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

Yuanyuan Deng; Mykola Pechenizkiy; Songgaojun Deng

cs.LGSep 28, 2026

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

Authors: Yuanyuan Deng, Mykola Pechenizkiy, Songgaojun Deng

Organizations: Eindhoven University of Technology Eindhoven, The Netherlands

Abstract

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

Figures & tables

Figure 1 : Direct error mixing in prediction-space cross-variate interaction. A conceptual illustration using variables from Weather (Tpot, VPmax, OT, and other variables). Cross-variate interaction applied to backbone predictions can directly mix their uncorrected errors across variates (Step 2). In TSF-TTA, ground-truth observations become available only after the prediction horizon, delaying corrective feedback (Step 3). Proposition 3.2 formalizes the direct error-mixing pathway, while Figure 5 empirically compares prediction-space and correction-space interaction across prediction horizons.

Figure 4 : Framework of CoRe for multivariate test-time adaptation. A frozen backbone produces a base forecast, while a base adapter produces per-variate corrections. SCR performs cross-variate interaction in the correction space . Spectral descriptor of the current input window provides per-variate gates that modulate the SCR refinement. The refined corrections are added to the unchanged backbone prediction to form the final forecast.

Table 4

Method	DLinear	PatchTST	MICN	Avg. MSE	vs. COSA
COSA	0.2981	0.2927	0.3241	0.3050	–
+SCR (no anchor; per-variate bottleneck)	0.2760	0.2721	0.3018	0.2833	$-7.1\%$
+SCR (no anchor/bottleneck; full $C{\times}C$ mixing)	0.2745	0.2706	0.3044	0.2832	$-7.2\%$
+SCR (loss-trend gate)	0.2712	0.2657	0.2963	0.2777	$-9.0\%$
+SCR (fixed gate, $g{=}1$ )	0.2703	0.2651	0.2954	0.2769	$-9.2\%$
CoRe	0.2675	0.2646	0.2942	0.2754	$-9.7\%$

Table 4 : Ablation study of CoRe components. Average MSE over three backbones (DLinear, PatchTST, MICN), six datasets, four horizons, and 10 seeds. vs. COSA : relative MSE reduction over COSA. Loss-trend gate : COSA’s adaptive learning-rate signal repurposed as the SCR gate. Fixed gate (

g{=}1

): uniform cross-variate refinement without input-dependent modulation.

\mathbf{s}_{t}

: Spectral descriptors for gating (Section 3.5 ).

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6 : Controlled comparison of correction-space and prediction-space interaction across all seven backbones.

Backbone	Dataset	$H=96$		$H=192$		$H=336$		$H=720$		Avg. Gain
Backbone	Dataset	TAFAS	+CoRe	TAFAS	+ CoRe	TAFAS	+ CoRe	TAFAS	+ CoRe	Avg. Gain
DLinear	ETTh1	0.4604	0.4604	0.5099	0.5020	0.5622	0.5515	0.6685	0.6548	$+$ 1.83%
	ETTh2	0.2302	0.2039	0.2836	0.2710	0.3193	0.3229	0.3939	0.4140	$+$ 2.41%
	ETTm1	0.3488	0.3342	0.4159	0.3924	0.4787	0.4681	0.5495	0.5539	$+$ 2.81%
	ETTm2	0.1588	0.1505	0.1927	0.1841	0.2337	0.2298	0.3053	0.3008	$+$ 3.21%
	Weather	0.1828	0.1531	0.2219	0.2136	0.2695	0.2454	0.3550	0.3164	$+$ 9.95%

Appendix

Table 7 : MSE of TAFAS and TAFAS+ CoRe (ours). Bold = better result. Avg. Gain = relative MSE reduction averaged over

H\in\{96,192,336,720\}

	ETTh1		ETTh2		ETTm1		ETTm2
Backbone	w/o SCR	CoRe -MLP	w/o SCR	CoRe -MLP	w/o SCR	CoRe -MLP	w/o SCR	CoRe -MLP
DLinear	0.4842	0.4767	0.2519	0.2378	0.5285	0.5101	0.2096	0.1949
FreTS	0.4722	0.4681	0.2556	0.2439	0.5243	0.4957	0.2056	0.1923
iTransformer	0.4685	0.4673	0.2710	0.2605	0.5337	0.5145	0.2296	0.2131
PatchTST	0.4595	0.4526	0.2534	0.2494	0.5265	0.5038	0.2121	0.1980
OLS	0.4706	0.4661	0.2503	0.2383	0.5270	0.5087	0.2086	0.1966

Appendix

Table 8: MSE of CoRe -MLP without and with SCR, averaged over 4 horizons and 10 seeds. w/o SCR : standalone MLP correction only. CoRe -MLP : MLP correction + SCR. Gain: relative MSE reduction from SCR.

