Causality-Aware Contrastive Learning for Robust Multivariate Time-Series Anomaly Detection

We would like to thank Reviewer J2ZH’s insightful and constructive comments. This response presents additional experiments and discussion to address the reviewer’s concerns, all of which will surely be integrated into the main paper.

Time series anomalies can arise from various sources… discuss specific types of time series anomalies their method is designed to address and identify scenarios in which their approach might fail.

As the reviewer notes, certain anomalies may result in anomalies while keeping the causal structure. CAROTS is designed to detect anomalies that violate inter-variable causal dependencies, including scenarios:

• A variable behaves inconsistently with its known causal parents.
• Structural dynamics deviate from the learned causal graph.
• Temporal or multi-variable patterns break causal relationships.

We humbly acknowledge that CAROTS may be less sensitive to anomalies that lie within the causal structure; however, our method consistently demonstrates strong anomaly detection performance across a wide range of datasets, which suggests that CAROTS remains effective in practice.

Computational cost of CAROTS

Even with causal modules, CAROTS is efficient due to a lightweight one-layer LSTM. We compare the total train time of the studied methods on SWaT:

Train Time (min):

The train time of CAROTS is comparable to baselines and cheaper than heavier models like TimesNet, indicating that leveraging causal structure can reduce reliance on deeper architectures.

We also profile the per-iteration time on MSL_P-14:

Per-Iter Time (seconds):

Each iteration takes less than 0.05 sec, and even the heaviest component (CPA) is lightweight.

# of sensors included in the discovered causal mode/graph

Discovered causal graphs include all sensors, even if they can include disjoint subgraphs and isolated nodes. Some sensors appear as isolated nodes, which is expected in complex systems with partially independent components.

CPA and CDA ensure that every sensor is incorporated in training and synthetic outlier generation. CDA randomly selects a node and performs DFS over causal graph to extract a local subgraph. If the selected node is isolated, it forms a single-node subgraph, and bias is directly injected into its values. This allows us to synthesize abnormal behavior for all sensors, regardless of connectivity. As node selection is random and repeated, all variables have equal chances to be selected and perturbed during training.

CPA similarly applies to all sensors, including isolated nodes. CPA perturbs parents of a selected variable and forecasts the target with the causal forecaster. For isolated nodes, we consider temporal self-dependence as the causal link. The node is perturbed through its past values, and forecasting is performed accordingly, enabling CPA to handle the absence of graph edges.

We also report # of non-trivial subgraphs (excluding isolated nodes) and isolated nodes in datasets:

CAROTS’ strong performance even on datasets with a large # of isolated nodes and fragmented subgraphs indicates that it can handle complex graph structures and partially disconnected systems.

More related works

• Related works will be updated to include discussion of [1, 2], a relevant early work connecting causality and anomaly detection.

• Correlation vs. Causality: Correlation-based methods, such as [3, 4], model dependencies with co-activation patterns across variables. While these methods are effective at capturing immediate statistical associations, they may fail to distinguish true dependencies from spurious correlations, particularly under distribution shifts or external interventions. In contrast, CAROTS is grounded in the causal perspective. It explicitly models directional relationships by learning a causal graph from train data using a causal discovery method. This enables our method to simulate both causality-preserving and causality-breaking augmentations, which serve as the foundation for contrastive learning. Anomalies are then interpreted as deviations from learned causal relationships, making CAROTS more robust to superficial variations that do not reflect structural disruptions.

[1] Yang et al., A Causal Approach to Detecting Multivariate Time-series Anomalies and Root Causes, 2022.
[2] Qiu et al., Granger Causality for Time-Series Anomaly Detection, 2012.
[3] Zheng et al., Correlation-aware Spatial-Temporal Graph Learning, 2023.
[4] Wang et al., Correlation-Based Anomaly Detection Method for Multi-sensor System, 2022.

We are grateful for Reviewer Xmpv’s detailed yet positive comments. Overall, the reviewer believes that “the paper’s claims are are supported by results on multiple datasets,” which “makes it suitable for real-world scenarios.” This response includes additional experiments and discussion to consolidate our contributions. We would love to answer further questions during the discussion period. Lastly, the “Interpretability analysis” will be updated later in the discussion period.

How inaccuracies in the learned causal graph impact CAROTS

To assess the robustness of CAROTS to inaccuracies in the causal graph, we study the performance of CAROTS on the SWaT dataset as diverse perturbations are introduced to the learned causal structure:

• random init: causal edges randomly initialized
• zero init: no causal edges (fully disconnected graph)
• flipped: cause-effect directions reversed
• noisy: Gaussian noise added to the causal matrix
• original: learned causal structure without perturbation

We report AUROC (mean ± std over 3 seeds) below:

These results show that CAROTS is robust under moderate graph perturbations. Notably, the combining CPA and A_CD helps preserve performance even when the causal structure is partially inaccurate.

