SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
L = L _ {\text {p r e d}} + L _ {\text {a l i g n}} + L _ {\text {s m o o t h}}. \tag {9}
$$
We find that the performance of BasisFormer is robust to the weights in front of the terms in (9). Therefore, we set the weights to be one in all our experiments. Sensitivity analysis of the weights in the loss function can be found in the Appendix A.4.
To assess the effectiveness of our model, we conducted comprehensive experiments on six datasets from real-world scenarios, using the experimental setup identical to that in [5, 7-9]. Below we summarize the experimental setup, datasets, models, and compared models.
Experimental setup: The length of the historical input sequence is maintained at 96 (or 36 for the illness dataset), whereas the length of the sequence to be predicted is selected from a range of values, i.e.,  $\{96, 192, 336, 720\}$  ( $\{24, 36, 48, 60\}$  for the illness dataset). Note that the input length is fixed to be 96 for all methods for a fair comparison.
Datasets: The six datasets used in this study comprise the following: 1) ETT [7], which consists of temperature data of electricity transformers; 2) Electricity, which includes power consumption data of several customers; 3) Exchange [24], containing financial exchange rate within a specific time range; 4) Traffic, comprising data related to road traffic; 5) Weather, which involves various weather indicators; and 6) Illness, consisting of recorded influenza-like illness data. Note that ETT is further divided into four sub-datasets: ETTh1, ETTh2, ETTm1, and ETTm2, and the results in Table 1 are based only on the ETTm2 sub-dataset. The results for the remaining three sub-datasets can be found in the Appendix.
Models for comparison: In this study, we compare our proposed model against the following state-of-the-art models: four transformer-based models, namely FEDformer [9], Autoformer [8], Pyraformer [13]; one MLP-based model, i.e., Dlinear [4]; and one CNN-based model, i.e., TCN [5]. We also consider two recently proposed models, e.g., N-Hits [2] and FiLM [3]. Due to space
Table 1: Multivariate results for six datasets were obtained using an input length of  $I = 96$  (or  $I = 36$  for the illness dataset) and output lengths of  $O \in \{96, 192, 336, 720\}$  (or  $O \in \{24, 36, 48, 60\}$  for the illness dataset). In all experiments, lower MSE values indicate better model performance, and we present the best results in boldface.
Models	Fedformer	Autoformer	N-HiTS	Film	Dlinear	TCN	Basisformer
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETT	96	0.203	0.287	0.255	0.339	0.192	0.265	0.183	0.266	0.193	0.292	3.041	1.330	0.184	0.266
192	0.269	0.328	0.281	0.340	0.287	0.329	0.247	0.305	0.284	0.362	3.072	1.339	0.248	0.307
336	0.325	0.366	0.339	0.372	0.389	0.389	0.309	0.343	0.369	0.554	3.105	1.348	0.321	0.355
720	0.421	0.415	0.422	0.419	0.591	0.491	0.407	0.399	0.554	0.522	3.135	1.354	0.410	0.404
electricity	96	0.193	0.308	0.201	0.317	1.748	1.020	0.199	0.276	0.199	0.284	0.985	0.813	0.165	0.259
192	0.201	0.315	0.222	0.334	1.743	1.018	0.198	0.279	0.198	0.287	0.996	0.821	0.178	0.272
336	0.214	0.329	0.231	0.338	-	-	0.217	0.301	0.210	0.302	1.000	0.824	0.189	0.282
720	0.246	0.355	0.254	0.361	-	-	0.280	0.358	0.245	0.335	1.438	0.784	0.223	0.311
exchange	96	0.148	0.278	0.197	0.323	1.685	1.049	0.083	0.201	0.088	0.218	3.004	1.432	0.085	0.205
192	0.271	0.380	0.300	0.369	1.658	1.023	0.179	0.300	0.176	0.315	3.048	1.444	0.177	0.299
336	0.460	0.500	0.509	0.524	1.566	0.988	0.337	0.416	0.313	0.427	3.113	1.459	0.336	0.421
720	1.195	0.841	1.447	0.941	1.809	1.055	0.642	0.610	0.839	0.695	3.150	1.458	0.854	0.670
