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Downside-Controlled Online Forecast Combination under Delayed and Revised Outcomes

Minkyoung Kim, Hyunjung Byun, Yohan Lee, Beakcheol Jang

arXiv:2609.29096v1cs.LG

TL;DR

Post-hoc correction can improve frozen forecasters, but a corrector fitted where errors are stable can hurt where they shift, and delayed or revised outcomes complicate online weighting. The paper combines a frozen forecast with static and online correctors using matured losses. Across benchmarks, downside remains small while gains reach 11.5%, and the combination improves every evaluated load zone on its learning outcome.

  • Problem

    A post-hoc corrector can hurt a frozen forecaster when error patterns shift, while forecast losses mature after the horizon and operational outcomes may be revised.

    Source: §1 · p. 1

  • Method

    The method combines a frozen forecaster with static and online correctors on the simplex, updating weights from matured outcomes.

    Source: §1 · p. 2

  • Results

    Across 28 benchmark pairs, worst deterioration is 0.15% and gains reach 11.5%; the combination improves all seven load zones on its learning outcome.

    Source: §8 · p. 27

  • Takeaways & Limitations

    The downside-controlled combination can preserve gains across benchmark models, including foundation models, and operational load zones where individual correctors can be harmful.

    Source: §8 · pp. 26–27

    Evaluation covers standard multivariate benchmarks and one operational dataset with one revision mechanism; streams with retrospective revisions and regime changes remain untested.

    Source: §7.3 · p. 26

Abstract

from arXiv · show

Post-hoc correction adjusts a forecaster that cannot be retrained, such as a foundation model, but a correction fitted where errors are stable can hurt where they shift. We aim for downside control: not much worse than the starting forecast. We combine the frozen forecaster, a static corrector and an online corrector on the simplex, using only losses that mature after the horizon. Across seven benchmarks and four base models, two of them foundation models, the worst deterioration over 28 pairs at the main horizon is 0.15% and gains reach 11.5%. On day-ahead load for seven European bidding zones it lowers mean MSE in all seven zones, while single correctors raise mean MSE by up to 102% where the published forecast is most accurate. Three empirical conditions on expert speed, stream length and outcome alignment, each fixed by a documented failure, delimit its scope. Learning from the provisional outcome improves four zones on the settled one; learning on the settled outcome restores all seven.

1. Introduction

Correcting a frozen forecaster can help where its errors are stable and hurt where they shift, while delayed and revised outcomes complicate online learning. The paper combines the frozen forecast with static and online correctors to limit downside and evaluates that approach on benchmarks and operational load forecasts.

2.2. Correcting a frozen forecaster

  • Post-hoc correction can reverse its benefit across datasets; recent-error adjustment can reduce bias after a shift but adds variance when no shift has occurred.The gate weights adjusted and unadjusted forecasts by realized losses rather than committing to either.Source: §1 · p. 1, §2.2 · p. 4

2.1. Forecast combination and expert aggregation

  • Online combination uses past realized losses, but multi-step forecast losses arrive after the horizon and operational outcomes may be revised from provisional to settled versions.Simplex constraints can reduce weight-estimation variance when experts are highly correlated, trading a small bias for that reduction (Radchenko et al., 2023).Source: §1 · p. 1, §2.1 · p. 3

