TimeFilter: Patch-Specific Spatial-Temporal Graph Filtration for Time Series Forecasting

Thank you for your insightful advice for polishing our manuscript. We have conducted sufficient experiments and analysis to dispel your concerns. The details can be found below.

Other Strengths And Weaknesses

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W1: Alalysis of theoretical complexity.

R1: We include a detailed comparison of the theoretical complexity of our method and other SOTA models. The Tab. below summarizes the theoretical computational complexity, where C is the number of channels, L is the input lengths, and P is the patch length.

Moreover, theoretical complexity alone cannot fully capture real-world performance due to implementation differences. We tested on one NVIDIA A100 GPUs, measuring training (1 epoch) and inference times for three datasets of increasing size, with results averaged over 5 runs in tha table below.

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W2: Description of the hyperparameter tuning process, especially the patch length parameter.

R2: We conducted an extensive hyperparameter search covering learning rates from $5\times 10^{-4}$ to $10^{-3}$, encoder layers from $1$ to $3$, $d_{model}$ values from $32$ to $512$, training epochs from $10$ to $30$, and patch lengths from $2$ to $96$.

In particular, the patch length $P$ has a significant impact on the results as it determines the contextual semantic information captured. We chose $P$ based on the characteristics of the dataset, experimenting with values of $L, \frac{L}{2}, \frac{L}{3}$, etc. This data-driven, result-oriented tuning ensures optimal performance. For transparency, we'll describe this process in detail in the appendix.

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W3: Difference with other masking (or pruning) dependencies methods.

R3: While previous work has explored dependency masking, our approach differs itself by filtering based on dependency types rather than binary causal relationships. Traditional methods often mask dependencies by assuming fixed causal relationships, which can be misleading due to spurious correlations in short-term dependencies. Our innovation lies in recognizing that different domains require distinct dependency types. For example: Traffic flow prediction benefits from spatial dependencies (e.g., interactions between adjacent road segments); financial markets often require filtering out short-term noise while preserving long-term trends; electricity demand forecasting relies heavily on periodic dependencies tied to daily/seasonal cycles. The significant performance gains in our experiments validate the effectiveness of this type of dependency filtering and demonstrate its superiority over generic causal masking strategies.

Questions For Authors

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Q1: Details of parallel pruning.

R1: Sorry for the confusion. For each patch i, its ego graph corresponds to a row in the overall adjacency matrix M. Pruning is achieved by directly modifying M. For the three filters, we pre-generate masks. Then, by element-wise multiplication of M with the selected mask via Einstein summation, we efficiently parallelize the filtering without any additional complexity. We'll clarify this in the revised manuscript.

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Q2: Why choose a pure GNN architecture rather than attention?

R2: We appreciate this insightful technical question. The choice of a GNN-based architecture over masked self-attention is motivated by both computational efficiency and inherent advantages in modeling temporal dependencies. GNNs offer lower complexity at $O(Qm)$ where m, pruned to O(kn), is the number of edges, compared to Transformers' quadratic $O(n^2)$ complexity (Q layers, m edges, n nodes). GNNs are inherently adept at modeling instance relationships through explicit graph structures, preserving inductive bias and offering better interpretability than the often opaque attention weights. We'll ensure these advantages are clearly articulated in the revised manuscript.

We hope our response addresses your concerns.

Thank you for your insightful advice. Here are responses to your questions:

Other Strengths And Weaknesses

W1: Some modules exhibit limited novelty. For instance, the idea of Dynamic Expert Allocation is not first proposed in this paper.

R1: We sincerely appreciate your deep expertise in identifying this important aspect. Dynamic MOE's adaptive token-wise expert allocation to address predetermined expert selection constraints, our TimeFilter does indeed build on this previous work[1,2], but we would like to clarify that our paper does not claim this as our primary contribution. Our key innovation lies in fine-grained dependency modeling beyond CD/CI, which can be realized through various mechanisms. Dynamic MOE serves as our implementation choice, but alternative approaches like attention-gated dependency routers or hierarchical graph convolutions with adaptive receptive fields could also instantiate our core paradigm. The novelty lies in the proposed dependency disentanglement framework rather than the specific routing mechanism.

[1] Harder Tasks Need More Experts: Dynamic Routing in MoE Models.

[2] Dynamic Mixture of Experts: An Auto-Tuning Approach for Efficient Transformer Models.