traffic	96	0.587	0.366	0.613	0.388	2.138	1.026	0.652	0.395	0.650	0.396	1.438	0.784	0.444	0.315
192	0.604	0.373	0.616	0.382	-	-	0.605	0.371	0.605	0.378	1.463	0.794	0.460	0.316
336	0.621	0.383	0.622	0.387	-	-	0.615	0.372	0.612	0.382	1.479	0.799	0.471	0.317
720	0.626	0.382	0.660	0.408	-	-	0.692	0.428	0.645	0.394	1.499	0.804	0.486	0.318
weather	96	0.217	0.296	0.266	0.336	0.648	0.492	0.193	0.234	0.196	0.255	0.615	0.589	0.173	0.214
192	0.276	0.336	0.307	0.367	0.616	0.479	0.238	0.270	0.237	0.296	0.629	0.600	0.223	0.257
336	0.339	0.380	0.359	0.395	0.579	0.462	0.288	0.304	0.283	0.335	0.639	0.608	0.278	0.298
720	0.403	0.428	0.419	0.428	0.541	0.447	0.358	0.350	0.343	0.383	0.639	0.610	0.355	0.347
illness	24	3.228	1.260	3.486	1.287	3.297	1.679	2.198	0.911	2.398	1.040	6.624	1.830	1.550	0.814
36	2.679	1.080	3.103	1.148	2.379	1.441	2.267	0.926	2.646	1.088	6.858	1.879	1.516	0.819
48	2.622	1.078	2.669	1.085	3.341	1.751	2.348	0.989	2.614	1.086	6.968	1.892	1.877	0.907
60	2.857	1.157	2.770	1.125	2.278	1.493	2.508	1.038	2.804	1.146	7.127	1.918	1.878	0.902
Experiment with  $-$  means it reported an out-of-memory error on a computer with 80G memory.
constraints, we present the results of a selected number of models in this paper, according to their performance and diversity. Interested readers can refer to the supplementary material for a more comprehensive comparison.
4.1 Main results
Multivariate results: The results of the multivariate time series forecasting are presented in Table 1. Very frequently, the proposed BasisFormer outperformed the comparison models across all six datasets, achieving the best results. Furthermore, Basisformer exhibited a  $21.79\%$  improvement over the state-of-the-art method Fedformer. We also observed that when compared to recently proposed models, such as FiLM [3] and Dlinear [4], BasisFormer significantly improved the average MSE performance by a large margin of  $10.78\%$  and  $14.78\%$ , respectively. It is worth noting that BasisFormer is vastly superior to the other methods for the traffic dataset, probably because Traffic dataset is a highly periodic dataset and our model can learn periodic representation well.
Univariate results: The results of the univariate time series forecasting are presented in Table 2. Our model is on par with the state-of-the-art methods. Specifically, compared to the sota method such as FEDformer [9], our proposed model improved the average MSE performance by  $15.36\%$ . Compared to the recent models such as FiLM [3] and Dlinear [4], we also achieved better performance with the increment of  $1.6\%$  and  $16.17\%$  separately. $^3$
Effect of learnable basis: In order to demonstrate the effectiveness of learnable bases, we replaced the learnable basis part in our model with three commonly used types of fixed bases: fixed sine/cosine basis that covers all possible frequencies within the input length, random sine/cosine basis selected in a wide range of frequencies, and covariate embedding which is generally used in Transformer-based models. The results are shown in Table 3. The substitution of the learnable bases resulted in, at minimum, a  $5\%$  average decrease in performance. It is pertinent to mention that although
3Additional experimental details and results are presented in the Appendix section.
Table 2: Univariate results for six datasets were obtained using an input length of  $I = {96}$  (or  $I = {36}$  for the illness dataset) and output lengths of  $O \in  \{ {96},{192},{336},{720}\}$  (or  $O \in  \{ {24},{36},{48},{60}\}$  for the illness dataset). In all experiments,lower MSE values indicate better model performance,and we present the best results in boldface.
