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

We appreciate your constructive suggestions, which will be fully reflected in the final version, including clarifying misreading/misunderstanding. Extra experiment results and the updated Figure 1 are attached in the one-page PDF within the global response for all reviewers.

Weakness (1)

There could be misreading/misunderstanding of the motivation and main contributions of our work. We redraw Figure 1 and revise Introduction as explained in Introduction and Figure 1 of the global response to clarify the workflow and compare our FBM with others. Our contribution is to extract more interpretable features to enable much more effective downstream mapping.

Thus, we disagree with the statement that “some key components are similar to existing works.” Instead, our FBM opens a new perspective for time series forecasting using the FBM with the time-frequency features, making the following main differences:

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

In fact, existing Fourier-based mapping methods (e.g. FEDformer) do not involve Fourier basis mapping like ours. They use real and imaginary parts as inputs to their mapping networks and conduct in frequency space. In contrast, FBM conducts mapping in both time and frequency space.

We provide a new perspective that real and imaginary parts can be interpreted as the coefficients of cosine and sine basis functions at different frequencies, as shown in Eq. (3). However, existing Fourier-based mappings do not involve such basis functions, thus failing to interpret these coefficients correctly. Eq. (4) shows that adding a cosine wave and a sine wave at the same frequency results in a shifted cosine wave of the same frequency. This means that the crucial information is embedded in the amplitude and phase of each cycle rather than in the real and imaginary parts. This leads to issues of inconsistent starting cycles and inconsistent series lengths in existing methods, addressed by our FBM.

In addition, a Fourier basis function is time-dependent when the input length is not divisible by a frequency level (see new Figure 1 in PDF ). Consequently, the mapping in the frequency space fails to capture this time-dependent frequency information. Our FBM addresses these issues. It decomposes an input time series into $\frac{T}{2}+1$ pieces at hierarchical frequency granularity, where T is the input sequence length. We show that time-frequency features enable simplified downstream mapping, and very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can beat complicated Transformers, etc. FBM achieves SOTA performance using either a vanilla linear layer or a three-layer MLP on eight real-world datasets for LTSF and PEMS datasets for STSF.

We sincerely appreciate if you could read our response to Weakness W2 for Reviewer akVH as well, as it is important to understand the unique contribution of time-frequency features.

Weakness (2)

Please refer to new Figure 1 and the updated Introduction in the global response.

As discussed in line 167, we use the normalization and denormalization same as the following reference:

Ref: Reversible Instance Normalization for Accurate Time-Series Forecasting against Distribution Shift.

Specifically, normalization subtracts the average value of the input time series and denormalization adds that back to the forecast time series at the end, as widely used in LTSF and demonstrated effective by the reference.

The Fourier basis expansion operation follows Eq. (3) and aims to solve the issues discussed in Section 3 of the paper.

The internal mechanisms of the three mapping methods are provided in Figure 6 in Appendix. Assuming the look-back window $T=336$ and forecast horizon $L=96$, FBM-L decomposes the time series into $\frac{T}{2}+1$ pieces at different frequency granularities by Eq (3). Thus, our FBM-L mapping uses a “nn.Linear (336 $\times$ 169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions. FBM-NL uses the same structures and hyperparameters for each dataset across different horizons for better reproducibility.

Weakness (3)

Four SOTA baseline methods Pathformer, iTransformer, TimeMixer and TimesNet [1] are compared in Table 2 of the PDF for the global response. Due to time constraints, we only finish LTSF experiments on datasets ETTh1 and ETTh2 but also add extra experiments for STSF on the PEMS dataset. FBM-L and FBM-NL consistently achieve the SOTA performance for both LTSF and STSF without tuning, as shown in the PDF.

Weakness (4)

Many thanks for pointing out the typos, we’ve revised the paper and thoroughly polished the paper for the final version.

Weakness (5)

Figure 1 is updated in the PDF to emphasize the distinctive features of FBM against existing methods, pointing out the issues of previous methods and highlighting our main contribution. All other figures are adjusted for better visualization. In Figure 5, the x-axis represents the frequency level (Hz), and the y-axis represents the amplitude of real and imaginary values. ``5’’ refers to the last value of 175, removed from all figures.

Limitations:

This paper on time series forecasting does not involve negative social impact.

Ref:

[1] TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis

We have thoroughly addressed all comments and produced a new version of the paper and thank you for any further kind suggestions.

