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Correction-space Cross-variate Interaction for Test-time Adaptation in Time Series Forecasting

Yuanyuan Deng, Mykola Pechenizkiy, Songgaojun Deng

arXiv:2609.34638v1cs.LG

TL;DR

Time-series forecasting accuracy can degrade as real-world data drift, while delayed supervision complicates cross-variate interaction through backbone predictions. CoRe performs interaction on adapter corrections; across 168 settings, it reduces MSE relative to frozen backbones and COSA, with stronger gains at medium-to-long horizons.

  • Problem

    Forecasting accuracy degrades under distribution drift, and delayed supervision makes direct cross-variate interaction on backbone predictions a concern because errors cannot be immediately corrected.

    Source: §1 · pp. 1–2

  • Method

    CoRe performs structured cross-variate interaction on adapter corrections using Shared-anchor Correction Refinement and input-conditioned spectral gating.

    Source: §1 · p. 2

  • Results

    Across 168 settings, CoRe reduces MSE by 25.82% relative to the frozen backbone and by 10.57% relative to COSA, with larger gains at H=336 and H=720 than H=96.

    Source: §4.2 · p. 7

  • Takeaways & Limitations

    Correction-space interaction enables information sharing across variates without directly mixing uncorrected backbone errors.

    Source: §1 · p. 2

    The structural analysis does not establish an accuracy ordering between interaction spaces, and adapter corrections are not necessarily independent of backbone errors.

    Source: §3.2 · p. 4

Abstract

from arXiv · show

Test-time adaptation (TTA) is a promising paradigm for handling distribution shift in time-series forecasting (TSF), where models adapt at inference time, often leveraging delayed observed data to refine predictions. In the multivariate setting, distribution shifts often exhibit cross-variate dependencies, yet existing TSF-TTA methods adapt each variate independently and ignore this cross-variate structure. Exploiting such structure motivates cross-variate interaction, but coupling variates through backbone predictions introduces direct pathways for mixing uncorrected errors across variates, a concern under the delayed supervision of TSF-TTA. We identify the \emph{interaction space} as a key design choice, and show that acting on adapter corrections that refine backbone outputs, the \emph{correction space}, rather than on the predictions themselves, avoids directly propagating backbone errors across variates. We build on this to propose \textsc{CoRe} (\textsc{Co}rrection-space Interaction \textsc{Re}finement), realizing correction-space interaction through (i) Shared-anchor Correction Refinement (SCR), which combines each variate's correction with a shared anchor through a parameter-efficient bottleneck, and (ii) input-conditioned spectral gating, which adaptively modulates the refinement from the current input window. Across seven backbones, six datasets, and four prediction horizons, \textsc{CoRe} reduces MSE by 25.82\% on average over backbones and 10.57\% over the state-of-the-art TSF-TTA method, with stronger gains at medium-to-long horizons and modest computational overhead. Data and code are available at: https://github.com/yyddou/CoReTTA

1 Introduction

Distribution shift exposes a gap in multivariate TSF-TTA: methods often adapt variates independently, while prediction-space interaction can mix uncorrected errors before delayed feedback arrives. CoRe instead refines adapter corrections through shared-anchor interaction and input-conditioned spectral gating, with experiments showing consistent improvements.

1 Introduction

  • Forecasting accuracy can degrade after deployment as real-world time series drift, while existing strategies may require labels, stable domain boundaries, or high computational costs.Normalization approaches transform inputs statically and do not support feedback-driven adaptation over time.Source: §1 · p. 1
  • Existing TSF-TTA methods often adapt variates independently despite joint shifts; the leading singular mode captures approximately 73% of correction energy on average, supporting shared correction structure.Prior cross-variable attention and shared prediction components had inconsistent benefits across datasets [Im and Kwon, 2026].Source: §1 · p. 2
  • Prediction-space interaction can mix uncorrected backbone errors across variates, while delayed supervision prevents immediate corrective feedback; correction-space interaction avoids directly mixing those errors.Ground-truth observations become available only after the prediction horizon.Source: §1 · p. 2
  • CoRe combines Shared-anchor Correction Refinement (SCR), which joins each variate’s correction with a shared anchor through a parameter-efficient bottleneck, and input-conditioned spectral gating.The gate uses the current input window to modulate the refinement.Source: §1 · p. 2
  • Across seven backbones, six datasets, and four horizons, CoRe shows consistent improvements; controlled comparisons favor correction-space interaction, and gains are not explained by added capacity alone.The experiments also find an advantage over prediction-space interaction.Source: §1 · p. 3

2 Related Work

Related work distinguishes channel-independent from channel-dependent forecasting models and reviews TSF-TTA methods that adapt variates independently. Prior findings also caution that cross-variate modeling can be harmful when dependencies are weak, noisy, or non-stationary.