Models	Fedformer	Autoformer	Informer	Dlinear	FiLM	Basisformer
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETT	96	0.072	0.206	0.065	0.189	0.080	0.217	0.070	0.191	0.071	0.193	0.070	0.191
192	0.102	0.245	0.118	0.256	0.112	0.259	0.104	0.238	0.101	0.236	0.101	0.238
336	0.130	0.279	0.154	0.305	0.166	0.314	0.135	0.278	0.129	0.273	0.131	0.276
720	0.178	0.325	0.182	0.335	0.228	0.380	0.188	0.332	0.179	0.329	0.181	0.330
electricity	96	0.253	0.370	0.341	0.438	0.258	0.367	0.374	0.439	0.400	0.453	0.333	0.408
192	0.282	0.386	0.345	0.428	0.285	0.388	0.352	0.423	0.380	0.438	0.371	0.427
336	0.346	0.431	0.406	0.470	0.336	0.423	0.378	0.441	0.419	0.464	0.413	0.455
720	0.422	0.484	0.565	0.581	0.607	0.599	0.419	0.479	0.472	0.506	0.471	0.498
exchange	96	0.154	0.304	0.241	0.387	1.327	0.944	0.094	0.236	0.103	0.249	0.108	0.242
192	0.286	0.420	0.300	0.369	1.258	0.924	0.185	0.342	0.238	0.386	0.229	0.364
336	0.511	0.555	0.509	0.524	2.179	1.296	0.340	0.458	0.355	0.489	0.436	0.499
720	1.301	0.879	1.260	0.867	1.280	0.953	0.618	0.624	0.589	0.610	1.130	0.815
traffic	96	0.207	0.312	0.246	0.346	0.257	0.353	0.303	0.398	0.246	0.320	0.175	0.265
192	0.205	0.312	0.266	0.370	0.299	0.376	0.256	0.345	0.208	0.284	0.169	0.259
336	0.219	0.323	0.263	0.371	0.312	0.387	0.242	0.330	0.203	0.284	0.162	0.254
720	0.244	0.344	0.269	0.372	0.366	0.436	0.288	0.368	0.231	0.316	0.183	0.272
weather	96	0.0062	0.062	0.0110	0.081	0.0040	0.044	0.0050	0.057	0.0013	0.026	0.0014	0.027
192	0.0060	0.062	0.0075	0.067	0.0020	0.040	0.0058	0.063	0.0015	0.029	0.0016	0.030
336	0.0041	0.050	0.0063	0.062	0.0040	0.049	0.0063	0.067	0.0017	0.030	0.0016	0.030
720	0.0055	0.059	0.0085	0.070	0.0030	0.042	0.0064	0.066	0.0022	0.035	0.0022	0.035
illness	24	0.708	0.627	0.948	0.732	5.282	2.050	0.815	0.736	0.674	0.643	0.798	0.620
36	0.584	0.717	0.634	0.650	4.554	1.916	0.914	0.817	0.758	0.731	0.563	0.605
48	0.717	0.697	0.791	0.752	4.273	1.846	0.956	0.854	0.774	0.753	0.596	0.632
60	0.855	0.774	0.874	0.797	5.214	2.057	1.065	0.907	0.913	0.828	0.695	0.686
Table 3: Comparison of learnable and other basis on the Electricity dataset. The best results are marked in bold.
basis	learnable(ours)	fixed sine/cosine	random sine/cosine	covariate embedding
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.165	0.259	+6.8%	+5.4%	+7.2%	+5.5%	+5.2%	+4.5%
192	0.178	0.272	+4.7%	+3.2%	+6.6%	+4.3%	+3.8%	+3.5%
336	0.189	0.282	+9.2%	+6.1%	+10.1%	+6.5%	+5.2%	+4.6%
720	0.223	0.311	+16.4%	+10.4%	+12.4%	+7.5%	+7.1%	+5.6%
avg	0.189	0.281	+9.3%	+6.3%	+9.1%	+6.0%	+5.3%	+4.6%
Table 4: The Impact of multi-head Operations on Basis. In this experiment,  $\mathrm{k}$  represents the number of heads, with  $k \in  \{ 4,8,{16},{32}\}$  . We used the Electricity dataset for this experiment. The best results are marked in bold.
heads	4	8	16	32
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.177	0.273	0.168	0.263	0.165	0.259	0.170	0.264
192	0.185	0.278	0.180	0.273	0.178	0.272	0.176	0.269
336	0.196	0.291	0.194	0.288	0.189	0.282	0.198	0.289
720	0.248	0.331	0.221	0.311	0.223	0.311	0.225	0.312
avg	0.201	0.293	0.191	0.284	0.189	0.281	0.199	0.283
sinusoidal types of bases have good generalization ability, they are inadequate in terms of adaptability for specific data. For covariate embeddings, despite having learnable parameters and containing additional sequence information, they do not account for the distinct correlation between the basis and different time series.
