MoFo: Empowering Long-term Time Series Forecasting with Periodic Pattern Modeling

Dear Reviewer Q3nE,

Thank you for your kind words. Your feedback and guidance have been instrumental in improving our paper, and we sincerely appreciate your support and constructive suggestions. Next, we will respond to your concerns point by point.

W1. Presentation Typos

We sincerely appreciate your constructive feedback and will thoroughly revise and proof-read the manuscript in the next version to address every point you raised.

W2.1. Padding Strategy

Our padding strategy is already described in detail in Section 3.1, lines 109–119 and formalized in Equation (1) of the manuscript. In fact, Given an input sequence $\mathbf{X} = [\mathbf{x}_1, …, \mathbf{x}_T]^\top$, we start from the current time step $\mathbf{x}_T$ and move past to delineate periods of length $P$. Any prefix that does not form a complete period is left-padded with the leftmost elements of the nearest full period on its right; this yields a padded input length of $\lceil T / P\rceil * P$ and exactly $\lceil T / P\rceil$ periods.

To evaluate the impact of our strategy, we introduce a variant "+ zero padding" that pads the value with zeros and conpare on ETTm1 and ETTm2 datasets. The results are shown below and the best-performing metric for each experiment is bolded.

Our padding strategy consistently outperforms zero padding, demonstrating that re-using the immediately preceding historical pattern enhances generalization rather than introducing redundancy.

W2.2.1. Determine the cycle length

We determine the period length by calculating the distance between the peaks of the autocorrelation coefficients of the time series data on the training set. The specific method for this calculating can be found in Appendix A.2.

W2.2.2. What is $\mathcal{H}$ ?

We apologize for omitting the precise definition. In our notation, $\mathcal{H}$ denotes the Heaviside step function satisfying

We will add this clarification to the next version.

W3.1. Transformer layers in MoFo

MoFo attains best performance with only one Transformer layer. The corresponding ablation is provided in Section 4.6 and Figure 5 of the manuscript, where we systematically vary the number of Transformer blocks. Across all settings, stacking additional layers yields negligible to zero gains, confirming that a single layer already saturates the model’s capacity.

W3.2. Period length of MoFo

Following your suggestion, we designed two variants—"+ Half" and "+ Double"—to investigate MoFo’s sensitivity to the chosen the half and the double period length. As shown the below Table, the results on three diverse datasets with different period reveal that a well-calibrated period is essential: an inappropriate choice disrupts the model’s ability to capture the intrinsic periodicity of the time series, whereas the original setting preserves this structure and retains optimal performance.

We appreciate your guidance—your suggestions have helped us produce a more precise, readable, and broadly accessible manuscript.

W1. More Baselines

Thank you for your valuable suggestion. The FITS [5] model is already included in Table 4 of Appendix E.3. To further address your concern, we have conducted additional experiments comparing FITS [5] and SparseTF [6] on the ETTm1 and ETTm2 datasets; the results are reported below. For clarity, the best performance in each experiment is highlighted in bold. Across all evaluations, MoFo consistently achieves the most competitive results, demonstrating superior performance.

Furthermore, as requested by Reviewer 7GM5, we have added comparisons with the TimesNet [1], WITRAN [2], PGN[3], and TimeKAN[4] models. Please refer to our response to W1 for details.

W2. Why does the time complexity of MoFo remain stable as the input time series becomes longer?

In MoFo, due to the unique patch strategy, the number of patches is equal to the period length and thus remains constant. Each patch contains exactly $\lceil T / P\rceil$ elements, where $T$ is the input sequence length and $P$ is the period length. Since the time complexity depends solely on the number of patches (i.e., the period length), MoFo is not sensitive to increases in the input sequence length.

W3. What is MACs

MACs (Multiply–Accumulate operations) quantify the total number of fused multiply-add operations executed during a single forward pass. This metric serves as a hardware-agnostic proxy for computational cost: fewer MACs generally translate to lower latency and reduced energy consumption on both CPUs and accelerators. By reporting MACs alongside accuracy, we provide a clear, hardware-independent view of the efficiency–performance trade-off offered by each model.

Ref:

[1] Wu, H., Hu, T., Liu, Y., Zhou, H., Wang, J., & Long, M. (2023). Timesnet: Temporal 2d-variation modeling for general time series analysis. In The Eleventh International Conference on Learning Representations.

[2] Jia, Y., Lin, Y., Hao, X., Lin, Y., Guo, S., & Wan, H. (2023). WITRAN: Water-wave Information Transmission and Recurrent Acceleration Network for Long-range Time Series Forecasting. In Thirty-seventh Annual Conference on Neural Information Processing Systems.

[3] Jia, Y., Lin, Y., Yu, J., Wang, S., Liu, T., & Wan, H. (2024). PGN: The RNN's New Successor is Effective for Long-Range Time Series Forecasting. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.

