Thank you for your insightful comments and valuable questions.
**Baselines**: Sorry we didn't write this before due to the limited space. Here is our detailed discussion.
Indeed, several baseline methods are designed to handle non-stationary temporal data, such as CD-NOD[1] and JIT-LiNGAM[2]. However, it's important to note that the outputs of these baselines are **summary graphs**, allowing them to identify the causal relationship between time series $X^j$ and $X^i$ without pinpointing the specific time lag $\tau$. In contrast, our method enables the detection of the precise variable $X^j_{t-\tau} \in X^j$ that causes the behavior in $X^i_t$, presenting a distinguishing feature. These approaches highlight distinct aspects from ours. While summary graphs offer a degree of insight, they are inherently less informative about the underlying causal structure in our specific case compared to the detailed window graphs produced by our method.
To showcase our method's capacity to handle periodicity in causal mechanisms with time-lag effect, a capability lacking in other approaches, we will introduce a related baseline called "Regime-PCMCI" in our new experiment. It's worth noting that this algorithm necessitates a linear model (while our method is non-parametric) and additionally mandates precise foreknowledge of the number of causal mechanisms inherent within the time series. Remarkably, based on several experiment results we have, our algorithm's superior performance is sustained even when the accurate count of causal mechanisms is supplied to the "Regime-PCMCI" method. Given the considerable time requirement of the "Regime-PCMCI" method, we regret to say that the result plot is currently unavailable. However, we intend to incorporate the said plot into the paper later.
The significance of the comparison depicted in Fig. 3(a) with the baselines we have is apparent when considering the following points:
1. Our algorithm excels in the reduction of False Positive edges, as indicated by the higher Precision, particularly when compared to the stationary algorithm employed in a semi-stationary setting. This outcome effectively validates **Lemma 3.4**, providing a solid foundation for our approach.
2. With the expansion of the sample size, a notable reduction in False Negative edges becomes evident, as evidenced by the recall rate approaching that of PCMCI, which stands out as the best-performing among the baseline methods.
3. In cases where the ground truth involves periodic causal mechanisms, the misuse of a stationary algorithm leads to a significant amount of error, emphasizing the critical importance of employing an appropriate method, such as ours, to account for such complexities.
As far as we know, our method is the first to address the semi-stationary Structural Causal Model (SCM) with the capability to accommodate periodicity. We hope that our approach can establish itself as a potential reference for individuals interested in periodic causal mechanisms.
**Case Study**:
Thank you for the comment. We will add the following discussion in the camera-ready paper to avoid potential misunderstanding regarding our conclusion in the case study.
We could not comment on whether the result of the case study is significant. We open a door for the related experts; if assumptions A1-A8 are satisfied, the stationary assumption may not hold in this real-world data, and such periodicity exists. However, if the finding is not correct from an expert's viewpoint, the following assumptions may be violated:
1. **Hard Mechanism Change** combined with limited power of CI tests: if there is a soft mechanism change in the dataset, the reliability of the CI test of two variables given their parents will be influenced by the skewed distribution of the parent variables. This effect will be exacerbated by the fact that the sample size will be shrunk by $\omega$.
2. **No Contemporaneous Causal Effects**: There's a possibility of potential causal effects from $X^{ta}_t$ to $X^{cp}_t$ that we're unable to capture in our analysis under this assumption.
Our method provides a sound and robust (shown as figures in the Rebutall pdf file) algorithm for experts in various fields who are interested in validating the presence of periodicity within the causal mechanisms specific to their domain.
**More details about $p(V)\neq 0$**: Suppose Y is a continuous random variable. p(Y) here is the probability density, which is always positive in the domain of Y. For example, this would be a case for any non-degenerate jointly Gaussian distribution.
Thank you again and we wholeheartedly invite further discussions.
[1]Huang, Biwei, et al. "Causal discovery from heterogeneous/nonstationary data." The Journal of Machine Learning Research 21.1 (2020): 3482-3534.
[2]Fujiwara, Daigo, et al. "Causal Discovery for Non-stationary Non-linear Time Series Data Using Just-In-Time Modeling." 2nd Conference on Causal Learning and Reasoning. 2023.