FreDF: Learning to Forecast in the Frequency Domain

[W1-2] Comparison with DTW-based loss to keep shape information of time series [3,4].

• Investigation.
  • Reference [3] focuses on point forecasting for non-stationary signals. It proposes DILATE, which aligns the shape of the forecast sequence with the label sequence using a differentiable Dynamic Time Warping (DTW) loss
  • Reference [4] focuses on the probabilistic forecast task for non-stationary time series. It proposes STRIPE which captures structured diversity based on shape and temporal features using DTW.
• Discussion.
  • Research problems are different. References [3-4] focuses on forecast tasks for non-stationary signals, specifically targeting the alignment of forecast and label sequence shapes, which improves performance in an intuitive manner. In contrast, FreDF is engineered for general forecasting tasks, specifically targeting the elimination of bias within the learning objective, which improves performance with theoretical guarantees.
  • Implementations are different. While references [3] and [4] rely on modified DTW loss, FreDF utilizes frequency-based loss, which operates independently of DTW loss. Furthermore, FreDF is adaptable to various loss functions, such as Legendre and Chebyshev losses (see Section 4.4), highlighting its flexibility and difference from [3-4].
  • Most importantly, references [3,4] neither recognize nor address the bias introduced by label autocorrelation. FreDF bridges this gap by formally defining the bias (see Theorem 1) and mitigating it with theoretical guarantees. The identification and handling of this bias constitute the core contribution of FreDF, as highlighted in the title Label correlation biases direct time-series forecast.

• Comparison.
  • Setup. To ensure a fair comparison, we integrated the official implementations of the loss functions [3,4] into the iTransformer model within the TSLib framework [7]. We adhered to the forecasting and input length settings specified by TSLib to perform long-term forecasts. The loss strength parameters were fine-tuned on the validation set within the range [0.1, 0.3, 0.5, 0.7, 1].
  • Observation. As shown in the table below, FreDF significantly outperforms the baseline methods [1,2] across both datasets. This result aligns with our expectations and the observations reported in References [3,4]. Specifically, as highlighted in Table 1 and the first paragraph of Reference [3]: DILATE is equivalent to MSE when evaluated on MSE on 3/6 experiments. Indeed, for MLP-like models incorporating DILATE, forecast accuracy measured by MSE decreases. In contrast, FreDF significantly outperforms DTW-based methods across both datasets. This improvement stems from FreDF’s unique ability to debias the learning objective, a capability that DILATE and DPP do not possess.

[W1-3] Comparison with methods utilizing multiresolution trends during training [5,6].

• Investigation.

  • Reference [5] focuses on architecture development. It proposes Timemixer that decomposes multiscale series and successively aggregates the microscopic and macroscopic information.
  • Reference [6] also focuses on architecture development. It proposes Scaleformer that refines a forecasted time series at multiple scales with shared weights.

• Discussion.

  • Research problems are different. References [5-6] targets utilizing the multi-scale information to enhance performance. In contrast, FreDF targets the elimination of bias within the learning objective**.
  • Implementations are different. References [5-6] develop new network architectures. In contrast, FreDF develops new loss function, which supports diverse network architectures (see results in Section 4.4).
  • Most importantly, references [5-6] neither recognize nor address the bias introduced by label autocorrelation. FreDF bridges this gap by formally defining the bias (see Theorem 1) and mitigating it with theoretical guarantees. The identification and handling of this bias constitute the core contribution of FreDF, as highlighted in the title Label correlation biases direct time-series forecast.

• Comparison.
  • Setup. To ensure a fair comparison, we utilized the official repositories of [5-6], downloading and configuring them according to their specified requirements. We modified their temporal MSE loss with the proposed loss in the FreDF. We processed the datasets using the unified protocol of TSLib [7] and adjusted the forecasting horizon lengths to perform long-term forecasts. The loss strength parameters were fine-tuned on the validation set within the range [0.1, 0.3, 0.5, 0.7, 1].
  • Observation. As shown in the table below, FreDF significantly enhances the performance of the architectures [5,6], demonstrating FreDF’s ability to support and improve existing models. These improvements underscore the independent and complementary nature of FreDF’s contributions.

