FreDF: Learning to Forecast in the Frequency Domain

Hao Wang; Licheng Pan; Zhichao Chen; Degui Yang; Sen Zhang; Yifei Yang; Xinggao Liu; Haoxuan Li

cs.LGFeb 4, 2024

FreDF: Learning to Forecast in the Frequency Domain

Authors: Hao Wang, Licheng Pan, Zhichao Chen, Degui Yang, Sen Zhang, Yifei Yang, Xinggao Liu, Haoxuan Li, +1 more

Organizations: Department of Control Science and Engineering, Zhejiang University · School of Automation, Central South University · Trust and Safety Team, TikTok Sydney, ByteDance Inc. · Department of Computer Science and Engineering, Shanghai Jiao Tong University · Center for Data Science, Peking University · Generative AI Lab, College of Computing and Data Science, Nanyang Technological University

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Visualizing label autocorrelation in time series forecasting. (a) shows the generation process of time series with dependencies depicted as arrows. (b) shows the label correlation in the time domain, where each element $\rho_{i,j}$ indicates the partial correlation between $Y_{i}$ and $Y_{j}$ given $L$ . (c-d) shows the label correlation in the frequency domain, where each element $\rho_{i,j}$ indicates the partial correlation between $F_{i}$ and $F_{j}$ given $L$ , shown with the real (c) and imaginary part (d). Due to the symmetry inherent in FFT, the forecast length in the frequency domain is halved.

Figure 2: The workflow of FreDF. Key operations in the time and frequency domains are highlighted in red and blue, respectively.

Models	FreDF		iTransformer		FreTS		TimesNet		MICN		TiDE		DLinear		FEDformer		Autoformer		Transformer		TCN
Models	(Ours)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2022)		(2021)		(2017)		(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	0.392	0.399	0.415	0.416	0.407	0.415	0.413	0.418	0.399	0.423	0.419	0.419	0.404	0.407	0.440	0.451	0.596	0.517	0.943	0.733	0.891	0.632
ETTm2	0.278	0.319	0.294	0.335	0.335	0.379	0.297	0.332	0.300	0.356	0.358	0.404	0.344	0.396	0.302	0.348	0.326	0.366	1.322	0.814	3.411	1.432
ETTh1	0.437	0.435	0.449	0.447	0.488	0.474	0.478	0.466	0.525	0.515	0.628	0.574	0.462	0.458	0.441	0.457	0.476	0.477	0.993	0.788	0.763	0.636
ETTh2	0.371	0.396	0.390	0.410	0.550	0.515	0.413	0.426	0.624	0.549	0.611	0.550	0.558	0.516	0.430	0.447	0.478	0.483	3.296	1.419	3.325	1.445

Table 1: Long-term forecasting performance.

Panel of Figure 3 — Figure 3: Visualization of forecast sequence generated with and without FreDF in the time (a-b) and frequency (c-d) domains, using the iTransformer as the backbone model.

Model	$\mathcal{L}^{\mathrm{(tmp)}}$	$\mathcal{L}^{\mathrm{(feq)}}$	Data	T=96		T=192		T=336		T=720		Avg
Model	$\mathcal{L}^{\mathrm{(tmp)}}$	$\mathcal{L}^{\mathrm{(feq)}}$	Data	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
DF	✓	✗	ETTm1	0.346	0.379	0.391	0.400	0.426	0.422	0.493	0.460	0.414	0.415
			ETTh1	0.390	0.409	0.442	0.440	0.479	0.457	0.483	0.479	0.449	0.446
			ECL	0.147	0.239	0.166	0.258	0.178	0.271	0.209	0.298	0.175	0.266
			Weather	0.201	0.246	0.250	0.282	0.302	0.317	0.370	0.361	0.280	0.302
FreDF †	✗	✓	ETTm1	0.324	0.361	0.374	0.387	0.403	0.405	0.468	0.443	0.392	0.399

Table 2: Ablation study results.