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W2: It would be better if the authors also visualize the dependency graphs of the other baselines.

R2: We've visualized PatchTST's and iTransformer's attention on Weather dataset in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png. TimeFilter achieves more precise dependency modeling by selectively focusing on specific types of relationships, unlike PatchTST's narrow focus on mutation points and iTransformer's overly generalized attention across periods. This selective approach allows TimeFilter to better identify meaningful interactions while filtering out noise.

Other Comments Or Suggestions

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C1: Explanation of the COMB operation in TimeFilter.

R1: Sorry for the confusion. COMB refers to the combination of ego and neighbour representations. We implement COMB using a simple FFN, similar to GCN [1]. We'll clarify this in the revised manuscript for better understanding.

[1] Semi-Supervised Classification with Graph Convolutional Networks.

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C2: Hyper-parameters explanation.

R2: In Equation 18, $\lambda_1$ and $\lambda_2$ are scaling factors for the loss functions. In our experiments, we set $\lambda_1$ to 0.05 and $\lambda_2$ to 0.005. These values ensure the three loss functions are balanced, facilitating gradient optimization.

Questions For Authors

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Q1: Why is the dependency graph of TimeFilter in Figure 2(c) not symmetric along the diagonal?

R1: The dependency modeling graph in Figure 2(c) is not symmetric because each patch has its own customized dependency relationships. Specifically, the filtering process for patch i's ego graph is independent of patch j's, and the selected filters may differ between them. This asymmetry is intuitive in real-world scenarios. For example, in financial systems, macroeconomic policy factors can directly influence price-volume factors (e.g., a positive policy change boosts market activity), but the reverse influence (e.g., price-volume changes trigger policy adjustments) is often weaker or absent. Thus, asymmetric dependencies naturally arise in such contexts.

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Once again, we are deeply grateful for your recognition of our work and your constructive feedback. If you have any further questions or suggestions, we would be more than happy to address them.

Thanks for your valuable comments. Here are detailed responses to your questions:

Claims And Evidence

E1: Claims of appropriate dependency modeling strategies.

R1: We conducted ablation studies in the table below with variants: tem.-only (T), spa.-only (S), spa.-tem.-only (ST), and their combinations without filtering. The results show that no single dependency combination outperforms others across datasets. The context-aware filtering mechanism reduces prediction errors by adaptively suppressing noise. This confirms that TimeFilter’s performance results from the principled fusion of complementary dependencies, not isolated components. We will expand this analysis in revision.

E2: Irrelevant dependency.

R2: "Irrelevant dependencies" refer to transient interactions caused by non-stationary patterns (extreme values, missing records, or noise) that lead to unstable statistical associations. As shown in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png, TimeFilter captures finer dependencies, distinguishing irrelevant ones. In contrast, PatchTST focuses on mutation points, and iTransformer spreads attention too broadly. This evidence shows how unfiltered dependencies propagate irrelevant ones, while our filtering mechanism preserves crucial context-aware relationships.

Methods And Evaluation Criteria

M1: Discussions exploring real-world scenarios.

R1: We provide real-world examples based on intuition: user behavior with low inter-channel correlation benefits from tem. dependencies, while spa. dependencies are key for real-time monitoring. In traffic flow prediction, spa.-tem. dependencies capture dynamic interactions across locations. For more details, refer to Section 4.2 (Routing Network, lines 205-233).

Weaknesses

W1: Parameter Sensitivity.

R1: The number of nodes $n$ is set as $n = C \times \lceil L/P \rceil$, where $P$ is the patch length. The result in table below shows that performance varies with $P$, as different patch lengths capture distinct temporal semantics based on the dataset's characteristics. The threshold $p$ influences the number of favored dependencies. Except for $p = 1.0$ (full retention), the model already removes irrelevant dependencies, making it less sensitive to $p$.

W2: Paradigm Transformation.

R2: Thank you for recognizing the novelty of our work. We’ll change "Paradigm Transformation" to "Novel Paradigm" and acknowledge prior channel strategies and GNNs.

Questions

Q1&Q3: Complexity.

R1: We compare the theoretical complexity of TimeFilter with other Transformer-based models in the table below. $C$ is the number of channels, $L$ is the input length, and $P$ is the patch length.