The Impact of multi-head mechanism: We further check the impact of the number of heads in the Coef module on performance. The results are displayed in Table 4. The results show that performance presents an upward trend when the number of heads is increased within a specific range. However, beyond a certain number, further augmentation of heads may result in a decline in performance. Consequently, we set  $H = 16$  in our experiments.
Ablation study of the Basis module: We finally employ ablation experiments to isolate the two different loss functions employed in the Basis module. The results are recorded in Table 5. It can be observed that the InfoNCE loss contributes significantly to the good performance of BasisFormer.
Table 5: The contribution of each loss term in the self-supervised module to performance. The dataset used in this experiment was Electricity. The best results are marked in bold.
supervision	none	infonce	smooth	infonce+smooth
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.177	0.273	-4.7%	-3.5%	-1.1%	-0.4%	-6.8%	-5.0%
192	0.188	0.282	-3.8%	-3.0%	-0.9%	-0.3%	-5.5%	-3.6%
336	0.205	0.298	-5.5%	-4.0%	-2.8%	-1.6%	-8.0%	-5.3%
720	0.243	0.327	-7.0%	-4.4%	-3.3%	-1.5%	-8.4%	-5.1%
avg	0.204	0.295	-5.2%	-3.7%	-2.0%	-0.9%	-7.2%	-4.8%
Table 6: The impact of the number of bases  $N$  on the performance of the model. The electricity dataset is employed in this experiment. We present the best results in boldface.
basis number	1	5	10	15	20
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.173	0.269	0.171	0.265	0.166	0.259	0.168	0.263	0.168	0.263
192	0.183	0.277	0.178	0.270	0.176	0.270	0.176	0.269	0.176	0.268
336	0.196	0.289	0.192	0.284	0.190	0.283	0.192	0.285	0.193	0.285
720	0.231	0.317	0.229	0.314	0.218	0.306	0.220	0.308	0.224	0.311
avg	0.196	0.288	0.192	0.283	0.187	0.279	0.189	0.281	0.190	0.282
(a) basis vector 1
(b) basis vector 2
Figure 2: Two highly correlated basis vectors when the number of basis vectors  $N$  is large.
The smoothness loss also makes positive contributions. Remarkably, the employment of both loss functions in conjunction yields the most optimal result. Specifically, the InfoNCE loss advances the bases' capability to acquire the representation of time series, while the smoothness loss function mitigates overfitting to noise within the data.
Impact of the number of basis vectors: We present the performance of the proposed model under varying numbers of basis vectors  $N$  in Table 6, where  $N$  is set to 1, 5, 10, 15, and 20. The results demonstrate that the model's performance remains stable over a wide range of  $N$ , indicating its ability to adaptively adjust to the number of basis vectors. Notably, when  $N$  increases beyond a certain threshold, some of the basis vectors may become redundant. To further explore this, we visualize a subset of the learned basis vectors when  $N = 20$  in Figure 2. Interestingly, we observe a high cosine similarity of  $-0.93$  between two of the bases, suggesting that some basis vectors may not be necessary for accurate prediction. Thus, in practical applications, we set  $N$  to 10 for all datasets to reduce computational complexity without compromising performance.
4.3 Other studies
The adaptability of the self-supervised module to other models: To demonstrate the generalizability of our self-supervised basis learning module, we treat the covariate used in the FEDformer [9] and Autoformer [8] as learnable bases and supervise them using our self-supervised framework. It is noteworthy that we only provide supervision to the covariate employed in these models, without introducing any additional parameters in the prediction pathway. The results are presented in Table 7. We can tell that the addition of self-supervision to the covariate exclusively causes a performance improvement of approximately  $5\%$ . This phenomenon suggests that the learnable basis provides a more reliable reference for the future than the given covariates.
