We sincerely appreciate the comprehensive review, detailed feedback, and valuable suggestions from the reviewer Bhsw.

Q1: The major issue with this paper is the lack of analysis and comparison with significant literature. Please compare your method with SegRNN and clarify your differences and advantages.

(1) Regarding information propagation paths: Shorter information propagation paths lead to less information loss, better captured dependencies, and lower training difficulty. In this regard, PGN can achieve an information propagation path of $O(1)$, while SegRNN, although capable of segment-wise iteration, still maintains an information propagation path of $O(L/W)$, where $W$ represents the number of segments. This is where PGN holds a greater advantage.

(2) In terms of time series forecasting design: Although SegRNN adopts parallel prediction in the decoder stage, successfully addressing the error accumulation issue of iterative RNN predictions, our approach, TPGN, draws inspiration from Informer's design, omitting the need for a decoder module. This not only resolves the error accumulation problem but also results in a more concise model design.

(3) Direct extraction of periodic semantic information: TPGN can directly extract semantic information related to the periodicity of time series data, while SegRNN segments the time series first and then feeds the representation of each segment into GRU units, which hinders direct capture of the periodicity within the time series.

The segmented iteration and parallel prediction of SegRNN are indeed thought-provoking. We have conducted preliminary experiments and observed that TPGN (PGN) has shown some improvement compared to SegRNN. In the revised version, we will incorporate the above analysis.

Q2: In Section 2, you should distinguish between Linear-based and MLP-based methods. Methods like DLinear and FITS should be classified as Linear-based methods.

Thanks again for your valuable suggestion. We will conduct a full check and modification in the revised version.

Q3: The description of HIE is unclear: (i) Figure 2(a) conflicts with Equation 1 in the paper, where H = HIE(Padding(X)), and the actual code. It is recommended to modify Figure 2 to make this clearer. (ii) Line 169 describes that “HIE(·) is a linear layer,” but in practice, the behavior of HIE is more like CNN (or TCN) rather than a sorely linear mapping. Given (ii), calling the proposed method RNN-based is debatable since it is more likely TCN-based.

(i) Thank you for pointing out the issue. We have made modifications to Figure 2 (a), and the new figure is shown as Figure A in the PDF file (newly submitted).

(ii) Thanks again for your detailed review. In fact, HIE is indeed a linear layer. However, during our implementation, we were inspired by the design of methods like Transformer and iTransformer for the FFN part, and we also used conv1D to replace Linear.

(iii) The function of the HIE layer is to replace the recurrent structure of RNN. In terms of specific code implementation, whether using conv1D or Linear, it does not affect its function of linear mapping. Additionally, the linear operation of HIE is just a part of the structure in PGN. Following HIE, there is the Gated Mechanism, and together they form the entirety of PGN. Therefore, PGN can be considered as the new successor to RNN-based methods.

Q4: You should include an ablation analysis of the normalization layer, explaining its impact on TPGN achieving state-of-the-art results.

Okay, since the normalization layer was not used in the ECL, ETTh2, and WTH datasets, we focused on conducting ablation experiments on the normalization layer for the Traffic and ETTh1 datasets. Due to space limitations, the experimental results have been placed in Table A of the PDF file. For details, please refer to Sec. A of the Global Rebuttal.

We observed a noticeable decrease in performance when the normalization layer was removed from the two datasets. This drop can be attributed to the inherent characteristics of the datasets. In Table 6, we calculated the distribution (Mean and STD) of the training and validation set for each dataset. The results indicate significant differences in the distributions of the Traffic dataset and the ETTh1 dataset. These differences can introduce disturbances to the model, hence the $norm$ should be set to 1 in this case. More details can be found in Appendix H.

Q5: It does not include the hyperparameter settings required to reproduce the key results in the paper. Are the hyperparameters in the main results all defaults? For instance, is TPGN_period=24? If not, providing a complete script file that can be run directly is necessary.

For the 'TPGN_period', we were inspired by WITRAN and aimed to partition time series based on natural period. Therefore, for each task in the five datasets, 'TPGN_period' was set to 24. The selection of $norm$ varied depending on the dataset, as detailed in Table 6. The value of $dm$ for each forecasting task was determined through the validation set, with specific details available in Appendix D. Due to space limitations, we have included the hyperparameters of our model for each task in Table B of the PDF file for result reproducibility, please refer to Sec. B of the Global Rebuttal for more details. Furthermore, we will also enhance the files in the source code moving forward.

We extend our sincere appreciation to Reviewer t9pC for providing valuable feedback and acknowledgment of our research.

Q1: For tables with a large amount of content, such as Table 1, it may be beneficial to consider using different colors for highlighting, as it could enhance clarity. Additionally, another option to consider is moving some of the experimental results to an appendix.

