ECHO: Correcting Time Series Foundation Model Forecasts with Historical Analogues and Errors

Boya Zhang; Xilin Dai; Yilin Wang; Qiang Xu; Luyao Dong; Xing He; Wen Wu

cs.LGOct 12, 2026

ECHO: Correcting Time Series Foundation Model Forecasts with Historical Analogues and Errors

Authors: Boya Zhang, Xilin Dai, Yilin Wang, Qiang Xu, Luyao Dong, Xing He, Wen Wu

Organizations: Shanghai Artificial Intelligence Laboratory · Shanghai Jiao Tong University · The Chinese University of Hong Kong

Abstract

Time Series Foundation Models (TSFMs) provide general-purpose forecasting capabilities, yet their forecasts can remain misaligned with the temporal characteristics of a target domain. This motivates lightweight forecast correction even when only black-box access to the TSFM is available. Our key observation is that two forms of historical information provide complementary correction signals: historical analogues reveal how similar contexts evolved, while past forecast errors reveal a TSFM's characteristic deviations in the target domain. Based on this observation, we propose ECHO, a black-box forecast correction framework that learns from both sources. ECHO uses AnalogE to assess retrieved historical analogues together with the current context and base forecast, learning transferable corrections through cross-domain pretraining. In parallel, RecentE learns target-specific residual corrections from past forecast errors and refines them as causally available feedback accumulates. ECHO dynamically combines the two corrections according to their observed forecasting performance. Across 5 representative TSFMs and 9 real-world datasets, ECHO reduces MSE in all 45 backbone--dataset settings, yielding an average relative reduction of 9.26% in macro-average MSE. Its gains also extend to long-horizon forecasting.

Figures & tables

Figure 1: Overview of ECHO. (a) Overall framework. (b) AnalogE constructs a weighted historical template and estimates a gate to control the correction of the base forecast. (c) RecentE predicts spectral residuals and updates its residual map using fully observed feedback.

Models	TimesFM 2.5		Chronos-2		Chronos-Bolt		Moirai 2.0		Sundial
MSE	Ori	ECHO	Ori	ECHO	Ori	ECHO	Ori	ECHO	Ori	ECHO
ETTh1	.355	.346	.377	.348	.362	.343	.367	.340	.337	.336
ETTh2	.257	.242	.266	.246	.252	.240	.266	.241	.242	.233
ETTm1	.280	.263	.309	.274	.311	.271	.322	.273	.268	.257
ETTm2	.148	.136	.149	.138	.149	.139	.155	.138	.149	.135
Exchange	.059	.056	.060	.058	.067	.064	.060	.058	.082	.067

Table 1: Cross-backbone forecasting MSE on 9 datasets with context length

L=512

and forecast horizon

H=64

. Ori denotes the original forecasts of each frozen TSFM, and ECHO denotes its corrected forecasts. The better result within each pair is bolded. Overall reports the relative change in macro-average MSE over the 9 datasets.

Dataset 96 192 336 720 Ori ECHO Ori ECHO Ori ECHO Ori ECHO ETTh Avg. .337 .320 .391 .367 .413 .390 .412 .399 ETTm Avg. .238 .222 .293 .273 .340 .322 .422 .400 Exchange .091 .085 .194 .181 .360 .326 1.046 .944 Weather .159 .144 .209 .189 .275 .246 .362 .332

Dataset 96 192 336 720 cont. Ori ECHO Ori ECHO Ori ECHO Ori ECHO PEMS08 .109 .098 .143 .126 .177 .156 .249 .199 Electricity .131 .127 .148 .146 .165 .166 .205 .209 Solar .222 .193 .255 .231 .292 .230 .315 .235 Overall -6.91% -7.11% -8.10% -8.50%

Table 2: Long-horizon MSE of TimesFM 2.5 with

L=512

and

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

. ETTh Avg. and ETTm Avg. average the corresponding ETT datasets. Lower is better; pairwise best results are bolded. Overall reports the relative change in macro-average MSE over the nine original datasets.

Figure 4: Role of the base forecast. Backbone-Blind predicts directly without the base forecast. Panels (a) and (b) compare the recent and historical analogue branches. Panel (c) compares ECHO. ECL denotes Electricity. Lower MSE indicates better forecasting performance.