The interpretability of the bases: In order to elucidate the impact of basis in a more comprehensible manner, we visualize the time series and the corresponding learned basis for the traffic dataset in Figure 3(a) and Figure 3(b). Specifically, we choose 4 time series and basis vectors at random. Upon observation of the figures, several key points can be discerned: Firstly, the traffic data have strong periodic patterns, with roughly four peak values in a sinusoidal shape within a length of 96. The positions of the peaks are not evenly distributed over time though. Correspondingly, the learned basis
Table 7: The performance comparison of the self-supervised module used in Transformer-based models. The 'origin' represents no modifications made to the original model and the  $(+)$  coef module' represents applying our designed self-supervised network to supervise the covariate in the models. The dataset used in this experiment was Electricity. The best results are marked in bold.
baseline	Fedformer	Autoformer
comparison	origin	(+) coef module	origin	(+) coef module
metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.193	0.308	0.183	0.301	0.201	0.317	0.193	0.305
192	0.201	0.315	0.196	0.312	0.222	0.334	0.200	0.312
336	0.214	0.329	0.209	0.324	0.231	0.338	0.211	0.325
720	0.246	0.355	0.229	0.340	0.254	0.361	0.252	0.359
avg	0.214	0.327	0.204	0.319	0.227	0.338	0.214	0.325
(a) Visualization of time series on the Traffic dataset
(b) Visualization of learned basis on the Traffic dataset
Figure 3: The visualization of time series and learned basis on the Traffic dataset: The solid line indicates the historical series and the dashed line indicates the future series. For this visualization, we set the input length  $I$  to 96 and the output length  $O$  to 96.
follows this periodic pattern and effectively capture the salient features of the data. Secondly, the basis vectors given by our approach are smooth, indicating that they are not corrupted by the noise in the data. Note that the noise is not predictable, and so the basis that drives the change in the future is preferred to be smooth. Thirdly, it is evident that the learned bases have varying heights and intervals, thereby providing diversity to characterize different features of the time series. Finally, the obtained bases are consistent from both past and future perspectives, thus facilitating the prediction of future trends based on the coefficient similarity between a time series and the basis in the historical part. More visualized results can be found in the Appendix E.
This paper presents BasisFormer, a novel solution to alleviate two significant limitations that impede the efficacy of the existing SOTA methods. Through the utilization of BasisFormer, automatic learning of a self-adjusting basis can be achieved. Moreover, given the learned basis, BasisFormer also allows different time series to be correlated with distinct subsets of basis vectors. Our experimental findings provide compelling evidence of the superiority of BasisFormer over existing methods.
The paper is supported in part by the following grants: National Key Research and Development Program of China Grant (No.2018AAA0100400), National Natural Science Foundation of China (No. 62325109, U21B2013, 61971277).
[18] Wei Fan, Shun Zheng, Xiaohan Yi, Wei Cao, Yanjie Fu, Jiang Bian, and Tie-Yan Liu. Depts: deep expansion learning for periodic time series forecasting. arXiv preprint arXiv:2203.07681, 2022.
[19] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
[20] Emadeldeen Eldele, Mohamed Ragab, Zhenghua Chen, Min Wu, Chee Keong Kwoh, Xiaoli Li, and Cuntai Guan. Time-series representation learning via temporal and contextual contrasting. arXiv preprint arXiv:2106.14112, 2021.
[21] Zhihan Yue, Yujing Wang, Juanyong Duan, Tianmeng Yang, Congrui Huang, Yunhai Tong, and Bixiong Xu. Ts2vec: Towards universal representation of time series. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 8980-8987, 2022.
[22] Petar Velicković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903, 2017.
[23] Hang Yu and Justin Dauwels. Modeling spatio-temporal extreme events using graphical models. IEEE Transactions on Signal Processing, 64(5):1101-1116, 2015.
[24] Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In The 41st international ACM SIGIR conference on research & development in information retrieval, pages 95-104, 2018.
[25] Gerald Woo, Chenghao Liu, Doyen Sahoo, Akshit Kumar, and Steven Hoi. Cost: Contrastive learning of disentangled seasonal-trend representations for time series forecasting. arXiv preprint arXiv:2202.01575, 2022.
[26] Patrick Schäfer. The boss is concerned with time series classification in the presence of noise. Data Mining and Knowledge Discovery, 29:1505-1530, 2015.