Model		ETTh1				ETTm1				ECL
Model	MSE	$\Delta$	MAE	$\Delta$	MSE	$\Delta$	MAE	$\Delta$	MSE	$\Delta$	MAE	$\Delta$
iTransformer	0.449	-	0.447	-	0.415	-	0.416	-	0.176	-	0.267	-
+ FreDF-T	0.437	$\downarrow$ 2.63%	0.435	$\downarrow$ 2.62%	0.392	$\downarrow$ 5.49%	0.399	$\downarrow$ 4.01%	0.170	$\downarrow$ 3.41%	0.259	$\downarrow$ 2.77%
+ FreDF-D	0.445	$\downarrow$ 0.92%	0.440	$\downarrow$ 1.42%	0.395	$\downarrow$ 4.77%	0.398	$\downarrow$ 4.33%	0.171	$\downarrow$ 2.51%	0.260	$\downarrow$ 2.52%
+ FreDF-2	0.432	$\downarrow$ 3.94%	0.431	$\downarrow$ 3.57%	0.392	$\downarrow$ 5.60%	0.399	$\downarrow$ 4.05%	0.166	$\downarrow$ 5.32%	0.256	$\downarrow$ 4.20%

Table 3: Varying FFT implementation results.

Panel of Figure 4 — Figure 4: Benefit of incorporating FreDF in varying models, shown with colored bars for means over forecast lengths (96, 192, 336, 720) and error bars for 99.9% confidence intervals.

Panel of Figure 5 — Figure 5: Varying projection bases results, shown with colored bars for means over forecast lengths (96, 192, 336, 720) and error bars for 99.9% confidence intervals.

Panel of Figure 6 — Figure 6: Varying strength of frequency loss ( $\alpha$ ) results, shown with colored lines for T=192, 336.

Panel of Figure 7 — Figure 7: Learning curve on ETTm1 dataset.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 8 — Figure 8: Visualization of partial correlation and DML approach for partial correlation quantification. (a) The correlation graph where the pseudo correlation is caused by the fork structure $\mathcal{T}\leftarrow\mathcal{X}\rightarrow\mathcal{Y}$ . (b) The implementation of DML, where $\beta$ is the identified strength of the partial correlation $\mathcal{T}\rightarrow\mathcal{Y}$ . (c) The partial correlation identified by DML.

Panel of Figure 9 — Figure 9: More comprehensive visualizations of label autocorrelation in different domains and datasets, with columns representing different datasets: Traffic, ETTh1, and ECL, from left to right. Panels (a-c) show the label correlation in the time domain, where each element $\rho_{i,j}$ indicates the partial correlation between $Y_{i}$ and $Y_{j}$ given $L$ . Panels (d-i) show the label correlation in the frequency domain, where each element $\rho_{i,j}$ indicates the partial correlation between $F_{i}$ and $F_{j}$ given $L$ , shown with the real (d-f) and imaginary part (g-i). Due to the symmetry inherent in FFT, the forecast length in the frequency domain is halved.

Panel of Figure 10 — Figure 10: More comprehensive visualizations of label autocorrelation in different domains and label lengths, with columns representing label lengths H=48, 96, 192, 336 from left to right. Panels (a-d) show the label correlation in the time domain, where each element $\rho_{i,j}$ indicates the partial correlation between $Y_{i}$ and $Y_{j}$ given $L$ . Panels (e-l) show the label correlation in the frequency domain, where each element $\rho_{i,j}$ indicates the partial correlation between $F_{i}$ and $F_{j}$ given $L$ , shown with the real (e-h) and imaginary part (i-l).

Panel of Figure 11 — Figure 11: The label sequences (black lines) and forecast sequences generated by DF (blue lines) and FreDF (red lines). The forecast model used is iTransformer, with experiments conducted on selected snapshots characterized by periodicity (a) and trend (b).

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 4: Dataset description.

Models		FreDF		iTransformer		FreTS		TimesNet		MICN		TiDE		DLinear		FEDformer		Autoformer		Transformer		TCN
Models		(Ours)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2022)		(2021)		(2017)		(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	0.324	0.362	0.346	0.379	0.339	0.374	0.338	0.379	0.318	0.366	0.364	0.387	0.345	0.372	0.389	0.427	0.468	0.463	0.591	0.549	0.887	0.613
	192	0.373	0.385	0.392	0.400	0.382	0.397	0.389	0.400	0.364	0.396	0.398	0.404	0.381	0.390	0.402	0.431	0.573	0.509	0.704	0.629	0.877	0.626
	336	0.402	0.404	0.427	0.422	0.421	0.426	0.429	0.428	0.398	0.428	0.428	0.425	0.414	0.414	0.438	0.451	0.596	0.527	1.171	0.861	0.890	0.636
	720	0.469	0.444	0.494	0.461	0.485	0.462	0.495	0.464	0.514	0.501	0.487	0.461	0.473	0.451	0.529	0.498	0.749	0.569	1.307	0.893	0.911	0.653

Appendix

Table 5: The comprehensive results on the long-term forecasting task.