Thank you for your suggestion. In the revised version, we will enhance the table presentation by utilizing color changes, bolding, and other methods. Additionally, we will thoroughly check and optimize the entire paper.

Q2: While TPGN exhibits some advantages in terms of efficiency, I have noticed that it still appears to be challenging to reach the optimal level. Specifically, I have noticed that as the input sequence size increases, the efficiency of TPGN may gradually become inferior to that of iTransformer.

We have noticed that in terms of efficiency, linear-based methods such as FITS do indeed perform better. And for iTransformer, since it maps the entire sequence into a patch for processing the time dimension, while TPGN requires long-term and short-term information extraction branches in the time dimension, the efficiency of TPGN may slightly lag behind iTransformer when the input sequence length varies. However, TPGN still holds significant advantages as it can maintain good efficiency while ensuring optimal performance.

Q3: Why was the Gated Mechanism designed in PGN this way in Figure 2 (a)? Can this part be replaced with GRU or other RNN variants?

In our design, only one gate is needed for information selection and fusion, which incurs smaller overhead compared to GRU's two gates and LSTM's three gates. While PGN, as a general architecture, can have its gated mechanism replaced with GRU and other RNN variants, this substitution would introduce higher costs. In terms of performance, we have observed that despite having only one gate, PGN still outperforms GRU or LSTM, more details of the experiments can be found in Table 2.

Q4: Is it necessary to have two Linear layers in the design of the short-term branch in TPGN?

Yes, because in the short-term branch, the semantic information extracted by the two linear layers carries different meanings. The first layer primarily aggregates short-term information, extracting localized information. The second layer further aggregates the representation from the first layer at a higher level, extracting global information. While it is possible to extract global semantic information using a single linear layer, it would overlook the extraction of information at different granularity levels. Additionally, in terms of efficiency, the theoretical complexity of using only a single linear layer is $O(L)$, which is larger than the current $O(\sqrt{L})$.

We express our sincere gratitude to Reviewer fqFX for providing comprehensive review, insightful perspectives, and thought-provoking questions.

Q1: I think the total amount of computation done in the PGN should be O(L^2) ... Can the authors kindly address this issue?

Sincerest gratitude for your detailed comments. Here we would like to provide more explanations about our model:

Firstly, regarding the complexity analysis:

(1) The computational complexity related to sequences in PGN mainly focuses on the HIE layer. In our specific implementation, we were inspired by the implementation details of methods like Transformer/iTransformer for the FFN part and used conv1D to replace Linear. Indeed, our actual convolution kernel (lookback window) size is set to $L-1$. However, typically in CNN computations, the convolution kernel size is considered a hyperparameter. For example, the convolution kernel in MICN[1] is actually linearly related to $L$, but the theoretical complexity of MICN remains $O(L)$. Therefore, when calculating the theoretical complexity in our case, we followed a similar approach.

(2) The HIE layer can still be implemented using Linear. In this scenario, due to parameter sharing, the way each time step processes information from its previous $L-1$ time steps is exactly the same. Hence, we can follow the approach of WITRAN[2] in handling slices by expanding the dimensions that need to be processed in parallel and merging them into the sample dimension. In this case, Linear reduces $L-1$ dimensions to 1, so its theoretical complexity remains $O(L)$.

(3) In our time series forecasting task, the lookback window size of PGN is set to $L-1$ to fully capture long-term information from the input signal. However, since PGN is a general paradigm, when applied to other tasks, the lookback window can be fixed to a certain value or adjusted. In such cases, regardless of whether implementation (1) or (2) is used, the theoretical complexity remains $O(L)$ or lower.

(4) For self-attention, its computation between sequences is entirely different from PGN because it requires matrix operations to obtain information between every two time steps. Therefore, the necessary $O(L^2)$ theoretical complexity cannot be reduced using the aforementioned ways. From this, we can see that the essence of PGN is fundamentally different from self-attention.

Secondly, In terms of innovation, we have focused on proposing PGN as the successor to RNN to address the issues stemming from RNN's excessive reliance on recurrent information propagation and overly long information pathways, including: (a) difficulty in capturing long-term dependencies in signals; (b) gradient vanishing/exploding; (c) low efficiency in serial computation.

(1) Analysis of information propagation paths: PGN, through the HIE layer, can capture historical information at each time step with an $O(1)$ information propagation path, followed by information selection and fusion through Gated Mechanism, thereby altering RNN's $O(L)$ information propagation path. A shorter information propagation path lead to less information loss, better captured dependencies, and lower training difficulty. Consequently, PGN can more effectively capture long-term dependencies of inputs signals. And, as the information path shortens, it naturally resolves the issue of gradient vanishing/exploding.