Revision actions.

• We have discussed reference [5-6] in the lines 91-93, and explained the distinction of FreDF from [1-4] in the lines 158-161.
• We have involved the empirical comparison results with [3-4] in Table 11 to highlight the advantage of FreDF.
• We have involved the empirical comparison results with [5-6] in Table 12 to highlight the generality of FreDF.

Reference

[1] Henning Lange, et al. From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction. JMLR 2021.

[2] Xinyu Yuan and Yan Qiao. Diffusion-TS: Interpretable Diffusion for General Time Series Generation. In ICLR 2024

[3] Vincent Le Guen and Nicolas Thome. Shape and Time Distortion Loss for Training Deep Time Series Forecasting Models. In NeurIPS 2019.

[4] Vincent Le Guen and Nicolas Thome. Probabilistic time series forecasting with shape and temporal diversity. In NeurIPS 2020.

[5] Shiyu Wang, et al. TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting. In ICLR 2024.

[6] Amin Shabani, et al. Scaleformer: Iterative Multi-scale Refining Transformers for Time Series Forecasting. In ICLR 2023.

[7] Wang, Yuxuan, et al. "Deep time series models: A comprehensive survey and benchmark." arXiv preprint arXiv:2407.13278 (2024).

[8] Chen, Zhichao, et al. "Rethinking the Diffusion Models for Missing Data Imputation: A Gradient Flow Perspective." In NeurIPS 2024.

[W1-1] Comparison with methods incorporating Fourier transformation in loss functions [1,2].

• Investigation.
  • Reference [1] achieves time series forecasting by identifying the parameters in a (linear) dynamic system model. FFT is used to accelerate the computation of a loss function tailored for dynamic systems. This approach is analogous to using Fourier analysis for solving ordinary differential equations (ODEs).
  • Reference [2] focuses on the time series generation task based on diffusion models. A frequency loss is used to recover the original series $x_0$ to enhance interpretability and accuracy of the generated time series.
• Discussion.

  • Research problems are different. Fourier-DS [1] seeks to enhance the efficiency of training dynamic system models for time series forecasting by leveraging FFT to accelerate loss computations. Diffusion-TS [2] aims to advance diffusion models for conditional time-series generation by utilizing FFT-based frequency loss to improve interpretability and reconstruction fidelity. In contrast, FreDF is specifically designed for forecasting tasks, employing FFT to eliminate the bias resulting from label autocorrelation, which is a new research problem.

  • Implementations are different. Fourier-DS requires the underlying model to be a dynamic system that performs recurrent inference for multi-step forecasts. Diffusion-TS relies on diffusion models, requiring training with score matching and inference through recurrent sampling. In contrast, FreDF is model-agnostic, allowing seamless integration with a variety of forecasting models. Additionally, it generates multi-step predictions simultaneously, thereby avoiding the recurrent generation processes in [1,2].

  • Most importantly, references [3,4] neither recognize nor address the bias introduced by label autocorrelation. FreDF bridges this gap by formally defining the bias (see Theorem 1) and mitigating it with theoretical guarantees. The identification and handling of this bias constitute the core contribution of FreDF, as highlighted in the title Label correlation biases direct time-series forecast.

• Comparison.
  • Setup. To ensure a fair comparison, we utilized the official repositories of [1,2], downloading and configuring them according to their specified requirements. We processed the datasets using the unified protocol of TSLib [7] and adjusted the forecasting horizon lengths to perform long-term forecasts.
  • Observation. As presented in the table below, FreDF significantly outperforms the baselines [1,2] across both datasets. This outcome aligns with our expectations based on prior discussions. Fourier-DS [1] is restricted to a (linear) dynamic system model, limiting its ability to capture complex temporal patterns compared to advanced deep models like iTransformer, which leads to suboptimal performance. Diffusion-TS [2] is not specifically designed for time-series forecasting; although it can be adapted for forecasting via conditional generation, the inherent diversity of diffusion model outputs prove to adversely affect accuracy metrics [8,9]. In contrast, FreDF supports modern forecasting models without relying on generative paradigms, effectively debiasing the learning objective and achieving superior performance.

