CrossAD: Time Series Anomaly Detection with Cross-scale Associations and Cross-window Modeling

We sincerely thank Reviewer bwo6 for the valuable comments and constructive suggestions, which have helped us improve our work. We appreciate your recognition of our novel contributions and will address each concern point-by-point.

Q1: Computational Efficiency and Parameter Analysis

Thank you for raising this important question. The design of CrossAD has been carefully considered from multiple perspectives to ensure computational efficiency:

1. Instead of 'copy', we use downsampling to obtain multi-scale sequences. Each downsampling step results in a sequence that is half the length of the previous one. Let the length of the original sequence be N. After m downsampling steps, the total length of multi-scale sequences is: $\frac{N}{2} + \frac{N}{4} + ... + \frac{N}{2^m}<N$. For the encoder, the attention computation cost is based on multi-scale sequences. The overall complexity of the model remains $O(N^2)$. This quadratic complexity is consistent with most transformer-based methods.
2. For cross-window modeling, we employ a prototype compression mechanism that compresses the global context into a fixed number of prototypes, effectively avoiding the storage of a massive number of sub-series. Moreover, during inference, we only need to use the pre-trained global multi-scale context to generate the next-scale output, without having to compute sub-series representations.
3. Detailed information on runtime and parameter count is provided in Appendix E.2. As shown in Table 9, CrossAD achieves great performance with 0.93M parameters. On the GECCO dataset, its inference time (0.8 seconds) outperforms multi-scale methods like TimesNet and transformer-based approaches such as AnomalyTransformer.

We will include these details in Appendix E.2 to provide readers with a clearer understanding of our method's efficiency.

Q2: About the period-aware router

The period-aware routing mechanism filters out high-frequency noise by retaining only the top-k magnitude components and then reconstructs the time-domain patterns through the inverse Fourier transform to enhance stability. Additionally, the Gumbel-Softmax gating further improves robustness to noise by injecting stochastic noise. Consequently, the Fourier-based period-aware router is robust when handling non-stationary or noisy time series. This capability is validated on non-stationary datasets like SMD and SWaT[ref1] in Table 1, where CrossAD achieves superior detection performance.

The core design objective of the top-k Router is to extract long-term periodic patterns from time-series data, enabling appropriate selection of sub-series queries for constructing a global multi-scale context. Crucially, during detection, it does not filter the multi-scale series input to the Encoder and Decoder. Therefore, the selection of top-k in the router does not directly impact the method's sensitivity to time series containing transient anomalies or irregular patterns. As seen in Fig. 4, CrossAD effectively detects both global and contextual anomaly points, confirming its sensitivity to transient anomalies.

Q3: Training imbalance or representation collapse

We employ scale-independent encoding to ensure that features at different scales are not dominated by coarser granularities, and use a masking mechanism (Scale Independent Mask M1 in Fig.2(2)) to isolate cross-scale interference. The decoder operates in a staged manner: it first independently upsamples each scale and then reconstructs the signal under the guidance of a cross-scale mask (Cross Scale Mask M2 in Fig.2(4)) to control the reconstruction conditions.

At the same time, we dynamically update the global multi-scale context, enabling the model to effectively integrate information across scales and prevent the loss of fine-grained features. As shown in the detection results in Fig. 6, the model maintains a strong detection capability for fine-grained global point anomalies.

The training curves show that the MSE errors across all scales decrease synchronously, validating the balanced optimization process. No representation collapse is observed during training.

Formatting Concerns

We sincerely thank the reviewer for the valuable suggestions. We carefully review and revise the formatting. Modifications include, but are not limited to, the following:

1. Standardized the fonts in Figure 2 (Arial for text and Times New Roman for equations).
2. Corrected the operator notation in equations.
3. Fixed vertical spacing issues and symbol errors.
4. Streamlined the description of the standard architecture for clarity and conciseness.

[ref1] Chengsen Wang, Zirui Zhuang, Qi Qi, et.al. Drift doesn't matter: Dynamic decomposition with diffusion reconstruction for unstable multivariate time series anomaly detection. In NeurIPS, 2023.

