Rethinking Fourier Transform from A Basis Functions Perspective for Long-term Time Series Forecasting

Authors: runze  yang
longbing  cao
jianxun  li
jie  yang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 08:34:29

Abstract: The interaction between Fourier transform and deep learning opens new avenues for long-term time series forecasting (LTSF). We propose a new perspective to reconsider the Fourier transform from a basis functions perspective. Specifically, the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels, respectively. We argue existing Fourier-based methods do not involve basis functions thus fail to interpret frequency coefficients precisely and consider the time-frequency relationship sufficiently, leading to inconsistent starting cycles and inconsistent series length issues. Accordingly, a novel Fourier basis mapping (FBM) method addresses these issues by mixing time and frequency domain features through Fourier basis expansion. Differing from existing approaches, FBM (i) embeds the discrete Fourier transform with basis functions, and then (ii) can enable plug-and-play in various types of neural networks for better performance. FBM extracts explicit frequency features while preserving temporal characteristics, enabling the mapping network to capture the time-frequency relationships. By incorporating our unique time-frequency features, the FBM variants can enhance any type of networks like linear, multilayer-perceptron-based, transformer-based, and Fourier-based networks, achieving state-of-the-art LTSF results on diverse real-world datasets with just one or three fully connected layers. The code is available at: https://github.com/runze1223/Fourier-Basis-Mapping.

