B.1

  • With temporal bias set to zero, synthetic examples show loss-implied estimands diverging on the same pathological marginal, while deviation from the clean signal rises with pathology intensity.In the zero-dominant case, MSE and MAE estimates are 10 and 0, respectively; the experiments use closed-form optimal predictors rather than trained forecasting models.Source: §B.7 · pp. 19–20, Figure 5 · p. 20

B.8 BOUNDARY CONDITIONS FOR SATURATION VS. SUPER-ADDITIVE DEGRADATION

  • Under intermediate pathology intensity, coupled ZIR–CV degradation exceeds the isolated-pathology linear baseline across total MSE, burst-sample MSE, and pinball loss at τ=0.75.At high intensity, the isolated pathologies approach a near-random floor, so additional distortion yields diminishing degradation and eventual sub-additive tendencies; the authors do not claim generalization to all pathology pairs or real-world settings.Source: §B.8 · p. 21

B.1 LOSS–MARGINAL CONFLICT VERIFICATION & PROPOSITION DERIVATIONS

The appendix verifies that zero-inflation, skewness, and within-series scale variability violate conditions under which a global loss prior has an unambiguous statistical target. These violations make loss-implied optima dependent on the pathological marginal.

B.1 LOSS–MARGINAL CONFLICT VERIFICATION & PROPOSITION DERIVATIONS

  • Under severe skewness, MSE’s mean-centered target and MAE’s median-based target diverge on the same marginal because the symmetric-residual assumption fails.The resulting distinct optima reflect loss-prior dependence.Source: §B.1 · p. 14
  • High zero-inflation violates continuous-support assumptions: median-based optimization can collapse toward dominant zeros and discard high-demand burst mass.Low quantile targets can also degenerate to zero.Source: §B.1 · p. 14
  • High CV makes uniform sample weighting dilute high-impact burst gradients, under-fitting critical cases while prioritizing average accuracy over tail fidelity.The imbalance is between trivial low-magnitude samples and heavy-tailed demand bursts.Source: §B.1 · p. 14
  • Each foundational pathology breaks at least one classical condition for an unambiguous global loss prior, producing a systematic loss–prior/marginal conflict.The conditions concern symmetric marginals, continuous non-degenerate support, and homogeneous samples under uniform weighting.Source: §B.1 · p. 14

B.2 STANDARD REGULARITY ASSUMPTIONS FOR REGIME TREND ANALYSIS

The section separates loss–target mismatch from temporal-fitting error and formalizes how evaluation risk changes as pathological mass increases. It also reports that the regime-bias findings are robust to threshold perturbations.

B.2 STANDARD REGULARITY ASSUMPTIONS FOR REGIME TREND ANALYSIS

  • The formal analysis assumes bounded demand support, Lipschitz-smooth fixed-form losses, and bounded absolutely continuous transformations from a benign baseline.The contamination family mixes benign and fixed pathological distributions with intensity λ from 0 to 1.Source: §B.2 · p. 14
  • A loss elicits a statistical functional, and its deviation from the deployment target remains as bias even under perfect temporal fitting.The forecast deviation separates into loss-relative bias and residual temporal-fitting error.Source: §B.2 · p. 15
  • For a mean target, MSE has no estimand-mismatch component, but pooled training shifts its shared optimum away from each regime’s mean; MAE can have intrinsic and pooling-induced mismatch.The pooled optimum is a demand-weighted cross-regime compromise.Source: §B.2 · p. 15
  • Under pathology-as-contamination, fixed-predictor risk is affine in mixture weight λ, increasing when pathological risk exceeds benign risk.The slope is Rpath − Rbenign.Source: §B.2 · pp. 14–15

B.4 THRESHOLD PERTURBATION SENSITIVITY ANALYSIS

  • Across 39 threshold settings, loss rankings, bias-sign patterns, and U-shaped RBV(τ) remain invariant, while RBV magnitude is most sensitive to the CV threshold.The CV-threshold scan produces a 0.25–2.48 RBV spread; ql60 minimizes RBV under ±20% threshold shifts.Source: §B.4 · p. 18

C CROSS-DOMAIN GENERALIZATION OF THE PATHOLOGY TAXONOMY

The pathology taxonomy is defined at the statistical level across industrial domains, with domain-specific threshold recalibration rather than changes to the regimes themselves. The benchmark audit identifies which datasets provide pathological coverage for industrial stress testing.

