Time-o1: Time-Series Forecasting Needs Transformed Label Alignment

Hao Wang; Licheng Pan; Zhichao Chen; Xu Chen; Qingyang Dai; Lei Wang; Haoxuan Li; Zhouchen Lin

cs.LGMay 23, 2025

Time-o1: Time-Series Forecasting Needs Transformed Label Alignment

Authors: Hao Wang, Licheng Pan, Zhichao Chen, Xu Chen, Qingyang Dai, Lei Wang, Haoxuan Li, Zhouchen Lin

Organizations: Xiaohongshu Inc. · State Key Lab of General AI, School of Intelligence Science and Technology, Peking University · Gaoling School of Artificial Intelligence, Renmin University of China · Department of Control Science and Engineering, Zhejiang University · Center for Data Science, Peking University · Institute for Artificial Intelligence, Peking University · Pazhou Laboratory (Huangpu), Guangzhou, Guangdong, China

Abstract

Training time-series forecasting models poses unique challenges in loss function design. Most existing approaches adopt temporal mean squared error, but this study reveals two critical limitations: (1) it ignores the presence of label autocorrelation, which biases it from the true label sequence likelihood; (2) it involves excessive number of tasks, which complicates optimization, especially for long-term forecasting. To address these issues, we introduce Time-o1, a transform-enhanced loss function for time-series forecasting. The central idea is to transform the label sequence into decorrelated components with discriminated significance. Models are then trained to align the most significant components, thereby effectively mitigating label autocorrelation and reducing task amount. Experiments demonstrate that Time-o1 achieves state-of-the-art performance and is compatible with various forecast models. Code is available at https://github.com/Master-PLC/Time-o1.

Figures & tables

Panel of Figure 1 — Figure 1 : The autocorrelations and volumes in the label sequence $\mathbf{Y}$ and latent components $\mathbf{Z}$ .

Models	Time-o1		Fredformer		iTransformer		FreTS		TimesNet		MICN		TiDE		DLinear		FEDformer		Autoformer		Transformer
Models	(Ours)		(2024)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2022)		(2021)		(2017)
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	0.380	0.393	0.387	0.398	0.411	0.414	0.414	0.421	0.438	0.430	0.396	0.421	0.413	0.407	0.403	0.407	0.442	0.457	0.526	0.491	0.799	0.648
ETTm2	0.272	0.317	0.280	0.324	0.295	0.336	0.316	0.365	0.302	0.334	0.308	0.364	0.286	0.328	0.342	0.392	0.308	0.354	0.315	0.358	1.662	0.917
ETTh1	0.431	0.429	0.447	0.434	0.452	0.448	0.489	0.474	0.472	0.463	0.533	0.519	0.448	0.435	0.456	0.453	0.447	0.470	0.477	0.483	0.983	0.774
ETTh2	0.359	0.388	0.377	0.402	0.386	0.407	0.524	0.496	0.409	0.420	0.620	0.546	0.378	0.401	0.529	0.499	0.452	0.461	0.448	0.460	2.688	1.291

Table 1 : Long-term forecasting performance.

Panel of Figure 2 — Figure 2 : The visualization of label and forecast sequences generated by models trained with TMSE versus Time-o1. In both (a) and (b), the left panels display the time-domain sequences ( $\mathbf{Y}$ and $\hat{\mathbf{Y}}$ ), while the right panels illustrate their corresponding latent components ( $\mathbf{Z}$ and $\hat{\mathbf{Z}}$ ).

Loss		Time-o1		FreDF		Koopman		Dilate		Soft-DTW		DPTA		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Fredformer	ETTm1	0.379	0.393	0.384	0.394	0.389	0.400	0.389	0.400	0.397	0.402	0.396	0.402	0.387	0.398
	ETTh1	0.431	0.429	0.438	0.434	0.452	0.443	0.453	0.442	0.460	0.449	0.460	0.449	0.447	0.434
	ECL	0.178	0.270	0.179	0.272	0.190	0.282	0.187	0.280	0.206	0.298	0.202	0.294	0.191	0.284
	Weather	0.255	0.276	0.256	0.277	0.257	0.279	0.258	0.280	0.261	0.280	0.260	0.280	0.261	0.282
iTransformer	ETTm1	0.395	0.401	0.405	0.405	0.413	0.416	0.407	0.412	0.417	0.415	0.416	0.415	0.411	0.414

Table 2 : Comparable results with other loss functions for time-series forecast.