Main paper: 
 introduction: 
 Long-term time series forecasting (LTSF) is essential for applications with long-range sequences, which presents significant challenges including long-range temporal dependencies, and frequencyoriented global dynamics. Recently, deep neural networks (DNNs) have thrived to tackle LTSF challenges for the presence of hierarchical effects, varied outliers, and nonlinear dynamics. They use various DNN architectures including recurrent neural networks (RNNs) [1, 2, 3, 4, 5, 6, 7, 8] , convolution neural networks (CNNs) [9, 10, 11, 12, 13, 14] , multi-layer perceptron (MLP) based networks [15, 16, 17, 18] , Transformer-based networks [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] . However, a recent study NLinear [34] shows that an embarrassingly simple normalized linear model can outperform most of the DNN-based methods. This raises the question: Would a complex Transformer-like architecture necessarily lead to better LTSF performance? According to CrossGNN [35] , DNN-based methods like Transformers suffer from the influence of noise signals and produce high attention scores for unexpected noises. Existing methods primarily operate in the frequency domain, with the mapping focusing on frequency features. Our FBM simultaneously operates in both the time and frequency domains, with the mapping focusing on time-frequency features. Thus, Fourier-based time series modeling emerges as a new paradigm to remove noise signals by considering diverse effects hierarchically at different frequency levels. If we rethink the Fourier transform from a basis functions perspective, the real and imaginary parts can be interpreted as the coefficients of cosine and sine basis functions at tiered frequencies, respectively. However, existing Fourier-based methods do not involve basis functions, thus failing to interpret frequency coefficients precisely and do not consider the time-frequency relationships sufficiently, as shown in Figure 1 . They face two main issues: inconsistent starting cycles and inconsistent series length issues. For instance, FEDformer [22] , FreTS [18] , FiLM [36] , FITS [37] , FGNet [28] , and FL-Net [38] use the real and imaginary parts of frequency components as the input of the mapping network and conduct the mapping in the frequency space. From our new perspective, we find that the amplitude and arctangent of the real and imaginary components carry more explicit meanings than the components themselves. Furthermore, it is worth noting that the meanings of frequency components are bounded by the series length, and failing to establish the connections will lead to ambiguity in understanding the precise frequency in these models, e.g., 2 Hz sine and cosine wave with series lengths 96 and 336 has distinct meaning. More importantly, Figure 1 shows that a Fourier basis function is time-dependent when the input length is not divisible by a certain frequency level, making it even more challenging for the model to correctly interpret those frequency components without the basis functions. Consequently, constructing the mapping in the frequency space is not enough and cannot capture time-frequency relationships, which has been ignored by all existing Fourier-based methods. Although some methods like TimesNet [39] introduce time-frequency features, these features are non-interpretable and complex, as the summation over the frequency dimension does not recover the original time series. We provide a more detailed discussion of their limitations in Section 2.Accordingly, we propose the Fourier Basis Mapping (FBM) to address the aforementioned issues, involving learnable and non-learnable features. In the first stage, we embed the discrete Fourier transform with basis functions, referred to as Fourier basis expansion, to extract time-frequency features. In the second stage, we validate its effectiveness using three types of mapping networks through three FBM variants (FBM-L, FBM-NL, FBM-NP) against four categories of LTSF baseline methods: (1) Linear method: NLinear; (2) Transformer-based methods: iTransformer [33] and PatchTST [20] ; (3) MLP-based methods: N-BEATS [16] , FreTS, and TimeMixer [15] ; and (4) Fourier-based methods: N-BEATS, FITS, FreTS, and CrossGNN. The experimental results reveal that Fourier basis expansion can enhance any types of DNNs. With time-frequency features, simple mapping networks (e.g, L-vanilla linear networks and NL-three-layer MLPs) can achieve the SOTA of LTSF on eight real-world datasets and of STSF on PEMS datasets. 
related work: 
 Recent studies have investigated the effect of the Fourier transform in addressing LTSF challenges. Consequently, we categorize the relevant studies into two groups: (1) Fourier-based architectures, and (2) DNN-based architectures. In the first category, we discuss the applications of the Fourier transform, while the second addresses innovative model architectures.Fourier transform has been integrated into a wide range of network architectures, including CNNs [39] , recurrent networks [36] , graph neural networks (GNNs) [35, 40] , Transformer [22] , and MLP networks [18, 16, 17] . However, by rethinking the Fourier transform from a basis functions perspective, we identify the inconsistent starting cycles and series length issues. In path (i) of Figure 1 , methods like FEDformer [22] , FreTS [18] , FiLM [36] , FITS [37] , FGNet [28] , and FL-Net [38] use real and imaginary parts as inputs to their mapping networks but the networks cannot easily interpret those coefficients because the crucial information is stored in the amplitude, phase and length of each cycle. In path (ii) of Figure 1 , methods like CrossGNN [35] , and TimesNet [39] use the discrete Fourier transform (DFT) to select the top-k amplitudes for noise filtering. However, a higher amplitude does not necessarily indicate a useful frequency and a lower amplitude is not necessarily useless. In addition, CrossGNN only stores the values of the first cycles without considering the effects of phase shifts and time-dependent basis, while TimesNet introduces the multi-window Fourier transform, which compromises the integrity of the temporal information. In path (iii) of Figure 1 , methods like N-BEATS [16] and N-Hits [17] can similarly be viewed as forecasting through the inverse discrete Fourier transform (IDFT) by computing the output frequency spectrum of the time series, but their networks fail to capture time-dependent effects. In contrast, our method distinguishes itself from existing approaches by leveraging the Fourier basis expansion to provide a mixture of time and frequency domain features, thus avoiding the aforementioned issues.Firstly, we introduce key techniques for innovative model architectures. In [41] , initial normalization is introduced to enhance the robustness. Autoformer [24] advocates the trend and seasonal decomposition with the moving average kernel, consistent with other methods like DLinear [34] and TimeMixer [15] . However, the effectiveness of the decomposition relies on the choice of kernel size, which is improved by our methods. Methods like VH-NBEATS [42] and BasisFormer [21] involve the pre-trained basis functions to account for hourly, daily, and weekly effects. In contrast, our method uses the Fourier basis to capture those effects automatically. Transformer-based methods are one of the most popular LTSF architectures. Some transformer variants improve efficiency by reducing computational complexity and memory cost, such as LogTrans [43] , Pyraformer [25] , Informer [23] , and Autoformer [24] . On the other hand, PatchTST [20] introduces patching to treat a period of local time steps as a semantic vector, effectively reducing complexity and achieving state-of-the-art results over Transformer-based methods. In a recent study, iTransformer [33] suggests that employing attention layers to capture relationships between variables may yield better results. In this paper, we validate the effectiveness of Fourier basis expansion on linear, MLP, and Transformer-based networks, proving that it can enhance any types of networks. The Fourier basis expansion can also be viewed as an enhanced version of trend and seasonal decomposition, as the basis functions, shown in Figure 1 , automatically decompose various effects by frequency levels. the effect of the IDFT for frequency-based mapping can be seen as analogous to the design of the N-BEATS seasonal block. As such, we identify the inconsistent starting cycles and inconsistent series length issues in existing studies.Let X and Y be the input and output time series, respectively, and T and L represent the look-back window and forecast horizon, assuming both of them are even numbers. H X and H Y refer to the frequency spectrum of the input and the output, respectively. Then, DFT and IDFT of the input time series X can be expressed as follows:EQUATIONEQUATIONFrom the perspective of basis functions, both DFT and IDFT can be expressed by T 2 + 1 orthogonal cosine basis functions and T 2 -1 orthogonal sine basis functions. This is because the DFT of a real-valued signal is Hermitian symmetric. Their proofs and further explanations can be found in Appendices B.1 and B.2. Subsequently, we can rewrite the IDFT w.r.t. basis functions, and the connection between X and H X can be expressed as follows:EQUATIONH R [k] and H I [k] represent the real and imaginary parts of H[k] respectively, where k refers to the frequency level, defined byH[k] = H R [k] + iH I [k].Eq. ( 4) provides an essential insight that the real and imaginary parts of the frequency spectrum can be interpreted as the coefficients of cosine and sine basis functions, respectively. Hence, computing the frequency spectrum of the output time series is equivalent to computing the coefficients between cosine and sine basis functions, which can be considered analogous to the N-BEATS-seasonal-block design. Further explanation and proofs can be found in Appendix B.3.Consequently, we identify two issues in existing Fourier-based studies: inconsistent starting cycles and inconsistent series length issues. The inconsistent starting cycles issue arises since the real and imaginary parts do not carry explicit meanings without the basis functions. This is because that adding a sine wave and a cosine wave with the same frequency results in a shifted cosine wave with the same frequency, which can be expressed as follows:EQUATIONTherefore, the key information is embedded in the amplitude and arctangent of the real and imaginary values rather than the values themselves. The inconsistent series length issue arises since the meaning of frequency in hertz (Hz) is bound by the series length.Figure 2 illustrates these two issues as Cases I and II through a manually generated time series with pure sine and cosine basis functions, assuming using time series X to forecast time series Y. In Case I, X and Y have the same frequency and series length but different starting cycles. If the real and imaginary parts of the frequency spectrum of the input are used to compute that of the output through independent channels, we cannot find such a mapping as there is no mathematical solution. However, their relationships can be easily identified through the amplitude and arctangent of the real and imaginary values, which are embedded in basis functions. As shown in Figure 2 , a mathematical solution exists which is achievable through Eq. ( 4).In Case II, X and Y have almost the same frequency and starting cycles but different series lengths.Although the landscapes of the frequency spectrum look similar, their components (in Hz) have distinct meanings. For instance, an 8Hz cosine function with a series length of 192 is equivalent to a 14Hz cosine function with a series length of 336. Unfortunately, such a frequency connection is not provided by existing models. Hence, the precise interpretation of frequency components is challenging for the model due to varying series lengths. The presence of time-dependent basis functions makes it even harder for the model to accurately interpret the frequency components. Instead, these issues can be solved easily by the incorporation of basis functions, which will be discussed in Section 4.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: Our proposed model, FBM, is more suitable for time-series datasets that exhibit seasonality and periodicity, rather than monotonic trends.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). 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 Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. Justification: The social impact is discussed in introduction. Time series forecasting plays an increasingly significant role spanning various industries, from traffic flow estimation, energy inventory management, weather forecasting to financial investment, critical for businesses such as risk management, resource optimization, and strategic planning. There is no negative societal impacts of the work performed.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
fbm: fourier basis mapping: 
 To address two aforementioned issues, we introduce the Fourier basis mapping (FBM) . Figure 3 shows the architecture of FBM. The primary strength of FBM lies in constructing time-frequency features, which capture explicit frequency information while preserving temporal characteristics. Subsequently, the downstream mapping considers the time-frequency space rather than solely the time or frequency space for forecasting.First, we apply data normalization to enhance the robustness of FBM, the same as the implementation suggested in [41] . Then, we generate the frequency spectrum H using DFT. Next, we multiply the real part of H (denoted as H R ) with the orthogonal cosine basis C and the imaginary part of H (denoted as H I ) with the orthogonal sine basis S to obtain the mixture of frequency and temporal features. Let N = [0, 1, . . . , T -1], then C and S can be expressed as follows: Further, we combine the cosine and sine basis functions to obtain the shifted starting cycles of the time series, forming G. The scalar 2 is derived in Eq. ( 4) for Hermitian symmetric. Thus, by summing over the T 2 + 1 frequency domain, the model can recover the original time series without compromising the integrity of the temporal information. The process is analogous to decomposing the original time series into T 2 + 1 components with tiered frequency levels. Finally, we use a decoder to map the features G to the output time series. Our time-frequency features automatically decompose hierarchical effects by frequency levels, accounting for their interactive effects while filtering out associated noise through the mapping network. The Fourier basis expansion provides more effective and interpretable initial features, facilitating a simpler downstream mapping process. To this end, we further propose three mapping methods: linear (L), nonlinear three-layer perception (NL), and nonlinear PatchTST (NP). Consequently, three FBM variants are generated: FBM-L with linear network, FBM-NL with MLP network, and FBM-NP with transformer-based network with patching. The first one includes a single vanilla linear layer, the second one consists of three fully connected layers with activation functions, and the last one follows the same structure as PatchTST, but we conduct the patching after the Fourier basis expansion. It is worth noting that FBM-NP employs fewer patches than PacthTST to reduce complexity, making it more efficient while delivering superior performance. Details of our three mapping networks, including hyperparameters and layer specifications, are provided in Figure 6 and Table 5 in Appendix C.C = 1 T [1, 
baselines and experiment settings: 
 We compare FBM with eight baseline methods: NLinear, PatchTST, N-BEATS, CrossGNN, FEDformer, FreTS, FiLM, and BasisFormer. These methods are chosen because they represent four categories of LTSF modeling methods: (1) Linear method: NLinear; (2) Transformer-based methods: iTransformer and PatchTST; (3) MLP-based methods: N-BEATS, FreTS, and TimeMixer; and (4) Fourier-based methods: N-BEATS, FITS, FreTS, and CrossGNN. Furthermore, FBM is evaluated w.r.t. its three variants: FBM-L, FBM-NL, and FBM-NP. The mean squared error (MSE) and mean absolute error (MAE) are used as evaluation metrics. The look-back window is set to 336, and the forecasting horizons are set to 96, 192, 336, 720. In Appendix A.3, we also show that a look-back window of 336 is always associated with better results than that of 96 for most baseline methods. Eight datasets are used for the evaluation: ETTh1, ETTh2, ETTm1, ETTm2, Electricity, Traffic, Weather, and Exchange. In addition, we also evaluate the STSF of four PEMS datasets: PEMS03, PEMS04, PEMS07, PEMS08. More details about these baselines and datasets can be found in Appendix A.1. We split the ETT dataset into 12/4/4 months and the other datasets into training, validation, and test sets by the ratio of 6.5/1.5/2. Here, we use a smaller training set and a larger validation set compared to baseline methods to prevent early stopping and ensure a fairer comparison. With time-frequency features, the model can achieve state-of-the-art performance for both LTSF and STSF most of the time using either one linear layer or three non-linear layers.We evaluate the forecasting performance of three FBM variants against eight diversified baselines incorporated by different architectures: Linear, MLP, Transformer, and Fourier-based architectures. Table 1 shows the quantitative results on eight datasets. While preserving a simple structure, FBM variants achieve the best performance on most datasets across most prediction horizons. FBM variants vs. Fourier-based network: We also observe that FBM variants make a significant improvement over all Fourier-based methods, including N-BEATS, FreTS, CrossGNN, and FITS. This discrepancy may largely be attributed to the inconsistent starting cycles and series length issues discussed in Section 3. Notably, FITS and CrossGNN also emphasize the importance of the amplitude of real and imaginary values, similar to FBM. However, they overlook the fact that Fourier basis functions are time-dependent when the input length is not divisible by a certain frequency level. Consequently, mapping in the frequency space cannot capture the time-frequency relationship effectively. For instance, CrossGNN retains only the values from the first cycle but fails to account for variations in the subsequent cycles as well as shifts in the phase of the cycles. In contrast, FBM variants demonstrate a better representation of time-frequency domain features, as evidenced by their superior performance. In the following sections, we empirically discuss the reasons for their effectiveness through visualization.Although our model primarily focuses on addressing LTSF challenges, we also evaluate the performance of STSF on PEMS dataset in Table 2 , under the same settings as previous baseline methods.Our FBM variants consistently achieve the best performance in most cases.In Figure 4 , we visualize the effect of the weights of the one-layer vanilla linear network to show how FBM-L considers the time-frequency relationship. We decompose the weights W into L timespecific parts, denoted as W = [W 0 , W 1 , . . . , W L-1 ], where W i represents the impact of the time-frequency features on the i-th time step of Ŷ. We flatten the weights into an array similar to G in Figure 3 , where the horizontal axis represents time, and the vertical axis represents frequency. Specifically, we visualize the heat maps of W 0 for the ETTh1, Electricity, and Traffic datasets, comparing them with the Fourier basis. Since the datasets Electricity and Traffic are more stable than ETTh1, the weights of the linear layer for the former datasets are closer to the Fourier basis than those for the latter datasets. This implies that FBM-L considers more time-frequency relationships in ETTh1 than in Electricity and Traffic, leading to more significant improvement. The further visualization of the weights W 0 , W 10 , W 20 , W 30 , W 40 , and W 50 for Electricity is provided in Figure 4 in Appendix C. It is not surprising to see that the heatmap shifts over time. In conclusion, our approach allows the model to eliminate noises across a mixture of time and frequency domains and consider time-frequency relationships. In contrast, existing studies only eliminate noises in the frequency domain. ETTh1 Electricity Traffic BasisWe compare the performance of three FBM-variants on eight datasets, as shown in Table 1 . FBP-NL and FBM-NP perform similarly across the datasets, as both are nonlinear. In contrast, FBM-L and FBM-NL reveal great performance differences on distinct datasets, which may be attributed to their varied data characteristics. To further explain such differences, Figure 5 shows the visualization of the frequency spectrum for a random input on the last seven dimensions across eight datasets. It shows that trending effects usually appear on high-frequency levels, seasonal effects usually fall on intermediate frequency levels, and noise signals typically manifest on low-frequency levels. This explains why the Fourier transform is beneficial for LTSF, as it hierarchically separates various effects, with specific effects (e.g., hourly, daily, weekly) aligning with their corresponding frequency levels. For instance, there is consistently high energy at the multiples of 14 on all hourly-level data ETTh1, ETTh2, Electricity, and Traffic in Figure 5 . This is attributed to the day effect falling into these frequency levels, given the repeating cycle of 24 with a look-back window of 336.By examining the frequency spectrums of all hourly-level datasets, we also find that ETTh1 and ETTh2 display a much richer frequency spectrum compared to other datasets. Figures 8 and 9 in Appendix C further show that the frequency spectrums of Electricity and Traffic are more stable than those of ETTh1 and ETTh2. This suggests that there is a larger proportion of noise signals in datasets ETTh1 and ETTh2, but less in datasets Electricity and Traffic. The model is more likely to overfit when the data is mixed with a significant amount of noise. This is why FBM-L achieves better performance on datasets ETTh1 and ETTh2, while FBM-NL performs better on datasets Electricity and Traffic. On the other hand, the frequency spectrums for ETTm1 and ETTm2 shift to the left, contrasting with those for ETTh1 and ETTh2. This is because the granularity of ETTh1 and ETTh2 is four times larger than that of ETTm1 and ETTm2. This leads to the actual forecast horizon of the time being reduced, resulting in less noise in the data. This explains why FBM-NL performs slightly better than FBM-L on ETTm1 when the granularity changes. This also applies to datasets with other levels of granularity, such as the Exchange and Weather data. This observation validates our original hypothesis that providing the frequency spectrum is insufficient, as the real-world frequency can be ambiguous for the model when changes in granularity or series length alter the meanings of the spectrum. In conclusion, when a dataset consists of a large proportion of noise signals with a limited amount of data, a simpler network is preferred, and vice versa.  
conclusion: 
 We make the first attempt to theoretically and empirically discuss several issues associated with the existing research on the deep Fourier transform for LTSF from the perspective of basis functions.Our insights and findings reveal that the real and imaginary parts of the frequency spectrum can be interpreted as the coefficients of cosine and sine basis functions, disclosing two issues commonly appearing in existing studies: inconsistent starting cycles and inconsistent series length issues. We further propose a Fourier basis mapping model Fourier basis mapping (FBM) to address these issues by extracting more efficient time-frequency features that carry explicit frequency information while retaining temporal patterns. Extensive experiments empirically verify the effectiveness of the FBM approach, to enhance various types of mapping networks and achieve state-of-the-art performance with simple one or three layers. The visualization analysis also reveals that deep neural mapping can replace the effect of inverse discrete Fourier transform and allow for more flexible consideration of time-frequency relationships. The ablation study further shows that extracting time-frequency features is more valuable than simply increasing the depth of the network. The primary focus of this study is to refine Fourier-based methods in a way that is more reasonable and interpretable. The time-frequency features hold great potential for future time series analysis tasks by enhancing various types of mapping networks (e.g., linear, MLP, Transformer networks). For example, Fourier basis expansion can enhance the performance of Transformer-based networks by only adjusting the initial projection layer after Fourier basis expansion. In future, we will also consider applying Fourier basis expansion to CNN-and RNN-based networks, as well as to tasks such as anomaly detection.A Supplement materials for experiment results 
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 Appendix: We conduct our experiments on twelve real-world datasets: ETT2 (ETTh1, ETTh2, ETTm1, ETTm2), Electricity3 , Traffic4 , Weather5 , Exchange rate (Exchange) 6 NLinear [34] is a simple one-layer linear network with data normalization [41] . PatchTST [20] introduces a Transformer-based model for time series forecasting with two crucial improvements: patching and channel independence. FEDformer [22] adopts a structure similar to Transformer but converts time-domain features into frequency-domain features through the Fourier transform. CrossGNN [35] creates a multi-scale identifier to address noise in time series by applying the Fourier transform and utilizes graph neural network (GNN) to study cross-variable interactions. FreTS [18] projects the time series into high dimensions and characterizes the frequency space by utilizing interaction channels for the real and imaginary parts with a multilayer-perceptron-based network. TimesNet [39] transforms one-dimensional time domain information into a 2D tensor based on multiple time scales through Fourier transform, and then utilizes the convolution neural network to analyze the transformed information. N-BEATS [16] and N-HiTS [17] forecast by computing the coefficients of sine and cosine basis functions using a multilayer perceptron network. FiLM [36] combines the power of the Fourier transform with the recursive memory unit (LMU) [44] , where LMU is introduced to provide a robust representation for long time series through the Legendre projection, a variant version of recurrent neural network. BasisFormer [21] computes basis functions through adaptive self-supervised contrastive learning and employs a Transformer-based architecture to study the connection between input time series and its basis functions. Autoformer [24] is a model based on Transformer that employs a decomposition architecture along with an auto-correlation mechanism to capture dependencies across time. Informer [23] introduces a probability sparse self-attention mechanism to study the temporal relationship. LogTrans [43] develops a log sparse attention mechanism to improve the efficiency of Transformer-based networks. Pyraformer [25] is a Transformer-based model that learns a multi-resolution representation of time series data using a pyramidal attention module to capture cross-time dependencies for forecasting. iTransformer [33] applies attention to study the inverted dimensions of the input sequence, capturing the dependencies between time series. TimeMixer [15] introduces a multi-scale decomposition using a bottom-up and top-down MLP network. FITS [37] proposes complex frequency linear interpolation to study the frequency domain.In Table 3 and 4, we evaluate the complexity of our method in comparison to other sophisticated approaches and discuss the influence of the look-back window on the final prediction results. In Table 3 , we provide a comparison of the training time for one epoch cycle on the ETTh1 dataset, as well as an increase in complexity in the layer. Our FBM model achieves the best prediction results at a relatively fast training speed. In contrast, attention mechanisms typically suffer from higher memory costs and slower calculation speeds. All the experiments are conducted on a GeForce GTX 3090 GPU and an Intel(R) Core(TM) i7-6850K CPU. In the past literature, a substantial amount of work selects a look-back window of 96. In Table 4 , we conduct a comparison of prediction results with look-back windows of 96 and 336 on the ETTh1 dataset. Remarkably, the look-back window of 336 consistently yields better performance for almost every baseline method. While certain methods, such as iTransformer, and FreTS, exhibit lower sensitivity to the look-back window, the majority experiences a significant increase in both MSE and MAE when a look-back window of 96 is used rather than that of 336.O( T 2 2 ) O( T 2 P ) O(T ) O( T 2 P ) O(T ) O(T D) O(T D) O(T ) O(T ) O(T 2 ) O(T ) O(T 2 )The discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) can be expressed by T 2 + 1 cosine basis functions and T 2 -1 sine basis functions. We provide the proof that the real-valued discrete Fourier transform can be represented by sine and cosine functions as follows:EQUATIONEQUATIONEQUATIONThen, we can deduce that the frequency spectrum is Hermitian symmetric, whereH(k) = H(T -k).The real-valued inverse discrete Fourier transform can be expressed by sine and cosine basis functions as follows:EQUATIONEQUATIONEQUATIONEQUATIONB.3 Equivalence between the IDFT and N-BEATS-Seasonal-Block design for frequency side mapping.We provide further explanation of why IDFT and N-BEATS-Seasonal-Block are equivalent for the frequency space mapping. The general structure of a Fourier-based method can be summarized as follows:EQUATIONThe encoder can refer to any deep neural architecture, such as a CNN-based network, RNN-based network, GNN-based network, Transformer-based network, or MLP-based network. Therefore, the model aims to make forecasting by mapping H X to H Y . However, if we rewrite the IDFT in Equation ( 8) by Equation ( 4) and assume M = [0, . . . , L -1], we can obtain the following equation:EQUATIONThis is equivalent to the N-BEATS-Seasonal-Block design, as their structure can be expressed as follows: In the N-BEATS model, they use a multilayer perceptron to compute the coefficients of sine and cosine waves, denoted as θ. Although the dimension of θ ∈ R L+2 is twice as large as the dimension of H Y ∈ C L/2+1 , if we consider the real part and imaginary part of H Y separately, then the dimension is exactly the same. As a result, the objective of the deep Fourier-based method and the N-BEATS model in Equation ( 9) and Equation ( 10) are equivalent: to compute the coefficients of sine and cosine basis functions with tiered frequency, as the scalar of a basis function doesn't provide any difference in deep learning. The only difference is that the former one maps the frequency domain to the frequency domain, and the latter one maps the time domain to the frequency domain.EQUATIONFigure 6 is provided as supplementary material for Section 4. In Figure 6 , FBM-L represents a linear network, FBM-NL is a three-layer MLP network with dropout and ReLU as the activation function, and FBM-NP has the same structure as PatchTST but differs in the number of patches and initial projection layer. All the rest hyperparameters for FBM-NP are the same as those for PatchTST. The key hyperparameters are provided in Table 5 . You can always tune them to achieve better results than us. Since validation set is used to determine when to early stop the training, adjusting hyperparameters based on the test set can undermine generalization. Additionally, tuning random seeds to achieve better results may compromise reproducibility across different devices. Therefore, we use fixed hyperparameters in most cases and maintain the same random seed throughout the entire experiment.Figure 7 is provided as supplementary material for Section 5.3. In Figure 7 , we visualize the weights W 0 , W 10 , W 20 , W 30 , W 40 , and W 50 for the Electricity dataset and observe that the heatmaps of weights shift over time.Figures 8 and 9 are provided as supplementary material for Section 5.4. In these figures, we provide the frequency spectrums of three random inputs for each dataset. We observe that the frequency spectrums of the Electricity and Traffic datasets are quite stable, followed by the ETTm1, ETTm2, and Weather datasets. In contrast, the frequency spectrums of ETTh1 and ETTh2 are not stable, as reflected by the changing landscape over time with less similarity at certain levels. Justification: The abstract and introduction accurately reflect the paper's contributions and scope.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.Rethinking Fourier transform w.r.t. basis functionsIn this section, we offer a new perspective of the Fourier Transform for LTSF. Firstly, we discuss the mathematical reasoning behind the DFT and IDFT from the perspective of basis functions. From this new perceptive, we find that the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels. Consequently,https://github.com/zhouhaoyi/ETDatasethttps://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014http://pems.dot.ca.govhttps://www.bgc-jena.mpg.de/wetter/https://github.com/laiguokun/multivariate-time-series-datahttps://www.kaggle.com/datasets/elmahy/pems-dataset 