Thank you for taking the time to review our work and for appreciating our clarifications.

We highly value your comments and have invested six working days to thoroughly investigate, discuss, and compare the related works [1-6]. This effort involved familiarizing ourselves with various environments and debugging the corresponding codes. We appreciate your understanding regarding our delayed response.

The conclusion is that FreDF is distinctly differentiated from these works in multiple aspects, making unique contributions.

[W1] The proposed approach lacks novelty. Measuring time series structure in the frequency domain has been well-explored. Several prior works have investigated capturing time series characteristics or enhancing robustness in loss functions. These approaches are all relevant to this study. However, this paper does not adequately investigate, discuss, or compare these related methods.

Response. Thank you for your critical comment on related works. To clearly address your concerns, we have structured our response as follows, aligning with your demand of investigate, discuss, and compare.

Dear Reviewer 6T6T,

As the discussion deadline approaches, we would like to inquire whether our responses have adequately addressed your concerns. As the remaining opposing reviewer, your feedback is invaluable to us. During the rebuttal stage, our team has dedicated six working days to conduct additional experiments comparing all the referenced methods.

To date, we have thoroughly investigated, discussed, and compared these methods in both our response letter and the revised manuscript.

Should you have any further comments or questions, we would greatly appreciate the opportunity to address them.

Thank you in advance,

13602 Authors

The author has thoroughly addressed the distinctions between the relevant methods discussed and effectively emphasized their contributions. I am satisfied with the revisions made to the manuscript and have updated my rating accordingly.

Results with random seeds on the ETTh1 dataset.

Results with random seeds on the Weather dataset.

[W4] The short-term forecasting task results in the appendix report only qualitative time spans. It would be good to specify the results in terms of forecasting lengths, similar to the Table 1 for long-term forecasting.

Response. Thank you very much for your kind suggestion in this regard.

• In our experiments, each qualitative term corresponds to specific subsets within the M4 dataset, each with distinct input and forecasting lengths as detailed in the table below. These configurations follow the standard setups defined by the Time Series Library [1].

• Since in the standard protocol [1], different trials in short-time forecast experiment indicates different subset of dataset, input length and label length, we used the qualitative term to specify the results, in order to encapsulate these variations concisely and align with the standard setting.

[1] Wang, Yuxuan, et al. "Deep time series models: A comprehensive survey and benchmark." arXiv preprint arXiv:2407.13278 (2024).

[Q1] Are there any settings where defining the errors by transferring the predictions to the frequency domain is detrimental to the prediction accuracy?

Response. Yes, there are specific scenarios where FFT can adversely affect prediction accuracy, particularly when dealing with irregular time series (although it is not the focus of this paper).

• Irregular time series are characterized by non-uniform sampling intervals (e.g., observations recorded at varying frequencies such as days versus hours). FFT inherently assumes uniform sampling intervals, which means it cannot accurately capture the underlying frequency components of irregularly sampled data. Applying FFT directly to such time series disregards their temporal irregularities, leading to distorted frequency representations and, consequently, suboptimal prediction performance.
• To address this limitation, a promising approach is extending FreDF by involving the Non-uniform Discrete Fourier Transform (NDFT). Unlike FFT, NDFT is designed to handle data sampled at irregular intervals. The NDFT is defined as: $X_k = \sum_{n}^{N-1} x_n e^{-2\pi i p_n f_k}$, where $k$ (ranging from $0$ to $N-1$ ) denotes the frequency index, $p_0, \dots, p_{N-1}$ are the scaled time stamps of the samples, and $f_0, \dots, f_{N-1}$ are the corresponding frequencies. By leveraging NDFT, we can compute spectral distances for irregular time series, thereby extending FreDF to irregular settings.
• We acknowledge that handling irregular time series is a limitation of the current FreDF implementation and represents a plausible direction for future efforts. It has been introduced in the final paragraph of our conclusion. Thank you again for your fruitful advice.