We sincerely thank Reviewer fjUj for the valuable comments and constructive suggestions, which have helped us improve our work. We appreciate your recognition of our novel contributions and will address each concern point-by-point.

W1: Our motivation for cross-scale association

We sincerely apologize for the misunderstanding, and we will refine the logical flow in the introduction. The cross-scale association emphasizes the mapping relationship across different scales of the same subseries.

We use mathematical functions to further clarify. Consider three subseries: $A$, $B$, and $C$, where $A$ and $B$ are normal (blue parts in Fig. 1), and $C$ is anomalous (red part). Take scale 2 and scale 3 as examples, we denote $f$ as the cross-scale association of normal series.

We construct the model to learns the cross-scale association $f$. During training, we learn the normal cross-scale association patterns $f$ by reconstructing the fine-grained series from the coarse-grained series, e.g., $\hat A_{scale2}=f(A_{scale3})$, $\hat B_{scale2}=f(B_{scale3})$, and minimizing between $A_{scale2}$ and $\hat A_{scale2}$, and between $B_{scale2}$ and $\hat B_{scale2}$. However, during inference, when an anomalous series $C$ is encountered, the model reconstructs $\hat C_{scale2}=f(C_{scale3})$ using the cross-scale association $f$ and expects $\hat C_{scale2}=C_{scale2}$. As the associations are different between normal and abnormal series, the cross-scale association $f$ can only reconstruct normal subseries, leading to $C_{scale2} \neq \hat C_{scale2}$. This discrepancy enables effective anomaly detection (corresponding to lines 40–42: "when anomalies occur .. in the normal state").

Three questions in W1:

(1) In line 42, we say, "This discrepancy...abnormal behavior". The "discrepancy" refers to the different associations between normal and abnormal series, as we defined above. We do not detect anomalies by comparing the similarity among scales 1, 2, and 3.

(2) If all the normal peaks simulate the anomalies in the training data, the model will capture such associations and can't detect the peaks as anomalies in the test. If these peaks only appear in test data, it can be detected as the model does not learn this in the normal association.

The cross-scale association discrepancy enables the detection of anomalies by learning mappings across different scales and allows for more precise localization of the specific scale at which an anomaly occurs. Moreover, as shown in Fig. 4, we find that the cross-scale association discrepancy is more effective, particularly in identifying complex anomalies like shapelet or seasonal. From an experimental perspective, the results in Tables 1, 2, and 3 demonstrate that the next-scale reconstruction approach indeed achieves better detection performance.

(3) We can visualize the reconstruction results at each scale for both normal and abnormal subseries to illustrate the differences between normal and abnormal associations. As shown in Fig. 6 (Appendix E.3), when an anomaly occurs, the cross-scale association deviates from the normal pattern, leading to reconstruction failure across all scales. This observation is consistent with our original motivation.

If there are any parts that we have not explained clearly or if you have any questions, please feel free to let us know. We would be more than happy to provide further clarification and engage in deeper discussion.

W2: Justifications for design choices

(1) Scale-independent encoding: While the two implementations are functionally equivalent, we agree that your approach is indeed more computationally efficient. When we wrote the paper, we chose to concatenate the multi-scale representations before the encoder to simplify the illustration in Fig. 2, since these representations are later used for cross-attention. In practice, your suggested approach is more efficient and can be adopted during actual implementation.

(2) The query library is randomly initialized and learnable. We will include this clarification in the revised version of the paper.

(3) About the Next-scale generation: Since the output of the encoder needs to query a more relevant context from the global multi-scale context, attention mechanisms have a natural advantage in computing similarity.

(4) Ablation study on the query library: Thank you for the valuable suggestion. We add experiments that directly use a trainable context, and the results further support our design choices.

(5) About dataset MSL and SMAP: As a next-scale reconstruction-based approach, our model relies on learning continuous latent representations through autoencoding. Discrete variables inherently lack the smooth and structured latent space required for effective reconstruction.

W3: About "Up-interpolation"

We correct the "up-interpolation" to "interpolation" in the revised version of the paper.