1. Introduction

  • Residuals are serially dependent, but tested window summaries do not capture that dependence; the conditioning interface offers under 5% ceiling at the prespecified budget on all five datasets.The proposed alternative reads the error sequence as matured outcomes become observable.Source: §1 · pp. 1–2
  • Across 28 dataset-model pairs, worst deterioration is 0.15% and gains reach 11.5%; the combination improves all seven load zones on the outcome it learns from.Where the published forecast is most accurate, held-out and online correctors alone raise mean MSE by 102% and 85%, respectively.Source: §1 · p. 2
  • Adaptive conformal intervals beat a single split-calibrated width by Winkler score on 13 of 14 trained-base-model pairs and in every load zone, at the price of wider benchmark intervals.The intervals tune width online from realized coverage (Gibbs and Candès, 2021).Source: §1 · p. 2
  • Learning from provisional outcomes and judging on settled outcomes can leave a shared bias that no simplex combination can remove; learning on settled outcomes restores improvement in every zone.The paper presents this mismatch as evidence for its outcome-alignment condition.Source: §1 · p. 2
  • The gate combines the frozen model, a static corrector and an online corrector using matured losses, with applicability bounded by expert speed, stream length and outcome alignment.The two correctors address different risks: the static expert cannot react to changing patterns, while the online expert can overreact; weights are warm-started on held-out data.Source: §1 · p. 2

2.3. Base models, normalization and conditioning

  • The conditioning ceiling offers a mechanism for limited gains from language-model components in forecasting pipelines, but applies only to the shared conditioning route at the tested budget.Correctors that read the input window directly are unaffected by this ceiling (Dai et al., 2026; Liu et al., 2025b).Source: §2.3 · p. 5

3. Residual audit and ceiling test

The audit finds time dependence in frozen-model residuals but little usable signal in tested summaries of a single input window. A per-sample oracle ceiling test measures how much a conditioning interface could reach under a specified optimization budget.

3. Residual audit and ceiling test

  • The audit finds serial dependence in residuals but negative cross-validated R2 for predicting horizon-mean residuals from tested input-window summaries on all five datasets.The summaries therefore do not recover usable information about the residuals in this audit.Source: §3 · p. 5
  • Per-sample oracle probing optimizes a conditioning vector against each probe sample’s own outcome to estimate the route’s ceiling, rather than testing a deployable controller.The vector is optimized on the held-out probe region after the corrector is fitted on a separate portion of the fit region.Source: §3 · p. 5
  • At 2,000 steps, some mean ceilings exceed 5%, but the oracle uses outcome information unavailable to a controller acting before the outcome.The extended-budget headroom is not claimable by a controller because tested window summaries do not predict the residual.Source: §3 · p. 6
  • At 50 oracle optimization steps per sample, the ceiling stays at or below 5% on all five datasets, reaching 3.76% on ETTh1.Figure 1 plots the conditioning ceiling against oracle optimization steps per sample; bars show five-run means and standard deviations, with open markers for individual runs.Source: Figure 1 · p. 6, §3 · p. 6

4. Method

The method combines a frozen forecast with complementary static and online correctors, while restricting adaptation to matured outcomes. A Hedge gate allocates their weights, and an adaptive conformal layer separately tracks forecast intervals.

4.1. Setting and maturation protocol

  • Only fully observed outcomes released after the H-step horizon update adaptive components; the protocol excludes partially observed horizons and routes feedback through the maturation buffer.At release, the gate scores forecasts stored at their issue origins, while online-corrector updates affect only later forecasts.Source: Figure 2 · p. 7, §4.1 · p. 7

4.2. Expert library

  • The static corrector learns frozen in-sample residual patterns, while the online corrector learns from fewer matured out-of-sample errors and can track drift.The two correctors tend to suit static versus shifting error structures, reducing the need to choose between them in advance.Source: §4.2 · pp. 7–8

4.3. Hedge gate with held-out warm start

  • The Hedge gate forms a convex combination by raising weights on experts with smaller matured losses and lowering the others.Its regret bound is cumulative and conditional: it relies on bounded losses and a horizon-chosen learning rate, and gives no floor for any single stream.Source: §4.3 · p. 8
  • A held-out warm start replays the Hedge updates before testing, avoiding costly uniform-weight burn-in on short streams.On Exchange with PatchTST, the unwarmed gate ends 5.8% above the base model on the five-run mean, whereas the warm-started gate matches it from the first prediction.Source: §4.3 · pp. 8–9, Figure 3 · p. 9

4.4. Calibrated intervals via online conformal tracking

  • An adaptive conformal tracker wraps the gated point forecast, updates its interval radius from matured outcomes, and does not feed back into the gate.Its coverage guarantee is long-run only, not a guarantee for any finite stretch; finite-sample conformal guarantees require exchangeable data.Source: §4.4 · pp. 9–10

5. Experiments

The experiments evaluate a gate combining static and online correctors across error regimes, benchmarks, and outcome protocols, and examine how its learning rate, layout, and early-stopping settings affect performance.