Thank you very much for your positive comments and appreciation of our novelty, generality and empirical performance. Below are our responses to the specific concerns and queries.

[W1] Limited analysis of a bias in Theorem 1. Theorem 1 considers only univariate sequences, cross-correlation terms are not accounted for in the Bias formula.

Response. We understand that Theorem 1 can be extended to multivariate cases. We have made theoretical and empirical efforts as follows.

• Firstly, we have added Corollary B.3 that extends Theorem 1 to multivariate case. It immediately follows from Theorem 1 by denoting the multivariate label sequence $Z$ as an augmented univariate sequence. It demonstrates that the label correlation between different time steps and covariates leads to bias. The theorem is formulated as follows.

Given a input sequence $L$ and a multivariate label sequence $Y\in\mathbb{R}^\mathrm{T\times D}$, suppose $Z\in\mathbb{R}^\mathrm{T\times D}$ be the flattened version of $Y$ obtained by concatenating the rows, the learning objective of the DF paradigm is biased against the practical NLL, expressed as:

where $\hat{Z}\_i$ indicates the prediction of $Z\_i$, $\rho\_{ij}$ denotes the partial correlation between $Z\_i$ and $Z\_j$, $\rho\_i^2 = \sum\_{j=1}\^{i-1} \rho\_{ij}^2.$

• Secondly, to counteract the bias of cross-correlation terms, we perform FFT along the covariate dimension (FreDF-D). The results and observations are presented below.
  • In general, FreDF-D brings similar performance gain with FreDF, which showcases the efficacy of FreDF-D to counteract the bias of cross-correlation terms. In particular, FreDF-T slightly outperforms FreDF-D, which underscores the relative importance of auto-correlation in the label sequence. Finally,
  • Notably, a strategic approach is viewing the multivariate sequence as an image, performing 2-dimensional FFT on both time and covariate axes (FreDF-2), which accommodates the correlations between both time steps and covariates simultaneously and further improves performance.

[W2] Notational inconsistency and formula errors in the paper: The use of 'L' is overloaded; Y and Y_hat in Eq.(1) should be compared on the same indexes; Definition 3.2, uses 'j' instead of 'i'.

Response. Thank you very much for your meticulous comment sincerely! We have revised these typological errors in the revised manuscript.

• We have replaced $\mathrm{L}$ with $\mathrm{H}$ to denote the input length
• We have revised Eq.(1) as $\mathcal{L}^{(\mathrm{tmp})} := \sum_{t=1}^\mathrm{T} || Y_t - \hat{Y}_t ||_2^2.$
• We have revised Definition 3.2, as well as Theorem B.4, where we use $j$ to denote imaginary unit.

[W3] The main results in the paper do not report performance over distinct random seeds.

Response. We acknowledge the importance of evaluating performance variability across different random seeds.

• Additional experiment. We report the mean and standard deviation of the results by conducting experiments using 5 random seeds (2020, 2021, 2022, 2023, 2024). We investigate (1) iTransformer, which is the best baseline model; (2) FreDF, which is applied to refine iTransformer.
• Result analysis. The results indicate minimal performance variation, with standard deviations below 0.005 in 7 out of 8 Avg cases. This showcases the insensitivity of models in time series forecast to different random seed specifications.
• Revision. We have added a new section (see Appendix E.3) that include these analysis and results.

Thank you very much for reviewing our response and for your dedication in adjusting the evaluation score! We truly appreciate your increased support.

[Q3,W3] Applicability of frequency domain features: How effective is the FreDF method for data with unclear frequency domain features? Are there any relevant experimental results or analysis?

