DistDF: Time-Series Forecasting Needs Joint-Distribution Wasserstein Alignment

Hao Wang; Licheng Pan; Yuan Lu; Zhixuan Chu; Xiaoxi Li; Shuting He; Zhichao Chen; Haoxuan Li

cs.LGOct 28, 2025

DistDF: Time-Series Forecasting Needs Joint-Distribution Wasserstein Alignment

Authors: Hao Wang, Licheng Pan, Yuan Lu, Zhixuan Chu, Xiaoxi Li, Shuting He, Zhichao Chen, Haoxuan Li, +2 more

Organizations: Xiaohongshu Inc. · College of Control Science and Technology, Zhejiang University · College of Computer Science and Technology, Zhejiang University · School of Computing and Artificial Intelligence, Shanghai University of Finance and Economics · State Key Lab of General AI, School of Intelligence Science and Technology, Peking University · Squirrel AI · Institute for Artificial Intelligence, Peking University

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: The conditional correlation of label components given $\mathbf{x}$ , where the forecast length is set to $\mathrm{T}=192$ . The correlation matrices are computed for the raw labels (a), the frequency components in FreDF (b) ( Wang et al., 2025d ) and the principal components in Time-o1 (c) ( Wang et al., 2025c ) .

Loss		DistDF		Time-o1		FreDF		Koopman		Dilate		Soft-DTW		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
TimeBridge	ETTm1	0.383	0.397	0.383	0.397	0.386	0.398	0.460	0.438	0.387	0.400	0.395	0.402	0.387	0.400
	ETTh1	0.434	0.436	0.439	0.438	0.439	0.436	0.459	0.449	0.464	0.452	0.452	0.445	0.442	0.440
	ECL	0.172	0.267	0.175	0.268	0.175	0.267	0.182	0.277	0.176	0.271	0.173	0.268	0.176	0.271
	Weather	0.248	0.275	0.250	0.275	0.254	0.276	0.269	0.293	0.252	0.277	0.260	0.280	0.252	0.277
Fredformer	ETTm1	0.378	0.394	0.379	0.393	0.384	0.394	0.389	0.400	0.389	0.400	0.397	0.402	0.387	0.398

Table 1: Comparative results with other objectives for time-series forecasting.

Panel of Figure 2 — Figure 2: The forecast sequence of DF (in blue) and DistDF (in red), with history length $\mathrm{H}=96$ .

Model	Align $\mu$	Align $\mathbf{\Sigma}$	Data	T=96		T=192		T=336		T=720		Avg
Model	Align $\mu$	Align $\mathbf{\Sigma}$	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	0.142	0.239	0.161	0.257	0.182	0.278	0.217	0.309	0.176	0.271
			Weather	0.168	0.211	0.214	0.254	0.273	0.297	0.353	0.347	0.252	0.277
DistDF †	✓	✗	ETTm1	0.318	0.359	0.361	0.382	0.393	0.404	0.453	0.440	0.381	0.396

Table 2: Ablation study results.

Discrepancy		Ours		EMD		MMD@Linear		MMD@RBF		KL		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
TimeBridge	ETTm1	0.383	0.398	0.388	0.400	0.385	0.400	0.387	0.399	0.387	0.400	0.387	0.400
	ETTh1	0.433	0.437	0.441	0.439	0.438	0.437	0.441	0.440	0.437	0.438	0.442	0.440
	ECL	0.172	0.267	0.177	0.272	0.174	0.269	0.172	0.266	0.176	0.271	0.176	0.271
	Weather	0.248	0.275	0.251	0.276	0.253	0.278	0.250	0.276	0.253	0.277	0.252	0.277
Fredformer	ETTm1	0.379	0.395	0.386	0.397	0.380	0.395	0.385	0.397	0.385	0.397	0.387	0.398

Table 3: Comparative results with other discrepancies for aligning the joint distributions.

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

Table 7

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Models		DistDF		TimeBridge		Fredformer		iTransformer		FreTS		TimesNet		MICN		TiDE		PatchTST		DLinear
Models		(Ours)		(2025)		(2024)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2023)
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.316	0.357	0.323	0.361	0.326	0.361	0.338	0.372	0.342	0.375	0.368	0.394	0.319	0.366	0.353	0.374	0.325	0.364	0.346	0.373
	192	0.358	0.380	0.366	0.385	0.365	0.382	0.382	0.396	0.385	0.400	0.406	0.409	0.364	0.395	0.391	0.393	0.363	0.383	0.380	0.390
	336	0.392	0.404	0.398	0.408	0.396	0.404	0.427	0.424	0.416	0.421	0.454	0.444	0.395	0.425	0.423	0.414	0.404	0.413	0.413	0.414
	720	0.448	0.437	0.461	0.445	0.459	0.444	0.496	0.463	0.513	0.489	0.527	0.474	0.505	0.499	0.486	0.448	0.463	0.442	0.472	0.450

Appendix

Table 6: Comparative results with other established methods for time-series forecasting.

Panel of Figure 4 — Figure 4: The forecast sequences generated with DF and DistDF. The forecast length is set to 336 and the experiment is conducted on ETTm2.

Panel of Figure 5 — Figure 5: The forecast sequences generated with DF and DistDF. The forecast length is set to 192 and the experiment is conducted on ECL.

Forecast model: TimeBridge
Loss		DistDF		Time-o1		FreDF		Koopman		Dilate		Soft-DTW		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.319	0.358	0.318	0.356	0.325	0.361	0.572	0.493	0.321	0.360	0.321	0.359	0.323	0.361
	192	0.363	0.383	0.363	0.382	0.373	0.385	0.410	0.407	0.366	0.386	0.368	0.385	0.366	0.385
	336	0.394	0.405	0.396	0.407	0.398	0.406	0.397	0.408	0.397	0.409	0.405	0.410	0.398	0.408
	720	0.455	0.442	0.456	0.443	0.450	0.438	0.460	0.445	0.462	0.447	0.486	0.453	0.461	0.445

Appendix

Table 7: Comparable results with different learning objectives.