5.1. Setup

  • Experiments use seven benchmarks and four frozen forecasters at horizon 96, with test-stream state updated only from matured outcomes and no test-stream parameter estimation.The test stream sets no parameters; layout constants and the gate’s learning rate stay fixed across datasets, models, horizons, and experiments.Source: §5.1 · pp. 10–11, Table 1 · p. 11

5.2. Downside control on trained and foundation base models

  • The static corrector deteriorates on no benchmark pair (worst change −0.09%), but gains less than 1% on ETTm2 where the gate gains 6–8%.The static corrector is better on stable-error Weather, Electricity, and Exchange; the online corrector is better on shifting-error ETTm1 and ETTm2. The gate’s largest mean weight matches the regime on six of seven datasets.Source: §5.2 · pp. 11–12
  • Across 28 pairs, the gate’s worst change is +0.15%; it gains 11.5% on Chronos-Bolt and 10.5% on TimesFM, while 13 of 14 trained-base pairs improve.The 0.15% deterioration is the worst pair in the study. The static corrector alone improves all 14 foundation-model pairs and leads the combination on six, by at most 0.63 points.Source: §5.2 · pp. 11, 14, Table 2 · p. 12, Table 3 · p. 14

5.3. Comparison with test-time adaptation methods

  • On DLinear, the combination deteriorates on no dataset and updates 37,252 add-on parameters; TAFAS uses up to 50.4 million parameters and deteriorates by as much as 7.3%.Both baselines have lower MSE than the combination on three of seven pairs. PatchTST comparisons are incompatible because the adaptation implementations include their own PatchTST models; outcome protocol also affects reported gains.Source: Table 4 · p. 15, §5.3 · pp. 15–16

5.4. Ranks, scale and statistical base models

  • On 28 pairs, the Friedman/Nemenyi test is significant; the gate ranks first at 1.786 versus 1.893 for the static corrector, but separates from no other method.Using datasets as the seven blocks, the stricter test still rejects at 5% (p=0.008 and p=0.038); the surviving separation is the three-expert gate against the frozen base model.Source: §5.4 · p. 16, Table 5 · p. 16
  • The combination lowers MASE on 13 of 14 pairs; on Weather with PatchTST, MSE falls 1.26% while absolute error rises 1.81%.ETTh2 forecasts are worse than a seasonal rule, and Exchange forecasts are far from competitive against a random-walk reference, so gains there do not imply improvement to a strong forecast.Source: §5.4 · pp. 16–17
  • On statistical base models, the combination improves MSE by 21.4–91.3%, and none of 120 corrected cells deteriorates.The gate selects the online expert, assigning it mean weight of at least 0.989 on every row; the static corrector improves every row but falls 21–81 points short of the combination.Source: §5.4 · p. 17

