UNIFYING LONG AND SHORT SPATIO-TEMPORAL FORECASTING WITH SPECTRAL GRAPH NEURAL NETWORKS

Dear Reviewer,

We’d like to make sure we’ve fully answered your questions. If there’s anything else you need clarification on, please feel free to let us know. Thanks again for your time!

Thank you for taking the time to review our paper. As the discussion period is approaching its end, we would like to check whether our responses have addressed your questions.

We have revised the manuscript, including improvements to the figures, adding iTransformer as a baseline model, and modifying several sentences to improve the overall coherence of the manuscript.

We hope our responses can adequately address the your concerns and questions.

Q5 More baselines

To ensure a fair comparison, we implemented our method and all baselines on the open-source platform BasicTS with a batch size equal to 64. Incorporating feedback from other reviewers, we include additional baselines (iTransformer [2] and Fredformer [3]) to show our approach's competitiveness (as shown in Table 3). Due to out-of-memory (OOM) errors, we only report ForecastGrapher[4]’s results for the 96-96 task on the Weather dataset.

Table 3. More baselines.

【2】iTransformer: Inverted Transformers Are Effective for Time Series Forecasting， ICLR 2024

【3】Fredformer: Frequency Debiased Transformer for Time Series Forecasting， KDD 2024

【4】ForecastGrapher: Redefining Multivariate Time Series Forecasting with Graph Neural Networks， arXiv 2024

We thank the reviewer for offering the valuable feedback. We have addressed each of the concerns raised by the reviewer as outlined below.

W1. Motivation

The efficiency improvement of S2GNN primarily comes from its adoption of a decoupled GNN structure, which eliminates the need for inter-layer parameter matrices, thereby reducing computational complexity. Additionally, by addressing sample indistinguishability, we can use an MLP-based model instead of the commonly used complex sequence structures. This substitution maintains predictive accuracy while significantly lowering computational costs.

W2. Originality

Our primary contribution lies in utilizing a decoupled GNN + MLP-based structure to address the efficiency issues common in most existing spatiotemporal prediction models. This novel approach provides an efficient alternative for STGNN-type models to handle long input sequences, offering significant advantages over traditional GCN+CNN-based, GCN+RNN-based, or attention-based models. S2GNN is not only structurally more efficient but also maintains strong performance for both short-sequence and long-sequence spatiotemporal forecasting tasks.

Regarding the embeddings, the cross-correlation embedding is not simply a typical temporal embedding with an added node ID. Instead, all the feature embedding $H$ of the input sequence is obtained through matrix indexing, where $W$ is the feature matrix and $X$ is the input sequence. Specifically, the feature embedding is computed as: $H = W[f(X)]$, where $f(X)$ represents the indexing function applied to the input sequence $X$. The distinction between various feature embeddings lies in the method of computing this index. In the case of cross-correlation embedding, the index is derived from the similarity between the input sequence and multiple convolutional kernels. On the other hand, the indices for time embeddings and node embeddings are based on the inherent information present within the input sequence itself.

W3. Quality

Thank you for pointing out these issues. We promise to thoroughly review and revise the manuscript in the next stage.

Q0.

The improvement in model efficiency comes from employing a decoupled GNN + MLP-based architecture. This combination reduces computational requirements by eliminating inter-layer parameter matrices, making the model more scalable and efficient.

Addressing sample indistinguishability is crucial primarily for improving accuracy. By enhancing the model's ability to distinguish between similar samples, we can maintain high predictive performance even with a simpler MLP-based model. This is significant because it allows us to replace the commonly used, computationally expensive sequence structures (such as RNNs or attention mechanisms) with a more efficient MLP-based model. As a result, we reduce the overall computational cost while still achieving strong accuracy in spatiotemporal prediction tasks.

Q1.

The variation in the number of nodes refers to different datasets having different numbers of nodes. In existing STGNN models, fixed-size embeddings are used to learn node similarity, which becomes problematic when the number of nodes varies across datasets. For example, PEMS08 has 170 nodes, while PEMS07 has 883 nodes, yet the model uses the same embedding size for both datasets. This inconsistency makes it difficult to adapt the embeddings efficiently. Our proposed method mitigates this issue. For instance, assuming $c = 10$, $r = 0.5$, and $d = 32$ , existing methods need to adjust the distances between 170 and 883 vectors of length 32 in the feature space. In contrast, our approach only requires handling 29 and 37 vectors, respectively, making adaptation more efficient. Therefore, our model is inductive and designed to handle variations in the number of nodes across different datasets. However, scenarios involving missing or additional nodes within the same dataset, or when the number of nodes in the test dataset exceeds that in the training dataset, are outside the scope of this work.

Q2.

We believe that many factors determine the convergence speed of a model, such as the learning rate, dataset size, and the influence of random seeds. While some transformer-based models converge in fewer epochs, they often require more time per training epoch. In contrast, our method may need more epochs to converge, but trains faster per epoch, resulting in a reasonable total running time, as shown in Table 1. Note that although Fredformer converges in fewer epochs, its performance on spatiotemporal prediction tasks is suboptimal, as highlighted in Table 3 of Q5.

Additionally, Figure 2 illustrates the structural changes in the adaptive graphs of two existing models that use adaptive graph structures, not our proposed model. The stabilization of the adaptive graph structure indicates that the GNN part of the model has converged. We describe this from the perspective of graph structure to explain that applying spectral graph models to adaptive graphs is feasible.

Table 1 Comparison of training time. Best refers to the epoch when val loss is minimum.

Q3.

I’m not entirely sure what you mean by "a typical temporal embedding with an added node ID." To clarify, we are not adding node IDs to the input sequence redundantly.

