Quadratic Direct Forecast for Training Multi-Step Time-Series Forecast Models

Hao Wang; Licheng Pan; Yuan Lu; Zhichao Chen; Tianqiao Liu; Shuting He; Zhixuan Chu; Qingsong Wen

cs.LGOct 28, 2025

Quadratic Direct Forecast for Training Multi-Step Time-Series Forecast Models

Authors: Hao Wang, Licheng Pan, Yuan Lu, Zhichao Chen, Tianqiao Liu, Shuting He, Zhixuan Chu, Qingsong Wen, +2 more

Organizations: Xiaohongshu Inc. · State Key Lab of General AI, School of Intelligence Science and Technology, Peking University · College of Engineering, Purdue University · School of Computing and Artificial Intelligence, Shanghai University of Finance and Economics · College of Computer Science and Technology, Zhejiang University · Squirrel AI · Center for Data Science, Peking University · Institute for Artificial Intelligence, Peking University · Pazhou Laboratory (Huangpu), Guangzhou, Guangdong, China

Abstract

The design of learning objectives is central to training time-series forecasting models. Existing learning objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which leads to the following two challenges: (1) they overlook the label autocorrelation effect among future steps, leading to biased learning objectives; (2) they fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted learning objective, addressing both issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the preferred weights of the forecasting tasks with varying future steps. On this basis, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves the performance of various forecast models, achieving state-of-the-art results. Code is available at https://github.com/Master-PLC/QDF.

Figures & tables

Panel of Figure 1 — Figure 1: Statistics of label components conditioned on $\boldsymbol{X}$ , with a forecast horizon of $\mathrm{T}=96$ . (a) Partial correlation and conditional variance estimated from the raw label sequence $Y$ , with colors indicating different $\boldsymbol{X}$ . (b) Partial correlation matrices of label components extracted by FreDF and Time-o1 ( Wang et al., 2025e ; Wang et al., 2025d ) . Calculation details are provided in Appendix A .

Figure 2

Models	QDF		TQNet		PDF		Fredformer		iTransformer		FreTS		TimesNet		MICN		TiDE		PatchTST		DLinear
Models	(Ours)		(2025)		(2024)		(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	MSE	MAE
ETTm1	0.371	0.389	0.376	0.391	0.387	0.396	0.387	0.398	0.411	0.414	0.414	0.421	0.438	0.430	0.396	0.421	0.413	0.407	0.389	0.400	0.403	0.407
ETTm2	0.270	0.317	0.277	0.321	0.283	0.331	0.280	0.324	0.295	0.336	0.316	0.365	0.302	0.334	0.308	0.364	0.286	0.328	0.303	0.344	0.342	0.392
ETTh1	0.431	0.431	0.449	0.439	0.452	0.440	0.447	0.434	0.452	0.448	0.489	0.474	0.472	0.463	0.533	0.519	0.448	0.435	0.459	0.451	0.456	0.453
ETTh2	0.368	0.397	0.375	0.400	0.375	0.399	0.377	0.402	0.386	0.407	0.524	0.496	0.409	0.420	0.620	0.546	0.378	0.401	0.390	0.413	0.529	0.499

Table 1: Long-term forecasting performance.

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

Loss		QDF		Time-o1		FreDF		Koopman		Soft-DTW		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
TQNet	ETTm1	0.371	0.389	0.372	0.390	0.375	0.390	0.595	0.499	0.387	0.394	0.376	0.391
	ETTh1	0.431	0.431	0.437	0.432	0.432	0.432	0.451	0.442	0.453	0.438	0.449	0.439
	ECL	0.165	0.257	0.167	0.257	0.168	0.257	0.166	0.258	0.623	0.524	0.175	0.265
	Weather	0.242	0.268	0.245	0.269	0.244	0.268	0.282	0.306	0.255	0.276	0.246	0.270
PDF	ETTm1	0.381	0.394	0.386	0.399	0.387	0.400	0.587	0.485	0.396	0.404	0.387	0.396

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

Model	Hetero.	Auto.	Data	T=96		T=192		T=336		T=720		Avg
Model	Hetero.	Auto.	Data	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
DF	✗	✗	ETTm1	0.310	0.352	0.356	0.377	0.388	0.400	0.450	0.437	0.376	0.391
			ETTh1	0.372	0.391	0.430	0.424	0.486	0.454	0.507	0.486	0.449	0.439
			ECL	0.143	0.237	0.161	0.252	0.178	0.270	0.218	0.303	0.175	0.265
			Weather	0.160	0.203	0.210	0.247	0.267	0.289	0.346	0.342	0.246	0.270
QDF †	✓	✗	ETTm1	0.309	0.351	0.354	0.378	0.387	0.401	0.450	0.439	0.375	0.392

Table 3: Ablation study results.

Panel of Figure 3 — Figure 3: Improvement of QDF 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.