5.5. Intervals and ablations

  • The adaptive interval tracker beats split calibration on all 14 pairs, and the full layer beats the split-calibrated frozen forecast on 13 of 14.Against Chronos-Bolt’s native head, tracker mean absolute coverage error is 0.020 versus 0.056, while the native head has lower mean pinball loss on six of seven datasets (0.112 versus 0.116).Source: §5.5 · p. 17
  • Across a sixfold range of η, the largest cell MSE spread is 1.14%; larger η shifts weight toward a single expert without changing which expert, and η=0.1 is fixed throughout.The warm start already lands close to that single-expert allocation, absorbing most of the learning-rate effect.Source: §5.5 · p. 18
  • A 5% early-stopping tail improves all 14 foundation-model pairs and shifts the worst case to a 0.17% gain; a 20% tail leaves Exchange with TimesFM at +0.56%, versus its 4.4% gain.The 20% tail produces the largest deterioration in the sweep.Source: §5.5 · p. 18
  • On ETTh2 with Chronos-Bolt, halving the trust-region radius or doubling the update interval leaves results worse than the reported +0.15%, while a halved warm slice and both alternative tail values turn deterioration into gains.The two worse settings differ from the reported configuration by less than a twentieth of a point; the reported setting is not best on its grid.Source: §5.5 · p. 18

6. Load forecasts with revised outcomes

The load study applies post-hoc correction to public TSO forecasts and tests both zone-wise downside control and learning from outcomes that are later revised. The gate improves all seven zones when learning and scoring align, but provisional-trained forecasts lose gains under settled scoring until learning uses the settled outcome.

6. Load forecasts with revised outcomes

  • The study uses published TSO day-ahead forecasts that outsiders cannot retrain; provisional load arrives within an hour, while settled values follow re-metering up to three months later.The two outcome versions can differ substantially by zone, from 0.6% mean absolute difference in Hungary to 9.4% in Italy.Source: §6 · p. 18

6.1. Data, zones and protocols

  • Seven zones pass completeness and plausibility screens; forecasts issue daily for 24-hour horizons, with independent per-zone correction and no cross-zone information or added covariates.The layer uses load history and the published forecast, not weather or calendar covariates.Source: §6.1 · p. 19
  • L1 learns and scores on provisional outcomes, L2 learns provisional but scores settled, and L3 learns settled after delay; a 90-day delay cannot fit the warm-start layout in four years.At 90 days, the warm slice alone needs 291 origins of 297 available, leaving almost nothing for early stopping and fitting.Source: §6.1 · p. 19

6.2. Correcting the load forecast

  • The gate uses realized losses to stay near the TSO forecast where correction harms while gaining where useful, including −57.3% in Hungary and −10.1% in Italy.In Hungary, the gate moves weight toward the online expert; on Denmark, Croatia, Italy and Belgium, the two-expert gate applies essentially no correction.Source: §6.2 · p. 20
  • On L1, the three-expert gate lowers mean MSE in all seven zones; its worst zone-run cell is +0.37%, versus +140.2% held-out and +117.4% online correction in Denmark.The single-corrector losses occur in Denmark, where the TSO forecast is already most accurate.Source: §6.2 · p. 19, Table 6 · p. 20

6.3. Intervals on the load forecast

  • Adaptive tracking on gated forecasts has the lowest Winkler score in all seven zones versus split calibration, with intervals 1.3% wider on average; Portugal coverage is 0.79 at nominal 0.90.The reported coverage is within 0.04 of nominal in five zones, but not in Portugal.Source: §6.3 · p. 21, Table 7 · p. 22

6.4. Learning and scoring on different outcome versions

  • A shared outcome bias passes unchanged through simplex mixing; per-hour ratio correction reaches −81.8% in Italy but costs 66% in Denmark and 33% in Hungary.The ratio baseline works where the outcome-version ratio is stable, but is harmful where it is small or drifts.Source: §6.4 · pp. 22–23
  • Scoring L1 forecasts on settled outcomes leaves only four of seven zones improved; learning on settled outcomes restores all seven at delays of zero and 30 days.Italy changes from −10.1% to +6.2% when L1 predictions are scored on the settled outcome.Source: §6.4 · p. 22, Figure 8 · p. 23, Table 8 · p. 23

7. Discussion

Two failed extensions show that downside control depends on stable experts and a reliable warm-start allocation, while short streams and mismatched outcome versions further constrain where the comparison holds. The discussion identifies these boundaries through empirical failures and notes additional limits in the evaluation and interval method.