Models	FreDF			FreTS			iTransformer			MICN			DLinear			Fedformer			Autoformer
Models	(Ours)			(2023)			(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	13.556	3.046	0.798	13.576	3.068	0.801	13.797	3.143	0.818	14.594	3.392	0.873	14.307	3.094	0.827	13.648	3.089	0.806	18.477	4.26	1.101
Quarterly	10.374	1.229	0.919	10.361	1.223	0.916	10.503	1.248	0.932	11.417	1.385	1.023	10.500	1.237	0.928	10.612	1.246	0.936	14.254	1.829	1.314
Monthly	12.999	0.983	0.913	13.088	0.99	0.919	13.227	1.013	0.935	13.834	1.080	0.987	13.362	1.007	0.937	14.181	1.105	1.011	18.421	1.616	1.398
Others	5.294	3.614	1.127	5.563	3.71	1.17	5.101	3.419	1.076	6.137	4.201	1.308	5.12	3.649	1.114	4.823	3.243	1.019	6.772	4.963	1.495

Appendix

Table 6: The comprehensive results on the short-term forecasting task.

Models		FreDF		iTransformer		FreTS		TimesNet		MICN		TiDE		DLinear		FEDformer		Autoformer
Models		(Ours)		(2024)		(2023)		(2023)		(2023)		(2023)		(2023)		(2022)		(2021)
	$p_{\mathrm{miss}}$	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	0.125	0.00153	0.02790	0.00213	0.03307	0.01102	0.07843	0.01152	0.07267	0.00236	0.03371	0.45052	0.45514	0.00148	0.02380	0.68262	0.38111	0.37654	0.35378
	0.25	0.00287	0.03801	0.00402	0.04434	0.01089	0.07753	0.01245	0.07946	0.00284	0.03691	0.41777	0.45884	0.00154	0.02351	0.68235	0.38116	0.37059	0.35261
	0.375	0.00256	0.03669	0.00458	0.04663	0.01100	0.07812	0.01407	0.08673	0.00323	0.03900	0.62935	0.55570	0.00175	0.02385	0.68191	0.38105	0.37877	0.36093
	0.5	0.00152	0.02739	0.00363	0.04359	0.01102	0.07818	0.01676	0.09610	0.00352	0.04028	0.29342	0.39320	0.00192	0.02219	0.68119	0.38085	0.38052	0.36462

Appendix

Table 7: The comprehensive results on the missing data imputation task.

Panel of Figure 12 — Figure 12: The forecast sequences generated with DF and FreDF. The forecast length is set to 336 and the experiment is conducted on a snapshot of ETTm2.

Panel of Figure 13 — Figure 13: The spectrum of forecast sequences generated with DF and FreDF. The forecast length is set to 336 and the experiment is conducted on a snapshot of ETTm2. Only the first 24 frequencies of the spectrum are presented.

Panel of Figure 14 — Figure 14: The forecast sequences generated with DF and FreDF. The forecast length is set to 336 and the experiment is conducted on another snapshot of ETTm2.

Panel of Figure 15 — Figure 15: The spectrum of forecast sequences generated with DF and FreDF. The forecast length is set to 336 and the experiment is conducted on another snapshot of ETTm2. Only the first 24 frequencies of the spectrum are presented.

Panel of Figure 16 — Figure 16: Performance of different forecast models with and without FreDF. The forecast errors are averaged over forecast lengths and the error bars represent 95% confidence intervals.

Panel of Figure 17 — Figure 17: FreDF improves iTransformer performance given a wide range of frequency loss weight $\alpha$ . These experiments are conducted on ETTh1 (a), ETTh2 (b), ETTm1 (c), ETTm2 (d), ECL (e), Traffic (f) and Weather (g) datasets. Different columns correspond to different forecast lengths (from left to right: 96, 192, 336, 720, and their average with shaded areas being 50% confidence intervals).

Panel of Figure 18 — Figure 18: FreDF improves Autoformer performance given a wide range of frequency loss weight $\alpha$ . These experiments are conducted on ETTh1 (a), ETTh2 (b), ETTm1 (c), ETTm2 (d), ECL (e), Traffic (f) and Weather (g) datasets. Different columns correspond to different forecast lengths (from left to right: 96, 192, 336, 720, and their average with shaded areas being 50% confidence intervals).