W4 & W5: SOTA detection methods

We included TimesNet and TimeMixer as they are multi-scale methods, and they show great detection performance in our experiments. We add the anomaly detection models D3R[ref1] and the latest SOTA model Catch[ref2].

W6: Grammatical errors

We have fixed the error.

W7: About the words "generation", "(down/up)-sampling", and "interpolation"

From the perspective of module naming, we use the term "generation" to refer to the process of creating multi-scale series. In practice, we implement this via "downsampling". Thank you for your suggestion. We will review the entire paper and unify the term consistently.

W8: About the pooling size and period of time series

To enhance the model's generality, we use a fixed pooling size across all datasets. If prior knowledge is available, finer adjustments can be made. Adaptively selecting the pooling size based on each dataset is indeed a valuable direction for multi-scale methods, and we would like to leave it as a direction for future research and discussion.

W9: About the window size

Since CrossAD is not highly sensitive to the window size parameter, we selected a window size that achieves relatively good performance. We include results for various window sizes (96, 128, 192) for completeness. From the results, CrossAD demonstrates great detection performance across various window sizes.

Q1: About channel-independent

In our method, the query library and global multi-scale context store global information. Extending to multi-channel sub-series and global context would introduce additional complexity, making it more challenging to extract meaningful patterns. Therefore, modeling multi-channel series requires further specialized optimization. As the current work focuses on temporal dependency modeling, we leave the exploration of channel-wise correlations as future work.

The final anomaly score is calculated as the average of the scores from all channels.

Q2: About the correlation among multiple channels

Although CrossAD does not explicitly model inter-channel dependencies, it demonstrates strong capability in capturing temporal modeling. We conduct a visualization analysis of the anomaly scores on the SWAT dataset, and the results show that CrossAD is able to detect most of the anomalies without noticeable delay because we promptly detect the anomalies in each channel. We will include these visualization results in the revised manuscript.

Additionally, we add experimental results of Catch, an anomaly detection model that captures inter-channel correlations (The results are in W4&W5). However, in practice, Catch does not show a significant performance advantage. The primary reason is that CrossAD's strong capability in modeling temporal dependencies compensates for the lack of channel correlation modeling.

Q3: About patch embedding

Given a time series of length $T_i$, it can be divided into $P_i$ patches by splitting the sequence into $P_i$ non-overlapping segments of equal length. Each patch has a length of $T_i/P_i$. We will include additional explanations regarding the patch embedding in the revised manuscript.

[ref1] Chengsen Wang, Zirui Zhuang, Qi Qi, et.al. Drift doesn’t matter: Dynamic decomposition with diffusion reconstruction for unstable multivariate time series anomaly detection. In NeurIPS, 2023.

[ref2] Wu Xingjian, Qiu Xiangfei, Li Zhengyu, et.al. Catch: Channel-aware multivariate time series anomaly detection via frequency patching. In ICLR, 2025.

We sincerely thank Reviewer WTgi for the valuable comments and constructive suggestions, which have helped us improve our work. We appreciate your recognition of our novel contributions and will address each concern point-by-point.

Q1: Clarify the Novelty

We are grateful to the reviewer for bringing this relevant work to our attention. In the revised manuscript, we will add a discussion of [Ref1] in the Related Work section and highlight the key distinctions.

In [Ref1], a hierarchical RNN processes the raw time series to generate multi-scale representations, with each scale reconstructed independently. To improve reconstruction at the current scale, the method fuses coarser-level features into the current scale's representation. As a result, the reconstruction in [Ref1] is fundamentally reconstruction-based: it relies on the current scale's own features, enhanced by higher-level contextual information, mainly focusing on the modeling of temporal dependencies and ignoring cross-scale associations.

In contrast, CrossAD adopts a prediction-like mechanism. Instead of depending on the current scale, our method reconstructs (predicts) the fine-grained series solely from coarser-grained representations. By the next-scale reconstruction, the model captures the cross-scale associations. This design shift—from refining the current scale using multi-scale fusion, to generating it from higher-level abstractions—reflects a fundamental difference in modeling philosophy.