7.1. Two failed extensions

  • The fourth expert must be quasi-static over the maturation delay; with a full fit region, the quasi-static added expert captures value, including −43.4% mean under L3 with no delay.The paper uses update cadence as a proxy for movement, not as a general measure of how far an expert changes.Source: §7.1 · pp. 24–25
  • The 120-origin Exchange fit region yields a warm-start allocation that transfers poorly, leaving the combination 23.0% worse than its base model.At 303 origins the allocation transferred; at horizon 336 the pair was refused because the slice alone required more origins than the held-out split supplied.Source: §7.1 · pp. 24–25
  • A fourth, rapidly learning expert breaks downside control: Denmark incurs +5.0% before losses mature, and the worst benchmark pair moves from +0.15% to +0.47%.The gate later reduces the fourth expert’s weight, but delayed feedback cannot reject the warm-start allocation before test losses arrive.Source: §7.1 · pp. 24–25

7.2. Three applicability conditions

  • The regret bound is a meaningful comparison to the frozen base only when the stream is sufficient, experts are stable, and learning and scoring outcomes align.The bound itself holds without these empirical conditions; they keep the frozen base model a useful comparator over the delay.Source: §7.2 · p. 25, §7.2.1 · p. 25, §7.2.2 · p. 25

7.2.2. Stream sufficiency

  • Short streams may lack a reliable fit region and enough matured test origins; the 966-row influenza-like-illness series fails the layout requirement in all 12 tested configurations.With 98 to 146 matured test origins, the held-out corrector beat the gate in all 50 matched comparisons, by up to 5.46 points.Source: §7.2.2 · p. 25
  • On load data, the ninety-day settled-outcome delay leaves only 34 to 66 daily fit origins after the warm slice and tail, while the documented layout still transfers across zones.Exchange with DLinear at horizon 192 is the exception: its 120-origin fit region produces a 23.0% cost.Source: §7.2.2 · p. 25

7.2.3. Outcome alignment

  • When learning uses provisional outcomes but scoring uses settled outcomes, shared revision bias cannot be removed by simplex weights; the regret bound carries across versions only as far as they agree.The improvement survives on Hungary, where the two versions nearly agree.Source: §7.2.3 · p. 26

7.3. Limitations

  • The method also leaves partial-horizon scoring open, does not model conditional error scale, and evaluates only standard benchmarks plus one re-metered load dataset with one revision mechanism.One load zone under-covers by 0.11 at the 90% level within a four-year window, so the long-run guarantee is not per-zone.Source: §7.3 · p. 26

8. Conclusions

The paper treats improving a forecaster that cannot be retrained as a deployment problem and combines it with static and online correctors using delayed losses. The combination limits benchmark downside while improving load forecasts, with applicability bounded by expert stability, stream sufficiency, and outcome alignment.

8. Conclusions

  • The method keeps the base model frozen and combines it with static and online correctors using non-negative weights that sum to one, updated from losses after the horizon elapses.The weights are warm-started by replaying the update on a disjoint held-out slice.Source: §8 · p. 26
  • At the main horizon, the 28 benchmark pairs have a worst deterioration of 0.15% and gains reach 11.5%.The four frozen base models include two zero-shot foundation models, against which the correctors were not designed.Source: §8 · p. 27
  • The combination improves every day-ahead load bidding zone on the outcome it learns from, while individual correctors harm at least one zone and cost 102% and 85% in the most accurate zone.The limited downside comes from allocation among experts, not from any single corrector.Source: §8 · p. 27
  • Learning from provisional outcomes improves only where those outcomes nearly agree with settled outcomes; learning on settled outcomes at zero- and thirty-day delays restores improvement in every zone.Simplex weights cannot remove a bias shared by all experts.Source: §8 · p. 27
  • Three empirical conditions delimit the results: experts must be stable over the maturation delay, streams must support a reliable warm start and learning, and learning outcomes must align with scoring outcomes.The paper proposes delay-aware weighting, a fit-region layout guard, and estimating the shared revision term as possible extensions.Source: §8 · p. 27
  • An intercept correction improves six of seven load zones and has a better load mean than the gate, but admitting it as an expert sacrifices downside control on its harmful zone.A warm start that assigns little weight to experts whose slice performance does not transfer is proposed as a way to admit such an expert.Source: §8 · p. 27