Backbone	CoRe vs. PETSA
DLinear	$22.64\%$
FreTS	$21.90\%$
iTransformer	$20.86\%$
MICN	$24.83\%$
OLS	$21.71\%$
PatchTST	$24.03\%$

Appendix

Table 9 : Comparison with PETSA [ Medeiros et al., 2025 ] . We report the MSE reduction of CoRe relative to PETSA for each backbone, averaged over six datasets and four prediction horizons. Both methods use the same frozen backbone checkpoints. Higher is better.

		DLinear			FreTS			OLS			iTransformer			MICN
	$H$	DynaTTA	COSA	CoRe	DynaTTA	COSA	CoRe	DynaTTA	COSA	CoRe	DynaTTA	COSA	CoRe	DynaTTA	COSA	CoRe
ETTh1	96	0.4708	0.4922	0.4916	0.4511	0.4623	0.4625	0.4486	0.4729	0.4723	0.4509	0.4638	0.4545	0.5804	0.5837	0.5863
	192	0.5321	0.5371	0.4961	0.5138	0.5084	0.4692	0.5082	0.5173	0.4763	0.5156	0.5024	0.4589	0.6032	0.5802	0.5203
	336	0.5792	0.5208	0.4722	0.5838	0.4970	0.4596	0.5626	0.5041	0.4568	0.6052	0.4956	0.4661	0.7037	0.5889	0.5386
	720	0.7047	0.5433	0.4891	0.7086	0.5616	0.5052	0.6933	0.5443	0.4886	0.7118	0.5381	0.5055	0.8282	0.6177	0.5591
ETTh2	96	0.2338	0.2552	0.2426	0.2397	0.2560	0.2462	0.2325	0.2443	0.2389	0.2767	0.3050	0.2803	0.2612	0.2538	0.2476

Appendix

Table 10 : Prediction MSE comparison with DynaTTA across five backbones (lower is better). Bold : best among TTA methods. Underline : second-best among TTA methods. DynaTTA results use the COSA re-implementation with prediction-drift signal.

Dataset	$H$	RevIN	FAN	CoRe
ETTh1	96	0.4591	0.4620	0.4916
	192	0.5121	0.5159	0.4961
	336	0.5587	0.5427	0.4722
	720	0.7063	0.6593	0.4891
ETTh2	96	0.2302	0.2512	0.2426
	192	0.2834	0.2973	0.2275

Appendix

Table 11 : Prediction MSE comparison with RevIN and FAN on DLinear (lower is better).

Dataset	$H$	Base (TimeXer)	TAFAS	COSA	CoRe	PETSA
ETTh1	96	0.4301	0.4304 $\pm$ 0.0004	0.4407 $\pm$ 0.0001	0.4367 $\pm$ 0.0019	0.4349 $\pm$ 0.0002
	192	0.4895	0.4996 $\pm$ 0.0001	0.4867 $\pm$ 0.0004	0.4513 $\pm$ 0.0049	0.4922 $\pm$ 0.0003
	336	0.5605	0.5795 $\pm$ 0.0113	0.4776 $\pm$ 0.0003	0.4329 $\pm$ 0.0025	0.5544 $\pm$ 0.0005
	720	0.6931	0.6810 $\pm$ 0.0010	0.5195 $\pm$ 0.0001	0.4813 $\pm$ 0.0046	0.6698 $\pm$ 0.0014
ETTh2	96	0.2367	0.2317 $\pm$ 0.0001	0.2492 $\pm$ 0.0008	0.2469 $\pm$ 0.0034	0.2367 $\pm$ 0.0003
	192	0.3000	0.3021 $\pm$ 0.0001	0.2907 $\pm$ 0.0035	0.2420 $\pm$ 0.0039	0.2986 $\pm$ 0.0002

Appendix

Table 12 : Evaluation on TimeXer [ Wang et al., 2024 ] . MSE is reported for the frozen TimeXer backbone and four TTA methods over six datasets and four horizons. TTA results are mean

\pm

standard deviation over 10 seeds. Lower is better. Bold denotes the best TTA result in each setting.

Family	Backbones	MSE reduction vs. COSA
CI	DLinear, FreTS, OLS, PatchTST	10.87%
CD	iTransformer, Informer, MICN	10.18%

Appendix

Table 13 : Average MSE reduction of CoRe over COSA by backbone family, averaged over six datasets and four horizons.

	Dense	SCR $r{=}C$	SCR $r{=}16$
Avg params	339,024	9,922	16,480
Avg imp. vs. baseline	$+23.09\%$	$+25.82\%$	$+20.37\%$
Win rate vs. baseline	128/168	162/168	147/168

Appendix

Table 14: Adapter-only parameter counts and accuracy for three cross-variate interaction designs, averaged over 168 settings.