Decision: The paper leverages the Fourier Transform to frequency domain information in predictive models for temporal data. Reviewers Z6Ax and akVH appreciated the novelty of the method, its combination of learnable and non-learnable components, the model selection tailored to the data, and the extensive experiments showing the potential of the method. However, both reviewers indicated lapses in the presentation. Additionally, reviewer Z6Ax recommended comparative studies with the same decoder, while Reviewer akVH requested more comparisons to the state of the art. The authors have provided extensive additional experiments and have greatly improved their introduction and figure. The changes were appreciated by the two reviewers, who agreed to accept the paper. Moreover, reviewer akVH has replicated some of the results reported in the paper (comparison of the model with patchTST) and found that the numbers reported are reliable. Reviewer 8ZRG considered the technical contribution to be incremental and the method not clearly explained, while also pointing out some s.o.t.a models for comparison. The authors have provided an extensive response, which, by my assessment, resolves at least the issues with the comparisons against s.o.t.a., and clarifies the contributions. Notably, Reviewer Z6Ax also pointed Reviewer 8ZRG to a part of the response that appears to clarify the contribution and address a concern. Reviewer 8ZRG has chosen to decrease their score, which is extremely unusual at this stage, and I did not find the justification for it convincing. From their statement "this paper is not ready for being published in NeurIPS", one might gather that the decrease in score is meant to prevent this paper from being published. Reviewer 8ZRG's final set of concerns was also addressed by the authors. It seems that at least some of the results were reproduced by the other reviewer (relevant to point 3), while MAE and MAPE were indeed used by the contender method (point 1), then two of the 3 final points are addressed.
Overall, my assessment is that this paper constitutes a valuable contribution to the field and warrants acceptance. However, the authors should be extremely mindful when preparing the camera-ready and incorporate all the reviewers' suggestions to improve the clarity of the presentation and present the new experiments.

Reviews: review: paper_summary: paper_summary: The paper introduces a novel approach that rethinks the application of the Fourier Transform for time series prediction, proposing a unique Fourier Basis Mapping (FBM) method. It combines both learnable and non-learnable components and demonstrates potential improvements through comprehensive experiments. This paper provides cool solutions to the inconsistent starting cycle and series length issues practically and theoretically. However, significant concerns remain. The paper does not clearly link the identified problems with the proposed solutions, resulting in fragmented logic. Figure 1 and the Introduction section lack clarity and focus. The methods section's reliance on non-learnable structures without clear integration with learnable components raises questions about their effectiveness. The unclear title and purpose of Section 5.3, along with insufficient explanation of denoising capabilities, further weaken the paper. Additionally, the absence of comparative studies and lack of ablation experiments undermine the validation of the proposed method's effectiveness. Addressing these issues could potentially render the paper more acceptable (7 scores).
strength_weakness: strength_weakness: 1 The Introduction section inadequately discusses the issues with existing methods. The descriptions are obscure and hard to follow. It is recommended to replace these with more understandable explanations. Two separate paragraphs to elaborate on the identified problems and the corresponding proposed solutions are expected. The current version does not clearly articulate how the method is specifically designed to address each problem, leading to a fragmented logic. Readers may find it difficult to directly correlate the proposed method with the problem-solving benefits claimed.

2 Figure 1 lacks focus and merely replicates descriptions from the text without emphasizing the unique aspects of the proposed method. The distinctive features of the method are not prominently highlighted.

3 The methods section primarily consists of non-learnable structures, which raises significant concerns. The paper does not explain how these non-learnable components integrate with learnable structures to enhance feature extraction, leading to questions about their effectiveness. Therefore,  the title "Effect of Weights for Linear Mapping" in Section 5.3 becomes unclear and feels disjointed, making it difficult to grasp the purpose of this section at a glance. The paper fails to clearly explain why this visualization was conducted and how it supports the stated conclusions. Additionally, the explanation of the model's denoising capabilities is sparse and lacks clarity, seeming somewhat self-serving.

4 A major issue in the experiments is the absence of comparative studies with the same decoder, making it hard to confirm the effectiveness of the proposed frequency domain method. Furthermore, for the key contributions such as denoising and problem-solving, it is suggested to design ablation experiments to validate these effects.
questions: questions: At first, I want to give this paper an Accept score. However,  I have the following three main concerns after reading this paper:
1. Eq. (3) concludes the main technical contribution towards the long-term time series prediction problem. I agree that we should represent a time series with both the Time and Frequency domains to avoid the inconsistent starting cycle and length issue. However, when I look at the overall framework of the proposed method, I find that the function of this Time-Frequency joint representation is somewhat like extracting features. In this case, I want to know the relationship between the non-parameter feature extractor and the learnable decoder. As your ablation study in Sec 5.3 claims, the decoder has a noise-tuning effect for this module. I doubt this and expect the authors to provide a more comprehensive analysis, or the real effectiveness of the proposed module should be viewed as not fully explored.

2. Experiments are not fair. The comparative experiment should be built with the same architecture. FBM has three variants, and they should be compared with the baseline models separately. The most important two comparisons are FBM-L v.s. NLinear; FBM-NP v.s. PatchTST.  I hope a more fair comparisons can be conducted.

3. The writing should be improved, see Weekness for details.

With the following issues all addressed, this paper can score 7.
limitations: limitations: Experiments are not fair, and the proposed methods are not well elaborated.
paper_summary: paper_summary: This work rethinks the discrete Fourier transform from a basis functions perspective, identifying two key issues in existing Fourier-based methods: inconsistent starting cycles and series lengths. To address these, the paper proposes the Fourier basis mapping model (FBM), which leverages Fourier basis expansion to obtain a mixture of time and frequency features. Experiments show FBM outperforms diverse baselines on long-term forecasting tasks, with the linear FBM-L performing better for noisy data and nonlinear FBM-NL/NP better for less noisy data, highlighting the importance of combining time and frequency information.
strength_weakness: strength_weakness: W1: The method is primarily validated on long-term time series forecasting tasks, and its applicability to short-term forecasting tasks requires further exploration.

W2: While the FBM model exhibits better interpretability compared to more complex models like Transformers, further analysis of its internal mechanisms is needed to better understand the role of the time-frequency mixed features.