[4] Huang, S., Zhao, Z., Li, C., & Bai, L. (2025). TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting. In The Thirteenth International Conference on Learning Representations.

[5] Xu, Z., Zeng, A., & Xu, Q. (2023). FITS: Modeling time series with $10 k$ parameters. arXiv preprint arXiv:2307.03756.

[6] Lin, S., Lin, W., Wu, W., Chen, H., & Yang, J. (2024, July). SparseTSF: Modeling Long-term Time Series Forecasting with* 1k* Parameters. In International Conference on Machine Learning (pp. 30211-30226). PMLR.

Thank you for your detailed response and the additional experimental results. Regarding the additional experiments and analysis you provided, I appreciate the efforts of the authors in the rebuttal phase, and as a result, I conservatively keep the rating. I will keep tracking the discussion with 7gm5, who provide actionable discussions.

Dear Reviewer XTBU,

Thank you for your insightful feedback during the review phase, and we appreciate your support of our submission and your attention during the discussion phase between us and Reviewer 7gm5.

In response to Reviewer 7gm5's actionable discussions on two points: (1) the specific impact of model layers, and (2) evaluation on larger-scale datasets, we have conducted corresponding experiments and have addressed Reviewer 7gm5's concerns, as detailed in our subsequent response to Reviewer 7gm5 at [https://openreview.net/forum?id=sbvLts2HqR&noteId=fAooVNcIEf].

Thank you once again for your valuable feedback. Your thorough review has been truly encouraging for us.

Best regards,

The Authors of Submission 2224

Thank you very much for your time and effort; they are crucial for improving the quality of our manuscript.

W1. More baselines

We have already compared TimesNet in Table 4. Next, we added comparative experiments with PGN, TimeKAN, and WITRAN. The performance results are shown in the table below.

Fairness statement. Although the TFB-benchmark setting we adopted may differ from your expectations, we would like to clarify that all models are run under the same experimental environment and unified configuration to ensure basic fairness in evaluation. We chose to conduct experiments on the latest TFB mainly because it has been extensively validated and is academically recognized—the benchmark was nominated for the Best Paper Award at VLDB 2024 and has been cited over 62 times on Google Scholar.

W2. Historical look-back length

Please allow us to clarify TFB’s approach regarding historical window selection. According to the TFB GitHub guidelines, multiple look-back window lengths are preset in the experiments, covering typical settings commonly used in time series forecasting tasks. For each model, tests are conducted independently on all candidate historical window lengths, and the best-performing result is ultimately reported.

We believe this approach does not introduce fairness issues; on the contrary, it is fairer to models with varying sensitivities. Forcing all models to use exactly the same look-back length might disadvantage those that are particularly sensitive to this parameter, causing them to be unfairly regarded as “underperforming”. In contrast, as long as the device conditions and the amount of historical data are consistent, allowing each model to flexibly choose its optimal historical window according to its structural characteristics both ensures a fairer evaluation of predictive performance and better reflects the flexible practices found in real-world applications.

Despite having a different perspective, in response to your suggestion, we conducted experiments using several commonly adopted predefined look-back window lengths. Specifically, for forecasting with a future window length of 720, we used {96, 336, 512} as the look-back windows. Due to time constraints, partial experimental results are shown below.

W3. Hyperparameter search

(1) According to the TFB guidelines, the authors follow the hyperparameters specified in the original papers. And they further performed additional hyperparameter searches on 16 configuration sets to achieve a more comprehensive evaluation, which covers the key hyperparameters sufficiently.

(2) TFB has fully open-sourced the necessary environment configuration, including detailed version information, along with a fixed running protocol, enabling users to reproduce the same performance.

(3) Please refer to W1.

W4. Complexity

Our efficiency comparison, as stated in the caption of Table 3, is conducted under the condition that each model achieves its optimal predictive performance—a standard and widely accepted practice in academic research. Since the ultimate goal of time series forecasting is to enhance prediction accuracy, efficiency comparisons that consider performance are both meaningful and practical. For instance, if all models are configured with extreme parameter settings, like compressing the feature dimension to 1, computational efficiency might seem significantly improved, but prediction performance would suffer greatly. Therefore, a fair and informative efficiency evaluation must be conducted at comparable performance levels to ensure both validity and practical significance.

In response to your suggestion, we used the ETTh1 dataset as an example and optimized this configuration for the best performance, while keeping certain hyperparameters such as batch size and learning rate fixed. We report the efficiency for fixed batch sizes (specifically 4 and 128), learning rate 1e-4 and look-back window 512, as well as for each model under its best-performing setting. Please refer to the table below for details (K: Kilo-$10^3$, M: Million-$10^6$). We observe that our model achieves significant performance improvements with only a slight increase in complexity.