Results when partial anomalies are in the training set

To evaluate the robustness of CAROTS in more realistic settings, we conduct additional experiments where synthetic anomalies are injected into the train set at varying ratios (0% to 20%). Synthetic anomalies are generated by injecting point-level global anomalies (same as the outlier generation synthesis process in the main paper).

The results below (AUROC, mean ± std over 3 seeds on SWaT) show that CAROTS maintains strong performance even when the train data is partially contaminated.

Discussion of neural causal discovery works

While our method builds on recent causal discovery tools like CUTS+, we acknowledge that referencing earlier approaches like Neural Granger Causality [1] would help contextualize the development of CAROTS. We will revise the related work section to include and cite this line of research.

[1] Tank et al., Neural Granger Causality, 2018

Hyperparameter settings

The dynamic similarity threshold (from 0.5 to 0.9) is not dataset-specific but follows a fixed schedule. This threshold schedule is kept constant across all datasets and does not undergo dataset-specific tuning. Likewise, other hyperparameters such as temperature are selected based on standard practices from prior contrastive learning literature [2] and kept fixed throughout all experiments.

[2] Kim et al., Contrastive Time-series Anomaly Detection, 2024.

Interpretability analysis

CAROTS is inherently interpretable, as it leverages explicitly learned causal graphs and performs anomaly detection by identifying violations of these relationships. We are currently conducting following interpretability analyses:

• Forecasting-based attribution: identifying variables with high causal forecasting error and tracing them back to their parents in the graph to localize disrupted relationships.
• CDA attribution: comparing real anomalies with synthetic ones generated via CDA to identify which causal subgraphs were likely disturbed.

Other comments

• Tighter phrasing and broad statements: We will surely improve the writing of our paper and clarify broader statements in the revised version.
• “How inaccuracies in the learned causal graph impact CAROTS” presents results of incomplete causal graphs (in which CAROTS may underperform).

Questions

• Handing Imperfect Causal Graphs: included in “How inaccuracies in the learned causal graph impact CAROTS.”
• Similarity filter threshold: included in “Hyperparameter settings.”
• Computational Overhead: Due to the character limit, we would greatly appreciate it if the reviewer could refer to “Computational cost of our method” section to Reviewer J2ZH.
• Reusable Code: We fully agree that releasing code enhances reproducibility and transparency. We plan to make the implementation publicly available upon acceptance.

We would like to thank Reviewer aXkV for helpful comments, which we believe will enrich the depth of our work. We are delighted that the reviewer commends that the proposed method is intuitively convincing, outperforms existing approaches, and exhibits robustness across diverse datasets. We tried our best to answer all of the questions during the initial response period, and we are happy to engage in further discussion during the author-reviewer discussion period. Also, the “results of using other causal discovery methods” will be updated later in the discussion period.

Do causality relationships in normal multivariate time-series remain consistent over time?

While we do not assume strict stationarity, previous works in both classical [1,2] and deep learning-based causal discovery [3] has observed that causal structures often remain stable over time in real-world time-series.

To empirically assess whether this statement holds in our setting, we further analyze the evolution of causal structures in three benchmark datasets: SWaT, WADI, and PSM. For each dataset, we split the normal training data into four disjoint, time-ordered segments (Quarter 1 to 4), train a causal discovery model on each, and compute pairwise cosine similarities between the resulting graphs.

Causality Matrix Consistency across Time Segments (Cosine Similarity by Dataset)

The consistently high similarity indicates that the learned causal relationships remain stable across time segments, supporting the validity of our approach.

[1] Spirtes et al., Causation, Prediction, and Search, 2000.
[2] Peters et al., Elements of Causal Inference, 2017.
[3] Kong et al., CausalFormer:..., 2024.

Comparison with CARLA

According to the reviewer’s suggestion, we compare CAROTS with CARLA [4] under the same settings and datasets and report the mean ± std results over three seeds. The results below show that CAROTS outperforms CARLA on most datasets and metrics.

[4] Darban et al., CARLA:..., 2025

Ablation study on sigma

Results of ablation study on σ in CPA over a wide range (0 to 0.4) are presented below (SWaT dataset; mean ± std over 3 seeds):

Performance remains stable across different σ’s, indicating that the generated samples do not degrade model quality, even with higher noise levels. These results suggest that CPA is robust to the choice of σ within a reasonable range.