Figure 5: Effects of AnalogE Pretraining and RecentE Online Updates. (a) Relative MSE changes from cross-domain pretraining of AnalogE compared with training only on the target domain. (b) Relative MSE changes from online updates of RecentE compared with keeping the residual map frozen. Bar lengths represent relative reduction magnitudes.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Dim.	Statistics
Historical support quality	9	Log-transformed minimum, weighted mean, weighted standard deviation, and nearest-neighbor margin of retrieval distances; weighted context RMSE, correlation, and last-value mismatch; bounded context-volatility mismatch and signed log context-volatility ratio.
Correction evidence ambiguity	6	Normalized effective support; normalized top-4 template discrepancy; absolute and relative transition dispersion; directional agreement; and mean pairwise transition cosine.
Foundation–experience disagreement	4	Mean absolute difference, symmetric relative RMS difference, correlation, and maximum absolute difference between the historical transition template and the base forecast transition.

Appendix

Table 4: Explicit evidence statistics used by AnalogE. The three groups form the 19-dimensional vector

q_{t}

Dataset	TimesFM 2.5				Chronos-2				Chronos-Bolt				Moirai 2.0				Sundial
	Ori		+ECHO		Ori		+ECHO		Ori		+ECHO		Ori		+ECHO		Ori		+ECHO
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	.355	.369	.346	.371	.377	.366	.348	.366	.362	.365	.343	.365	.367	.367	.340	.365	.337	.369	.336	.370
ETTh2	.257	.315	.242	.310	.266	.301	.246	.302	.252	.299	.240	.302	.266	.306	.241	.305	.242	.308	.233	.306
ETTm1	.280	.316	.263	.314	.309	.310	.274	.303	.311	.318	.271	.312	.322	.321	.273	.314	.268	.309	.257	.310
ETTm2	.148	.232	.136	.225	.149	.219	.138	.218	.149	.224	.139	.222	.155	.228	.138	.223	.149	.234	.135	.224

Appendix

Table 5: Complete cross-backbone forecasting results on the nine main datasets and the additional PEMS04 dataset. Ori denotes the frozen backbone, and +ECHO denotes its correction by ECHO. Lower values are better. The better result within each Ori/+ECHO pair is bolded, including ties at the displayed precision.

Dataset	$H=96$				$H=192$				$H=336$				$H=720$
	Ori		+ECHO		Ori		+ECHO		Ori		+ECHO		Ori		+ECHO
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	.3753	.3826	.3637	.3825	.4149	.4071	.3939	.4035	.4344	.4183	.4103	.4140	.4191	.4269	.4062	.4313
ETTh2	.2989	.3429	.2763	.3339	.3681	.3876	.3408	.3779	.3907	.4101	.3691	.4018	.4043	.4297	.3924	.4261
ETTm1	.3011	.3299	.2842	.3298	.3457	.3596	.3271	.3602	.3773	.3826	.3662	.3899	.4405	.4185	.4316	.4323
ETTm2	.1741	.2519	.1607	.2437	.2398	.2979	.2187	.2855	.3018	.3388	.2780	.3264	.4039	.3989	.3682	.3846

Appendix

Table 6: Complete long-horizon forecasting results of TimesFM 2.5 on nine datasets using black-box forecasting. The context length is

L=512

, with forecasting horizons

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

. Ori denotes the original TimesFM 2.5 forecasts, and +ECHO denotes their correction using ECHO-Long. MSE and MAE are reported, with lower values indicating better performance. The better result within each Ori/+ECHO pair is bolded. The last row reports the relative change in the unweighted macro average over the nine datasets. Aggregates and relative changes are computed from unrounded values.