W3: There is a lack of visual showcase demonstrations, making it difficult to provide a qualitative evaluation of the experiments. Additionally, there is a lack of efficiency analysis regarding the computational complexity.

W4: More comparisons with state-of-the-art models are missing, which is crucial for evaluating the quality of the work.
questions: questions: Q1: How does your proposed FBM model perform in short-term forecasting tasks, especially in comparison to its performance in long-term time series prediction? Are there significant differences? Do you have plans to explore the applicability of FBM in short-term forecasting tasks in future research, such as in evaluations on the M4 dataset and PEMS dataset?

Q2: You mentioned that FBM has better interpretability compared to complex Transformer models. Could you further analyze the internal mechanism of FBM and elaborate on how the spatiotemporal mixed features affect the model's predictive performance? This is crucial for a better understanding of the advantages of FBM.

Q3: It would be helpful to provide more analysis on model parameter sensitivity, efficiency analysis of model computational complexity (including runtime and memory overhead), detailed settings of hyperparameters, and visual showcases.

Q4: Could more comparative experiments with state-of-the-art models such as N-Hits[1], iTransformer[1], TimeMixer[2], and TimesNet[3] be provided? This is crucial for a comprehensive evaluation of this work.

[1]N-Hits: Neural Hierarchical Interpolation for Time Series Forecasting

[2]iTransformer: Inverted Transformers Are Effective for Time Series Forecasting

[3]TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting

[4]TimesNet: TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis
limitations: limitations: The authors have not addressed limitation of their work. I would suggest the authors to include such section in the final version of their work.
paper_summary: paper_summary: The authors propose a Fourier basis mapping model, named FBM, for long-term time series forecasting. FBM embeds the discrete Fourier transform with basis functions, and then introduces a mapping network to replace the inverse discrete Fourier transform. This approach allows FBM to capture implicit frequency features while preserving temporal characteristics. Experiments demonstrate the effectiveness of the proposed framework.
strength_weakness: strength_weakness: (1)The technical contributions are incremental and the overall impact is not significant enough, as some key components are similar to existing works. For example, the Fourier basis mapping is similar to FEDformer. In addition, the three mapping methods used in the decoder look like a simple combination of linear network, nonlinear MLP, and PatchTST.

(2)The introduction of the model architecture is confusing, as many implementation details of the modules in Figure 1 and Figure 3 are not clearly explained or even introduced (e.g., the Fourier basis expansion operation, the normalization operation, and the de-normalization operation).

(3)Several groups of comparative experiments are carried out, but some SOTA works [1, 2, 3] are not compared or mentioned. The authors should compare their model with SOTA works to demonstrate its effectiveness.

(4)There are many typos and writing mistakes in the manuscript. For example, on page 2, "Huange et. al." should be "CrossGNN". On page 6, "Table 2" should be "Table 1". On page 12, "Crossgnn" should be "CrossGNN", "N-beats" should be "N-BEATS", and "Nhits" should be "N-HITS". The manuscript requires a thorough proofreading.

(5)Some figures provided are too small to allow for a thorough investigation of the details. The authors should use vector figures instead. In addition, in Figure 5, the meaning of the x- and y-axes need to be explained and the meaning of the number "5" in the right part should also be clarified.

[1] Liu Y, Hu T, Zhang H, et al. iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. ICLR 2024.
[2] Wang S, Wu H, Shi X, et al. Timemixer: Decomposable multiscale mixing for time series forecasting. ICLR 2024.
[3] Chen P, Zhang Y, Cheng Y, et al. Pathformer: Multi-scale transformers with Adaptive Pathways for Time Series Forecasting. ICLR 2024.
questions: questions: See Weaknesses.
limitations: limitations: No limitations or potential negative societal impacts are discussed in the paper.

Comments: title: New Further Comparison
Response to full evaluation on Electricity and Traffic datasets
Thanks for your positive feedback and your valuable suggestions
Reply for further improvements
Response to Your New Points (1), (2), and (3)
Reponses to your points - 3
Reponses to your points - 2
Reponses to your points - 1
Regarding performance deviation by different lengths and measures
Further comparison
Further Responses to Your New Questions Q4
Further Responses to Your New Questions Q2 and Q3
Further Responses to Your New Questions Q1 and Q5
Intuitive explanation to the FBM model's performance
More clarification
Revised introduction
Extra ablation study and empirical thinking
comment: Thank you for waiting. We have utilized all available GPU resources to run the experiments, but due to time constraints, we can only report results on the Electricity dataset with some efficient baseline methods at this stage. Pathformer is very slow, even with an input sequence length of 96, and running the Traffic dataset would take approximately three times longer than Electricity dataset. We will include the full set of experiments on the Electricity and Traffic datasets with full dimensions in the final version of the paper. Since FEDformer and FiLM are not efficient, especially with an input sequence length of 336, and do not achieve SOTA performance, they will be replaced with the latest SOTA baseline methods—TimeMixer, iTransformer, Pathformer, and FITS—in the final paper.

Model| FBM-NL|FBM-NL|FBM-NP|FBM-NP|PatchTST|PatchTST|iTransformer|iTransformer|TimeMixer|TimeMixer
-|-|-|-|-|-|-|-|-|-|-
Error|MSE| MAE|MSE|MAE|MSE|MAE|MSE|MAE| MSE|MAE
Electricity(ECL) - L=96|   0.132 | 0.227 | 0.133| 0.227 | 0.133   | 0.227    | 0.137   | 0.232  |  0.134  |0.230 
Electricity(ECL) - L=192| 0.149 |0.243 | 0.149 | 0.242 |  0.151 | 0.244  | 0.156 | 0.249  | 0.153 |0.245
Electricity(ECL) - L=336|0.167 | 0.261 | 0.167 | 0.261 |0. 167   |0.261 |  0.171 | 0.266 | 0.172 |0.267 
Electricity(ECL) - L=720|   0.207 | 0.295 | 0.208 | 0.295 |  0.210   | 0.297   | 0.195   | 0.288  | 0.212 | 0.298 

Our new results align closely with our previous findings (see the table in our response to you under 'Further Comparison' ) and also show lower MSE and MAE values than those reported in the iTransformer and TimeMixer papers. This further demonstrates that a sequence length of 336 is a more robust setting. You are welcome to compare your results with ours by running the official code for each baseline method and FBM for our codes using the following experimental settings: input sequence length T=336 and a train/validation/test ratio of 6.5/1.5/2. 

 Many thanks for your valuable suggestions again.
Thanks for your valuable suggestions. We will definitely evaluate the full dimensions of the Electricity and Traffic datasets in the final version. We will start to run the code right now and report as much as we can on the Electricity and Traffic datasets.
Thank you for your prompt feedback, and positive acknowledgement of our efforts. 

Firstly, we will use a uniform input length of 336 for all baseline methods in the final version and work out the appropriate hyperparameters for Pathformer. Secondly, we will establish a consistent range for the learning rate to ensure that the training epochs for each baseline method are tested within a unified range. Thirdly, we will enhance the presentation of the paper to help readers better understand the motivation and workflow of our method. Lastly, we will include a limitations section to discuss the shortcomings of the current work and potential directions for future improvements. All your valuable suggestions will be reflected in the final version of the paper. Many thanks!
There’s one more point I must address. While reviewing your code, I noticed that you only selected the last 20 dimensions of variables for testing on the Electricity and Traffic datasets. This does not align with current benchmark tests. The reasoning for speeding up training does not hold water. During the lengthy paper preparation period and a two-week rebuttal period, I cannot think of any valid reason for not conducting a complete test. This is a critical flaw, and I strongly urge the authors to conduct full experiments according to benchmark standards; otherwise, the value of this work will be significantly diminished.
Additionally, I suggest that the authors include a limitations section to discuss the shortcomings of the current work and potential directions for future improvements, which could enhance the impact of the study. Furthermore, regarding the contributions in the time-frequency domain, a more intuitive presentation for the readers should be employed, as the current format appears overly simplistic and makes it difficult for readers to grasp the value of the authors' contributions.
Thank you for your timely and detailed response. I have indeed seen the authors' efforts to improve their paper's quality; in light of this, I have updated my score accordingly. Regarding suggestion 2, I still strongly recommend that the authors use a uniform input length for testing across different models, for example, by using an input length of 336 for all models. Additionally, concerning the learning rate, as previously mentioned, I suggest that the authors test within a unified range (for example, from 0.001 to 0.01), and that the training epochs be tested within a unified range. This is crucial for fair comparisons between experiments; reasonable and fair experimental settings are necessary to obtain valuable comparative results. Finally, I look forward to witnessing the improvements and advancements in your work.
We thank you for taking your valuable time to read all the feedback and discussions, and we truly appreciate that you see the potential in our work. During the rebuttal process, we have done our best to address all the comments and continuously improve the quality of the paper. I will address each of your three suggestions individually to further enhance the quality of the paper.

**Suggestion 1**

We have included all the experimental settings in Section A.1 of the Appendix and visualized the implementation details of the internal mapping mechanism in Figure 6 of the Appendix. To further enhance readability, we will generate a table to better illustrate all the settings. We acknowledge that one experiment setting has caused misunderstandings. Therefore, in the final version, we will evaluate the full dimensions of the Electricity and Traffic datasets by replacing less efficient baseline methods, such as FEDformer and FiLM, with state-of-the-art methods, including TimeMixer, iTransformer, Pathformer, and FITS.

**Suggestion 2**

We have used the optimal hyperparameters reported in the official codes to implement all the baseline methods, ensuring a fair comparison by training them with the same number of epochs. However, we challenge the suggestion to use the similar learning rate for each baseline method. Each baseline model requires a different learning rate, and there is no universal rule for determining whether a larger or smaller learning rate is better for a specific model. For example, our FBM-L model decomposes the time series into pieces, which requires a smaller learning rate.

Since changing the input sequence length from 96 to 336 can affect the optimal learning rate for each baseline method, we decided to report the optimal learning rate we identified for each method in a table, specifying our experimental settings. This will benefit future research and enhance the reproducibility of the baseline methods.

**Suggestion 3**

We have included a new Figure 1 in the global response PDF to illustrate the uniqueness of our proposed method in time-frequency modeling. Figure 1 demonstrates that the uniqueness of our method lies in the inclusion of the basis function, specifically the Fourier Basis Expansion, which helps extract more efficient time-frequency features, thereby making the downstream mapping much easier. In Section 4, we have also provided additional explanations, such as:

In Line 163: “The process is analogous to decomposing the original time series into $\frac{T}{2}+1$ components with tiered frequency levels, which allows the model to separate various effects hierarchically but identify the affiliated noises.”

In line 174: ”if we sum along the $\frac{T}{2}+1$ frequency domain, we can obtain the original time series without corrupting the time domain information.”

These are all unique advantages of our method, and we will add more explanations to the paper.   

Finally, we sincerely thank you and the other reviewers for your valuable comments and suggestions. We have also done our best to improve the quality of our work based on the feedback.
Thanks again for your detailed response. I have read all the feedback and discussions, and I executed your code. The results of ETT met expectations, so I believe this work has potential. However, just as Reviewer 8ZRG, I think there is still significant room for improvement:

1. I strongly recommend that the authors clearly describe the experimental settings and implementation details. For example, selecting different dimensions from the dataset for evaluation leads to results that differ significantly from other papers, which can easily cause misunderstandings and ambiguities.

2. I suggest standardizing all experimental settings for the experiments. If you are using an input length of 336, I recommend reporting the results of other baselines under unified parameter settings. Additionally, parameters such as learning rate and epoch should also be kept within the same range to ensure a fair comparison.

3. Furthermore, I hope the authors can provide more visual examples to illustrate the uniqueness of your proposed method in time-frequency modeling. Currently, there are many similar works, but this has not been addressed in your paper.