Q1.

The Human Connectome Project (HCP) is a large-scale dataset with over 1TB of data, requiring mandatory institutional approval for access. Due to time constraints, we were unable to use and study it during the rebuttal period. To address your concerns, we opted to utilize the Influenza-Like Illness (ILI) dataset for evaluation instead. The ILI dataset, published by the US Centers for Disease Control and Prevention (CDC) from 2002 to 2021, comprises health-related data that record the weekly proportion of ILI patients relative to the total number of visits, without any obvious seasonality. We predict future values for four windows: $L\in$ {24,36,48,60}.

To emphasize our modeling of non-periodic time series, we follow the approach of TimesNet by computing the main pseudo-period as the reciprocal of the dominant frequency obtained via Fast Fourier Transform (FFT) on each input time series. Although MoFo was designed with periodic patterns in mind, the results below demonstrate that it still achieves remarkably high accuracy on non-periodic data, highlighting its strong capability in capturing complex temporal dynamics even in the absence of clear periodicity.

Thank you very much for your response. As the rebuttal period is nearing its end, we have done our utmost to address your concerns and ensure your satisfaction.

1. Model Layers

1.1. Model Layer of MoFo

MoFo attains best performance with only one Transformer layer. The corresponding ablation is provided in Section 4.6 and Figure 5 of the submission, where we systematically vary the number of Transformer blocks. In all configurations, adding more layers does not provide additional performance gains.

1.2. Model Layer of Baselines

To further address your concerns regarding the baseline model layer parameters, we conducted an investigation using DUET, TimeKAN, and TimesNet. Due to time constraints, we were only able to perform hyperparameter experiments on a single dataset, ETTh1, with prediction lengths of 96, 192, and 336. The optimal layer choices for each prediction length are listed in parentheses. The experimental results are as follows:

2. Larger-scale Experiments

To further address concerns about efficiency and performance on large-scale datasets, we conducted experiments on two large-scale time series datasets: XXLTraffic [1] and XTraffic [2]. The long-period dataset XXLTraffic includes 12 years of traffic flow data from 2012 to 2024, while the large-scale dataset XTraffic contains traffic data recorded since 2023 across 16,972 feature channels (nodes), showcasing a significantly larger number of channels.

Due to time constraints, we compared the models you are concerned with (TimeKAN, DLinear, and the current SOTA time series model DUET). The input length was uniformly set to 512, and the output length to 720. The performance results and training efficiency comparisons on two datasets are as follows:

In larger-scale scenarios, MoFo demonstrates the ability to achieve the highest performance while maintaining competitive efficiency. Specifically, for XTraffic dataset with a large number of nodes (channels), TimeKAN and DUBT are too complex to run. Compared to DLinear, MoFo achieves a 22.58% performance improvement on the large-scale dataset. Furthermore, for the long time-span dataset, MoFo achieves competitive performance while maintaining high efficiency.

Ref:

[1] Yin D, Xue H, Prabowo A, et al. Xxltraffic: Expanding and extremely long traffic dataset for ultra-dynamic forecasting challenges[J]. arXiv e-prints, 2024: arXiv: 2406.12693.

[2] Gou X, Li Z, Lan T, et al. Xtraffic: A dataset where traffic meets incidents with explainability and more[J]. arXiv preprint arXiv:2407.11477, 2024.

We sincerely thank you for your precious time and insightful comments. We will now address your concerns point by point below.

W1.

W1.1. Confusion about period

In Figure 1(a), the “Successive Patching” does not represent the coexistence of mixed periods; rather, it includes multiple peaks, which are dynamic components within a single period and together constitute a complete period.

We cannot simply assume that a single peak corresponds to one complete period. For instance, in traffic data, there are distinct morning and evening peak periods; however, the periodicity of traffic is typically considered to be one full day rather than 12 hours. To accurately determine the period length, we employ the autocorrelation function (ACF). As shown in Figure 1(b), the autocorrelation coefficients of the sequence in Figure 1(a) indicate that a truly complete period often contains multiple peaks.

W1.2. Accurate period selection

As stated in Appendix Section A.2 of the manuscript, we utilize three steps to accurately determine periodicity.

Firstly, we identify potential period lengths of repeating patterns within the sequence using the autocorrelation function (ACF). The ACF is a widely used tool in the field of time series analysis for examining periodic patterns [1-3].

Secondly, we verify the existence of these periods through statistical significance testing, such as utilizing a 95% confidence interval, to eliminate any noise interference.

Finally, we also conduct hyperparameter experiments to assess the sensitivity of model performance with respect to different period lengths.