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.

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:

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.

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.

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.

We thank you for your constructive suggestions, which will be fully reflected in the final version. Please refer to the new experiment results in the one-page PDF attached to the global response.

Weakness W1 and Question Q1

Table 3 in the PDF shows new short time-series forecasting (STSF) results on the PEMS dataset, but time constrains completing experiments on the M4 dataset (LTSF methods including Pathformer, iTransformer, CrossGNN, FiLM, PatchTST, NLinear and FreTS do not test on M4). FBM variants continuously achieve the SOTA performance for STSF on the PEMS datasets.

Weakness W2 and Question Q2

Yes, the main contribution of our work is on extracting more interpretable time-frequency features enabling more effective downstream mapping. Consequently, we can use very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) to achieve the SOTA.

The internal mechanisms of FBM variants are provided in Figure 6 in Appendix. Assume the look-back window $T=336$ and forecast horizon $L=96$, FBM-L decomposes a time series into $\frac{T}{2}+1$ pieces at different frequency granularities by Eq. (3). Thus, our FBM-L mapping uses a “nn.Linear (336 $\times$ 169, 96)”, and FBM-NL includes two additional nn.Linear layers with activation functions.

1. Performance: Our ablation study evaluates the comparisons between FBM-L vs NLinear and FBM-NP vs PatchTST in Table 1 and FBM-NL vs TimeMixer in Table 2 in the attached PDF, showing the effectiveness of the time-frequency features. The first compares two linear networks, the second compares two PatchTST networks, and the last compares two MLP networks. FBM-L outperforms NLinear on all datasets and forecast horizons. The average MSE and MAE of NLinear are 0.3135 and 0.3395, respectively, which drops to 0.3034 and 0.3297 for FBM-L. For PatchTST, the average MSE and MAE are 0.3079 and 0.3360, respectively, which drops to 0.3058 and 0.3340 for FBM-NL. FBM-NP performs better than PatchTST in most cases. The improvement on PatchTST is less significant because PatchTST primarily considers the time domain. It is worth mentioning that TimeMixer's structure and hyperparameters are well-tuned for each dataset across different forecast horizons, but FBM-NL uses the same structure and hyperparameters all the time for better reproducibility. However, FBM-NL still performs better than TimeMixer in most cases (6 out of 8). With our time-frequency features, our model achieves SOTA performance on all datasets for both LTSF and STSF simply with one or three layers even without tuning.

2. Interpretability: We empirically explain why the time-frequency features are better. The previous frequency mapping suffers from issues of inconsistent starting cycles and series lengths. This is because crucial information is stored in the amplitude and phase of the cycle, validated in Eq. (3) and (4). Additionally, a Fourier basis function is time-dependent when the input length is not divisible by the frequency level (see Figure 1 in the PDF). The mapping in frequency space cannot capture time-frequency relationships. Thus, Section 5.3 visualizes how FBM-L considers time-frequency relationships. Since the datasets Electricity and Traffic are more stable than ETTh1 and ETTh2, the weights 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 and ETTh2 than in Electricity and Traffic, leading to more significant improvement.

Weakness W3 and Question Q3

We revise the paper for better visual showcase. FBM-L and FBM-NL are listed as two independent columns in Table 1, with a new baseline TimeMixer added.

The efficiency analysis is conducted on ETTh1 with the same setting, here is the training speed for one epoch:

FBM-L and FBM-NL rank second and fifth in terms of training speed, respectively, because they just need to train one and three nn.Linear layers, respectively. The memory is $\mathbf{O}(L^2/2)$, but one can easily combine some frequency levels to reduce it. For FBM-NL, we use the same hyperparameters for each dataset across different horizons for better reproducibility, see Figure 6 in Appendix. FBM-NP uses the same optimal hyperparameters as PatchTST.

Weakness W4 and Question W4

Table 2 in the PDF shows the experiment results of new baselines iTransformer, TimeMixer, TimesNet, and Pathformer [1]. Due to time constraints, we cannot finish experiments on N-Hits. However, we can refer to experiment results on the same dataset reported in their paper for comparison. For example, they report 0.401 MSE and 0.413 MAE on ETTm2 (L=720), our FBM-L achieves 0.364 MSE and 0.381 MAE. Table 2 shows our FBM variants continuously achieve the SOTA performance against these new baselines.