	ETTh1		ETTh2		ETTm1		ETTm2		Exchange		Weather		Electricity		PEMS08		Solar
Method	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Backbone	.355	.369	.257	.315	.280	.316	.148	.232	.059	.168	.135	.171	.120	.212	.097	.196	.208	.217
AnalogE Only	.347	.369	.246	.313	.268	.318	.143	.231	.056	.165	.131	.172	.119	.212	.103	.209	.181	.217
RecentE Only	.364	.389	.249	.319	.270	.319	.135	.225	.070	.188	.122	.165	.118	.214	.089	.193	.186	.246
ECHO	.346	.371	.242	.310	.263	.314	.136	.225	.056	.164	.124	.164	.116	.209	.089	.191	.180	.217

Appendix

Table 7: Ablation study of ECHO on TimesFM 2.5 across nine datasets with context length 512 and forecasting horizon 64. AnalogE Only and RecentE Only retain Historical Analog Evidence and Adaptive Recent Evidence, respectively. The corresponding radar visualization uses the same MSE results, with ETTh1/ETTh2 and ETTm1/ETTm2 averaged as ETTh Avg. and ETTm Avg. Each entry reports MSE/MAE; lower values are better. Best and second-best results are shown in bold and underlined , respectively, according to unrounded values.

Method	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Weather	Electricity	PEMS08	Solar
TimesFM 2.5	.355 / .369	.257 / .315	.280 / .316	.148 / .232	.059 / .168	.135 / .171	.120 / .212	.097 / .196	.208 / .217
BB-TS-RAG	.351 / .369	.247 / .310	.272 / .315	.152 / .235	.074 / .183	.139 / .180	.127 / .214	.101 / .202	.196 / .221
Naive RAF	.350 / .372	.248 / .306	.280 / .319	.146 / .230	.058 / .166	.130 / .167	.127 / .221	.137 / .238	.202 / .208
ECHO AnalogE	.347 / .369	.246 / .313	.268 / .318	.143 / .231	.056 / .165	.131 / .172	.119 / .212	.103 / .209	.181 / .217

Appendix

Table 8: Comparison of historical evidence integration methods on a frozen TimesFM 2.5 backbone. Naive RAF uses a frozen backbone. ECHO AnalogE denotes our Historical Analog Evidence module. Each result is reported as MSE/MAE; lower values are better. Best results are bolded according to unrounded values.

Family	Variant	$X$	$Y^{\mathrm{base}}$	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Weather
ELF	–	✓	–	.342/.367	.247/.314	.276/.315	.146/.232	.067/.183	.133/.171
ELF	+ $Y^{\mathrm{base}}$	✓	✓	.342/.367	.250/.316	.276/.316	.147/.232	.067/.184	.133/.171
Ada-Y	–	–	✓	.357/.370	.257/.319	.279/.316	.149/.233	.059/.171	.134/.170
Ada-Y	+ $X$	✓	✓	.357/.371	.258/.319	.280/.317	.148/.232	.057/.167	.133/.171
RecentE	–	✓	✓	.364/.389	.249/.319	.270/.319	.135/.225	.070/.188	.122/.165

Appendix

Table 9: MSE/MAE comparison of recent-evidence methods on six datasets with TimesFM 2.5.

X

and

Y^{\mathrm{base}}

indicate predictor inputs. ELF retains its original forecaster and ensemble weighting. All methods are evaluated without AnalogE or TempRA. Lower is better.

Dataset	TimesFM 2.5				Chronos-2
	Fixed Fusion		ECHO		Fixed Fusion		ECHO
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	.352	.380	.346	.371	.355	.380	.348	.366
ETTh2	.243	.312	.242	.310	.241	.305	.246	.302
ETTm1	.264	.315	.263	.314	.266	.311	.274	.303
ETTm2	.133	.223	.136	.225	.135	.218	.138	.218

Appendix

Table 10: Comparison of TempRA with validation-selected fixed fusion. Macro reports the average over the six datasets. Relative Change reports the change in macro-average MSE/MAE from fixed fusion to ECHO. Lower values are better; the best results within each backbone are bolded, including ties at the displayed precision. Aggregates and relative changes are computed from unrounded values.

Figure 6: Ablation of temporal views in TempRA. Panels (a)–(c) report test MSE on ETTm1, Exchange, and Weather, respectively, comparing single-view, leave-one-out, and full TempRA configurations. Long, Short, and Recent denote the long, short, and recent views, with Recent using dense feedback. Each panel uses a different vertical axis range; lower MSE is better.