C CROSS-DOMAIN GENERALIZATION OF THE PATHOLOGY TAXONOMY

  • RetailShiftBench’s 89-step series have stable weekly periodicity without trend breaks or long-range drift, while M5 supplies external validation across scale, window length, and pathology mix.Removing temporal non-stationarity from RetailShiftBench is a deliberate design choice to isolate distributional misspecification.Source: C CROSS-DOMAIN GENERALIZATION OF THE… · p. 21, Figure 7 · p. 22
  • The taxonomy’s three dimensions—skewness, zero-inflation, and within-series scale variability—retain the same statistical definitions across domains; only thresholds are domain-adaptive.Table 6 maps the regimes and coupled forms to domain-specific manifestations.Source: C CROSS-DOMAIN GENERALIZATION OF THE… · p. 22, Table 6 · p. 23
  • RetailShiftBench has full-spectrum coverage by construction, M4 is insufficient as an industrial stress test, and M5 provides adequate coverage for external validation.The comparison uses the Pathological Coverage Score (PCS).Source: C CROSS-DOMAIN GENERALIZATION OF THE… · p. 22, Table 7 · p. 23
  • Domain adoption requires monotonic threshold recalibration, and PCS is directly transferable without a domain-specific objective.The paper states that a low-PCS dataset in any domain is unqualified for industrial stress testing.Source: C CROSS-DOMAIN GENERALIZATION OF THE… · p. 23

D DECOMPOSITION RESPONSE ANALYSIS

  • The appendix’s decomposition-response analysis uses matched train/test splits and hyperparameters across losses, with RetailShiftBench as its default experimental benchmark.It covers table expansions, attribution controls, and quantile-calibration analysis.Source: D DECOMPOSITION RESPONSE ANALYSIS · p. 23

D.1 READING TABLES 2–3

Regime-level diagnosis exposes bias patterns that aggregate metrics can mask, while attribution controls distinguish pathology-informed decomposition from mere partition granularity. Capacity scaling reduces but does not eliminate pooled bias for MSE; decomposition does, whereas MAE’s within-cell mismatch can be exposed by routing.

D.1 READING TABLES 2–3

  • The All row can pass while pathological-regime errors are an order of magnitude larger, and bias-sign fingerprints distinguish loss families across regimes.Mean-targeting losses are positive on sparse regimes, median-type losses strongly negative, and quantile heads show step behavior.Source: §D.1 · pp. 23–24
  • RBV remains elevated under both raw XGBoost and internally normalized TFT, although mean-loss signs flip on high-CV and zero-inflated cells under normalization.Median-type losses preserve their negative-bias fingerprint across both preprocessing pipelines.Source: §D.1 · p. 24
  • Decomposition improves all three instruments for mean-targeting losses, but median-type losses deteriorate in bias structure as per-regime fitting exposes their estimand-level mismatch.Huber remains in the worsen family across a 40-fold δ range; Poisson’s structural and squared-error gains coincide with degradation on the linear metric.Source: §D.1 · p. 24

D.2 ATTRIBUTION CONTROLS AND QUANTILE CALIBRATION SPECTRA

  • At high quantiles, random-label cells reproduce pooled calibration, while pathology-informed decomposition moves the bundle toward the diagonal; low-quantile coverage reflects the zero-inflated estimand floor.The false-zero rate falls only in the ZIR-dominated band where pooling inflates it.Source: §D.2 · pp. 24–25, Figure 8 · p. 25