In summary, I hope the authors can further improve their work for better achievements.
We appreciate your response and thank you for taking your valuable time to read our responses again. 

**Response to your new point (1)**

TimeMixer does use MAE and MAPE as the evaluation metrics for the PEMS04 dataset (see Table 3 in their paper).

**Response to your new point (2)**

We don't know whether Reviewer akVH ran his/her experiments with an input length of 336 or 96, as he/she hasn’t stated the experimental settings or provided the exact scores. This makes it difficult for us to fully address this question. However, we have done our best to reproduce every method, and it’s common to encounter reproduction errors, especially when experiments are conducted on different devices and settings. Even a tiny, unnoticed variation in the experimental setup can lead to significant differences in results. Additionally, since all baseline methods were run only once, some variation is to be expected. It’s also worth mentioning that even if we had achieved the results reported in the iTransformer paper, they are still not as good as ours (ETTh1-L=96: iTransformer reports MSE: 0.386, MAE: 0.405; Our FBM-L reports MSE: 0.366, MAE: 0.390).

**Response to your new point (3)**

In line 442 of our paper, we stated that we split the ETT dataset into 12/4/4 months and the other datasets into training, validation, and test sets with a ratio of 6.5/1.5/2. However, PatchTST splits the Weather dataset with a ratio of 7/1/2. We chose our splitting method because a validation set that is only half the size of the test set is relatively too small, particularly in comparison to the ETT dataset. Consequently, the experimental settings for the Weather dataset with a 96-step forecasting horizon differ as well. Although reproduction errors are common under different experimental settings, when running on different devices, and due to error bar in baseline methods, we have made every effort to reproduce each baseline method as accurately as possible and ensure a fair comparison. In the final version of the paper, we will run multiple iterations for each baseline method and report the average values.

We hope our explanation has addressed your concerns, and we appreciate your valuable time spent reviewing our response. We welcome any further comments or suggestions to help improve our work. Thank you very much.
Thanks again for your detailed rebuttal. I read all of the authors' responses and the discussions with other reviewers.

(1) The authors claim that they use MAPE as the evaluation metric on PEMS04 dataset. However, **the baseline methods (e.g., iTransformer and TimeMixer) do not use MAPE as the evaluation metric.** Simply stating that 'MAE and MAPE are the most reliable metrics for the evaluation on the PEMS04 dataset' lacks convincing.

(2) The authors claim that their input length is 336. I set the input length to 336 and rerun the source code of iTransformer with the default parameters on ETTh1 dataset. However, as reviewer **akVH** points out, **the results still show a significant difference from what the authors reported.**

(3) The authors claim that their input length is 336, which is similar to PatchTST. Why the authors do not directly use the experimental results from PatchTST? In addition, I rerun the source code of PatchTST and the results are generally consistent with those reported in the paper. **However, the results still differ significantly from those in the authors' paper**. For example, on Weather dataset under 96 forecasting horizon, the PatchTST results in the original paper are $\underline{\text{0.152 0.199}}$, while in the authors' paper, the PatchTST results are $\underline{\text{0.176 0.226}}$, and the authors' own results are $\underline{\text{0.152 0.199}}$. As reviewer **akvH** points out, this may lead to ambiguity.
**Here is the revised last two paragraphs of Introduction:**

Lastly, Fourier-based time series modeling emerges as a new paradigm to remove noise signals by considering diverse effects hierarchically at different frequency levels. However, methods like FEDformer, FreTS, FiLM, FGNet, and FL-Net use real and imaginary parts as inputs to their mapping networks but cannot easily interpret their coefficients because the crucial information is stored in the amplitude and phase of each cycle. This leads to inconsistent starting periods and series lengths, which are often ignored in existing research. Other methods e.g., CrossGNN and TimesNet use the top-k amplitudes to filter noises. However, a higher amplitude does not necessarily indicate useful frequency, and a lower amplitude is not necessarily useless. More importantly, Figure 1 shows that a Fourier basis function is time-dependent when the input length is not divisible by a certain frequency level. Consequently, the mapping in the frequency space is not enough and fails to capture time-frequency relationships.

We provide a new perspective that real and imaginary parts can be interpreted as the coefficients of cosine and sine basis functions at different granularities of frequencies. However, existing Fourier-based methods do not involve such basis functions, thus failing to interpret these coefficients correctly, as shown in Figure 1. Accordingly, we propose the Fourier Basis Mapping (FBM) by incorporating basis functions to extract more efficient time-frequency features to solve two inconsistency issues, making the downstream mapping much easier. With time-frequency features, very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstances. We evaluate our insights through three FBM variants against four categories of LTSF methods: (1) Linear method: NLinear; (2) Transformer-based methods: FEDformer, BasisFormer, iTransformer, Pathformer, and PatchTST; (3) Fourier-based methods: FEDformer, FreTS, N-BEATS, CrossGNN, TimesNet, and FiLM; and (4) MLP-based methods: N-BEATS, FreTS, and TimeMixer. Both FBM-L and FBM-NL achieves the SOTA performance of LTSF on eight real-world datasets or of STSF on datasets PEMS.

We will further revise the paper to thoroughly clarify your raised concerns.


**Response to your point (3)**

We thank you for your careful reading, and we will fix these spelling inconsistencies in the final version. 

Please be noted, FBM is our method.


**Concluding remarks**

As you would notice from our intensive responses to other reviewers, during this tight rebuttal period, we have been working day and night and making our unreserved efforts to address all comments and suggestions. Thanks to the constructive comments of all reviewers including you, this rebuttal has substantially improved our work quality, with major revisions and improvements including:
1)	Clarifying the motivation and insights of our method, in particular, redrawing Figure 1 in the PDF file, and rewriting the introduction, as shown in the response to Reviewer Z6Ax;
2)	Adding new baseline methods including Pathformer, iTransformer, TimeMixer, and TimesNet;
3)	Adding new datasets, including PEMS04 and PEMS08.

**As Reviewer Z6Ax concluded, and the above added experiments have shown, our work discloses a new perspective that can outperform the SOTA baselines with very simple models,** which is quite valuable for machine-efficient learning of complex time series.

We appreciate that both Reviewers Z6Ax and akVH acknowledge our responses. We understand you substantially downgraded your score based on your above three points. We hope our intensive clarifications in this response have clarified your comments, and we do appreciate your cross-reference with other reviewer’s comments and our responses to their comments. 

You are more than welcome to verify our codes as well. 
We appreciate your valuable time on reading this lengthy response, and any further comments or suggestions to further improve our work. Many thanks.
Lastly, **you are welcome to verify our codes for the above clarification of your misreading**

Regarding the above different evaluation settings and resultant performance, you are more than welcome to verify our shared codes in the **Supplementary Material** (in line 524 in the paper and our global response to all reviewers) to verify our results. Our codes follow the same structure as iTransformer, so it is easy to run if you are familiar with it. Once you have chance to test our codes, you would verify our above clarification and identify your misreading as well. You could verify FBM-L on the ETTh1 and ETTh2 datasets first, as it runs very quickly.


**Response to your point (2)**

Regarding the design for the ‘time-frequency space’, this is in fact the key innovation of our work, and we have substantially explained how to design and implement it in different ways and places. 

First, **Figure 3 explains the FBM structure**
 
In Section 4, we provide Figure 3 to summarize the framework of our proposed Fourier basis mapping (FBM), and explain its working mechanism, copied below for your reading convenience (Lines 160-166):

To address the two issues, we introduce the Fourier basis mapping model FBM. Figure 1 shows the architecture of FBM. The key strength of FBM lies in its designs of generating more implicit frequency features while retaining temporal features, as the actual frequency is inferred by the model through basis functions. The process is analogous to decomposing the original time series into $\frac{T}{2}+1$ components with tiered frequency levels, which allows the model to separate various effects hierarchically but identify the affiliated noises. Consequently, FBM emerges as a more natural method to extract the mixture of both time and frequency domain features.

Second, this section further **elaborates the design for capturing time-frequency features**, e.g., as shown in Lines 170-172, and Lines 173-176:

**Next, we multiply the real part of $\mathbf{H}$ (denoted as $\mathbf{H_R}$) with the cosine basis $\mathbf{C}$ and the imaginary part of $\mathbf{H}$ (denoted as $\mathbf{H_I}$) with the sine basis $\mathbf{S}$ to obtain the mixture of frequency and temporal features.
…
Further, we combine the cosine and sine basis functions to obtain the shifted starting cycle of the time series, forming $\mathbf{G}$. The scalar $2$ is derived in Eq. (4) to ensure that, if we sum along the $\frac{T}{2}+1$ frequency domain, we can obtain the original time series without corrupting the time domain information.**

Third, we further **explain the mechanism in this section**, e.g.: 

In Line 163: “The process is analogous to decomposing the original time series into $\frac{T}{2}+1$ components with tiered frequency levels, which allows the model to separate various effects hierarchically but identify the affiliated noises.” 

In Line 165: “Consequently, FBM emerges as a more natural method to extract the mixture of both time and frequency domain features.” 

In Line 173: “Further, we combine the cosine and sine basis functions to obtain the shifted starting cycle of the time series, forming G.” 

In Line 177, “…it allows the model to eliminate noise along both time and frequency domains. 
Finally, we use a decoder to map the features G to the output time series.“ 

Lastly but not least, we have **updated Figure 1 in the PDF file** to all reviewers, where the time-frequency space is shown in the lower middle panel, extracted from input and further fed to the mapping networks.

In fact, we have revised the introduction per the suggestion by **Reviewer Z6Ax** to explain the main insights and innovation of our work, who has accepted our revision. For your reading convenience, we copy the two paragraphs in our next response.
We appreciate your response and thank you for taking your valuable time to read our previous responses. Below, we address your three new points, and clarify potential misreadings. The response is a bit lengthy, please read this multiple-page response titled **Reponses to your points – 1** to **Reponses to your points – 3** due to the space limit, to substantially address your comments. We appreciate your valuable time and patience in reading this response.

**Response to your point (1)**

First, **misreading of different settings and evaluation measures**

You misread or overlook the different input sequence lengths used by our method (**with length 336**) vs that by iTransformer (**with length 96**) on dataset ETTh1, and the different performance measures for ours (**per MAE and MAPE**) vs iTransformer (**per MSE and MAE**) on dataset PEMS04.  

The result of ‘0.407 0.423’ corresponds to the input sequence length of 336, while ‘0.386 0.405’ corresponds to the sequence length of 96 for iTransformer on ETTh1 under 96 forecasting horizons. This setting difference applies to all new experimental comparisons in Table 2 in the PDF file to all reviewers.

The result of '0.176 0.354' refers to the value of MAE and MAPE, respectively; in contrast, '0.078 0.183' refers to the values of MSE and MAE, respectively for iTransformer on dataset PEMS04 under 12 forecasting horizons. This setting difference applies to all experiment results in Table 3 in the PDF file. 
 
In conclusion, 1) the performance difference you misread for iTransformer on ETTh1 corresponds to different input sequence lengths, i.e., we use 336 versus 96 used in the original paper; 2) the performance difference you misread for iTransformer on PEMS04 correspond to different evaluation measures, i.e., MAE and MAPE for ours vs MSE and MAE for iTransformer.

Please also refer to our previous response **Q1** in the second round of response titled **Further Responses to Your New Questions Q1 and Q5** to Reviewer akVH.


Second, **we explain why we use length 336 rather than 96 as in iTransformer.**

The input sequence length of 336 always produces better results than 96. This has been verified by the following studies, where methods in Group A with sequence length 336 consistently report lower MSE and MAE values than those in Group B with sequence length 96. 