Multivariate Time-Series Forecasting from a Channel Perspective

Test-Time Adaptation for Time-Series Forecasting

  • TSF-TTA methods include TAFAS [Kim et al., 2025], PETSA [Medeiros et al., 2025], DynaTTA, and COSA.Recent work also introduces a matured-ground-truth protocol [Wang et al., 2026].Source: §2 · p. 3
  • Existing TSF-TTA methods typically adapt variates independently, while prediction-space interaction introduces a direct pathway for mixing uncorrected backbone errors across variates.CoRe instead performs cross-variate interaction in correction space, removing this direct error-mixing pathway.Source: §2 · p. 3

3 Methodology

The method frames TSF-TTA as a frozen forecast plus per-variate adapter corrections updated when delayed targets arrive, then places cross-variate refinement on corrections rather than predictions. CoRe implements this with shared-anchor refinement, input-conditioned spectral gating, and joint online adaptation.

3.1 Problem Formulation

  • TSF-TTA forecasts from a frozen backbone and applies per-variate corrections, updating the adapter only when the corresponding targets arrive after H steps.Each prediction uses only past observations, making the setting leakage-free.Source: §3.1 · p. 4

3.2 Cross-Variate Interaction in TSF-TTA

  • Correction-space interaction keeps frozen-backbone error outside the cross-variate operator, unlike prediction-space interaction, which directly spreads that error across variates.This is a structural distinction, not a claim that adapter corrections are independent of backbone errors or that one interaction space is more accurate.Source: §3.2 · p. 4

3.3 The CoRe Framework

  • CoRe refines base-adapter corrections with SCR and input-conditioned spectral gates, then adds the refined corrections to the unchanged backbone forecast.The adaptation procedure updates the base-adapter and CoRe parameters jointly online.Source: §3.3 · p. 5, Figure 4 · p. 5, §3.6 · p. 6

3.4 Shared-anchor Correction Refinement (SCR)

  • SCR combines each variate’s correction with an anchor averaged across variates, then maps the joint representation through a shared bottleneck.With default r=C, SCR uses O(HC) parameters rather than O(H^2) for unconstrained dense mixing over the horizon.Source: §3.4 · pp. 5–6

3.5 Input-Conditioned Spectral Gating

  • Input-conditioned spectral gating derives shared window-level spectral statistics and maps them to per-variate gates that modulate SCR refinement.The descriptor uses spectral entropy and relative low-, mid-, and high-frequency band powers.Source: §3.5 · p. 6

3.6 Adaptation Objective

  • The base-adapter and CoRe parameters are updated jointly online using available-target batches, with weight decay controlled by λreg.The regularization term is intended to mitigate overfitting under limited supervision.Source: §3.6 · p. 6

4 Experiments

Across seven backbones, six datasets, and four horizons, CoRe improves forecasting MSE over the frozen backbone and COSA, with larger gains at medium-to-long horizons. The experiments also compare interaction spaces, parameter-matched capacity, and runtime overhead.

4.1 Experimental Setup

  • Evaluation covers six multivariate benchmarks, four horizons, seven frozen forecasting backbones, and TAFAS and COSA baselines; high-dimensional Electricity and Traffic assess scalability.The reported MSE is measured over the full test stream, with adaptation under delayed supervision.Source: §4.1 · p. 7

4.2 Overall Effectiveness

  • With DLinear, CoRe increases per-adaptation-step runtime from 6.28 ms to 6.76 ms over COSA, an increase of 0.48 ms (+7.7%).The average covers six datasets and four horizons.Source: §4.2 · p. 7

4.3 Why Correction-Space Interaction?

  • At H=336, correction-space interaction improves DLinear by 33.3%, compared with 19.8% for prediction-space interaction; the horizon-dependent pattern holds across seven backbones.Figure 5 controls for interaction architecture, changing only the interaction space.Source: §4.3 · p. 7
  • CoRe’s gain is not explained by added trainable capacity alone: it improves over COSA by 10.55% on average, versus 0.97% for parameter-matched COSA+MLP.The COSA+MLP comparison uses a single seed.Source: §4.3 · p. 7