Method	T=96		T=192		T=336		T=720
Method	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
DF	0.143	0.237	0.161	0.252	0.178	0.270	0.218	0.303
iMAML	0.135 5.74%↓	0.230 3.26%↓	0.154 4.31%↓	0.246 2.55%↓	0.170 4.48%↓	0.263 2.47%↓	0.205 5.90%↓	0.293 3.36%↓
MAML	0.136 5.54%↓	0.230 3.20%↓	0.154 4.24%↓	0.246 2.47%↓	0.170 4.71%↓	0.263 2.56%↓	0.205 5.65%↓	0.293 3.09%↓
MAML++	0.135 5.76%↓	0.229 3.33%↓	0.154 4.22%↓	0.246 2.49%↓	0.170 4.72%↓	0.263 2.65%↓	0.204 6.41%↓	0.292 3.67%↓
Reptile	0.136 5.06%↓	0.230 2.90%↓	0.155 3.73%↓	0.247 2.14%↓	0.171 3.91%↓	0.264 2.07%↓	0.206 5.36%↓	0.294 2.96%↓

Table 4: Comparison with meta-learning methods on ECL dataset.

Panel of Figure 4 — Figure 4: Impact of hyperparameters on the performance of QDF.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: The label autocorrelation effect on the original label sequence and the components extracted by FreDF and Time-o1 [ Wang et al., 2025d , Wang et al., 2025e ] . The datasets are ETTh1, ETTh2, ECL, and Weather from left to right. The forecast length is uniformly set to 96.

Dataset	D	Forecast length	Train / validation / test	Frequency	Domain
ETTh1	7	96, 192, 336, 720	8545/2881/2881	Hourly	Health
ETTh2	7	96, 192, 336, 720	8545/2881/2881	Hourly	Health
ETTm1	7	96, 192, 336, 720	34465/11521/11521	15min	Health
ETTm2	7	96, 192, 336, 720	34465/11521/11521	15min	Health
Weather	21	96, 192, 336, 720	36792/5271/10540	10min	Weather
ECL	321	96, 192, 336, 720	18317/2633/5261	Hourly	Electricity

Appendix

Table 5: Dataset description.

Models		QDF		TQNet		PDF		Fredformer		iTransformer		FreTS		TimesNet		MICN		TiDE		PatchTST		DLinear
Models		(Ours)		(2025)		(2024)		(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	MSE	MAE
ETTm1	96	0.307	0.349	0.310	0.352	0.326	0.363	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	0.352	0.376	0.356	0.377	0.365	0.381	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	0.383	0.398	0.388	0.400	0.397	0.402	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	0.441	0.434	0.450	0.437	0.458	0.437	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: Full results on the multi-step forecasting task. The length of history window is set to 96 for all baselines. Avg indicates the results averaged over forecasting lengths: T=96, 192, 336 and 720.

Panel of Figure 6 — Figure 6: The forecast sequences generated with DF and QDF. 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 QDF. The forecast length is set to 192 and the experiment is conducted on ECL.

Forecast model:TQNet
Loss		QDF		Time-o1		FreDF		Koopman		Soft-DTW		DF
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.307	0.349	0.309	0.351	0.314	0.355	0.806	0.578	0.315	0.353	0.310	0.352
	192	0.352	0.376	0.353	0.375	0.359	0.378	0.619	0.515	0.360	0.377	0.356	0.377
	336	0.383	0.398	0.383	0.398	0.382	0.396	0.507	0.468	0.398	0.402	0.388	0.400
	720	0.441	0.434	0.444	0.436	0.444	0.432	0.450	0.437	0.476	0.446	0.450	0.437

Appendix

Table 7: Comparable results with different learning objectives.

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

Models			QDF		TQNet		QDF		PatchTST
Metrics			MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Input sequence length	96	96	0.158	0.201	0.160	0.203	0.180	0.224	0.189	0.230
		192	0.207	0.245	0.210	0.247	0.226	0.262	0.228	0.262
		336	0.263	0.286	0.267	0.289	0.279	0.300	0.288	0.305
		720	0.342	0.339	0.346	0.342	0.354	0.347	0.362	0.354
		Avg	0.242	0.268	0.246	0.270	0.260	0.283	0.267	0.288

Appendix

Table 8: Varying input sequence length results on the Weather dataset.

Dataset	ECL				Weather
Models	QDF		DF		QDF		DF
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.135 ±0.000	0.229 ±0.000	0.143 ±0.000	0.237 ±0.000	0.160 ±0.001	0.203 ±0.001	0.160 ±0.001	0.203 ±0.001
192	0.153 ±0.000	0.245 ±0.000	0.161 ±0.000	0.252 ±0.000	0.208 ±0.001	0.246 ±0.001	0.211 ±0.001	0.248 ±0.001
336	0.169 ±0.000	0.262 ±0.000	0.178 ±0.000	0.270 ±0.000	0.264 ±0.001	0.287 ±0.001	0.266 ±0.001	0.289 ±0.001
720	0.202 ±0.002	0.291 ±0.002	0.218 ±0.000	0.303 ±0.000	0.343 ±0.001	0.340 ±0.001	0.345 ±0.001	0.342 ±0.000

Appendix

Table 9: Experimental results (

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

) with varying seeds (2021-2025).