Group A:

1.	PatchTST: A Time Series is Worth 64 Words: Long-term Forecasting with Transformers
2.	NLinear: Are Transformers Effective for Time Series Forecasting

Group B: 

1.	Pathformer: Multi-scale Transformers with Adaptive Pathways for Time Series Forecasting
2.	iTransformer: Inverted Transformers Are Effective for Time Series Forecasting
3.	TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting
4.	TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis

The above difference between our reported results and that in iTransformer is actually consistent with what we can identify in the results of iTransformer in comparison with PatchTST. 

For example, if you compare the MSE and MAE scores reported in PacthTST with the scores reported by iTransformer for PatchTST, you will find even more significant difference. We collect and present their results in the following table: 

PatchTST’s results | In PatchTST's paper with T=336 | In PatchTST 's paper with T=336 | In iTransformer's paper with T=96 | In iTransformer's paper with T=96
-|-|-|-|-
Error|MSE| MAE| MSE| MAE|
ETTh1 L= 96|0.375|0.399|0.414|0.419 
ETTh1 L=192|0.414|0.421|0.460|0.445 
ETTh1 L=336|0.431|0.436|0.501|0.466 
ETTh1 L=720|0.449|0.466|0.500|0.488
 

Therefore, we set the input sequence length to 336. Accordingly, our experiment settings are the same as that of PatchTST for better performance, which are different from Pathformer, iTransformer, TimeMixer, and TimesNet using the input sequence length of 96. 

Third, **we explain why we use MAE and MAPE rather than MSE and MAE as in iTransformer.**

For the PEMS04 dataset, MAE and MAPE are more suitable metrics than MSE and MAE, as shown in other baselines like TimeMixer. The difference in MAPE between our results and TimeMixer's is due to their filtering of extremely large values, while we retain the original values. However, both comparisons are reasonable.

In fact, our experiments also verify the above observation with a lower MAE score for the input sequence length of 336 than that reported in the iTransformer paper for length 96 (**0.176 vs 0.183**). This result further demonstrates that the input sequence length of 336 is more robust.
We thank you for your follow-up of our response and further comments during your busy agenda.

Regarding the performance difference concerning iTransformer on the ETTh1 dataset mentioned by Reviewer 8ZRG, this reviewer actually misread or overlook the different input sequence lengths used by our method (with length 336) vs that by iTransformer (with length 96) on dataset ETTh1, and the different performance measures for ours (per MAE and MAPE) vs iTransformer (per MSE and MAE) on dataset PEMS04. Please find our responses titled **Reponses to your points –1** to **Reponses to your points –3** to Reviewer 8ZRG, where we comprehensively clarified the misreading and justified why different lengths and measures were used.

In addition, please refer to our previous response **Q1** in the second round of response titled **Further Responses to Your New Questions Q1 and Q5** to you. We have thoroughly explained that this discrepancy is due to the different input sequence lengths used in the iTransformer paper and ours, i.e., 336 is applied in our work versus 96 in theirs. 

For example, if you compare the MSE and MAE scores reported in PacthTST with the scores reported by iTransformer for PatchTST, you will find even more significant difference. We collect and present their results in the following table: 

PatchTST’s results | In PatchTST's paper with T=336 | In PatchTST 's paper with T=336 | In iTransformer's paper with T=96 | In iTransformer's paper with T=96
-|-|-|-|-
Error|MSE| MAE| MSE| MAE|
ETTh1 L= 96|0.375|0.399|0.414|0.419 
ETTh1 L=192|0.414|0.421|0.460|0.445 
ETTh1 L=336|0.431|0.436|0.501|0.466 
ETTh1 L=720|0.449|0.466|0.500|0.488

Second, Reviewer 8ZRG overlooked the different performance measures used in evaluating iTransformer on the PEMS04 dataset: we report the MAE and MAPE results, as MAE and MAPE are the most reliable metrics for the evaluation on the PEMS04 dataset. In contrast, iTransformer reports the MSE and MAE results, which are less appropriate for that dataset. In fact, our experiments report a lower MAE score than that reported in the iTransformer paper because we use the input sequence length of 336 rather than 96. This result further demonstrates that the input sequence length of 336 is more robust.

Regarding the above different evaluation settings and results, you are more than welcome to test our shared codes in the Supplementary Material (mentioned in line 524 in our paper  and our global response to all reviewers) to verify our settings, evaluation measures and results. Our codes follow the same structure as iTransformer, so it is easy to run if you are familiar with it. Once you have chance to test our codes, you would verify our above clarification and Reviewer 8ZRG’s misreading as well. You could verify FBM-L on the ETTh1 and ETTh2 datasets first, as it runs very quickly.  

Finally, while we acknowledge your point that TimeMixer and iTransformer are efficient, other baseline methods, such as Fedformer, Pathformer, and FiLM, do not achieve the same level of efficiency. This is precisely why most papers only verify their methods using the input sequence length of 96 rather than 336, even though 336 has been demonstrated by NLinear and PatchTST to be a better choice. While the ETTh1 and ETTh2 datasets are quick to train, the training on the Electricity and Traffic datasets is much slower, especially since most baseline methods significantly increase the dimensionality of hidden states on those datasets.

We sincerely hope our further explanations have addressed all of your concerns relating to Reviewer 8ZRG’s feedback as well as your previous concerns. If you have any trouble in verifying our codes, finding any issues there, or having any further comments, please feel free to let us know. We are endeavoured to address them as much as we can. Many thanks.
Thank you for your further response. I have tried running the baseline models and datasets you mentioned, and I was able to complete the tests within a limited timeframe. Therefore, I don't understand why certain dimensions must be selectively chosen for acceleration. TimeMixer and iTransformer are relatively efficient models, and there shouldn't be any efficiency issues when testing well-known datasets. Moreover, Reviewer 8ZRG pointed out that the results you reported differ significantly from those in the paper, which may lead to ambiguity. For example, I also attempted to run tests using the code of iTransformer and found that the results were generally consistent with those reported in the paper. Therefore, I don't quite understand why your reported results deviate significantly. Based on these two points, I have decided to maintain my score.
Thanks for the rebuttal. I read all of the authors' responses and the discussions with other reviewers.

(1) Although the authors have added several comparative experiments with latest baselines, almost all the reproduced results differ significantly from those in the original paper, making the results unreliable. For example, '0.407 0.423' should be '0.386 0.405' for iTransformer on ETTh1 dataset under 96 forecasting horizon and '0.176 0.354' should be '0.078 0.183' for iTransformer on PEMS04 dataset under 12 forecasting horizon.

(2) The authors have claimed that the main difference between FBM and existing Fourier-based mapping is that "FBM is conducted in the time-frequency space." However, I did not see any targeted design towards "time-frequency space" in Section 4. In addition, there are many key concepts (e.g., Time Dependent, L is not divisible by K, and Time duration) in **Figure 1** of $\underline{\text{global response}}$ that are not adequately described or even explained in the main paper, which makes it difficult to understand in some places.

(3) The authors have promised to "revise the paper and thoroughly polish the paper for the final version", however, in **Figure 1** of $\underline{\text{global response}}$, we still find many inconsistency and informal expressions, which reflect a lack of rigor and carelessness of authors in the scientific research:

* "Frequency level" should be "Frequency Level";
  
* "Time duration" should be "Time Duration";
  
* "FBM's" should be "Our Method".
  

Given the above question and the writing, I think this paper is not ready for being published in NeurIPS, and I have updated my scores.
Thanks for your careful examination.

In line 441 of the paper, we mentioned that we selected the last 20 dimensions of the Traffic and Electricity datasets to accelerate training speed. This decision was made because most baseline methods were originally designed for the sequence length of 96 rather than 336. They did not account for the quadratic increase in both MACs and the approximately fourfold increase of training time when the sequence length is expanded, making them extremely slow to train with a sequence length of 336—approximately ten times slower or even more.

To ensure a fair comparison, we also tune their learning rates, as their experimental settings change with the increase of input sequence length from 96 to 336. Without this adjustment, these baselines would not perform anywhere close to our methods. However, it would be impossible to fully optimize their models, given that training for once could take days or even weeks on these datasets for each baseline. For example, Pathformer takes almost a day to train with the input sequence length of 96 under our settings. Thus, if we used the full dimension in our experiment, we would not be able to complete the training and present you new results before this rebuttal ends. Furthermore, if we expanded the input sequence length to 336 for Pathformer, assuming their hyperparameters would work for 336, it would take a few weeks to train just one baseline model using our computational facilities. It would have been impossible to help them tune the learning rate, making a fair comparison unattainable. Therefore, we slightly modified the experimental settings to preserve the characteristics of the data while ensuring that a fair comparison could be conducted.

We hope this makes sense to you, otherwise we would be continuing our experiments as well even though we can't provide more comparisons before this rebuttal ends. We hope our explanation addresses your concern. If you have any other concerns, please let us know. We will continuously try our best to address them. Many Thanks.
Thanks for your detailed response. After careful examination, I found some issues. The results reported on the Electricity and Traffic datasets differ significantly from those in other papers. What could be the reason for this?
**Further comparison**

Thank you for reading our responses carefully and positive acknowledgment of our responses. Definitely, the changes will be reflected in the final version. Below, we respond to your new enquiry.

Due to time constraints, we have not conducted experiments on TimesNet, as it is not the latest model. Below are the results of the newly introduced baselines with FBM-NL and FBM-NP on the Traffic, Electricity, and Weather datasets.

Model | FBM-NL | FBM-NL | FBM -NP | FBM-NP |iTransformer|iTransformer|TimeMixer|TimeMixer|Pathformer|Pathformer
-|-|-|-|-|-|-|-|-|-|-
Error| MSE | MAE| MSE | MAE|MSE|MAE| MSE|MAE|MSE|MAE
Electricity L=96| 0.106 | 0.199 | 0.107 | 0.199 | 0.109 | 0.205 | 0.108 | 0.200 | 0.113 | 0.199
Electricity L=192| 0.122 | 0.215 | 0.121 | 0.210 | 0.125 | 0.222 | 0.122 | 0.212 | 0.127 | 0.211
Electricity L=336| 0.141 | 0.231 | 0.140 | 0.229 | 0.141 | 0.236 | 0.143 | 0.233 | 0.147 | 0.231
Electricity L=720| 0.175 | 0.265 | 0.175 | 0.264 | 0.163 | 0.262 | 0.183 | 0.270 | 0.176 | 0.260
Traffic L=96| 0.283 | 0.227 | 0.288 | 0.231 | 0.292 | 0.244 | 0.290 | 0.240 | 0.362 | 0.268
Traffic L=192| 0.289 | 0.231 | 0.293 | 0.234 | 0.301 | 0.246 | 0.304 | 0.350 | 0.371 | 0.263 
Traffic L=336| 0.293 | 0.236 | 0.298 | 0.239 | 0.306 | 0.250 | 0.313 | 0.254 | 0.374 | 0.267
Traffic L=720| 0.306 | 0.247 | 0.309 | 0.248 | 0.316 | 0.264 | 0.320 | 0.267 | 0.384 | 0.290
Weather L=96| 0.152 | 0.200 | 0.156 | 0.204 | 0.162 | 0.211 | 0.158 | 0.204 | 0.151 | 0.191
Weather L=192| 0.194 | 0.242 | 0.198 | 0.245 | 0.204 | 0.249 | 0.197 | 0.246 | 0.202 | 0.242
Weather L=336| 0.244 | 0.282 | 0.248 | 0.285 | 0.248 | 0.285 | 0.242 | 0.281 | 0.260 | 0.283
Weather L=720| 0.317 | 0.334 | 0.319 | 0.337 | 0.322 | 0.335 | 0.319 | 0.335 | 0.337 | 0.336

Both FBM-NL and FBM-NP achieve better performance than all newly introduced baseline methods in most cases, except for three: (1) Electricity L=720, where iTransformer achieves the best MSE, and Pathformer achieves the best MAE; and (2) Weather L=336, where TimeMixer achieves the best MSE and MAE; and (3) Weather L=96, where Pathformer achieves the best MSE and MAE. It's worth noting that there is still potential to improve the performance of our FBM variants, as we used the fixed hyperparameters. 