In addition, the number of scales in [Ref1] is strictly determined by the number of model layers, whereas in CrossAD， it can be set adaptively with flexible numbers of scales. This demonstrates that CrossAD has a more flexible and general architecture.

The novelty of CrossAD compared to fusion-based methods such as [Ref1] can be highlighted in the following aspects: (1) Modeling objective is novel. Fusion-based methods aim to enhance the modeling of temporal dependencies within each scale by integrating multi-scale features. In contrast, CrossAD not only captures temporal dependencies but also explicitly models the associations across scales. (2) Modeling methodology is novel. To achieve the objective, we novelly reconstruct fine-grained series exclusively from coarse-grained inputs, without relying on the current scale, which is a design principle that has not been explored in existing fusion-based approaches. (3) The criterion is novel. Our method explicitly models the cross-scale associations and, in a novel way, leverages these associations as an anomaly criterion for anomaly detection.

Q2: Dataset usage

Thank you very much for your suggestion. Due to considerations of evaluation methodology and paper length, we have not included the UCR dataset in the main text, but have provided a detailed discussion in Appendix E.5. Unlike other datasets, UCR consists of 250 distinct univariate sub-datasets, each containing only one continuous anomalous sequence. Importantly, the UCR dataset focuses on whether a model can successfully detect anomalies in each individual sub-dataset. Therefore, we follow the evaluation protocol used by Timer [ref3] and DADA [ref4]. As shown in Figure 7 of Appendix E.5, when alpha quantile = 3%, CrossAD successfully detects anomalies in 186 sub-datasets, outperforming Timer with 110 and DADA with 166. Moreover, CrossAD achieves a lower average quantile of 4.40% across all sub-datasets, compared to 12.5% for Timer and 7.60% for DADA. These results demonstrate the superior performance of CrossAD. We have also included visualization results for several sub-datasets.

We will strengthen the empirical evaluation by including additional benchmarks in Figure 7, and we will move this analysis to a more prominent location in the main text. Additionally, since the TSB-AD benchmark is more comprehensive and includes the UCR datasets, we will include the TSB-AD results in Table 1.

Q3: Benchmark coverage

Thank you for your suggestion. The TSB-AD benchmark is a collection of multiple open-source datasets after preprocessing and integration. We add experimental results on TSB-AD-U and TSB-AD-M and provide a comparison with the state-of-the-art methods from TSB-AD [Ref2]. CrossAD achieves excellent detection performance on this benchmark. Further comparative results can be found in Table 5 of [Ref2].

Q4: Statistical significance testing

Thank you for the suggestion. We conduct the Wilcoxon signed-rank test on the VUS-PR results in TSB-AD-U (in Q3), which has 350 sub-datasets. CrossAD achieves the highest average rank among all models, which is 9.54.

We include the p-values between CrossAD and other baseline models, indicating that the performance improvement of CrossAD over other models is statistically significant and not due to random chance.

As visualizations are not permitted in the rebuttal, we will include the critical difference diagram in the revised version of the paper.

Q5: Ablation study

Thank you very much for your suggestion. We include additional datasets as follows. Due to time constraints, we will include results on a larger benchmark in the final version of the paper.

Q6: Execution time analysis

We sincerely thank you for your valuable suggestion. We will include further discussion along the following directions:

1. The effect of varying sequence length and dimensionality on inference time (in Q7).
2. A 2D scatter plot comparing execution time versus accuracy across different methods.

We will place this analysis in a more visible location in the revised manuscript to better highlight it.

Q7: Scalability evaluation

(1) About series length.

We provide a theoretical complexity analysis of how sequence length affects computational cost. Let the length of the original sequence be N. After m downsampling steps, the total length of multi-scale sequences becomes:

$\frac{N}{2} + \frac{N}{4} + ... + \frac{N}{2^m}<N$

For the encoder, the attention computation cost is based on these sequences. $O((\frac{N}{2}+ ...+ \frac{N}{2^m})^2)<O(N^2)$

Since the length S of the sub-series queries and context prototypes is smaller than N, the overall complexity of the model remains $O(N^2)$. This quadratic complexity is consistent with most transformer-based methods.