4.2 Overall Effectiveness

  • 10.57%: CoRe reduces MSE relative to COSA across 168 settings, with larger gains at H=336 and H=720 than at H=96.It also reduces MSE by 25.82% relative to the frozen backbone; gains by horizon over COSA are 5.27% at H=96, 13.34% at H=336, and 11.53% at H=720.Source: §4.2 · p. 7, Table 1 · p. 8

H Base TAFAS COSA CoRe Base TAFAS COSA CoRe Base TAFAS COSA CoRe

The ablations support shared-anchor refinement and input-conditioned spectral gating, while CoRe also improves online-adaptation reliability relative to COSA.

H Base TAFAS COSA CoRe Base TAFAS COSA CoRe Base TAFAS COSA CoRe

  • Structured cross-variate refinement supports gains under a comparable parameter budget, with Exchange Rate highlighted as a setting of substantial temporal variation in cross-variate relationships.Table 2 is a parameter-matched comparison on DLinear across six datasets and four horizons, with Exchange Rate reported separately.Source: H Base TAFAS COSA CoRe Base TAFAS COSA… · p. 8, Table 2 · p. 9

4.4 Structured and Selective Refinement

  • The full SCR design improves over COSA by 9.7%, versus 7.1% without the shared anchor and 7.2% with unconstrained C × C mixing.The leading singular mode captures 73% of correction energy on average, and rank-16 SCR retains improvements on Electricity and Traffic.Source: §4.4 · p. 9
  • Spectral gating improves over COSA by 9.7%, compared with 9.2% for a fixed gate and 9.0% for loss-trend gating.The spectral signal comes from the current input window and is available from the first test batch; most of the improvement comes from SCR.Source: §4.4 · p. 9

4.5 Adaptation Reliability

  • CoRe improves over COSA across all four online-adaptation reliability measures and makes adaptation more consistently beneficial relative to the frozen backbone.The evaluation covers three backbones, six datasets, and four measures; lower is better in Table 5.Source: §4.5 · pp. 9–10, Table 5 · p. 10

5 Conclusion

CoRe applies cross-variate interaction to adapter corrections rather than backbone predictions, using structured refinement and spectral gating. Comparisons across alternative adapters and horizons support the design's effectiveness.

5 Conclusion

  • CoRe avoids directly mixing uncorrected backbone errors by performing cross-variate interaction on adapter-produced corrections, with SCR and input-conditioned spectral gating.The conclusion reports improved accuracy in most evaluated settings across seven backbones, six datasets, and four horizons, with modest computational overhead.Source: §5 · p. 10

A Extended Related Work

  • Existing TSF-TTA methods typically adapt variates independently, while multivariate forecasting approaches also include channel-dependent and partially coupled designs.The cited related-work passage describes cross-variate modeling effectiveness as varying with data characteristics.Source: A Extended Related Work · p. 14

B.1 Spectral Descriptor Computation

  • The spectral descriptor combines normalized spectral entropy with low-, middle-, and high-frequency band-energy ratios.The entropy summarizes spectral concentration, while the band ratios describe coarse frequency allocation; mean subtraction removes the DC component.Source: §B.1 · pp. 14–15

B.2 Full Online Adaptation Procedure

  • Online updates use only forecast–target pairs whose targets have already been revealed, while the forecasting backbone remains frozen.The adaptation batch contains samples with available targets, maintaining the stated leakage-free delayed-supervision setting.Source: §B.2 · p. 15

B.4 Variance Across Random Seeds

  • Across the 168 main experimental settings, the observed variance over 10 random seeds is small relative to average performance differences.Table 6 reports standard deviations for CoRe MSE.Source: §B.4 · p. 17, Table 6 · p. 17

C.1 Full Correction-Space vs. Prediction-Space Comparison

  • At H = 192, H = 336, and H = 720, the largest observed correction-space advantages over prediction-space interaction are 16.2%, 15.0%, and 9.5%, respectively.The schemes are generally comparable at H = 96; the controlled comparison holds the refinement architecture and base-adapter correction unchanged.Source: §C.1 · p. 17

C.2 Generality across Base Adapters

  • Adding SCR reduces MSE by 4.49% on TAFAS and 3.70% on a standalone MLP adapter on average.The MLP comparison improves 156 of 168 settings; the remaining negative differences are small in absolute magnitude (|∆MSE| < 0.005).Source: §C.2.2 · pp. 18–19, §C.2.1 · p. 18, Table 7 · p. 18

C.3 Backbone-Error Diagnostics

Across the evaluated settings, larger frozen-backbone errors are associated with stronger corrections and greater relative improvement.

