Tensor Time-Series Forecasting and Anomaly Detection with Augmented Causality

Thanks for the review and paying attention to our forecasting module. Your concerns are listed and addressed below.

• First of all, all the suggested literature [1-6] is added and discussed in the revised paper and colored blue. Furthermore, paper [2] is a very new paper accepted by NeurIPS 2023 and has not been released to the public yet. Then, we discussed its content based on the previous version on arXiv.

• Second, we add state-of-the-art baselines since one of our baselines, DCRNN, is 2018. On the one hand, we would like to mention our other baseline GTS is in ICLR 2021, which is newer than the suggested baselines [3] in IJCAI 2018, [4] in AAAI 2020, and [5] in AAAI 2021. On the other hand, in the suggested SOTA algorithm [6] in AAAI 2023, baselines [3-5] are included and compared. Therefore, we did the new experiments for the SOTA baseline ST-SSL [6]. The new table is shown below and has also been included in the revised paper (colored blue). In general, the SOTA baseline ST-SSL achieves a very competitive forecasting performance through contrastive learning, which is superior to one of our proposed methods TacSas, and slightly lower than our TacSas++, which is newly proposed by not only discovering the fine-grained temporal structures but also getting calibrated by contrasting the nearest time window for persistence forecasting.

ST-SSL relies on the off-the-shelf input structure like what DCRNN does, but its input is usually time-evolving and hardly acquired in most real-world settings (and usually gets replaced by random ones). It is this very problem that motivates our paper.

• Third, thanks again for your raised concern on our forecasting module. They are very actionable and improve our paper. Please also do not overlook our theoretical and empirical efforts in terms of fine-grained temporal-spatial structures discovery, its leverage for anomaly detection, the end-to-end generative model for realizing them all, and the new data source.

Thanks so much for your questions and concerns, they are listed and addressed below.

• First, all your suggested related work is added and discussed in the updated paper and colored blue.

• Second, the reason why we did not compare with your suggested SOTA baseline in the case of forecasting "Li et al., Geophysical Research Letters 50" and "Lira et al, Sensors 22" is that we searched online but did find the published code for those methods. But, on the one hand, they are now added to the revised paper and discussed. On the other hand, we added a new sota baseline suggested by Reviewer TMqL for forecasting, which is a new paper this year "Ji et al., AAAI 2023. Spatio-Temporal Self-Supervised Learning for Traffic Flow Prediction.". In short, our proposed method can still outperform the baselines as follows. More details are added in the revised paper.

• Third, "why did not consider other kinds of anomalies, like inter-variable relationship anomalies"? Thanks for the question. To the best of our knowledge, facing various types of anomalies, different methods are proposed for different anomaly patterns. On the one hand, in our paper, the proposed model did the value anomaly detection, and the selected baselines and discussed related work are also the methods that only did the value anomaly detection. On the other hand, the structural anomaly detection method you suggested also targets one specific type of anomaly among various anomaly patterns, which is also unable to do value anomaly detection. Each aspect has its own value, like value anomaly detection can help with extreme weather detection, and structural anomaly detection can help with high-tech equipment deployment. Proposing a universal anomaly detection method for all possible anomaly patterns is very interesting, we would like to explore it as a future direction. But for now, it is beyond the scope of our submission.

• Finally, thanks again for your raised concerns. Please also do not overlook our theoretical and empirical efforts in terms of fine-grained temporal-spatial structures discovery, its leverage for anomaly detection, the end-to-end generative model for realizing them all, and the new data source.

Thanks for your raised concern.

• First, we further formalized and polished the language of the paper, and uploaded the revised paper.

• Second, the reason why we use the term "proof sketch" is that the original paper gives the proof, that's why we included the citation in our Thm 3.1.

• Third, we could not agree that the "tensor time series" is meaningless. The reason why a toy example is called a toy is because it is simply created to illustrate the key idea. Take our data sets as an example, 238 (spatial locations) * 365 (days) * 45 (weather features), each dimension has its unique meaning. Of course, you can flatten it into a vector, then the information is lost. If doing so, let alone tensor time series, multivariate time series can be represented as vectors.

• Finally, we add the SOTA time series forecasting baseline ST-SSL (Ji et al., 2023) in the below table, which outperformed many simple forecasting baselines like ARIMA (Kumar and Vanaiakshi 2015) and SVR (Castro-Neto et al. 2009) in their paper.

Thanks so much for your time and review, your questions are addressed as follows.

• First, regarding how TacSas can capture the ground-truth causality, we prepared the ground-truth benchmark Lorenz96 and compared TacSas with the SOTA baseline GVAR (Marcinke-vics and Vogt, 2021). More details can be found in Appendix B.1. In short, our TacSas can discover the ground-truth causal structure defined by Lonrenz96 with competitive accuracy compared with GVAR.

• Second, for the scenario that causal structures are hardly available, in the climate datasets we contributed as a new time series source (where the time series items are rich but ground causal structures are not available), the visualization plotted in Figure 3 quantitatively shows the causal structures captured by TacSas, which is the potential reason why TacSas can outperform DCRNN (Li et al., 2018) and GTS (Shang et al., 2021) for capturing fine-grained temporal-spatial structures in the hardly-available scenario.

• In addition, our paper gets further improved during this rebuttal period. We compared with another state-of-the-art baseline ST-SSL (Ji et al., 2023), proposed a more accurate forecasting module TacSas++, and extended the literature discussion for various related works. All the updates are included in the revised paper and colored blue.