We will also reflect these comparisons in the final version. Please let us know if you have any other constructive suggestions, we will continuously try our best to address them and improve the paper quality. Many thanks.
I would like to express my gratitude to the author for the comprehensive response, which has addressed several of my initial concerns. Consequently, I have increased my assessment score. I strongly encourage the author to incorporate this information into the final version of the manuscript. Nevertheless, I still have some inquiries. Could the author provide a comprehensive comparison of the ETT dataset alongside the Traffic, Electricity, and Weather datasets in relation to all the newly introduced baselines? I anticipate the author's further elaboration on this matter.
**Q4**

In fact, we have provided empirical visual analysis of the time-frequency features:

1. The updated Figure 1 in the PDF with the global response to all reviewers visualizes the Fourier basis expansion, where Fourier basis can be categorized into time dependent basis and time independent basis. 

2. Section 5.3 visualizes the weight of FBM-L to explain how FBM-L considers the time-frequency relationships. 

First, we find that a Fourier basis function is time-dependent when its input length is not divisible by the frequency level (see time dependent basis in Figure 1 in the PDF). Mapping in the frequency space cannot capture the time-frequency relationships. 

Second, Section 5.3 visualizes how FBM-L considers the time-frequency relationships, addressing the interpretability of FBM in the response to **Weakness W2**.

We thank you for the detailed comments and hope we have clarified your concerns. We will reflect these responses in the final version. Please let us know if you have any other constructive suggestions. Many thanks.
**Q2** 

FITS reports their results w.r.t. the Parameters and MACs, we follow this to compare FBM with it for your convenience as our model’s parameters and memory are calculable by hand. The internal mechanisms of FBM variants with their hyperparameters are provided in Figure 6 in Appendix. 

Assuming $T=96, L=720$, the same settings as FITS, the number of parameters of FBM-L is (96$\times$49) $\times$720=3.38M. 

FBM-NL has three layers, totalling (96$\times$49) $\times$1440+1440$\times$1440+1440$\times$720=9.88M. The MACs of FBM-L and FBM-NL are 49 and 96 times larger than that of NLinear, respectively. The hidden states in the second and third layer of FBM-NL are fixed, which is 1440, see Figure 6 in Appendix. FBM-NL uses the same structure and hyperparameters all the time for better reproducibility.

We also use the hyperparameters of FBM-NP as the optimal hyperparameters for PatchTST. Thus, the only difference lies in the first initial projection layer, and all the downstream structures are completely the same. In PatchTST, the initial projection layer is nn.Linear($\frac{2T}{P}$, K), while ours is nn.Linear($\frac{T^2}{2P}$, K). The optimal P and K of PatchTST for data Electricity is 16 and 512 respectively, reported in their official codes. Thus, the difference is $(\frac{96}{2}-2)\times512 \times \frac{96}{16}$=0.14M. FBM-NP has 0.14M more parameters, but the same MACs as PatchTST.

The following compares our FBM with the results in Table 3 in FITS:
 
Model|Parameter|MACs
-|-|-
TimesNet|301.7M| 1226G
FiLM|14.91M |5.97G
FEDformer|20.68M|4.41G
PatchTST|1.5M|5.07G
DLinear|0.14M|0.04G
NLinear|0.07M|0.02G
FBM-L|3.38M|0.96G
FBM-NL|9.88M|1.82G |
FBM-NP|1.64M|5.07G

FBM has lower MACs than every transformer-based models. The number of total parameters is also lower than most Transformer-based methods. It is worth mentioning that FBM-NP improves the performance of PatchTST without substantially increasing its complexity, using the same optimal hyperparameter as PatchTST without tuning, which further demonstrates the effectiveness of time-frequency features. If the sequence length increases, the number of parameters and MACs will increase quadratically for FBM-L and increase quadratically in the first layer but remain the same for the rest two layers of FBM-NL. However, all transformer-based methods suffer from a quadratic increase, thus our FBM variants always have lower MACs compared with transformer-based models. 

In addition, FBM-L and FBM-NL rank second and fifth in terms of training speed, respectively, with T=336 and L=336. This is not a large language model, the training speeds and MACs play a more significant role rather than model parameters, which demonstrates that FBM variants are efficient comparing to existing LTSF methods. 

We hope the above explanation helps you capture the internal mechanisms of FBM and address your concern about efficiency. We’ll reflect the above efficiency analysis in the appendix of the final paper, due to limitation of space and time in rebuttal. 

**Q3**

The internal mechanisms of FBM variants are provided in Figure 6 in Appendix with all the hyperparameters and layers provided, making the model’s parameters calculable by hand, as shown in the response to your question Q2. 

The hyperparameters for FBM-NL are fixed, the same as those used in PatchTST for FBM-NP, to ensure better reproducibility. While a sensitivity analysis is not meaningful here, we address your concern by analyzing the hidden states from 720 to 1440 for FBM-NL on ETTh1:

FBM-NL|hidden=1440|hidden=1440|hidden=720|hidden=720 
-|-|-|-|-
Error|MSE| MAE| MSE| MAE|
ETTh1-L=96|0.368|0.395|0.370|0.397 
ETTh1-L=192|0.408|0.418|0.409|0.420 
ETTh1-L=336|0.425|0.430|0.427|0.433 
ETTh1-L=720|0.456|0.466|0.462|0.470 

The performance of varying hidden states can differ across datasets, but tuning the hyperparameters will undermine reproducibility with changing settings (e.g., sequence length). Therefore, we have included a sensitivity analysis for sequence lengths of 96 and 336 in Table 4 of Appendix, which shows that a sequence length of 336 is more robust and produces better results for most baseline methods.

The MSE and MAE results of FBM variants in the tables are calculated by averaging the results of two runs. Since we didn’t store the initial error bar in the previous experiments, we cannot provide the full reports here due to limit of space and time, but we will add the error bar analysis of FBM variants to update the Appendix. Here is the error bar analysis conducted on ETTh1:

Model| FBM-L|FBM-L|FBM-NL|FBM-NL|FBM-NP|FBM-NP
-|-|-|-|-|-|-
Error|MSE| MAE|MSE|MAE|MSE|MAE
ETTh1-L=96|0.367±0.01| 0.391±0.01 | 0.369±0.01| 0.395±0.01 | 0.367±0.01|0.394±0.01
ETTh1-L=192| 0.403±0.01 |0.410±0.01 | 0.408±0.01 | 0.418 ±0.01|0.407±0.01|0.417±0.01
ETTh1-L=336|0.420±0.03| 0.420±0.01 | 0.428±0.04| 0.432 ±0.03 |0.429±0.05|0.435±0.05
ETTh1-L=720|0.416±0.03|0.439±0.01 | 0.456±0.02 |0.466±0.02 | 0.442±0.04 |0.462±0.03
Thank you for reading our responses and raising further questions. Below, we respond to each of your questions.

**Q1**

The ETTh1 and ETTh2 datasets are used for two reasons. First, they are widely used for LTSF, appearing in almost all relevant methods as cited in their experiments. The second reason is that the experiment settings on this data are always the same except for the input sequence length, making the results reported in their papers comparable with each other. 

Accordingly, in our global response to all reviewers, we point out that all the experiment results are conducted with an input sequence length of 336, rather than 96, except for Pathformer, as its hyperparameters do not work with 336. This is consistent with the settings in our paper. A length of 336 always produces better results than 96. This is also evidenced by the following papers, where Group A (with sequence length 336) consistently reports lower MSE and MAE than Group B (sequence length 96).

Group A:

1.	PatchTST: A Time Series is Worth 64 Words: Long-term Forecasting with Transformers
2.	NLinear: Are Transformers Effective for Time Series Forecasting

Group B: 

1.	Pathformer: Multi-scale Transformers with Adaptive Pathways for Time Series Forecasting
2.	iTransformer: Inverted Transformers Are Effective for Time Series Forecasting
3.	TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting
4.	TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis

Therefore, we set the input sequence length to 336. Thus, our experiment settings are the same as that of PatchTST and NLinear, but slightly different from Pathformer, iTransformer, TimeMixer, and TimesNet, as they use the input sequence length of 96. As the same hyperparameters in their official codes are used to reproduce their results, our reported performance DOES NOT differ significantly from those appearing in their papers. In fact, our reported MSE and MAE scores are close to theirs with an acceptable difference, which is very common in reproducing existing methods. In fact, if you compare the MSE and MAE scores reported in PacthTST with the scores of those papers in Group B reported for PatchTST, you would find a more significant difference. In addition, as TimesNet uses 0 and 1 normalization, it is not directly comparable.

**Q5**

Let’s answer your question Q5 firstly before addressing others, as this is the most important question. 

As suggested by Reviewer Z6Ax in the first round of review, we generated a new **Figure 1 shown in the PDF for global response** to all reviewers. The new figure better illustrates, compares, and summarizes existing Fourier-based methods with our FBM and emphasizes our main contributions. It also shows visual use cases to illustrate their differences.

You may misread/misunderstand the motivation and main contributions of our work. Please let us explain below. In fact, we revised the last two paragraphs of the introduction to clarify the contributions, requested by Reviewer Z6Ax, who has adjusted his/her misreading.

Please refer to the response titled **Here is the revised last two paragraphs of Introduction** to Reviewer Z6Ax 

Thus, our FBM has a new insight into deep time series forecasting using the basis functions with time-frequency features, making the following main differences:

1.	Existing Fourier-based mapping is conducted in the frequency space, while FBM is conducted in the time-frequency space. 
2.	Existing methods face issues of inconsistent starting cycles and inconsistent series lengths, but addressed by FBM.
3.	Existing methods ignore that Fourier basis could be time dependent, but addressed by FBM. 

We find that FITS you suggested is a very interesting and relevant model to compare, so we’ll cite this paper in the related work and take it as a new baseline to update results in Table 1. FITS also addresses the issue of inconsistent starting cycles but does not resolve the issue of inconsistent series lengths and fails to consider time-frequency relationships, as its mapping is conducted in the frequency space. However, the existing mapping in the frequency space completely ignores the time dependent frequency information. This is because a Fourier basis function is time-dependent when the input length is not divisible by the frequency level (**See the time-dependent basis in Figure 1 of the PDF**). FITS primarily focuses on reducing the size of model parameters. In contrast, the main contributions of our paper lie in disclosing the issues of inconsistent starting cycles and inconsistent series lengths with existing Fourier-based methods and demonstrating the effectiveness of the time-frequency features. With time-frequency features, remarkably simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstance.

Pls read the following responses to Q2-Q4 as well.
Thank you for your prompt feedback and positive acknowledgement of our responses.

Definitely the changes will be reflected in the final version. In fact, we have revised the introduction, as illustrated in the various responses and the two last paragraphs shown to you.

We thank you for referring to our responses to the other reviewers. We'll timely work on the responses to any further questions and suggestions by other reviewers.

Please be noted that we have responded to your comments following **Reviewer akVH**'s questions on visual assessment and computational complexity, and we are further working on additional update on **akVH**'s new feedback, which mostly has been addressed in the attached PDF file for all reviewers already.

Do you still have any comments or suggestions? Please kindly let us know, we will continuously try our best to address them and improve the paper quality.
cool response. Please really add our suggestion to your manuscript. Here is your rate of 6.
Thank your suggestion. Similar to your perspective, I recognize the potential in the paper; however, I continue to have several inquiries. The author's responses have not fully addressed my concerns. I will take the time to thoroughly evaluate the author's forthcoming replies before concluding.
Thank you for the author's response. I have carefully read your reply, but my concerns have not been alleviated. I still have the following questions:

Q1. Why are only the ETTh1 and ETTh2 datasets included in the new baseline comparison experiments? Additionally, why do the reported results of the new baselines differ significantly from those in their paper? If the experimental settings were modified, please describe the parameter settings for each model in detail.