Model	Decorrelation	Reduction	Data	T=96		T=192		T=336		T=720		Avg
Model	Decorrelation	Reduction	Data	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
DF	✗	✗	ETTm1	0.326	0.361	0.365	0.382	0.396	0.404	0.459	0.444	0.387	0.398
			ETTh1	0.377	0.396	0.437	0.425	0.486	0.449	0.488	0.467	0.447	0.434
			ECL	0.150	0.242	0.168	0.259	0.182	0.274	0.214	0.304	0.179	0.270
			Weather	0.174	0.228	0.213	0.266	0.270	0.316	0.337	0.362	0.249	0.293
Time-o1 †	✗	✓	ETTm1	0.338	0.366	0.369	0.383	0.397	0.403	0.458	0.441	0.391	0.398

Table 3 : Ablation study results.

	ECL				Weather
Transformation	MSE	$\Delta$	MAE	$\Delta$	MSE	$\Delta$	MAE	$\Delta$
None	0.179	-	0.270	-	0.249	-	0.293	-
RPCA	0.171	4.31% $\downarrow$	0.261	3.16% $\downarrow$	0.244	1.78% $\downarrow$	0.286	2.38% $\downarrow$
SVD	0.175	2.24% $\downarrow$	0.264	2.18% $\downarrow$	0.248	0.34% $\downarrow$	0.290	0.93% $\downarrow$
FA	0.175	2.35% $\downarrow$	0.265	1.82% $\downarrow$	0.245	1.35% $\downarrow$	0.287	1.97% $\downarrow$
Ours	0.170	4.86% $\downarrow$	0.260	3.57% $\downarrow$	0.241	2.94% $\downarrow$	0.280	4.28% $\downarrow$

Table 4 : Varying transformations results.

Panel of Figure 3 — Figure 3 : Improvement of Time-o1 applied to different forecast models, shown with colored bars for means over forecast lengths (96, 192, 336, 720) and error bars for 50% confidence intervals.

Table 8

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4 : The label autocorrelation in the original label sequence and the extracted components. The datasets are ETTh1, ETTh2, ETTm1, and Weather from left to right. The forecast length is set to 96.

Panel of Figure 5 — Figure 5 : Running cost for projection matrix calculation (left panel with varying number of samples, right panel with varying prediction length) and sequence transformation (left panel for forward pass, right panel for backward pass, with average and shaded areas for 95% confidence intervals).

Models		Time-o1		Fredformer		iTransformer		FreTS		TimesNet		MICN		TiDE		DLinear		FEDformer		Autoformer		Transformer
Models		(Ours)		(2024)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2022)		(2021)		(2017)
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.321	0.357	0.326	0.361	0.338	0.372	0.342	0.375	0.368	0.394	0.319	0.366	0.353	0.374	0.346	0.373	0.401	0.434	0.485	0.468	0.503	0.482
	192	0.360	0.378	0.365	0.382	0.382	0.396	0.385	0.400	0.406	0.409	0.364	0.395	0.391	0.393	0.380	0.390	0.415	0.446	0.504	0.482	0.807	0.664
	336	0.389	0.400	0.396	0.404	0.427	0.424	0.416	0.421	0.454	0.444	0.395	0.425	0.423	0.414	0.413	0.414	0.432	0.450	0.520	0.489	0.847	0.678
	720	0.447	0.435	0.459	0.444	0.496	0.463	0.513	0.489	0.527	0.474	0.505	0.499	0.486	0.448	0.472	0.450	0.522	0.500	0.594	0.523	1.037	0.771

Appendix

Table 7 : The comprehensive results on the long-term forecasting task.

Models	Time-o1			Fredformer			iTransformer			FreTS			MICN			DLinear			Fedformer
Models	(Ours)			(2024)			(2024)			(2023)			(2023)			(2023)			(2023)
Metric	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA
Yearly	13.485	3.010	0.791	13.509	3.028	0.794	13.797	3.143	0.818	13.576	3.068	0.801	14.594	3.392	0.873	14.307	3.094	0.827	13.648	3.089	0.806
Quarterly	10.105	1.180	0.889	10.140	1.185	0.893	10.503	1.248	0.932	10.361	1.223	0.916	11.417	1.385	1.023	10.500	1.237	0.928	10.612	1.246	0.936
Monthly	12.649	0.930	0.875	12.696	0.931	0.878	13.227	1.013	0.935	13.088	0.990	0.919	13.834	1.080	0.987	13.362	1.007	0.937	14.181	1.105	1.011
Others	4.852	3.274	1.027	4.848	3.230	1.019	5.101	3.419	1.076	5.563	3.71	1.17	6.137	4.201	1.308	5.12	3.649	1.114	4.823	3.243	1.019

Appendix

Table 8 : The comprehensive results on the short-term forecasting task.

Panel of Figure 6 — Figure 6 : The forecast sequences generated with DF and Time-o1. The forecast length is set to 336 and the experiment is conducted on ETTm2.

Panel of Figure 7 — Figure 7 : The forecast sequences generated with DF and Time-o1. The forecast length is set to 192 and the experiment is conducted on ECL.

Panel of Figure 8 — Figure 8 : Performance of different forecast models with and without Time-o1. The forecast errors are averaged over forecast lengths and the error bars represent 50% confidence intervals.