Configuration	avg_imp
Mean pooling, $r=4$	+18.09%
Mean pooling, $r=8$	+18.84%
Mean pooling, $r=16$	+19.66%
Mean pooling, $r=32$	+20.24%
Mean pooling, $r=C$	+25.45%
Max pooling, $r=16$	+19.36%

Appendix

Table 16 : Ablation on SCR pooling strategy and bottleneck rank (DLinear backbone, 24 settings). avg_imp : average MSE improvement over backbone.

Model	Dataset	$H$	Backbone	COSA	CoRe	COSA $\to$ CoRe
DLinear	Electricity	96	0.2078	0.2024	0.1992	$+1.58\%$
		192	0.2081	0.1922	0.1771	$+7.86\%$
		336	0.2228	0.1947	0.1751	$+10.07\%$
		720	0.2644	0.2139	0.1926	$+9.96\%$
	Traffic	96	0.6710	0.6685	0.6648	$+0.55\%$
		192	0.6251	0.6127	0.6031	$+1.57\%$

Appendix

Table 17 : Results on large-scale datasets Electricity (

C{=}321

) and Traffic (

C{=}862

), averaged over 10 seeds. COSA

\to

CoRe : MSE improvement of CoRe over COSA. All methods use a fixed bottleneck rank

r{=}16

for these datasets to keep the parameter cost independent of

C

Method	DLinear	PatchTST	MICN	Avg. MSE	vs. COSA
COSA	0.2981	0.2927	0.3241	0.3050	–
+SCR (no anchor; per-variate bottleneck)	0.2760	0.2721	0.3018	0.2833	$-7.1\%$
+SCR (no anchor/bottleneck; full $C{\times}C$ mixing)	0.2745	0.2706	0.3044	0.2832	$-7.2\%$
+SCR (loss-trend gate)	0.2712	0.2657	0.2963	0.2777	$-9.0\%$
+SCR (fixed gate, $g{=}1$ )	0.2703	0.2651	0.2954	0.2769	$-9.2\%$
+SCR ( $\mathbf{s}_{t}{=}[\mathrm{SE}_{t}]$ )	0.2676	0.2638	0.2946	0.2753	$-9.7\%$

Appendix

Table 18 : Ablation study of CoRe components. Average MSE over three backbones (DLinear, PatchTST, MICN), six datasets, four horizons, and 10 seeds. vs. COSA : relative MSE reduction over COSA. Loss-trend gate : COSA’s adaptive learning-rate signal repurposed as the SCR gate. Fixed gate (

g{=}1

): uniform cross-variate refinement without input-dependent modulation.

\mathrm{SE}_{t}

: spectral entropy; LBR/MBR/HBR t : low/mid/high band energy ratios (Section 3.5 ).

Figure 9 : Example spectral-gate trajectories over adaptation steps on Weather and Exchange Rate.

Dataset	Mean	Std	Near zero ( $\|g\|{<}0.1$ )	Near init ( $g{<}{-}0.8$ )
ETTh1	$-0.583$	$0.325$	$5.9\%$	$34.5\%$
ETTh2	$-0.631$	$0.285$	$9.3\%$	$37.5\%$
ETTm1	$-0.357$	$0.421$	$9.9\%$	$15.1\%$
ETTm2	$-0.498$	$0.354$	$15.5\%$	$28.4\%$
Exchange	$-0.676$	$0.194$	$2.4\%$	$23.9\%$
Weather	$-0.352$	$0.371$	$20.3\%$	$9.0\%$

Appendix

Table 19 : SCR spectral gate statistics across datasets (DLinear backbone, averaged over 4 horizons and 10 seeds). Mean: time-averaged gate value across all variates and test batches; Std: standard deviation across variates and batches; Near zero (

|g|{<}0.1

): fraction of gate activations close to zero, indicating suppressed cross-variate refinement; Near init (

g{<}{-}0.8

): fraction of gate activations close to the initialisation bias

\mathbf{b}_{g}{=}{-1}

, indicating gates that have not moved significantly from initialisation. The gate is bounded in

(-1,1)

\tanh

$\mathbf{b}_{g}$	$-2.0$	$-1.0$	$0.0$
vs. default $\downarrow$	$+0.46\%$	$+0.00\%$	$+1.56\%$

Appendix

Table 20 : Gate bias sensitivity.