Q2. The analysis of model performance should encompass running time, model parameter size, and GPU memory usage. The current analysis is too simplistic, and it should also examine how the model's efficiency changes with increasing input length.

Q3. Moreover, a sensitivity analysis of model parameters, detailed hyperparameter settings, visual showcases, and error bars with confidence intervals for prediction results should be provided.

Q4. In addition to empirical analysis, is there a more intuitive visual analysis of the model's interpretability? This is crucial for enhancing our understanding of the model.

Q5. Furthermore, many works on time-frequency features in time series, such as FITS. What are the differences between the proposed model and those? Using Fourier transforms to process time series is a long-established topic; what are the novel contributions of our approach?

Resolving the above issues is crucial for improving the quality of this work. I will carefully consider the author's responses before making a final decision.
First, we thank you for checking the comments providing feedback to our rebuttal to **Reviewer akVH**. 

Regarding the **Weakness W3** and **Question Q2**, we have responded to them, please find the corresponding responses to **Weakness W3** and **Question Q3** in the rebuttal to **Reviewer akVH**.

Further, regarding the internal mechanism of our FBM, we provide the following intuitive explanation.

The effectiveness of the time-frequency feature lies on that it shares the similarity with methods like DLinear and Autoformer, where a model's performance can sometimes be improved using a moving average to separate trend and seasonal effects. However, this approach is not always effective because it requires determining an appropriate kernel size for the moving average. In our method, we use the Fourier basis to decompose all potential effects into pieces (e.g., 2-hour cycles, 24-hour cycles (daily), 168-hour cycles (weekly), and k-hour cycles) so that different effects can fall into different frequency levels. Then, FBM can consider all the potential effects hierarchically, making downstream mapping much easier. Additionally, the Fourier basis includes time-dependent and time-independent basis functions, which further help automatically separate trend and seasonal effects. The mapping allows the model to consider all the potential effects interactively and hierarchically and remove noises in both time and frequency spaces. You can think of this as an upgraded version of DLinear. While DLinear only performs better than NLinear in very small cases, our FBM-L consistently outperforms NLinear for all the time. In conclusion, FBM separates all potential effects and allows the neural network to consider those effects automatically and interactively, while DLinear only considers the trend and seasonal effects and whether it is a good separation largely depends on the kernel size you choose.
Thanks for your suggestions; the clarifications about motivation, internal mechanisms, contributions and performance evaluation, etc. will be reflected in the introduction and evaluation sections respectively.  In addition, a fairer evaluation will be conducted that our FBM variants are listed as independent columns in Table 1.  The details of the revised contents are included in our global reply to all reviewers.
Thank you for your further suggestions on adding the revised explanation to the introduction. 

Below, we show the last two revised paragraphs of the introduction addressing your comments and suggestions. The final version could be slightly different as more citations will be added and with our continuous improvement. 

***Here is the revised last two paragraphs of Introduction**

Lastly, Fourier-based time series modeling emerges as a new paradigm to remove noise signals by considering diverse effects hierarchically at different frequency levels. However, methods like FEDformer, FreTS, FiLM, FGNet, and FL-Net use real and imaginary parts as inputs to their mapping networks but cannot easily interpret their coefficients because the crucial information is stored in the amplitude and phase of each cycle. This leads to inconsistent starting periods and series lengths, which are often ignored in existing research. Other methods e.g., CrossGNN and TimesNet use the top-k amplitudes to filter noises. However, a higher amplitude does not necessarily indicate useful frequency, and a lower amplitude is not necessarily useless. More importantly, Figure 1 shows that a Fourier basis function is time-dependent when the input length is not divisible by a certain frequency level. Consequently, the mapping in the frequency space is not enough and fails to capture time-frequency relationships.

We provide a new perspective that real and imaginary parts can be interpreted as the coefficients of cosine and sine basis functions at different  granularities of frequencies. However, existing Fourier-based methods do not involve such basis functions, thus failing to interpret these coefficients correctly, as shown in Figure 1. Accordingly, we propose the Fourier Basis Mapping (FBM) by incorporating basis functions to extract more efficient time-frequency features to solve two inconsistency issues, making the downstream mapping much easier. With time-frequency features, very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstances. We evaluate our insights through three FBM variants against four categories of LTSF methods: (1) Linear method: NLinear; (2) Transformer-based methods: FEDformer, BasisFormer, iTransformer, Pathformer, and PatchTST; (3) Fourier-based methods: FEDformer, FreTS, N-BEATS, CrossGNN, TimesNet, and FiLM; and (4) MLP-based methods: N-BEATS, FreTS, and TimeMixer. Both FBM-L and FBM-NL achieves the SOTA performance of LTSF on eight real-world datasets or of STSF on datasets PEMS.
We thank you for your further suggestion. We add new results of the horizon of 96 and 720 to Table 1 in the PDF file to address your previous suggestion. Due to the space limit, we could not incorporate all results into that table.

However, the full experiment results of the horizon of 96 to 720 have already been conducted and are presented in Tables 1 and 2 of the paper, respectively. Below, we consolidate them into a single table for better comparison of FBM-L vs. NLinear and FBM-NP vs. PatchTST over the entire horizons, including the results at horizon L = 192 and L = 336 on the eight datasets.

|Model|ETTh1|ETTh1|ETTh2|ETTh2|ETTm1|ETTm1|ETTm2|ETTm2|Electricity|Electricity|Traffic|Traffic|Weather| Weather|Exchange|Exchange|
|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
|Length|192|336|192|336|192|336|192|336|192|336|192|336|192|336|192|336|
| FBM-L-MAE | 0.411| 0.420 |0.374|0.376 | 0.364| 0.384|  0.290|0.326 | 0.216 | 0.232| 0.237 |0.244 |  0.247| 0.285  |0.309 | 0.421| 
|NLinear-MAE|0.426| 0.435| 0.387| 0.395| 0.374| 0.390 | 0.294| 0.331| 0.219| 0.237| 0.245| 0.253| 0.262| 0.296| 0.316| 0.426| 
|FBM-L-MSE|0.403|0.418| 0.333|0.321|0.337|0.371|0.219|0.273| 0.129|0.147| 0.307 |0.314 | 0.203 | 0.252  |  0.195| 0.347 |
|NLinear-MSE |0.421| 0.435| 0.350 | 0.344|0.347| 0.377| 0.223| 0.277|0.128| 0.148| 0.311| 0.319| 0.220 | 0.265| 0.203| 0.356| 
|FBM-NP-MAE|0.416 |0.438|0.382 |0.411|0.368| 0.389|0.296 |0.331|   0.210 |0.229 | 0.234 | 0.239 |0.245 |0.285  |0.312 |0.425 |
|PatchTST-MAE| 0.422 | 0.436 | 0.378 | 0.385| 0.370 |0.394 | 0.306 |0.336 | 0.215|0.234 | 0.232 |  0.240  |0.246 | 0.285 | 0.325|0.435 | 
| FBM-NP-MSE |0.407 |0.433 |0.344| 0.374| 0.334| 0.371| 0.224 |0.277 | 0.121 |0.140 | 0.293 |0.298 |0.198 |0.248 |0.196 | 0.353|
|PatchTST-MSE|0.417 | 0.431 |0.341 | 0.332 |0.333 | 0.363| 0.255| 0.285 |0.123 |0.142 | 0.289 | 0.295  |0.200  | 0.252 |0.210  | 0.366|

Please be noted, all the average MSE and MAE results of FBM-L, NLinear, FBM-NP, and PatchTST reported in the rebuttal are based on full experiments (L=96,192,336,720). We use the same hyperparameters for FBM-NP as the optimal hyperparameters for PatchTST without tuning. Even then, FBM-NP still performs better than PatchTST at most of the time. It is important to note that FBM-L and NLinear don’t have any hyperparameters, as NLinear simply uses the layer nn.Linear(T,L), making it easier to compare. FBM-L always performs better than NLinear in all cases, as shown w.r.t. MAE, which demonstrates the effectiveness of time-frequency features.

**Empirically why FBM works well**

The empirical reason for the effectiveness of the time-frequency feature is that FBM shares the similarity with methods like DLinear and Autoformer, where modeling performance can sometimes be improved using a moving average to separate trend and seasonal effects.  However, this approach is not always effective because it requires determining the appropriate kernel size for the moving average. In our method, we use the Fourier bases to decompose all potential effects into pieces (e.g., 2-hour cycles, 24-hour cycles (daily), 168-hour cycles (weekly), and k-hour cycles), their different effects can fall into different frequency levels. Then, FBM can consider all the potential effects hierarchically, making downstream mapping much easier. Additionally, the Fourier bases includes time-dependent and time-independent basis functions, which further help automatically separate trend and seasonal effects. The mapping allows the model to consider all the potential effects interactively and hierarchically and remove noises in both time and frequency spaces. You can think of FBM as an upgraded version of DLinear, while DLinear only performs better than NLinear in a very small cases, but our FBM-L consistently outperforms NLinear at all the times. 

In conclusion, FBM separates all potential effects and allows neural networks to consider those effects automatically and interactively, while DLinear only considers the trend and seasonal effects and whether it is a good separation largely depends on the kernel size you choose.
I think the author's response is worth consideration. This is a cool paper because very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstance with the proposed time-frequency features.
I have similar concerns. 

W3: There is a lack of visual showcase demonstrations, making it difficult to provide a qualitative evaluation of the experiments. 

Q2: Additionally, there is a lack of efficiency analysis regarding the computational complexity.

You say that "The internal mechanisms of FBM variants are provided in Figure 6 ", but I mean that could you please provide some illustration to show how the proposed method can improve the performance? I need an intuitive explanation.

I know that you can no longer provide a Figure in the discussion phase. Please tell me your plan to improve the presentation.
I think you should see the author's response to my question. "**Our main contribution is on extracting better initial features enable more effective downstream mapping. With time-frequency features, very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstance. **The internal mechanisms of FBM variants are provided in Figure 6 in Appendix. Assuming the look-back window and forecast horizon , FBM-L decomposes the original time series into pieces at different frequency granularities. Thus, our FBM-L mapping uses a “nn.Linear (336 169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions, see Figure 6 in Appendix for details." **This paper is cool because it can improve the simple method's performance.**
If your comparative experiment is OK (missing horizon length of 192 and 336 now) and you prove that you revise the Introduction section. This manuscript's score can be updated to 6, otherwise it should be 4 for incomplete comparison and introduction.
2. Why only the horizon of 96 and 720 are provided in Table 1 in the attached file? Where are the horizon lengths of 192 and 336? Without a complete experiment result, the analysis will not be sound.
1. "We take your kind suggestion and revise the Introduction." Please tell me your revised content.

2. Please add your clarification "**Our main contribution is on extracting better initial features enable more effective downstream mapping. With time-frequency features, very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstance. **The internal mechanisms of FBM variants are provided in Figure 6 in Appendix. Assuming the look-back window  and forecast horizon , FBM-L decomposes the original time series into pieces at different frequency granularities. Thus, our FBM-L mapping uses a “nn.Linear (336  169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions, see Figure 6 in Appendix for details." to the revised Introduction.

3. Please add your clarification "The Fourier transform is to decompose a time series hierarchically to remove noise signals. For example, for a two-week hourly time series, with day effects and noises only, after Fourier transform, day effects fall in the frequency levels that are multiples of , and noises fall in all the other frequency levels, making mapping much easier. However, the existing mapping in the frequency space completely ignores the time dependent frequency information and faces two inconsistency issues. This is because a Fourier basis function is time-dependent when the input length is not divisible by the frequency level (see Figure 1 in the PDF). " to the revised Introduction