The graph construction and filtering modules have complexity $O((\frac{CL}{P})^2)$, and the GNN module is $O(CL)$. However, theoretical complexity alone doesn't fully capture real-world performance, as shown by testing on a NVIDIA A100 GPU with training (1 epoch) and inference times averaged over 5 runs:

Despite higher theoretical complexity, TimeFilter outperforms PatchTST in training and inference speed. This efficiency gain likely comes from structured sparsity in filtering, where einsum-based operations produce sparse matrices, reducing FLOPs and memory overhead.

Q2: Is $m=2$ sufficient for modeling?

R2: The value of $m$ is not a fixed hyperparameter. It represents the number of filter types, determined dynamically by the confidence threshold ($p$). Typically, one or two filters exceed $p$, so the dynamic routing mechanism selects one or two dependencies. When all three filters are active, it indicates a need for full channel dependency structures, though only a few patches require all dependencies.

Thank you again for your careful review and constructive suggestions, which have inspired us to improve our paper further.

Thanks for your valuable comments. Here are responses to your insightful concerns and questions:

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Q1: Novelty and Contribution. (The proposed idea is conceptually similar to existing Granger causality-based methods [1, 2] and instantaneous time series [3], yet the authors fail to discuss this connection.)

R1: We would like to clarify that our core innovation lies in proposing a novel fine-grained dependency modeling paradigm. Existing methods predominantly focus on channel-wise correlations at a coarse granularity, whereas our approach explicitly models time-varying fine-grained dependencies that cannot be effectively captured by channel-wise interactions alone.

The foundation motivation of TimeFilter stems from the following observation: Real-world time series often exhibit hybrid characteristics containing both causal signals (spa.-tem.&spa.) and inertial signals (tem.). However, the global time series signals do not conform to the causal, auto-generative nature assumed in modeling, but are influenced by unobserved factors [5]. Such forecasting are similar to the natural language understanding (NLU), where integrating all latent signals is more crucial. As the reviewer pointed out, Granger causality[1,2] only accounts for temporal inertia and spatio-temporal causality, while the instantaneous effects[3] only capture spatial causality. In contrast, TimeFilter integrates all potential signals during graph construction and then adaptively filters out invalid signals under specific segments, achieving superior forecasting capability.

To better demonstrate our novelty, we provide a comprehensive comparison across critical dimensions in the following table. (C channels, L lookback horizon, D hidden dimensions and K lead-lag steps).

[5] Time Series Prediction: Forecasting The Future And Understanding The Past

[6] MTEB: Massive Text Embedding Benchmark

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Q2: Analysis of why TimeFilter improve the performance. (The paper lacks theoretical grounding or in-depth analysis to explain why TimeFilter leads to performance improvements.)

R2: We appreciate this insightful question regarding the validation of the effectiveness of our method. The core mechanism can be explained from two complementary perspectives:

• Theoretical foundation: As noted in Q1, our filtering mechanism addresses the spurious regression artefacts prevalent in dependency modeling. Specifically, while some signals provide genuine predictive patterns, others exhibit localized spurious correlations arising from short-term segment analysis - a phenomenon particularly pronounced in non-stationary time series [7]. TimeFilter's dynamic pruning acts as a variance reduction operator, suppressing these transient noise signals while preserving true causal relationships.

• Empirical verification: As shown in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png, we quantitatively demonstrate that in the Weather dataset. The filtered adjacency matrix exhibits clearer cluster structures aligned with physical sensor relationships. This fine-grained opeartion allows it to focus on high-spatial and high-temporal-spatial correlations (e.g., Fea0-Fea1 and Fea1-Fea3) while eliminating irrelevant dependencies (e.g., Fea0-Fea3).

This dual validation by both causal theory and data-driven evidence confirms that our adaptive filtering successfully disentangles persistent patterns from transient noise, directly contributing to the observed performance gains.

[7] Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization.

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Q3: Compare with LIFT[4]. (The proposed method is very similar to [4], but does not consider it as a baseline.)

R3: LIFT[4] is added into the baseline. Our method still outperforms it (PatchTST+LIFT) and we will include LIFT in the revised version. While LIFT models lead-lag dependency modeling through lead estimation and refinement, it is sensitive to data distribution and less effective for synchronous data without significant lead-lag relationships. TimeFilter enhances such generalization ability and customizes dependency types for datasets from different domains.

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Overall, thanks again for your valuable comments. We hope the detailed experiments and clarifications provided above address your concerns. If you have any further questions, please feel free to reach out, and we would be happy to provide additional clarifications.