(2) Regarding efficiency, the thoughtful design of PGN allows for shared parameters in HIE, enabling information from subsequent time steps to be computed in parallel without waiting for the completion of computations from preceding time steps. This approach achieves good efficiency and resolves the inefficiency in serial computation of RNN.

In conclusion, through its innovative approach, PGN addresses all the limitations of RNN. Therefore, we claimed that PGN can be considered as the new successor to RNN.

Refs [1][2] details are in Sec. G of Global Rebuttal.

Q2: The above discussion can also be tested in the Efficiency of Execution experiment.

Okay, thank you for your suggestion. We replaced PGN with self-attention and conducted efficiency experiments. Due to space constraints, we have placed the results in newly submitted PDF file. More details please refer to Sec. C of the Global Rebuttal.

Q3: Why did the authors choose to fixate on input length of 168 instead of testing some other options?

This is because, on the one hand, considering factors such as the load capacity of training devices and data collection, forecasting longer futures with shorter sequence lengths is an important research problem, which has been the focus of much previous works, such as Informer, MICN, etc. On the other hand, in order for the model to have sufficient historical information for prediction from a limited sequence length, we aim for sequence length could be longer. Therefore, we chose 168 historical time steps (7 days) to predict the future (7 days, 14 days, 30 days, 60 days) as the forecasting tasks. This setting satisfies both considerations and serves as the basis for evaluating the performance of all models. Additionally, we also conducted experiments to demonstrate the reasonableness of our second consideration. Due to space constraints, more details can be found in Sec. D in the Global Rebuttal.

Q4: In the ablation test, some datasets ... Can the authors provide some analysis on these cases?

Thank you again for your detailed question. We have conducted relevant statistics and analysis, but due to space constraints, we regret to inform you that we present all the details in Sec. E of the Global Rebuttal.

Q5: I am also interested to ... what would happen if the PGN is replaced by self-attention.

Within the framework of TPGN, we replaced PGN with self-attention and conducted experiments on five datasets. Due to space limitations, for experimental results and more detailed analysis, please refer to Sec. F in the Global Rebuttal.

We sincerely appreciate Reviewer K1qz for offering valuable insights and recognizing of our work.

Q1: The computational complexity of TPGN is not well discussed in this paper, and it would be better if the inference efficiency was adequately discussed as the time series size increases.

The TPGN architecture is designed to capture information more effectively from historical sequences. The complexity of processing input time series can be referenced in Sec. 3.3. For the forecasting module, our approach involves concatenating the local long-term and globally aggregated short-term information obtained by TPGN to predict the future at different time points within a period. As a result, the dimensionality of the forecasting module changes from $P \times (2 \times dm)$ to $P \times R_\mathrm{f}$, with a complexity of $O(R_\mathrm{f})$. As $R_\mathrm{f} \times P = L_f (L)$, the complexity of the forecasting module is $O(\sqrt{L})$. Indeed, it is also important to note that since both the input and output serires must be fully stored in memory, the total memory complexity should be $O(L)$. However, this does not conflict with the $O(\sqrt{L})$ complexity of TPGN. As the input seriers grows linearly, the complexity of TPGN also increases in a square root manner, with the actual time and memory overhead reflected in Figures 4(c) and (d). When the forecasting series grows linearly, although theoretically the complexity of TPGN should also increase in a square root manner, when $d_\mathrm{m}$ is relatively large, the overhead of the forecasting module is much smaller compared to the encoder part. Therefore, as shown in Figures 4(a) and (b), the actual cost variation is not significant as the forecasting sequence grows.

Q2: Some presentation needs to be improved. For example, it is difficult for readers to quickly get important conclusions on Table 1 and Table 4.

Thank you for the reminder. In the revised version, we will carefully check the entire paper and enhance its presentation by adjusting colors, bolding text, particularly to emphasize the key conclusions in Table 1 and Table 4.

Q3: In table of experiment comparison, could you explain why TPGN-long outperform TPGN in some cases.

From Table 2, we observe that in most datasets across various forecasting tasks, TPGN outperforms TPGN-long, with the exception of a few instances where TPGN-long performs better than TPGN in the Traffic dataset. Observing Figure 12, the Traffic dataset primarily exhibits strong periodicity along with short-term variations in different patterns. In this context, the long-term module in TPGN is already adept at capturing the predominant strong periodicity in sequences. However, in certain cases where the short-term variation patterns exhibit strong randomness, it may introduce disturbances to the short-term module of TPGN. Hence, there are scenarios in specific forecasting tasks where TPGN-long outperforms TPGN in terms of the MAE metric. Nevertheless, we also observe that TPGN remains best performance in terms of the MSE metric in these forecasting tasks.