Trans		PCA		RPCA		SVD		FA		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ECL	96	0.1449	0.2348	0.1450	0.2349	0.1450	0.2350	0.1478	0.2385	0.1500	0.2415
	192	0.1592	0.2487	0.1594	0.2487	0.1595	0.2490	0.1619	0.2517	0.1681	0.2591
	336	0.1731	0.2645	0.1732	0.2646	0.1730	0.2643	0.1789	0.2711	0.1823	0.2744
	720	0.2033	0.2920	0.2066	0.2960	0.2214	0.3066	0.2095	0.2975	0.2145	0.3035
	Avg	0.1701	0.2600	0.1710	0.2611	0.1747	0.2637	0.1745	0.2647	0.1787	0.2696

Appendix

Table 9 : Varying transformation results.

Panel of Figure 9 — Figure 9 : Time-o1 improves Fredformer performance given a wide range of transformed loss strength $\alpha$ . These experiments are conducted on ETTh1 (a), ETTh2 (b), ETTm1 (c), ETTm2 (d), ECL (e), Weather (f) datasets. Different columns correspond to different forecast lengths (from left to right: 96, 192, 336, 720, and their average with shaded areas being 15% confidence intervals).

Panel of Figure 10 — Figure 10 : Time-o1 improves Fredformer performance given a wide range of rank ratio $\gamma$ . These experiments are conducted on ETTh1 (a), ETTh2 (b), ETTm1 (c), ETTm2 (d), ECL (e), and Weather (f) datasets. Different columns correspond to different forecast lengths (from left to right: 96, 192, 336, 720, and their average with shaded areas being 15% confidence intervals).

Forecast model: FredFormer
Loss		Time-o1		FreDF		Koopman		Dilate		Soft-DTW		DPTA		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.321	0.357	0.326	0.355	0.335	0.368	0.337	0.367	0.332	0.363	0.332	0.364	0.326	0.361
	192	0.360	0.378	0.363	0.380	0.366	0.384	0.364	0.384	0.370	0.386	0.370	0.386	0.365	0.382
	336	0.389	0.400	0.392	0.400	0.399	0.408	0.397	0.406	0.406	0.409	0.409	0.410	0.396	0.404
	720	0.447	0.435	0.455	0.440	0.456	0.441	0.457	0.443	0.478	0.450	0.476	0.448	0.459	0.444

Appendix

Table 10 : Comparable results with different loss functions.

Models			Time-o1		iTransformer		Time-o1		PatchTST
Metrics			MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Input sequence length	96	96	0.163	0.202	0.171	0.210	0.175	0.213	0.200	0.244
		192	0.214	0.248	0.246	0.278	0.224	0.257	0.229	0.263
		336	0.274	0.294	0.296	0.313	0.276	0.296	0.287	0.303
		720	0.351	0.344	0.362	0.353	0.353	0.346	0.363	0.353
		Avg	0.250	0.272	0.269	0.289	0.257	0.278	0.270	0.291

Appendix

Table 11 : Varying input sequence length results on the Weather dataset.

Dataset	ECL				Weather
Models	Time-o1		DF		Time-o1		DF
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.145 ±0.000	0.235 ±0.000	0.150 ±0.001	0.242 ±0.001	0.164 ±0.001	0.203 ±0.001	0.190 ±0.012	0.232 ±0.014
192	0.160 ±0.001	0.249 ±0.001	0.166 ±0.002	0.257 ±0.002	0.216 ±0.002	0.250 ±0.001	0.240 ±0.011	0.272 ±0.010
336	0.174 ±0.002	0.266 ±0.002	0.181 ±0.001	0.273 ±0.001	0.274 ±0.001	0.294 ±0.001	0.293 ±0.003	0.310 ±0.003
720	0.205 ±0.001	0.293 ±0.001	0.216 ±0.004	0.303 ±0.003	0.353 ±0.002	0.344 ±0.001	0.361 ±0.002	0.352 ±0.001

Appendix

Table 12 : Experimental results (

\mathrm{mean}_{\pm\mathrm{std}}

) with varying seeds (2021-2025).

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Training time-series forecasting models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach resorts to minimizing the conditional negative log-likelihood, typically estimated by the mean squared error. However, this estimation proves biased when the label sequence exhibits autocorrelation. In this paper, we propose DistDF, which achieves alignment by minimizing a distributional discrepancy between the conditional distributions of forecast and label sequences. Since such conditional discrepancies are difficult to estimate from finite time-series observations, we introduce a joint-distribution Wasserstein discrepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. The proposed discrepancy is tractable, differentiable, and readily compatible with gradient-based optimization. Extensive experiments show that DistDF improves diverse forecasting models and achieves leading performance. Code is available at https://anonymous.4open.science/r/DistDF-F66B.

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