Limitation

For the limitation part, there may be some misunderstanding. Our method does consider the monotonic trend effect, as the Fourier basis has time dependent and independent frequency basis, which helps separate the trend and seasonal effects like DLinear. However, how to use the previous trend to predict future trends is still a major challenge, which hasn’t been solved for every method, as real-world data usually don’t have a patternable trend effect.

Ref:

[1] Pathformer: Multi-scale Transformers with Adaptive Pathways for Time Series Forecasting

We have thoroughly addressed all comments and produced a new version of the paper, thank you for any other suggestions.

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.

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 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.

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:

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:

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:

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.

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.

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.

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.

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:

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.

We thank you for your constructive suggestions, which will be fully reflected in the final version. Please also refer to the one-page PDF attached and our reply within the global response to all reviewers for the major changes.

Weakness 1

We take your kind suggestion and revise the Introduction. As this is important and beneficial for all reviewers to better understand the paper, we have put our answer in Introduction and Figure 1 within the global response.

Weakness 2

We update Figure 1 in the PDF file attached to the response to all reviewers, illustrating, comparing, and summarizing existing Fourier-based methods with FBM and emphasis our main contributions.

Weakness 3

You may misread/misunderstand part of our work. 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 $T=336$ and forecast horizon $L=96$, FBM-L decomposes the original time series into $\frac{T}{2}+1$ pieces at different frequency granularities. Thus, our FBM-L mapping uses a “nn.Linear (336 $\times$ 169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions, see Figure 6 in Appendix for details.

The Fourier transform is to decompose a time series hierarchically to remove noise signals. For example, for a two-week hourly time series, $T=336$, with day effects and noises only, after Fourier transform, day effects fall in the frequency levels that are multiples of $14$, 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). The mapping in the frequency space cannot capture the time-frequency relationships. Thus, Section 5.3 visualizes how FBM-L considers the time-frequency relationships. Since the datasets Electricity and Traffic are more stable than ETTh1 and ETTh2, 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 and ETTh2 than in Electricity and Traffic, leading to more significant improvement. We add a noise attack example to further verify the denoising effect: random Gaussian noises are added to ETTh1 with a 0.1 probability at each input time point during training, resulting in the performance below:

Weakness 4

We add experiments for a fairer comparison of FBM-L, FBM-NL, and FBM-NP in the PDF:

1. FBM-L vs NLinear in Table 1
2. FBM-NP vs PatchTST in Table 1
3. FBM-NL vs TimeMixer [1] in Table 2

(1) both with linear networks, (2) both with PatchTST networks, and (3) both with MLP networks.

FBM-L outperforms NLinear on all datasets and forecast horizons. The average MSE and MAE of NLinear are 0.3135 and 0.3395, respectively, which drops to 0.3034 and 0.3297 for FBM-L. For PatchTST, the average MSE and MAE are 0.3079 and 0.3360, respectively, which drops to 0.3058 and 0.3340 for FBM-NL. FBM-NP performs better than PatchTST in most cases. The improvement on PatchTST is less significant because PatchTST primarily considers the time domain. It is worth mentioning that TimeMixer's structure and hyperparameters are well-tuned for each dataset across different forecast horizons, but FBM-NL uses the same structure and hyperparameters all the time for better reproducibility. However, FBM-NL still performs better than TimeMixer in most cases (6 out of 8). These three comparisons demonstrate the effectiveness of the time-frequency features.

Regarding the ablation test in the paper, three layers MLP are sufficient to capture deep time-frequency relationships because FBM-NL always achieves better or the same performance as FBM-NP when non-linear mapping is preferred for a dataset. With our time-frequency features, FBM achieves SOTA performance in almost all circumstances by simply using either a linear layer or a three-layer MLP without tuning.

Question 1

FBM considers time-frequency space mapping rather than frequency space mapping. For internal mechanism and empirical thinking, you can refer to our response to Weakness 3. For the effectiveness of time-frequency features, you can refer to our response to Weakness 4.

Question 2

Please refer to our response to Weakness 4.

Question 3

Please refer to our response to Weakness 1 and the global responses for major changes.

Limitation

See our response to Weakness 4 for a fairer comparison and to Weakness 3 for elaborating the proposed methods.

Ref:

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

We have substantially revised the paper addressing all of your comments, please let us know any other kind suggestions.

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.

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.

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

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.

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.

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.