Dataset	TimesFM 2.5	Chronos-2	Chronos-Bolt	Moirai 2.0	Sundial
ETTh1	$0.34601\pm 0.00080$	$0.34853\pm 0.00040$	$0.34358\pm 0.00092$	$0.33998\pm 0.00108$	$0.33314\pm 0.00225$
ETTh2	$0.24253\pm 0.00085$	$0.24634\pm 0.00084$	$0.23981\pm 0.00045$	$0.24126\pm 0.00053$	$0.23371\pm 0.00055$
ETTm1	$0.26401\pm 0.00076$	$0.27446\pm 0.00074$	$0.27301\pm 0.00356$	$0.27541\pm 0.00181$	$0.25828\pm 0.00152$
ETTm2	$0.13644\pm 0.00034$	$0.13789\pm 0.00029$	$0.13941\pm 0.00034$	$0.13801\pm 0.00015$	$0.13519\pm 0.00031$
Exchange	$0.05627\pm 0.00054$	$0.05847\pm 0.00012$	$0.06306\pm 0.00047$	$0.05785\pm 0.00024$	$0.06588\pm 0.00126$
Weather	$0.12380\pm 0.00028$	$0.12841\pm 0.00035$	$0.13400\pm 0.00046$	$0.12779\pm 0.00080$	$0.12290\pm 0.00015$

Appendix

Table 11: MSE of ECHO on six datasets across five forecasting backbones. Each entry reports the mean

\pm

sample standard deviation over three seeds (42, 2024, and 2025), with AnalogE independently pretrained for each seed. Lower values are better.

Dataset	TimesFM 2.5	Chronos-2	Chronos-Bolt	Moirai 2.0	Sundial
ETTh1	$0.37077\pm 0.00052$	$0.36652\pm 0.00013$	$0.36462\pm 0.00006$	$0.36536\pm 0.00028$	$0.36795\pm 0.00146$
ETTh2	$0.30948\pm 0.00132$	$0.30042\pm 0.00115$	$0.30017\pm 0.00149$	$0.30414\pm 0.00113$	$0.30505\pm 0.00065$
ETTm1	$0.31487\pm 0.00119$	$0.30361\pm 0.00054$	$0.31177\pm 0.00046$	$0.31358\pm 0.00003$	$0.30899\pm 0.00109$
ETTm2	$0.22421\pm 0.00049$	$0.21674\pm 0.00091$	$0.22051\pm 0.00120$	$0.22241\pm 0.00024$	$0.22374\pm 0.00054$
Exchange	$0.16422\pm 0.00075$	$0.16562\pm 0.00021$	$0.17289\pm 0.00067$	$0.16712\pm 0.00050$	$0.17544\pm 0.00183$
Weather	$0.16390\pm 0.00028$	$0.15913\pm 0.00085$	$0.17201\pm 0.00145$	$0.16418\pm 0.00026$	$0.16605\pm 0.00018$

Appendix

Table 12: MAE of ECHO on six datasets across five forecasting backbones. Each entry reports the mean

\pm

sample standard deviation over three seeds (42, 2024, and 2025), with AnalogE independently pretrained for each seed. Lower values are better.

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Time-series foundation models (TSFMs) enable zero-shot forecasting through large-scale cross-domain pretraining, while retrieval augmentation further improves their performance by leveraging historical information. However, existing methods typically correct TSFM forecasts using the ground-truth futures of similar historical windows, which contain both predictive components already captured by the foundation model and sample-specific random fluctuation that is difficult to transfer. In contrast, recurring systematic model bias within prediction errors more directly characterizes the failure modes of a frozen TSFM and therefore provides more valuable correction signals. Effectively exploiting such model bias, however, poses two challenges: prediction errors at different numerical levels are difficult to compare due to scale differences, and the recurring bias must be extracted from prediction errors contaminated by random fluctuation. To address these challenges, we propose GRAM, a general retrieval-augmented framework for frozen TSFMs. GRAM first introduces an Amplitude Memory Module (AMM) that scales prediction errors by amplitude and aggregates them into retrievable prototypes. It then employs a Prototype Graph Module (PGM) to model relations among prototypes to aggregate consistent bias information while suppressing random fluctuation. During online forecasting, GRAM retrieves and expands prototypes relevant to the current query and generates per-horizon corrections to refine the original TSFM forecast. Experiments across multiple datasets and foundation models demonstrate consistent forecasting improvements.

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