Panel of Figure 6 — Figure 6: Performance of different forecasting models with and without DistDF. The forecasting errors are averaged over forecast lengths and the error bars represent 50% confidence intervals.

Panel of Figure 7 — Figure 7: Running time (ms) with varying forecast length.

Models		TQNet		TQNet †		TimeBridge		TimeBridge †		Fredformer		Fredformer †		iTransformer		iTransformer †		FreTS		FreTS †
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.372	0.391	0.372	0.391	0.373	0.395	0.372	0.392	0.377	0.396	0.373	0.393	0.385	0.405	0.383	0.403	0.398	0.409	0.399	0.409
	192	0.430	0.424	0.430	0.422	0.428	0.426	0.424	0.429	0.437	0.425	0.428	0.425	0.440	0.437	0.438	0.434	0.451	0.442	0.457	0.447
	336	0.486	0.454	0.472	0.444	0.471	0.451	0.467	0.450	0.486	0.449	0.466	0.445	0.480	0.457	0.476	0.455	0.501	0.472	0.504	0.474
	720	0.507	0.486	0.477	0.468	0.495	0.487	0.472	0.471	0.488	0.467	0.453	0.453	0.504	0.492	0.492	0.483	0.608	0.571	0.557	0.537
	Avg	0.449	0.439	0.438	0.431	0.442	0.440	0.434	0.436	0.447	0.434	0.430	0.429	0.452	0.448	0.447	0.444	0.489	0.474	0.479	0.467

Appendix

Table 8: The performance comparison of DF and DistDF on different forecasting models.

Panel of Figure 8 — Figure 8: Evolution of training objectives and validation metrics across four datasets: ETTm1, ETTh1, and ECL (from left to right).

Models		TimeBridge		TimeBridge †		Fredformer		Fredformer †
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.405	0.402	0.395	0.391	0.391	0.396	0.386	0.390
	192	0.467	0.438	0.419	0.408	0.494	0.449	0.493	0.446
	336	0.518	0.467	0.460	0.437	0.572	0.500	0.579	0.486
	720	0.725	0.514	0.527	0.478	1.821	0.837	0.833	0.563
	Avg	0.528	0.455	0.450	0.428	0.820	0.546	0.573	0.471

Appendix

Table 9: The performance comparison of DF and DistDF on the autoregressive forecasting setting.

Models		D3U				D3U †
Metrics		MSE	MAE	CRPS	CRPS ${}_{\text{sum}}$	MSE	MAE	CRPS	CRPS ${}_{\text{sum}}$
ETTm1	96	0.317	0.357	0.263	0.723	0.316	0.357	0.265	0.720
	192	0.361	0.383	0.285	0.749	0.360	0.383	0.282	0.747
	336	0.394	0.404	0.299	0.742	0.390	0.402	0.298	0.731
	720	0.460	0.437	0.325	0.892	0.453	0.435	0.328	0.849
	Avg	0.383	0.395	0.293	0.776	0.380	0.394	0.293	0.762

Appendix

Table 10: The performance comparison of DF and DistDF on the probabilistic forecasting task.

Models		TimeMixer		TimeMixer †		SCINet		SCINet †
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.329	0.369	0.326	0.369	0.325	0.365	0.319	0.359
	192	0.371	0.391	0.373	0.392	0.383	0.397	0.367	0.385
	336	0.427	0.425	0.412	0.423	0.436	0.424	0.403	0.406
	720	0.564	0.506	0.491	0.459	0.528	0.476	0.469	0.444
	Avg	0.422	0.423	0.401	0.411	0.418	0.416	0.389	0.399

Appendix

Table 11: The performance comparison of DF and DistDF on the multi-scale architectures.

Models			DistDF		TimeBridge		DistDF		PatchTST
Metrics			MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Historical sequence length	96	96	0.164	0.209	0.168	0.211	0.179	0.220	0.189	0.230
		192	0.212	0.252	0.214	0.254	0.222	0.257	0.228	0.262
		336	0.270	0.295	0.273	0.297	0.278	0.298	0.288	0.305
		720	0.348	0.345	0.353	0.347	0.354	0.348	0.362	0.354
		Avg	0.248	0.275	0.252	0.277	0.258	0.281	0.267	0.288

Appendix

Table 12: Varying input sequence length results on the Weather dataset.

Dataset	ECL				Weather
Models	DistDF		DF		DistDF		DF
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.138 ±0.001	0.236 ±0.001	0.141 ±0.001	0.239 ±0.001	0.167 ±0.003	0.209 ±0.001	0.169 ±0.001	0.212 ±0.001
192	0.159 ±0.001	0.257 ±0.001	0.161 ±0.000	0.258 ±0.001	0.213 ±0.001	0.253 ±0.001	0.215 ±0.001	0.254 ±0.001
336	0.179 ±0.001	0.272 ±0.001	0.183 ±0.002	0.279 ±0.002	0.271 ±0.002	0.296 ±0.002	0.272 ±0.001	0.296 ±0.001
720	0.210 ±0.001	0.301 ±0.001	0.221 ±0.005	0.311 ±0.004	0.349 ±0.002	0.347 ±0.002	0.352 ±0.002	0.348 ±0.001

Appendix

Table 13: Experimental results (

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

) with varying seeds (2021-2025).

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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.

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