Data and code availability

  • The evaluation uses public benchmark archives and the public ENTSO-E load package; the authors say code, stored runs, and result files will be made public on publication.The supplementary material contains Sections S1–S14, Tables S1–S23, and Fig. S1.Source: Data and code availability · p. 27

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

Supplementary Material

The supplement provides conventions for terminology, numbering, and reporting variability in tables.

Supplementary Material

  • Table S1 defines terms used throughout, and supplementary section, table, and figure numbers carry an S prefix.Source: Supplementary Material · p. 31
  • References without an S prefix refer to the main text.Source: Supplementary Material · p. 31
  • For tables reporting mean ± standard deviation over five runs, the standard deviation uses one more decimal place than the mean, or enough decimals to show its first significant digit.Source: Supplementary Material · p. 31

S1 Terms, data regions, datasets and settings

The supplementary material clarifies terminology, split regions, and implementation settings. It specifies both the held-out layout and the configuration choices used in audits, ceiling tests, and the static corrector.

Terms and data regions

Decomposition parameters

  • Online-corrector decomposition settings vary by dataset; for example, Weather uses period 7 and kernel 48, capped at the decomposed sequence length.The residual audit and ceiling test instead take their trend kernel and seasonal period from one configuration file.Source: S1 Terms, data regions, datasets and… · p. 31
  • Under the specified budget, the ceiling test remains below 1% on ETTm1, ETTm2, and Weather when rerun at period 24: 0.06%, 0.01%, and 0.82%, respectively.At 2,000 steps, Weather reaches 6.80%, versus 6.88% at its native period.Source: S1 Terms, data regions, datasets and… · p. 31

Datasets and splits

  • The held-out split reserves, in temporal order, a corrector fit region, a warm slice of H + 200 origins for gate replay, and a 10% early-stopping tail.Third-party baselines use the same boundaries, so their reported numbers differ from published values.Source: S1 Terms, data regions, datasets and… · p. 31

Static corrector settings

  • Static-corrector settings follow the published trust-region output-adapter description and stay fixed across datasets, base models, and horizons except for δ, which differs between ETT and other datasets.The worst-deterioration pair is tested with δ halved and doubled.Source: S1 Terms, data regions, datasets and… · p. 31

S2 Residual predictability audit

The audit tests whether frozen DLinear residuals are white noise, predictable from input-window summaries, or serially dependent. Residuals reject whiteness across the datasets, but the tested window summaries fail to predict them.

S2 Residual predictability audit

  • The audit measures whiteness, horizon-mean residual predictability from input-window statistics, and lag-one autocorrelation on the held-out fit region.The fit region is reserved for corrector fitting and disjoint from the test stream.Source: S2 Residual predictability audit · pp. 31–33
  • The whiteness test uses one-step-ahead residuals at lags 10, 24, and 48, and the ridge regression uses five contiguous-in-time folds.The regression predicts horizon-mean residuals from per-sample window statistics.Source: S2 Residual predictability audit · pp. 32–33
  • Ljung–Box rejects whiteness on all five datasets, while ridge R2 from per-sample input-window statistics is negative everywhere.The table reports the maximum Ljung–Box p-value over channels and runs, and channel-median ridge R2 averaged over runs.Source: Table S5 · p. 33, S2 Residual predictability audit · p. 33

S3 Ceiling test: full results

The ceiling test probes how much an eight-dimensional conditioning interface can help a corrector. Its pre-specified budget stays below the 5% threshold on every dataset, while much larger, outcome-informed searches produce unstable higher ceilings.