Panel of Figure 19 — Figure 19: FreDF improves DLinear performance given a wide range of frequency loss weight $\alpha$ . These experiments are conducted on ETTh1 (a), ETTh2 (b), ETTm1 (c), ETTm2 (d), ECL (e), Traffic (f) and Weather (g) datasets. Different columns correspond to different forecast lengths (from left to right: 96, 192, 336, 720, and their average with shaded areas being 50% confidence intervals).

Dataset	ETTm1						ETTh1
Models	FreDF		Dilate		DPP		FreDF		Dilate		DPP
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.324	0.362	0.498	0.443	0.631	0.495	0.382	0.400	0.790	0.567	0.815	0.577
192	0.373	0.385	0.993	0.625	0.975	0.617	0.430	0.427	0.950	0.643	0.916	0.633
336	0.402	0.404	0.946	0.628	0.945	0.626	0.474	0.451	0.978	0.663	0.986	0.660
720	0.469	0.444	0.999	0.652	1.079	0.678	0.463	0.462	0.922	0.654	0.898	0.649

Appendix

Table 8: Comparable results with DTW-based loss.

Models			FreDF		iTransformer		FreDF		PatchTST
Metrics			MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Input sequence length	96	96	0.164	0.202	0.201	0.247	0.174	0.217	0.200	0.244
		192	0.220	0.253	0.250	0.283	0.230	0.266	0.234	0.268
		336	0.275	0.294	0.302	0.317	0.279	0.301	0.311	0.321
		720	0.356	0.347	0.370	0.362	0.355	0.351	0.365	0.353
		Avg	0.254	0.274	0.281	0.302	0.259	0.284	0.278	0.297

Appendix

Table 1: Varying input sequence length results on the Weather dataset.

Figure 1: Running time in the forward pass (left panel) and backward pass (right panel), shown with dashed lines for the average and shaded areas for 99.9% confidence intervals.

Dataset	ETTh1				Weather
Models	FreDF		iTransformer		FreDF		iTransformer
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.377 ±0.001	0.396 ±0.001	0.391 ±0.001	0.409 ±0.001	0.168 ±0.003	0.205 ±0.003	0.203 ±0.002	0.246 ±0.002
192	0.428 ±0.001	0.424 ±0.001	0.446 ±0.002	0.441 ±0.002	0.220 ±0.001	0.254 ±0.001	0.249 ±0.001	0.281 ±0.001
336	0.466 ±0.001	0.442 ±0.001	0.484 ±0.005	0.460 ±0.003	0.281 ±0.002	0.298 ±0.002	0.299 ±0.002	0.315 ±0.002
720	0.468 ±0.005	0.465 ±0.003	0.499 ±0.015	0.489 ±0.010	0.364 ±0.008	0.354 ±0.006	0.371 ±0.001	0.361 ±0.001

Appendix

Table 2: Experimental results (

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

) with varying seeds (2020-2024).

Amp.	Pha.	ECL		ETTm1		ETTh1
Amp.	Pha.	MSE	MAE	MSE	MAE	MSE	MAE
✓	✗	0.3356	0.4060	0.5936	0.5169	0.7303	0.5968
✗	✓	0.1836	0.2752	0.4204	0.4173	0.4751	0.4487
✓	✓	0.1698	0.2594	0.3920	0.3989	0.4374	0.4351

Appendix

Table 3: Impact of aligning the amplitude and phase characteristics.

Dataset	ETTm1								ETTh1
Models	FreDF		TimeMixer		FreDF		Scaleformer		FreDF		TimeMixer		FreDF		Scaleformer
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.316	0.354	0.322	0.361	0.365	0.391	0.393	0.417	0.364	0.393	0.375	0.445	0.375	0.415	0.407	0.445
192	0.360	0.377	0.362	0.382	0.417	0.436	0.435	0.439	0.422	0.424	0.441	0.431	0.414	0.440	0.430	0.455
336	0.383	0.399	0.392	0.405	0.478	0.461	0.541	0.500	0.454	0.432	0.490	0.458	0.463	0.468	0.462	0.475
720	0.447	0.440	0.453	0.441	0.575	0.533	0.608	0.530	0.467	0.460	0.481	0.469	0.484	0.499	0.545	0.551

Appendix

Table 4: Comparable results with baselines utilizing multiresolution trends.

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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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