Look Back, Correct Forward: Error Feedback Calibration for Time Series Forecasting

Haonan Yang; Yulin He; Zhuo Li

cs.AIOct 16, 2026

Look Back, Correct Forward: Error Feedback Calibration for Time Series Forecasting

Authors: Haonan Yang, Yulin He, Zhuo Li

Organizations: College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China.

Abstract

Time series forecasting models are increasingly deployed in continuously evolving environments. After a forecast is issued, future observations gradually arrive and reveal how the model has failed. Existing methods mainly treat such observations as supervision for parameter updates. However, it remains unclear whether verified forecasting errors can themselves provide causal feedback for correcting future predictions. To address this, we introduce Error Feedback Calibration (EFC), a fully test-time framework for deterministic post-hoc correction of time series forecasting. EFC constructs a Causal Error Diagonal (CED) from revealed errors of overlapping historical forecasts, providing a fresh view of the frozen forecaster's recent failure behavior. Then Cumulative Error Transport (CET) maps this error state to future residuals using cumulative estimation, while temporal out-of-sample validation controls the correction strength. Temporal Residual Learning (TRL) further predicts only the residual structure left after CET from recent CED trajectories, rather than relearning the entire forecasting error. Extensive experiments across forecasting backbones and datasets show that EFC broadly improves forecasting performance while substantially controlling harmful over correction.

Figures & tables

Fig. 1: Forecast errors as causal feedback. Historical point forecasts issued at different origins overlap at the same target time. Once its observation becomes available, their errors at different lead times are revealed. Rather than discarding these verified errors after evaluation, error feedback correction uses them to inform subsequent forecasts from a frozen forecaster.

Fig. 2: Overview of Error Feedback Calibration (EFC). Upon observing the current target, CED collects verified errors from overlapping historical forecasts at different origins and lead times. CET learns cumulative error transport from fully matured historical samples, while TRL models the remaining CET-deployment residual using recent CED states.

Dataset	Domain	Length	Variates	Forecast Horizon	Dataset Size	Interval
ETTh1	Temperature	14,400	7	{96,192,336,720}	(8545,2881,2881)	1 hour
ETTh2	Temperature	14,400	7	{96,192,336,720}	(8545,2881,2881)	1 hour
ETTm1	Temperature	57,600	7	{96,192,336,720}	(34465,11521,11521)	15 min
ETTm2	Temperature	57,600	7	{96,192,336,720}	(34465,11521,11521)	15 min
Electricity	Electricity	26,304	321	{96,192,336,720}	(18317,2633,5261)	1 hour
Exchange	Exchange Rate	7,588	8	{96,192,336,720}	(5120,665,1422)	1 day

TABLE I: Descriptions of conventional time series forecasting benchmarks. Length denotes the total number of timestamps, Variates denotes the number of observed variables, and Split represents the number of samples in (train, validation, test).

Model	Normalized MASE $\downarrow$		Reduction	Task Win
	Frozen	+EFC	(%)	/92
VisionTS	0.808	0.776	4.0	80/92
Sundial	0.746	0.731	1.9	78/92
Chronos-Bolt	0.728	0.701	3.7	71/92
Moirai	0.830	0.791	4.7	70/92
TimesFM-2.0	0.713	0.652	8.5	68/92

TABLE III: Average forecasting performance on time series foundation models under the TIME benchmark. Lower normalized MASE indicates better forecasting accuracy. Boldface denotes the better result.

Method	Metric	Electricity				ETTh1				Weather				Traffic
		96	192	336	720	96	192	336	720	96	192	336	720	96	192	336	720
Backbone	MSE	0.152	0.166	0.181	0.268	0.392	0.451	0.516	0.552	0.176	0.225	0.281	0.359	0.396	0.420	0.392	0.422
Backbone	MAE	0.243	0.256	0.273	0.330	0.407	0.442	0.471	0.507	0.214	0.256	0.296	0.349	0.271	0.281	0.316	0.333
PIR	MSE	0.152	0.169	0.181	0.223	0.396	0.464	0.522	0.601	0.171	0.218	0.279	0.362	0.389	0.414	0.427	0.455
PIR	MAE	0.245	0.261	0.277	0.312	0.402	0.456	0.501	0.597	0.212	0.253	0.300	0.356	0.267	0.278	0.323	0.336
COSA	MSE	0.162	0.175	0.188	0.212	0.439	0.499	0.557	0.639	0.168	0.215	0.272	0.342	0.415	0.439	0.454	0.482

TABLE IV: Comparison with representative post-hoc forecast revision and test-time adaptation methods on Electricity, ETTh1, Weather, Traffic datasets. We use iTransformer as the backbone. The best results are highlighted in bold, and the second-best results are underlined.