Panel of Figure 9 — Figure 9: The running time of the QDF algorithm given varying forecast horizons ( $\mathrm{T}$ ). In each subfigure, the left panel considers the complexity of each inner-loop update (i.e., step 4 in Algorithm 1 ), the right panel considers the complexity of each outer-loop update (i.e., step 7 in Algorithm 1 ).

Forecast model:TQNet
Dataset	QDF	Time-o1	FreDF	Koopman	Dilate	Soft-DTW	DF
ETTm1	2.305	2.315	2.313	2.783	2.338	2.319	2.338
ETTh1	9.619	9.697	9.875	10.488	10.036	10.283	10.290
ECL	2.509	2.540	2.534	2.601	2.554	5.316	2.578
Weather	3.054	3.098	3.086	3.573	3.121	3.276	3.121
Forecast model:PDF

Appendix

Table 10: The comprehensive results of different learning objectives on the MAPE metric.

Forecast model:TQNet
Loss	QDF			Time-o1			FreDF			Koopman			DF
Metric	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA	SMAPE	MASE	OWA
Yearly	13.355	3.015	0.788	13.377	3.004	0.787	13.404	3.022	0.790	22.588	5.512	1.385	13.502	3.074	0.800
Quarterly	10.018	1.174	0.883	10.174	1.200	0.899	10.116	1.196	0.895	17.713	2.415	1.685	10.132	1.192	0.895
Monthly	12.756	0.939	0.884	12.776	0.949	0.889	12.786	0.952	0.891	18.655	1.506	1.355	12.777	0.945	0.887
Others	4.909	3.203	1.022	5.039	3.285	1.048	4.908	3.219	1.024	7.478	5.365	1.633	5.048	3.292	1.050

Appendix

Table 11: The comprehensive results on the short-term forecasting task.

Panel of Figure 10 — Figure 10: The dynamics of loss functions on ETTm1, ETTm2, ETTh1, ETTh2, ECL, and Weather from left to right. The forecast horizon $\mathrm{T}=336$ .

Direction	Loop	T=64	T=96	T=128	T=192	T=256	T=336	T=512	T=720
Forward	Inner	1.196 ±0.007	1.175 ±0.011	1.176 ±0.009	1.409 ±0.012	1.431 ±0.015	1.590 ±0.011	1.523 ±0.012	1.763 ±0.011
Forward	Outer	1.161 ±0.010	1.168 ±0.014	1.162 ±0.009	1.350 ±0.019	1.366 ±0.011	1.357 ±0.013	1.361 ±0.012	1.672 ±0.012
Backward	Inner	1.147 ±0.005	1.144 ±0.005	1.137 ±0.006	1.487 ±0.007	1.492 ±0.008	1.647 ±0.007	1.591 ±0.010	1.770 ±0.010
Backward	Outer	0.954 ±0.006	0.967 ±0.006	0.950 ±0.008	1.190 ±0.009	1.194 ±0.009	1.343 ±0.008	1.296 ±0.007	1.477 ±0.008

Appendix

Table 12: Running complexity of QDF algorithm under task splits K=3.

Forecast model:TQNet
Rank		Rank=1			Rank=0.8			Rank=0.6			Rank=0.4			Rank=0.2			DF
Metrics		MSE	MAE	MAPE	MSE	MAE	MAPE	MSE	MAE	MAPE	MSE	MAE	MAPE	MSE	MAE	MAPE	MSE	MAE	MAPE
ETTm1	96	0.307	0.349	2.156	0.308	0.350	2.191	0.310	0.351	2.174	0.309	0.351	2.200	0.310	0.352	2.201	0.310	0.352	2.212
	192	0.352	0.376	2.281	0.354	0.377	2.288	0.355	0.379	2.308	0.354	0.378	2.298	0.352	0.377	2.295	0.356	0.377	2.288
	336	0.383	0.398	2.329	0.387	0.399	2.349	0.387	0.399	2.344	0.387	0.400	2.356	0.387	0.401	2.357	0.388	0.400	2.338
	720	0.441	0.434	2.522	0.445	0.437	2.534	0.446	0.437	2.546	0.443	0.437	2.530	0.445	0.437	2.538	0.450	0.437	2.516

Appendix

Table 13: The performance of varying ranks of the inverse covariance matrix

\bar{\boldsymbol{\Sigma}}

Panel of Figure 11 — Figure 11: The learned correlation matrix on different datasets: ETTm1, ETTm2 and Weather from left to right.

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FreDF: Learning to Forecast in the Frequency Domain

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Time series modeling presents unique challenges due to autocorrelation in both historical data and future sequences. While current research predominantly addresses autocorrelation within historical data, the correlations among future labels are often overlooked. Specifically, modern forecasting models primarily adhere to the Direct Forecast (DF) paradigm, generating multi-step forecasts independently and disregarding label autocorrelation over time. In this work, we demonstrate that the learning objective of DF is biased in the presence of label autocorrelation. To address this issue, we propose the Frequency-enhanced Direct Forecast (FreDF), which mitigates label autocorrelation by learning to forecast in the frequency domain, thereby reducing estimation bias. Our experiments show that FreDF significantly outperforms existing state-of-the-art methods and is compatible with a variety of forecast models. Code is available at https://github.com/Master-PLC/FreDF.

Abstract

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