Response. Thank you for your meticulous comment regarding the effectiveness of FreDF in cases where frequency features are unclear. Our analysis and experiments demonstrate that FreDF maintains effective even under these conditions.

• FreDF is fundamentally designed to mitigate label correlation by utilizing the FFT as a mathematical tool. This approach is agnostic to the clarity of inherent frequency features within the data, as demonstrated in Theorem 3.3. Consequently, FreDF effectively improves forecast performance by reducing label correlations regardless of whether the frequency components are clear.
• To empirically evaluate FreDF’s efficacy under these conditions, we introduced a variant named FreDF-D, which applies FFT along the variable dimension instead of the time dimension. The frequency features are unclear when performing FFT along the variable dimension. The results are shown in the table below.

• The results indicate that FreDF-D bring comparable performance gain with FreDF, demonstrating FreDF’s capability to handle label correlations even when frequency features are not explicitly clear. By the way, FreDF slightly outperforms FreDF-D, underscoring the importance of temporal auto-correlation in label sequences, which is exactly the focus of this paper.

[Q4] Model interpretability: Does the FreDF method have interpretability in the frequency domain? How do users understand and interpret these predicted results?

Response. Thank you for your insightful comment on interpretability. Yes, FreDF offers enhanced interpretability compared to traditional DF with temporal losses. We are pleased to communicate the key points with you as follows.

• Interpretability of labels and predictions. Transforming time series data into the frequency domain facilitates a clearer understanding of complex patterns. For instance:

  • A Linear Frequency Modulation (LFM) signal in the frequency domain appears as a linear function, explicitly illustrating the modulation process over time. This clear representation aids in comprehending the underlying signal dynamics.
  • Consider respiratory sounds from healthy individuals versus patients with pneumonia. In the time domain, both may appear as noisy sequences, making differentiation challenging. However, in the frequency domain, pneumonia-induced respiratory sounds exhibit increased energy at specific frequencies, enabling easier identification and interpretation of pathological conditions.

• Interpretability of loss components. Analyzing forecast errors across different frequency components provides valuable insights into the model’s learning process and guides the refinement of training strategies. For instance:

  • If the model exhibits high errors in low-frequency components, it indicates that the model fails to learn basic, slowly changing semantics in the dataset. In this case, we could filter out high-frequency noise, or increase the loss weight for low-frequency components, to prioritize the learning of low-frequency components.
  • If the model exhibits high errors in high-frequency components, while low-frequency components are accurately predicted, it suggests that the model has captured the fundamental semantics but struggles with finer details. In this case, we could increase the loss weight for high-frequency components to encourage the model to learn more detailed patterns.

• In conclusion, FreDF indeed has advantage of interpretability which enable users to understand dataset property and diagnose model performance. Nevertheless, in this work we contemporally focus on its efficacy to improve model performance by diminishing the bias from label correlation.

Thank you very much for your positive comments and appreciation of our novelty, generality and empirical performance. Below are our responses to the specific concerns and queries.

We observed that some comments in the weaknesses and questions sections convey similar points. We will selectively merge them to make the response concise and easy to follow.

[Q1,W1] Computational complexity of frequency domain conversion: Will the frequency domain conversion mentioned in the paper significantly increase computational complexity? How efficient is the FreDF method when dealing with large-scale datasets?

Response. Thank you for your insightful question. We acknowledge the importance of investigating the complexity of FreDF, typically the FFT for domain conversion. Since FFT is executed for each label sequence, it is the label sequence length, denoted as $T$, that mainly impacts the executive efficiency. Therefore, our discussion focuses on the complexity of FFT for varying values of $\mathrm{T}$.

• In the training stage, FFT operates with a computational complexity of $\mathcal{O}(\mathrm{T}\log \mathrm{T})$, ensuring scalability through its logarithmic scaling. To validate this, we measured FFT processing times across various sequence lengths. As shown in Table 1, the FFT operation introduces minimal overhead, with the maximum observed time being 0.076 ms for $T=720$. Therefore, FFT does not increase the computational complexity significantly of model training.