C.3 Backbone-Error Diagnostics

  • The analysis relates frozen-backbone error for each test window to correction magnitude and relative improvement across all 168 main settings.Together, the reported associations indicate that correction signals vary systematically with observed frozen-backbone error.Source: §C.3 · p. 19
  • Backbone error is positively associated with correction magnitude, with mean Pearson r = 0.59 and positive correlation in 167/168 settings.After normalization by local signal scale, the association remains similar (r = 0.60).Source: §C.3 · p. 19
  • Backbone error is positively associated with relative improvement, with mean r = 0.34 and positive correlation in 166/168 settings.For non-overlapping windows, mean r = 0.36 and is positive in 138/161 measurable settings; seven Exchange-Rate settings at H = 720 have too few such windows for reliable estimation.Source: §C.3 · p. 19

D.1 Comparison with Other TSF-TTA Methods

Across the evaluated comparisons, CoRe improves on several TSF-TTA methods and remains effective across a further backbone, backbone families, and diverse shifts. The rank and capacity analyses report both strong aggregate results and boundaries for fixed-rank use.

D.1 Comparison with Other TSF-TTA Methods

  • CoRe reduces MSE relative to PETSA by 23.39% on average and outperforms it in 145/168 settings across seven backbones, six datasets, and four horizons.The average gain is positive for all seven backbones, ranging from 20.86% on iTransformer to 27.74% on Informer.Source: §D.1 · p. 20, Table 9 · p. 20

D.2 Comparison with Normalization-Based Methods

  • CoRe outperforms DynaTTA and COSA in the majority of evaluated cases, with particularly large differences in several Exchange-Rate and Weather settings.For normalization-based comparisons on DLinear, RevIN and FAN improve several settings, while CoRe has stronger aggregate performance under the evaluated test-time adaptation protocol.Source: §D.1 · p. 20, §D.2 · p. 20

D.3 Generalization to an Additional Backbone: TimeXer

  • On TimeXer, CoRe improves over COSA in 23 of 24 settings, reducing MSE by 9.13% on average.It also reduces average MSE by 24.42% relative to frozen TimeXer and 23.62% relative to PETSA; the sole regression against COSA is ETTm1 at H = 96.Source: §D.3 · p. 22, Table 12 · p. 22

D.4 Channel-Independent and Channel-Dependent Backbones

  • CoRe improves channel-independent and channel-dependent backbone groups by comparable margins.The grouping treats DLinear, FreTS, OLS, and PatchTST as channel-independent, and iTransformer, Informer, and MICN as channel-dependent.Source: §D.4 · p. 22, Table 13 · p. 23

D.5 Robustness under Diverse Distribution Shifts

  • CoRe's MSE improvement over the corresponding variant without SCR remains positive across ten evaluated shift types and is generally larger at medium-to-long horizons.The robustness evaluation uses TTFBench on ETTh1 with DLinear.Source: §D.5 · p. 23, Figure 7 · p. 25

E.1 SCR Rank, Structure, and Scalability

  • The leading singular mode captures approximately 73% of correction energy on average, and rank-16 SCR retains positive aggregate improvements over COSA on Electricity and Traffic.On Traffic, improvements are smaller and several settings are negative, including DLinear at H = 720; among evaluated ranks on six standard datasets, r = C gives the strongest aggregate performance.Source: §E.1 · pp. 23, 27, Figure 8 · p. 25
  • Dense mixing uses approximately 34× more parameters than SCR with r = C, yet has a lower win rate (128/168 versus 162/168) and smaller average MSE reduction over the frozen backbone (23.09% versus 25.82%).The comparison covers the 168 main settings.Source: §E.1 · p. 23, Table 14 · p. 25, Table 15 · p. 26

E.2 Spectral Descriptor and Gate Ablations

The ablations examine spectral descriptors and gating choices, then characterize learned gate behavior and online adaptation reliability.