S3 Ceiling test: full results

  • The test fits a corrector with free per-sample embeddings, then freezes it and optimizes each probe sample’s embedding against that sample’s outcome.The probe region is unseen during fitting; the 8-dimensional bottleneck separates the input window from the corrector.Source: S3 Ceiling test: full results · p. 34
  • The ceiling is an outcome-informed diagnostic benchmark, not a reported method; direct-window correctors lie outside this benchmark.The oracle probes what a per-sample controller could deliver through the conditioning bottleneck.Source: S3 Ceiling test: full results · p. 34
  • At 2,000 steps, ceilings exceed 5% on all five ETTh1 runs, three of five ETTh2 and Weather runs, two of five ETTm1 runs, and none of five ETTm2 runs.The reported run-by-run outcomes vary with the extended optimization budget.Source: S3 Ceiling test: full results · p. 34
  • Table S6 reports per-sample oracle-probing ceilings at 50 and 2,000 steps, with means and standard deviations over five runs; the 5% threshold was fixed before running.Source: Table S6 · p. 34

S4 Reproducibility

The experiments are designed for bit-for-bit repeatability within the reference computing environment, with results regenerated from code, input data, and run sets.

S4 Reproducibility

  • Every run reproduces bit for bit within the reference computing environment, with random sources fixed and deterministic kernels enabled.Error-covariance files are canonicalized before writing.Source: S4 Reproducibility · p. 34
  • Every reported number regenerates from the code, input data, and run set alone, without stored checkpoints.Source: S4 Reproducibility · p. 34
  • Third-party baselines are pinned by commit and left unmodified; required split alignment is applied outside their code.Source: S4 Reproducibility · p. 34
  • Runs use NVIDIA L40S devices, with run records, cache keys, and cross-environment numerical tolerance documented alongside the code.Source: S4 Reproducibility · p. 34

S5 Gate variants and classical weighting

The section compares classical weighting rules with the gate and reports how weight-estimation differences relate to combination variance. Large weight disagreement can occur with little loss difference in a highly correlated case.

S5 Gate variants and classical weighting

  • The compared classical rules include equal weights, an intercept correction from the latest 200 matured errors, and an exponentially weighted correction with a 100-origin half-life.Both corrections return the forecast unchanged before any error matures and have no fitted parameter or random source.Source: S5 Gate variants and classical weighting · p. 35
  • The error-driven rule assigns simplex weights inversely proportional to each expert’s trailing-window mean squared error over matured origins.Its reported entry uses the quadratic form at time-averaged weights because per-origin cross terms are not stored.Source: S5 Gate variants and classical weighting · p. 35
  • Four cells have two correctors close enough that which receives the largest mean gate weight changes across runs.The two-expert gate table covers 14 foundation-model pairs.Source: Table S10 · p. 36, Table S9 · p. 36
  • The Spearman correlation between ρ(e1,e2) and the variance gap is −0.42; half the cells have weights over 0.2 from the in-sample optimum, at a median cost of 1.2% of combination variance.Source: S5 Gate variants and classical weighting · p. 36
  • In the extreme cell with ρ = 0.968, the weights differ by a factor of 30 while the losses differ by 0.09%.This is the case in which simplex trimming is predicted to do best and weight-estimation variance is largest.Source: S5 Gate variants and classical weighting · p. 36

S6 Scale-free measures and statistical base models

Across the evaluated frozen statistical models, every corrected cell improves and the gate places almost all its weight on the online expert. Gains are associated with starting error, rather than a difference in method behavior across base-model families.