D.3 CAPACITY SCALING CANNOT SUBSTITUTE FOR REGIME DECOMPOSITION

  • Decomposed RBV stays within 0.022–0.142 across the capacity grid and never exceeds its matched pooled value, while also improving MSE.The smallest decomposed configurations use 32× fewer trees than the pooled optimum; pooled RBV plateaus at 0.116.Source: Figure 9 · p. 26, §D.3 · p. 27
  • For MAE, decomposition raises RBV because routing exposes the within-cell mean–median gap that pooled training dilutes.Capacity acts on fitting error, whereas decomposition removes the pooling compromise; the two interventions address different exposures.Source: Figure 10 · p. 26, §D.3 · p. 27

E.1 M5 EXPERIMENT SETUP

On M5, regime decomposition distinguishes loss-induced bias from grouping artifacts and reveals how intrinsic estimand floors shape the observed bias. The appendix also tests intermittent-demand baselines and a frozen zero-shot foundation model, where estimand choice remains consequential.

E.1 M5 EXPERIMENT SETUP

  • Under a controlled single-origin protocol using 113 training days and forecasting horizons 1–7, the M5 comparison constructed 30,490 SKU×store series and scored 26,746; results are not comparable to official M5 leaderboard results.A 28-day minimum-history screen excluded 1,995 series, and 1,749 more lacked usable evaluation signal. Separately, 17.3% of series contained at least one censored edge run.Source: §E.1 · pp. 27–28
  • Figure 11 shows regime curves fanning out under global pooling, while regime decomposition exposes a High-CV mid-τ collapse and releases high-τ curves toward the diagonal.The aggregate curve can conceal these regime differences; low-τ elevations partly reflect the zero-inflation estimand floor, not miscalibration.Source: Figure 11 · p. 28, §E.1 · pp. 28–29
  • On M5, size-matched random splits change RBV by at most 0.04, while regime grouping drives non-linear-loss RBV near zero and exposes linear-loss bias consistent with intrinsic floors.For MAE, High-CV mean-relative bias moves from −0.60 to −0.95 on M5, versus −0.58 to −0.98 on the main benchmark; per-cell exceed rates also move toward τ.Source: §E.1 · p. 28, Table 9 · p. 28
  • For quantile losses, a model-independent estimand floor can explain regime-grouped RBV, while pooled training can push RBV below that floor despite cell-level miscalibration.For MAE, the empirical floor and observed RBV are 0.80 and 0.76; for QL60, they are 0.87 and 0.77. Quantile-loss RBV should therefore be read jointly with calibration.Source: §E.1 · p. 29
  • On the under-trained NHiTS backbone, regime-wise training sharply improves WMAPE but only modestly changes RBV, leaving a CV-cell dispersion collapse that per-cell training cannot repair.Predicted within-cell CV is 0.06–0.11 versus 0.47 realized; the contrast with XGBoost separates optimization-type from bias-type failure.Source: §E.1 · p. 29, Table 10 · p. 29

E.2 OR BASELINES AND ORACLE REFERENCE PREDICTORS

  • On M5, the oracle median achieves the best WMAPE (0.668) but the largest regime bias variation (RBV 0.695), whereas the OR methods’ low RBV reflects mean-rate calibration.The oracle median under-forecasts pathological regimes by 46–94%; this comparison is a calibration reference, not an accuracy ranking.Source: §E.2 · p. 30, Table 11 · p. 30

E.3 ZERO-SHOT FOUNDATION-MODEL VALIDATION

  • With a frozen Chronos-Bolt checkpoint, the median under-predicts globally while the quadrature mean is nearly unbiased, yet the median has better WMAPE and twice the regime-bias dispersion.The τ-scan has a U-shaped RBV curve, with WMAPE-optimal τ*=0.4 below 0.5; the median head’s exceed rate is 0.574 at nominal 0.5.Source: §E.3 · p. 30, Table 12 · p. 30, Figure 12 · p. 31
  • Chronos-Bolt’s unconstrained quantile heads can emit negative demand forecasts, mechanically depressing low-τ exceed rates and contributing to median under-prediction in pathological cells.Negative predictions occur for 65.2% of q0.1 and 5.0% of q0.5 (SKU, horizon) predictions; the quadrature mean is effectively non-negative.Source: §E.3 · pp. 30–31

Related papers

Loading 2609.40117v1…