W1.3. Time-varying periods

The electricity dataset (ETT) used in our experiments aligns perfectly with your requirements. The ETT dataset records electricity consumption from two counties in China over a continuous period from July 2016 to July 2018, thus encompassing both winter and summer patterns. However, it is worth noting that the periodic lengths for the two seasons do not differ significantly. For your convenience, we have included the relevant performance results below.

Furthermore, to address your concerns more comprehensively, we have created a synthetic dataset (Mixed-ETT) by systematically combining ETTh1 (with a period of 24) and ETTm1 (with a period of 96). We divided each dataset into four equal parts along the time axis. Then, we alternately sampled each segment to construct the new dataset Mixed-ETT, represented as {ETTh1[:$\frac{1}{4}$], ETTm1[$\frac{1}{4}$:$\frac{1}{2}$], ETTh1[$\frac{1}{2}$:$\frac{3}{4}$], ETTm1[$\frac{3}{4}$:1]}. Consequently, the Mixed-ETT dataset displays a mixed periodic length of (24, 96, 24, 96). We compare the performance of state-of-the-art models, DUET and DLinear, as shown below.

Even under time-varying periods, MoFo retains the competitive performance, underscoring the soundness of its architectural design and its robust capability to capture dynamic temporal patterns.

Q1.

Q1.1. Performance under perturbed period setting

To resolve any remaining concerns, we introduced four variants:

• "+5 %" and "+15 %": the period setting is increased by 5 % and 15 %, respectively.
• "–5 %" and "–15 %": the period setting is decreased by 5 % and 15 %, respectively.

We evaluated these perturbations on three datasets with distinct period lengths; the quantitative results are reported below.

Q1.2. Performance under multiple periods setting

We utilized the traffic dataset, which includes the multiple time intervals you mentioned: 1 day and 7 days. In practice, when dealing with multi-period data, we typically set the shorter period as our baseline configuration. Our experimental results were very impressive, as shown in Table 2 in the manuscript.

To address your concern regarding the performance when integrating multiple MoFo models to handle different periods, we introduced three models. One original model (MoFo-1) excels in sequences with a period length of 24 hours (1 day), another (MoFo-7) is designed for a weekly period of 7 days, and the final ensemble model (Mix-MoFo) learns and predicts in parallel using the two MoFo models, combining their outputs to produce the final prediction. Below are the results from our experiments on the Traffic dataset.

The experimental results indicate that using the shorter period as the configuration allows MoFo to capture more precise periodic patterns, leading to enhanced performance.

Experimental results demonstrate that our straightforward MoFo implementation achieved strong performance. When dealing with multi-period time series, we set the smallest period length as our baseline configuration. The “Mix-MoFo” variant added assumptions regarding additional periods, which increased the risk of overfitting and consequently led to a decline in performance.

Ref:

[1] Wu H, Xu J, Wang J, et al. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting[J]. Advances in neural information processing systems, 2021, 34: 22419-22430.

[2] Lin S, Lin W, Hu X, et al. Cyclenet: Enhancing time series forecasting through modeling periodic patterns[J]. Advances in Neural Information Processing Systems, 2024, 37: 106315-106345.

[3] Lin S, Lin W, Wu W, et al. SparseTSF: Modeling Long-term Time Series Forecasting with* 1k* Parameters[C]//International Conference on Machine Learning. PMLR, 2024: 30211-30226.

Hi Reviewer EWUL,

The authors have submitted their rebuttal. Do you have any further questions or comments based on their response?

Best regards, The AC

Dear Reviewer EWUL,

Thank you very much for your thorough review. If you have any further questions or suggestions, we would be delighted to discuss them with you.

To make our response clearer, we have summarized it as follows:

(1) We would like to clarify that the selection of the period length is determined using the autocorrelation function with Bartlett's Test under the Null Hypothesis, a widely adopted method for periodic analysis in time series data. Further details can be found in Appendix A.2 of the paper.

(2) We designed experiments to evaluate the model's performance in dynamic-, mixed-, and multi-period scenarios. The highly competitive results demonstrate the outstanding performance of our model under these complex conditions.

Additionally, we are happy to share the latest progress of our experiments.

In our response to Reviewer 7GM5, we utilized a 12-year-long time series dataset, XXLTraffic, which potentially contains complex periodic scenarios over such an extended time span. On this dataset, our model also achieved competitive performance. For convenience, we have reproduced the performance comparison below.

Thank you again for your professional and valuable review! Your feedback can significantly enhance the quality of our paper, and we sincerely appreciate it.

Sincerely,

The authors of the paper.

Dear Reviewer EWUL,

We are deeply grateful for your thoughtful and constructive feedback, as well as the time and effort you have dedicated to reviewing our manuscript. Your insightful suggestions have been invaluable in guiding us to enhance the quality of our research.

Once again, we sincerely thank you for your support and expertise, and we truly appreciate your valuable contribution to this work.

With warm regards,

The Authors of Submission 2224