[27] Juntang Zhuang, Tommy Tang, Yifan Ding, Sekhar C Tatikonda, Nicha Dvornek, Xenophon Papademetris, and James Duncan. Adbelief optimizer: Adapting step sizes by the belief in observed gradients. Advances in neural information processing systems, 33:18795-18806, 2020.
A.1 Experiments on the ETT datasets
In the main body, we present a comparison of the benchmark methods on the ETTm2 dataset. In this section, we extend our analysis to the remaining three ETT datasets, namely ETTh1, ETTh2, and ETTm1, as summarized in Table 8. Our experimental results reveal that Basisformer outperforms all other methods in terms of MSE and MAE. Specifically, Basisformer demonstrates a superior average MSE reduction of  $1.32\%$  ,  $6.74\%$  and  $9.23\%$  when compared to FiLM, Fedformer and DLinear, respectively.
Table 8: Multivariate results for the remaining three ETT datasets using an input length of  $I = 96$  (or  $I = 36$  for the illness dataset) and output lengths of  $O \in \{96,192,336,720\}$  (or  $O \in \{24,36,48,60\}$  for the illness dataset). In all experiments, lower MSE values indicate better model performance, and we present the best results in boldface.
Models	Fedformer	Autoformer	N-HiTS	FiLM	Dlinear	Informer	Basisformer
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.376	0.419	0.449	0.459	0.419	0.413	0.388	0.401	0.386	0.400	0.865	0.713	0.394	0.411
192	0.420	0.448	0.500	0.482	0.468	0.443	0.443	0.439	0.437	0.432	1.008	0.792	0.442	0.437
336	0.459	0.465	0.521	0.496	0.551	0.489	0.484	0.461	0.481	0.459	1.107	0.809	0.473	0.451
720	0.506	0.507	0.514	0.512	0.669	0.559	0.525	0.519	0.519	0.516	1.181	0.865	0.460	0.465
ETTh2	96	0.358	0.397	0.346	0.388	0.374	0.383	0.292	0.341	0.333	0.387	3.755	1.525	0.312	0.356
192	0.429	0.439	0.456	0.452	0.476	0.446	0.378	0.396	0.477	0.476	5.602	1.931	0.382	0.401
336	0.496	0.487	0.482	0.486	0.472	0.446	0.426	0.438	0.594	0.541	4.721	1.835	0.418	0.431
720	0.463	0.474	0.515	0.511	0.932	0.636	0.443	0.455	0.831	0.657	3.647	1.625	0.418	0.438
ETTm1	96	0.379	0.419	0.505	0.475	0.324	0.349	0.357	0.373	0.345	0.372	0.672	0.571	0.342	0.374
192	0.426	0.441	0.553	0.496	0.376	0.379	0.387	0.385	0.380	0.389	0.795	0.669	0.380	0.392
336	0.445	0.459	0.621	0.537	0.409	0.405	0.420	0.407	0.413	0.413	1.212	0.871	0.420	0.418
720	0.543	0.490	0.671	0.561	0.472	0.443	0.478	0.439	0.474	0.453	1.166	0.823	0.492	0.458
A.2 Experimental results with longer length input setting
Throughout our research, we maintain consistency in our experimental settings by fixing the input length to be 96 (with a reduced input length of 36 for the illness dataset), instead of using a longer length. The main rationale behind this decision is that, in practical scenarios where the model is deployed as an online service and tasked with predicting a long range of the future at a granular level of minutes or hours, collecting a lengthy history (i.e., spanning 720 timestamps) for a large number of time series in real-time can be quite challenging. Therefore, the adoption of an input length of 96 proves to be more practical and feasible.
Given that certain recent methods utilize longer input lengths to yield better performance, irrespective of the length, we present supplementary comparison outcomes with extended input lengths in Table 9. Specifically, Fedformer, Autoformer, and TCN exhibit a decline in performance with an increase in input length, and hence, we retain their original outcomes at an input length of 96. In contrast, Dlinear employs an input length of 336 (104 for the illness dataset) by default, FiLM utilizes an input length that is at most four times of the output length, and N-HiTS adopts an input length that is five times of the output length. To enable a fair comparison, we standardize our input length for longer inputs to 192 (72 for the illness dataset).