E.2 Spectral Descriptor and Gate Ablations

  • Differences among spectral-descriptor variants are small, and the four-dimensional descriptor is retained for jointly summarizing spectral concentration and coarse frequency allocation at negligible additional cost.The full spectral gate adds a smaller average gain beyond structured SCR interaction.Source: §E.2 · pp. 27–28

E.3 Spectral Gate Analysis

  • Per-variate and shared spectral descriptors perform comparably at average MSE 0.2752 versus 0.2754, so the shared descriptor is retained as the simpler default.Gate activations vary across datasets and test batches, with approximately 10.5% near zero, indicating input-dependent modulation.Source: §E.3 · pp. 28–29, Table 19 · p. 29

F Online Adaptation Reliability

  • Online adaptation reliability is characterized by mean regret, harmful-adaptation frequency and severity, and the number of backbone–dataset combinations with positive mean regret.The reported comparison covers 18 backbone–dataset combinations; negative mean regret indicates average improvement over the frozen backbone.Source: F Online Adaptation Reliability · pp. 29–30, Table 21 · p. 30

G.1 Dataset and Correction Statistics

The section describes cross-variate correlation and correction statistics, qualitative forecasts, and runtime, then relates correction structure to the interaction-space analysis under delayed supervision.

G.1 Dataset and Correction Statistics

  • The six main datasets have broadly comparable mean static absolute correlations, but Exchange Rate has the largest reported shifting standard deviation (0.479), while Traffic and Electricity are more stable.The measurements are descriptive context for cross-variate structure in the benchmark suite.Source: §G.1 · p. 30, Table 22 · p. 31
  • Corrections differ across variates in magnitude and direction, and the leading singular mode captures approximately 73% of correction energy on average.The statistics describe correction-space properties rather than causal explanations of gating results.Source: §G.1 · p. 30

G.2 Qualitative Forecast Visualization

  • Figures 10 and 11 compare COSA and CoRe on Exchange Rate and Weather; on Exchange Rate, illustrated windows show per-window MSE reductions up to 36% relative to COSA.The cited example is H = 192, Var. 6: 0.0160 → 0.0102.Source: §G.2 · p. 32

G.3 Runtime Breakdown

  • With DLinear, CoRe increases per-adaptation-step runtime over COSA from 6.28 ms to 6.76 ms (+0.48 ms; +7.7%), while full-stream runtime increases from 11.42 s to 12.17 s (+6.5%).Spectral descriptor computation and SCR account for 1.14 ms and 1.45 ms, respectively.Source: §G.3 · p. 32, Table 24 · p. 32

H.1 Theoretical Connections

  • The spectral descriptor uses FFT-derived spectra as empirical descriptors of the finite current window, not as estimates of a globally stationary process.The discussion connects power spectra to second-order temporal structure and spectral entropy to energy spread.Source: §H.1 · p. 32

H.2 Theoretical Analysis of Interaction Space

  • The analysis distinguishes prediction-space from correction-space interaction as two different places to model cross-variate structure in the backbone-plus-correction decomposition.The operators may be nonlinear; for CoRe, the SCR bottleneck and spectral modulation are included in the correction-space operator.Source: §H.2 · p. 33

H.2.1 Proof of Proposition 1

  • Prediction-space interaction is directly sensitive to frozen-backbone prediction error, whereas the cross-variate correction-space operator acts on the adapter correction rather than directly on that error.The correction itself may still depend on the backbone prediction through the base adapter.Source: §H.2.1 · p. 33

H.3 Implication under Delayed Supervision

  • Because target observations arrive only after the prediction horizon, prediction-space interaction operates on error-containing backbone outputs before corresponding supervised feedback is available.The authors frame this as mechanistic motivation, not a guarantee that the performance gap grows monotonically with horizon; correction-space interaction operates on the adapter correction while preserving the backbone prediction path.Source: §H.3 · p. 34

I Limitations and Future Work

The limitations concern short horizons, interaction capacity and adapter dependence, and the scope of the analysis and deployment setting.

I Limitations and Future Work

  • At H = 96, correction-space interaction has smaller benefits and slight degradation in a small number of settings.Adapting interaction capacity to the available correction signal may improve short-horizon performance.Source: I Limitations and Future Work · p. 34
  • CoRe depends on an underlying adapter that supplies corrections, and reduced-rank SCR has smaller gains in some high-dimensional settings.Adaptive rank selection and stronger correction interfaces remain future directions.Source: I Limitations and Future Work · p. 34
  • The analysis establishes a structural distinction rather than an accuracy guarantee, and experiments assume standard delayed supervision and an additive correction interface.Irregular feedback delays and alternative interfaces, including time-series foundation models, remain untested directions.Source: I Limitations and Future Work · p. 34

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