• In the inference stage, the frequency loss computation is not required, eliminating any additional computational overhead associated with FFT. Consequently, the FFT does not increase any computational complexity of model inference.
• In conclusion, the complexity of FreDF is ignorable in both training and inference stages. It ensures that FreDF remains scalable for long label sequence and large datasets. We have added these discussions in the revised Appendix E.2.

[Q2,W2] Generalization ability: How does the FreDF method generalize across different domains and datasets? Has it been tested and validated in more practical application scenarios?

Response. We appreciate the reviewer’s emphasis on the generalization capabilities of FreDF. To validate FreDF across diverse application domains, we undertook the following efforts.

• Firstly, we tested FreDF on nine datasets spanning various domains. The consistent performance of FreDF across these datasets demonstrates its applicability to different real-world scenarios.

• Secondly, to further validate FreDF’s generality, we have added experiments on the SRU dataset from the chemical process engineering domain. This dataset comprises monitoring logs from a Sulfur Recovery Unit, featuring complex correlations among multiple sensor readings over time. FreDF significantly outperformed baseline methods on this challenging dataset, underscoring its effectiveness in practical, high-stakes environments.

Dear Reviewer 3nNT,

We sincerely appreciate the time and effort you have dedicated to reviewing our manuscript. We are delighted that you recognize the significance of our work in identifying and eliminating bias, find our method novel and interesting, and consider our empirical results promising. In response to your valuable feedback, we have undertaken the following actions:

As the discussion draws to a close, we are wondering whether our responses have properly addressed your concerns? Your feedback would be extremely helpful to us. If you have further comments or questions, we hope for the opportunity to respond to them.

Many thanks,

13602 Authors

[Q2] According to the ablation study, the frequency term seems to be almost all that's needed. Is it correct?

Response. Thank you for your meticulous observation. Your point is valid, and we have also observed that the fusing loss $\mathcal{L}^\alpha$ produces marginal improvements over $\mathcal{L}^\mathrm{feq}$.

• This phenomenon, in our view, reinforces the strength of FreDF: the ability to achieve most performance gains, without intricate tuning of $\alpha$ (directly setting $\alpha=1$), makes FreDF easy to use in practice.
• The role of the temporal loss is two-fold. Firstly, the improvement from $\mathcal{L}^\mathrm{feq}$ to $\mathcal{L}^\alpha$ is sometimes not ignorable. For instance, an MAE improvement of 0.003 is observed on Weather, which is comparable to the gap between TiDE (2023) and iTransformer (2024). Secondly, the parameter $\alpha$ allows for adjusting the contribution from each loss flexibly, which makes it feasible to thoroughly evaluate the efficacy of the frequency loss by analyzing the forecast performance across different $\alpha$ values. The results show that as we incorporate $\alpha$ and moderately increase it, the forecast performance improves, which convincingly supports the efficacy of the proposed frequency loss.

[Q3] During my attempt to reproduce the experiments, I encountered an error.

Response. We sincerely apologize for this oversight. This error arises because we did not define self.seasonal_patterns before _build_model. We have corrected this issue in the updated repository. Additionally, we have conducted comprehensive tests on multiple devices to ensure the robustness of the revised codebase. We appreciate the reviewer for identifying this problem and regret any inconvenience it may have caused.

Thank you very much for your encouraging support and appreciation of our novelty, presentation, methodology and empirical results. Below are our responses to the specific query raised.

[W1] It seems that this work aims to penalize errors in a space that decorrelates labels.

Response. Thank you for your insightful question. We agree that exploring other transformations can enhance the technical quality of this manuscript.