Results of using other causal discovery methods

We agree that studying the impact of different causal discovery methods is important for assessing the generality of CAROTS. We are currently running experiments with alternative causal discovery methods, and we will upload the results as soon as the experiments are completed.

Explanation for the VAR results

CAROTS behaves differently on the VAR dataset because VAR has different characteristics from other datasets.

VAR is synthetically generated using a linear autoregressive process, where all variable relationships are linear and stable over time. As a result, methods that rely on modeling correlations or co-occurrence patterns (like TimesNet or USAD) are naturally well-suited for this setting.

In contrast, CAROTS is designed to detect anomalies that disrupt more complex or directional causal relationships, especially in non-linear or dynamic systems. This is why it performs particularly well on datasets like Lorenz96 or SWaT, which reflect those properties.

That said, CAROTS still achieves strong results on VAR for certain anomaly types, such as Point Contextual and Collective Global, where detecting multi-variable inconsistency is important. We believe this suggests that CAROTS complements correlation-based approaches and is especially useful when anomalies reflect deeper structural disruptions.

We thank Reviewer vFwM for constructive comments. We are encouraged that the reviewer acknowledges our work's potential to impact various domain that involves multivariate time-series. We hope our response addresses your concerns. Should the reviewer have more follow-up questions, we would be happy to answer them during the discussion period.

Extended ablation study from Table 4

The table below presents the extended ver. of Table 4 on 7 real datasets and 2 synthetic datasets. For Lorenz96 and VAR, the reported values represent the average performance across the four different synthetic anomaly types. In summary, the extended results are generally consistent with Table 4 in the main paper.

• The setting without data augmentation and contrastive learning corresponds to using only A_CD, the causal forecasting-based anomaly score. This result, included in Table 4, is also included in the table above (w/o A_CL). While A_CD alone performs competitively, full CAROTS, which includes CPA, CDA, SOC, and A_CL, further improves the detection results.

• Anomaly detection without both A_CL and A_CD is infeasible because, by definition, every detection method requires at least one anomaly score to score samples. Instead, in Table 4 and the extended table above, we report the results of using only one of the two scores (w/o A_CL and w/o A_CD) to demonstrate the efficacy of each score. CAROTS† (w/o A_CD) achieves the highest detection performance on 6 datasets, which highlights the effectiveness of the proposed causality-driven contrastive learning and A_CL. CAROTS, which additionally uses A_CD, further improves CAROTS† on 3 datasets, indicating that the auxiliary A_CD score offers complementary signals for anomaly detection.

• Table 4 and the extended table demonstrate that CDA and CPA are more effective than conventional data augmentation methods because replacing either one with a conventional method (w/o CPA & w/o CDA) results in a performance drop. More explanation on their effectiveness is detailed in the section below.

Validation for the effectiveness of each module

We hope our extended ablation study, which empirically evidences the effectiveness of each module, alleviates reviewer's concern about less theoretical validation. Below, we clarify the intuitive justification behind the design of each module.

• Novel Data Augmentation (CPA + CDA)

  • CPA (Causality-Preserving Augmentor) generates causality-preserving variations by perturbing causing variables and using the causal forecaster to reconstruct affected variables, ensuring that augmented samples retain the normal causal behavior. This encourages the model to learn representations that belong to diverse yet causally consistent normal patterns.
  • CDA (Causality-Disturbing Augmentor) breaks causal relationships to synthesize anomalies. By injecting perturbations into randomly extracted subgraphs of the causal graph, CDA produces samples that simulate how real anomalies disrupt inter-variable dynamics—something conventional augmentations cannot replicate.
  • Compared to conventional time-series augmentation methods that rely on surface-level distortions (e.g., time warping, noise injection), CPA and CDA leverage the underlying causal structure from the train data to obtain more semantically meaningful and task-relevant augmentations. Together, augmentations from CPA and CDA enable the contrastive learning process to discriminate samples based on causal consistency rather than superficial similarity.

• Novel SOC Loss guides contrastive learning to respect the semantic diversity within normal data by filtering out low-similarity positives early in training. In effect, it yields a more structured embedding space, grouped by semantic diversity.

• Novel Anomaly Scores (A_CL and A_CD): Once contrastive learning with CPA, CDA, and SOC yields a causality-informed embedding space, A_CL detects anomalies by measuring how much a test sample deviates from the causality-preserving embedding space. In addition, A_CD utilizes the causal forecaster to obtain an auxiliary causality-driven signal for anomaly detection; if the causal forecaster yields a high forecasting error, the sample is more likely to be an anomaly.

Ethics review flag

We noticed that an ethical review flag was raised for our paper. To our understanding, our work does not involve any ethical concerns, but we would be grateful for any clarification to help us address this appropriately.