		TAFAS		COSA		CoRe
Model	Dataset	$\overline{\mathcal{R}}_{T}$	$p_{\text{neg}}$	$\overline{\mathcal{R}}_{T}$	$p_{\text{neg}}$	$\overline{\mathcal{R}}_{T}$	$p_{\text{neg}}$
DLinear	ETTh1	$-0.003$	$0.519$	$-0.041$	$0.402$	$\mathbf{-0.080}$	$\mathbf{0.300}$
	ETTh2	$+0.003$	$0.570$	$-0.056$	$0.279$	$\mathbf{-0.089}$	$\mathbf{0.223}$
	ETTm1	$-0.024$	$\mathbf{0.330}$	$+0.013$	$0.383$	$\mathbf{-0.035}$	$0.382$
	ETTm2	$+0.001$	$0.546$	$-0.026$	$0.391$	$\mathbf{-0.055}$	$\mathbf{0.330}$
	Exchange	$-0.013$	$0.213$	$-0.262$	$0.142$	$\mathbf{-0.274}$	$\mathbf{0.088}$

Appendix

Table 21 : Online adaptation consistency over 18 backbone–dataset combinations (3 backbones

\times

6 datasets), each averaged over

H\in\{96,192,336,720\}

\overline{\mathcal{R}}_{T}

: mean per-batch regret relative to the frozen backbone (negative = better than backbone on average);

p_{\text{neg}}

: fraction of batches where the adapted model underperforms the backbone. Best per row in bold . Overall row also reports

\overline{\mathcal{R}}_{T}^{+}

(average regret on settings where the method is worse than the backbone) and

n_{\text{worse}}

: number of backbone–dataset combinations (out of 18) with positive mean regret.

Dataset	$N$	Window	Mean $\|r\|$	Shifting (std)	CV	Range	Strong	Weak
ETTh1	7	168	0.222	0.266	0.919	0.771	9.5%	81.0%
ETTh2	7	168	0.325	0.237	0.846	0.865	4.8%	57.1%
ETTm1	7	672	0.224	0.264	0.913	0.771	9.5%	81.0%
ETTm2	7	672	0.325	0.234	0.845	0.872	4.8%	57.1%
Weather	21	1008	0.296	0.180	0.736	0.637	21.0%	64.3%
Exchange Rate †	8	30	0.305	0.479	0.957	0.992	3.6%	42.9%

Appendix

Table 22 : Pairwise Pearson correlation statistics across variates.

N

: number of variates; Window: rolling window size (approx. 1 week in physical time). Mean

|r|

: time-averaged absolute pairwise correlation computed on stationary series after ADF test. Shifting (std) : standard deviation of windowed correlations over time, measuring temporal instability of cross-variate dependencies (computed on raw series to preserve trend-switching signals). CV : coefficient of variation (std/mean). Range : peak-to-trough variation of windowed correlations. Strong (

|r|{>}0.7

) and Weak (

|r|{\leq}0.3

): fraction of variate pairs by static

|r|

strength. Datasets above the divider are used in main experiments.

Dataset	CV	Cos-sim std	Fixed gate	Spectral gate	Gain
ETTh1	$0.217$	$0.097$	$0.4889$	$\mathbf{0.4873}$	$+0.34\%$
Exchange	$0.322$	$0.223$	$0.0890$	$\mathbf{0.0871}$	$+2.19\%$
ETTm1	$0.428$	$0.117$	$0.4598$	$\mathbf{0.4563}$	$+0.76\%$
ETTm2	$0.514$	$0.137$	$0.1716$	$\mathbf{0.1700}$	$+0.95\%$
ETTh2	$0.554$	$0.119$	$0.2329$	$\mathbf{0.2299}$	$+1.29\%$
Weather	$0.696$	$0.151$	$0.1797$	$\mathbf{0.1747}$	$+2.77\%$

Appendix

Table 23 : Per-variate correction heterogeneity and the benefit of spectral gating over fixed gating (DLinear backbone). CV : coefficient of variation of per-variate correction norms across variates; Cos-sim std : standard deviation of cosine similarity between each variate’s correction and the cross-variate mean anchor; Fixed gate and Spectral gate : average MSE under each gating strategy (10 seeds); Gain : relative improvement from spectral over fixed gating. Datasets ordered by CV.

Figure 10 : Qualitative comparison between COSA and CoRe on Exchange Rate across four prediction horizons and three selected variates.

Figure 11 : Qualitative comparison between COSA and CoRe on Weather across four prediction horizons and three selected variates.

Component	COSA(ms)	CoRe (ms)	Overhead(ms)
Backbone prediction	1.27	1.04	$-0.22$
Spectral FFT	-	1.14	$+1.14$
SCR forward	-	1.45	$+1.45$
Loss computation	0.84	1.00	$+0.16$
Backward + update	4.42	4.11	$-0.31$
Per adaptation step	6.28	6.76	$+0.48\ (+7.7\%)$

Appendix

Table 24 : Detailed per-step timing breakdown averaged over six datasets and four horizons (DLinear). Overhead = CoRe

-

COSA; "-" indicates a component absent in that method.

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