Simply speaking, sample indistinguishability occurs when dynamics from different factors overlap, making input sequence features too similar to distinguish. To address this, we feed the model with more information to help distinguish these inputs. The essence of node IDs or temporal features is to provide an index for the input sequence and then concatenate a feature embedding vector to the input sequence based on the index value.

In our method, we compute the cross-correlation between the input sequence and a set of feature sequences to derive an index value. Then, we concatenate the corresponding feature embedding vector to the input vector. Figure 5 shows what these feature sequences look like ultimately. Specifically, we calculate the similarity between the input sequence and these feature sequences, assign the feature embedding vector corresponding to the most similar feature sequence, and concatenate it with the input vector. This does not involve directly concatenating the feature sequence itself.

This process is the same as how the previous three features (Node ID, time of the day, day of the week) are handled.

Q4

Addressing sample indistinguishability is important to maintaining performance across varying forecasting horizons. As demonstrated in [1], it is effective for enhancing TSF outcomes. We extend this by strengthening the model's ability to handle inter-variable relationships, allowing it to excel in both short- and long-term forecasting tasks.

To illustrate this, we conducted an experiment comparing three models: the base MLP, MLP+embs, and our full S2GNN model. As shown in Table 2, our full S2GNN model achieves the best results, emphasizing that integrating embeddings, alongside the decoupled GNN architecture, is crucial for achieving superior LTSF performance.

Table 2 The impact of enhanced distinguishability.

【1】Spatial-Temporal Identity: A Simple yet Effective Baseline for Multivariate Time Series Forecasting，CIKM 2022（short）

I regret that the quality of this work is not satisfactory. Many essential revisions to the manuscript will need to wait until the next stage. Additionally, the reliability of the proposed S2GNN has not been compared with recent SOTA in the revised paper. The presentation of the work is also poor. I apologize to the authors for the time and effort invested in preparing the rebuttal.

Consequently, I am inclined to reject the submission and I will maintain my current score.

I appreciate your review and feedback. However, could you please detail the necessary changes and clarify which parts of the manuscript were unclear or poorly presented? Additionally, we have included the results of iTransformer and ForecastGrapher. It is evident that their performance is not satisfactory in terms of efficiency or accuracy.

R1

We respectfully disagree with the assessment that S2GNN merely optimizes the input embedding. S2GNN employs a decoupled GNN + MLP-based architecture, which effectively addresses the challenge of handling long input sequences in spatiotemporal forecasting tasks. We believe that the value of a novel method lies not in its complexity but in its simplicity and effectiveness.

R2

Thank you for your advice. We have updated Figures 1 and 2 in the revised PDF, enlarging the text to improve readability and ensure better visualization. If you have any further suggestions, please let us know.

R3

We have conducted extensive experiments, including evaluations on eight real-world datasets, to demonstrate S2GNN's capabilities in handling both short- and long-term spatiotemporal forecasting tasks. Additionally, we performed ablation studies to validate our design choices. Considering feedback from all reviewers, we will include more baselines and add comparisons of training efficiency in the next version to provide a more comprehensive evaluation. We believe these experiments sufficiently demonstrate the superior performance of S2GNN, especially in spatiotemporal forecasting tasks. If you have further suggestions for additional experiments, please let us know.

R4

Thank you for your feedback. To ensure fair comparisons and reproducible results, we implemented our method using the open-source platform BasicTS. If you have trouble reproducing the results in our paper, we can offer assistance.

Thank you for taking the time to review our paper. As the discussion period is approaching its end, we would like to confirm whether our responses have adequately addressed your questions. In response to your comments, we have:

• optimized most of the figures in the manuscript and added more examples in the appendix.
• added iTransformer as a baseline model and included more ablation studies to demonstrate the effectiveness of our proposed approach.
• improved the overall coherence of the manuscript in the revised PDF.

Thank you once again for your valuable comments and suggestions to improve our paper.

Dear Reviewer,

We’d like to ensure that we’ve answered all of your questions. Please let us know if our response has resolved your concerns, or if you require any additional clarification. Thank you for your time!

Thank you for taking the time to review our paper. As the discussion period is approaching its end, we hope that you will reconsider our work for the following reasons:

• The STGNN model excels at handling the relationships between variables in sequential data but generally suffers from inefficiency. We propose a solution to this issue by removing inter-layer weights to improve efficiency.
• Furthermore, most existing approaches rely on the homophily assumption, such as using similarity measures based on attention mechanisms to assess the influence between variables. In contrast, our work directly demonstrates the existence of heterophily relationships between temporal variables by visualizing the weights of filters, highlighting that considering only homophily relationships is insufficient.
• In the revised manuscript, we emphasize that our work structurally ensures an improvement in spatiotemporal prediction efficiency, and we have also made adjustments to the figures in the paper.

We again thank you for your insightful review and suggestions to improve our paper.

My concerns are addressed, and I will maintain my score and confidence.

Dear Reviewer,

We have added additional visual results similar to those in Figure 8 in the appendix (Figure 11). In addition, we promise to include iTransformer as a baseline model and provide a visualization of its results for comparison. As the discussion period is nearing its conclusion, we welcome any further comments on our rebuttal or the revised paper. Thank you for your time!

Thank you for taking the time to review our paper. As the discussion period is approaching its end, we would like to check whether our responses have addressed your questions. Following your comments, we:

• We have updated Figures 1, 2, and 8, and added several sentences to improve the overall coherence of the manuscript.
• In our ablation study (Table 2), we provide more feature embeddings that relate to the model's predictive performance, further emphasizing the necessity of handling indistinguishability among samples.
• We have included iTransformer as a baseline model and visualized the results in Figure 8.

Thank you again for your valuable comments and suggestions to improve our paper. We look forward to your reply.