• To investigate the generality with different transformations, motivated by the fact that FFT can be viewed as projections onto Fourier polynomials, we extended the implementation of FreDF by replacing FFT with projections onto other established polynomials (Legendre, Chebyshev, Laguerre). Each of these polynomials is adept at capturing specific temporal patterns such as trends (Legendre polynomial) and periodicity (Fourier polynomial).
• The results are summarized in Figure 5, where we observed that Legendre and Fourier bases demonstrate superior performance. This superiority is attributed to the orthogonality between polynomials, a feature not guaranteed by others, as analyzed in Appendix C. It underscores the importance of orthogonality when selecting polynomials for implementing FreDF.

[W2] While the introduction discusses various forecasting models, it could benefit from a stronger focus on established forecasting paradigms (e.g., direct forecasting, iterative forecasting).

Response. Thank you for your valuable advice. We agree that forecasting models are orthogonal to the contribution of FreDF. Nevertheless, we believe that including a discussion of these models in the introduction is crucial for two reasons. Firstly, the development of forecast models (iTransformer, DLinear, etc.) is indeed a recent focus in time series modeling. Introducing them could engage readers in this field and contextualize our work. Secondly, after introducing forecast models, we can directly clarify the distinct contribution of FreDF in addressing label correlation, independent of the input correlation managed by forecast models.

[W3] The source code should be refined.

Response. We are pleased that you are willing to read our code and reproduce our results.

• Firstly, we clarify that all experimental scripts are located in ./scripts/. Typically, the main results of FreDF can be reproduced by executing the bash script in ./scripts/fredf_exp. For example, you can reproduce the long-term forecast, short-term forecast and imputation results by executing:

  # long-term forecast

  bash ./scripts/fredf_exp/ltf_overall/ETTh1_script/iTransformer.sh

  # short-term forecast

  bash ./scripts/fredf_exp/stf_overall/FreTS_M4.sh

  # imputation

  bash ./scripts/fredf_exp/imp_autoencoder/ETTh1_script/iTransformer.sh

• Secondly, we agree that a file for constructing environments could be important for reproduction. We have uploaded a requirement.txt file in the updated repository, which encapsulates all packages and dependencies in our environment.

• Action. We have updated the README.md file to include these examples, which should help users understand how to navigate the code. Furthermore, we have uploaded a requirements.txt file for environment configuration. We hope these efforts will assist you in reproducing our results.

[Q1] In the experiments, how to determine the hyperparameters of FreDF and baselines?

Response. Thank you very much for your kind query. The implementation of FreDF is built upon the repository Time Series Library (TSLib). Therefore, in our experiments, we adhere to the hyperparameters suggested by TSLib for the baseline methods. For FreDF, we fix the hyperparameters associated with the forecast models, such as the iTransformer and TimesNet, and only fine-tune the frequency loss strength $\alpha$ and the learning rate conservatively on the validation set. This approach allows us to isolate the contribution of FreDF without introducing gains from tuning the forecast models, thereby making the empirical advantages of FreDF persuasive.

Dear ErxY,

We are pleased that our responses address your concerns and help your verification of our empirical results.

Your clarified suggestion is valuable and we promise conducting the corresponding experiments.

Big thanks!

13062 Authors

We express our sincere appreciation to the helpful comments from reviewers and the area chair. Currently, the camera-ready of this work has been available. In preparing the final camera-ready version, we have meticulously incorporated all the changes suggested during the rebuttal phase and have undertaken additional revisions to enhance the presentation of our manuscript.

• In alignment with the insightful suggestion from the area chair in the metareview, we have revised the title to "FreDF: Learning to Forecast in the Frequency Domain" to better reflect the core contribution of our work.
• We have carefully addressed typographical errors and refined the narrative to improve readability. Additionally, we have adjusted the overall layout of the paper to ensure a more coherent and logical flow of ideas.
• Recognizing that a lengthy appendix attached to the main text might hinder readers from efficiently locating relevant information, we have reorganized the appendix into two distinct parts. The components deemed less critical for core understanding (but necessary for comprehensiveness) of this work have been moved to the supplementary appendix, to make the main text focused.

Once again, we sincerely thank the reviewers and the area chair for their comments and guidance, which has been instrumental in shaping this work.