Experimental results on GECCO (5 epochs) demonstrate that the model scales efficiently, with computational time growing acceptably as sequence length increases.

(2) About data dimension.

Due to its channel-independent design, the theoretical complexity scales linearly with dimensionality. For $c$ channels, the complexity becomes $O(cN^2)$, aligning with existing channel-independent approaches.

Reference

[ref1] Qingning, L., Wenzhong, L., Chuanze, Z., Yizhou, C., Yinke, W., Zhijie, Z., Linshan, S, Sanglu, L. (2023). Multi-Scale Anomaly Detection for Time Series with Attention-based Recurrent Autoencoders. Proceedings of The 14th Asian Conference on Machine Learning, in Proceedings of Machine Learning Research 189:674-689

[ref2] Qinghua Liu and John Paparrizos. 2025. The elephant in the room: towards a reliable time-series anomaly detection benchmark. In NeurIPS, 2024.

[ref3] Yong Liu, Haoran Zhang, Chenyu Li, Xiangdong Huang, Jianmin Wang, and Mingsheng Long. Timer: Transformers for time series analysis at scale. In ICML, 2024.

[ref4] Qichao Shentu, Beibu Li, Kai Zhao, Yang Shu, Zhongwen Rao, Lujia Pan, Bin Yang, and Chenjuan Guo. Towards a general time series anomaly detector with adaptive bottlenecks and dual adversarial decoders. In ICLR, 2024.

We sincerely thank Reviewer sbi4 for the valuable comments and constructive suggestions, which have helped us improve our work. We appreciate your recognition of our novel contributions and will address each concern point-by-point.

W1 & Q1: The pseudocode of the full algorithm

We sincerely thank the reviewer for the valuable suggestion. To enhance the clarity and reproducibility of our work, we will include complete pseudocode in the revised manuscript, which will clearly illustrate the logical flow of each step in the algorithm, enabling readers to easily reproduce our methodology.

W2 & Q2: Supplement to the function definitions

We will provide clear and detailed definitions of these functions, such as LayerNorm, Attention, and FeedForward, in the corresponding section of the methodology to help readers gain a deeper understanding of every technical aspect of our research.

W3: Synthetic Anomalies

Thank you very much for your suggestion. Our method is comprehensively validated across multiple real-world datasets containing diverse anomalies, where it significantly outperforms baseline approaches. Furthermore, as evidenced by the detection results in Fig. 4 in the main text, our method effectively handles various synthetic anomalies.

To better test the limits of CrossAD, we add an additional 40 anomalies of different degrees to the test dataset. These anomalies affect each scale to varying degrees. We divided these anomalies into 4 groups according to the degree of impact, with 10 in each group. From the following results, it can be found that our method detects a total of 33 synthetic anomalies, which is more than the 26 of the baseline method, TimeNets.

Q3 & Q4 in weakness: Variable Calculation and Parameter Selection

We sincerely apologize for the confusion. S denotes the length of the learnable query, P represents the patch length, and T stands for the length of the input series to the model. S is a hyperparameter of the model. The learnable query is employed to extract sub-sequence representations from the input sequence; hence, S is smaller than T. We will elaborate on this in the corresponding parts of the paper.

K indicates the number of prototypes in the global multi-scale context. We performed a grid search on two validation datasets to select a better-performing K and used this K value for all the test datasets. Selecting K is quite open-ended. Here are some of our thoughts:

• We can assess the dataset complexity to determine a suitable K. For simple and small-scale datasets, a small K suffices.
• Dynamic prototypes can be considered. Instead of being fixed, the number of prototypes dynamically adjusts during data processing to determine the optimal K value for retaining relevant information.
• Neural architecture search techniques can be used to find the optimal K.

Q5 in weakness: The Relationship between Training Time and Training Set Size

Thank you for your care about computational resource consumption. When the sliding window size is fixed, the training time per epoch mainly depends on the number of window sequences (i.e., samples) in the training set. Under this condition, the training time grows roughly linearly with the size of the training set. The following result is the time cost required for training 5 epochs on training data sets with varying numbers of time points, which increases linearly.