Error Representation	Delay	Multi-lead	MSE $\downarrow$	$\Delta$ MSE $\downarrow$
①w/o CED	–	$\times$	0.195	–
②Matured error $E_{t-H}$	$h$	$\checkmark$	0.164	-15.90%
③One-step error	0	$\times$	0.171	-12.31%
④CED	0	$\checkmark$	0.159	-18.46%

TABLE V: Ablation study of Causal Error Diagonal (CED).

Variant	Transport	OOS $\alpha$	EMA	MSE $\downarrow$	$\Delta$ MSE $\downarrow$
①Direct CED	$\times$	–	–	0.317	62.56%
②Raw Transport	$\checkmark$	$\times$	$\times$	0.166	-14.87%
③+ OOS Reliability	$\checkmark$	$\checkmark$	$\times$	0.161	-17.44%
④ Full CET	$\checkmark$	$\checkmark$	$\checkmark$	0.159	-18.46%

TABLE VI: Ablation study of Cumulative Error Transport (CET).

Variant	Temporal Context	Residual Target	OOS $\beta$	MSE $\downarrow$	$\Delta$ over CET $\downarrow$
①CET only	–	–	–	0.170	–
② $R=1$	$\times$	$\checkmark$	$\checkmark$	0.162	-4.71%
③Full-error target	$\checkmark$	$\times$	$\checkmark$	0.166	-2.35%
④ $\beta=1$	$\checkmark$	$\checkmark$	$\times$	0.160	-5.88%
⑤ Full TRL	$\checkmark$	$\checkmark$	$\checkmark$	0.159	-6.47%

TABLE VII: Ablation study of Temporal Residual Learning (TRL).

Fig. 3: Method analysis of error feedback signal. We use DLinear as backbone, with the left is on Traffic dataset, and right is on ECL dataset. Normalized residual MSE is computed relative to zero correction, with a reference value of 1.0. Lower values indicate better prediction of future forecasting errors.

Transport Estimator	Correction Gain $\Delta$ MSE $\downarrow$	Win Rate $\uparrow$	Numerical Stability
Online SGD	+26.40 (med +14.20)	0.000	Unstable
Sliding RLS	+7.11 (med +1.66)	0.458	Oscillating
Forgetting RLS	-5.39 (med -5.15)	0.583	Stable
Cumulative RLS(Ours)	-9.72 (med -11.43)	0.809	Stable

TABLE VIII: Stability analysis of different error transport estimators. Results are evaluated across three forecasting backbones and four datasets. Values in parentheses denote median correction gain across deployment steps.

Fig. 4: Analysis of residual predictability after CET. (a) Residual error reduction brought by TRL. Residual MSE normalized by CET-only correction (CET only = 1.0), comparing zero correction, CET only, and CET+TRL. (b) Correlation between CET reliability and TRL gains.

Backbone	Parameters			Inference Time
Backbone	Base	+EFC	$\Delta$	Base	+EFC	$\Delta$
	(M)	(M)	(%)	(ms/iter)	(ms/iter)	(%)
DLinear	0.074	0.108	45.9	0.630	0.917	45.6
PatchTST	2.189	2.240	2.3	2.151	2.529	17.6
iTransformer	25.647	25.698	0.2	1.451	1.714	18.1
TimeXer	34.933	34.984	0.1	6.400	7.122	11.3

TABLE IX: Efficiency analysis of EFC under online inference on Traffic dataset. The overhead is computed relative to the frozen backbone.

Panel of Figure 13 — Fig. 5: Hyperparameter sensitivity analysis of the error history length $R$ , transport window size $z$ , forgetting factor $\rho$ , and hidden dimension.

Panel of Figure 14 — Fig. 6: Forecasting visualization for backbone models with and without EFC. Representative forecasting samples are on Electricity, Traffic, and Weather datasets using DLiner, iTransformer, and PatchTST. Pred denotes the frozen backbone forecast, while +EFC denotes the corrected forecast.

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Abstract

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