S6 Scale-free measures and statistical base models

  • All 120 corrected cells improve, with almost all gate weight on the online expert.The warm start converges to the same allocation vertex as the test stream.Source: S6 Scale-free measures and statistical… · p. 36
  • On ETTh1, base-model error spans 2.86× while combined error spans 1.15×; on Weather, the factors are 2.20× and 1.30×.Two statistical cells finish above every neural cell’s combined error on the same dataset.Source: S6 Scale-free measures and statistical… · p. 36
  • MASE is the mean of five per-run values, while RMSSE is the root of run-mean MSE.For RMSSE, the printed standard deviation is across five per-run roots; its mean is up to 0.013 lower on Exchange and differs by less than the last digit elsewhere.Source: Table S11 · p. 37
  • The results support an association between base-model error and correction gain, not different behavior across base-model families.On four datasets, seasonal-naive and damped-trend exponential-smoothing models are frozen; three datasets are omitted because seasonal-period conventions disagree.Source: S6 Scale-free measures and statistical… · pp. 36–37

S7 Sensitivity of the layout constants and the learning rate

Sensitivity checks find that the reported gate configuration is among the better tested settings, while the interval comparison shows under-coverage by native quantiles and closer calibration by the tracker. The load interval methods are also compared by mean pinball loss.

S7 Sensitivity of the layout constants and the learning rate

S8 Interval scores: native quantiles and load pinball

S9 Zone selection and the load layout

Seven of 36 bidding zones pass the data-quality screens, and the two outcome versions can reverse the relative accuracy of a published forecast. On short streams, the gate can lack enough matured outcomes to learn its allocation.

S9 Zone selection and the load layout

  • Seven of 36 zones pass the missing-data, gap-length, and forecast-defect screens; the settled outcome is a binding rejection reason for 16 of the 29 rejected zones.It is the only binding reason for eight rejected zones; the common grid spans 35,710 hours from 2015-01-05 to 2019-01-31.Source: S9 Zone selection and the load layout · p. 40, Table S16 · p. 40
  • Italy’s TSO forecast ranks second-most accurate against provisional load but second-least accurate against settled load; settled load exceeds provisional load almost everywhere.Every zone except Hungary scores materially worse against the settled outcome.Source: S9 Zone selection and the load layout · pp. 40–41, Table S18 · p. 41

S10 Short series

  • The warm-start and uniform allocations are indistinguishable on a 66-to-77-origin slice, but the largest allowed slice produces a different allocation after 129 replay updates.No variant deteriorates on run means; the largest per-run deterioration across 150 variant cells is 0.24%.Source: S10 Short series · p. 41, Table S19 · p. 41
  • In all 50 matched short-stream comparisons, the plain held-out corrector beats the gate by 1.01 to 5.46 points.With 98 to 146 matured test origins, the gate lacks feedback to learn the allocation; the 90-day load-delay configuration also skips cells for insufficient origins.Source: S10 Short series · p. 41

S11 Longer horizons

Longer-horizon results are mixed at horizon 192 and show no deterioration at 336. Additional diagnostics address expert speed, alternative aggregation rules, and how rank-test blocks affect method comparisons.

S11 Longer horizons

  • At horizon 336, no pair deteriorates; at horizon 192, four pairs deteriorate, three by less than their own run spread.At horizon 336, Exchange is absent because the layout guard refuses that pair; the same run-spread criterion marks three pairs at horizon 96.Source: S11 Longer horizons · p. 42, Table S20 · p. 42

S12 Expert speed and the fourth expert

S13 Alternative aggregation rules over the same experts

  • Against the replayed Hedge gate, fixed-share at α = 1/Tm differs by −0.05 percentage points and Bernstein aggregation by +0.15 on benchmark pairs.On load cells, the Bernstein gap is +0.01 points under L1 and +0.98 under L3 at D = 30, with a worst zone of +3.29; these time-averaged-weight prices are not comparable to streaming results.Source: S13 Alternative aggregation rules over… · p. 43, Table S22 · p. 43

S14 Rank test with the dataset as the block

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