TrendDiff: Decoupling Intrinsic and Measurement Trends for Enhanced Time Series Causal Discovery

As pointed out by other reviewers, the novelty of the paper is lacking, since many fundamental aspects come from the CD-NOD paper.

I think that the paper should be more transparent about not being able to simultaneously handle intrinsic and measurement trends for the same variable. This is only discussed at the very end. I would recommend clarifying it as a model assumption within Section 3. In general, I think that the model assumptions can be stated more clearly.

Most trends in the experiments section are sinuosoidal. There is a lack of evaluation for non-periodic trends (ex. Growth or decay).

I found some notations/definitions ambigious:

Although reasonably intuitive, intrinsic and measurement trend should be defined more rigorously in Definitions 2-3, given the importance to the paper.

I think that the phrasing "X2 and X3 are not observable" (line 63) should be adjusted, since it can easily be misinterpreted to mean that they are completely unobserved, rather than their true values not being observed.

What is C precisely? Is it the set of measurement times (modeled as a r.v.)?

Figure 6a) legend typo "Guassian" -> "Gaussian"

1. The authors seem to confuse the concept of trend with nonstationarity. Nonstationarity means the causal mechanism $f_i$ that generates $X_i$ from its parent $PA_i$ changes with time. For trend, such a change should has a direction (increasing or decreasing) with time, see Def. 1 of (White and Granger, 2011) in your reference. However, Def. 1 in the paper fails to signify this.

In a similar way, measurement trend should not be confused with meansurement error along time. According to Def. 3 in the paper, a white noise along time or a Brownian motion can be considered as measurement trend, however, there is no tendency in them.

2. The novelty of the proposed techniques is limited. The use of a time indexing variable $C$ to identify nonstationary variables is a classic approach established by CD-NOD (Huang et al., 2020). I believe that trend warrants special attention, particularly regarding their properties of increasing or decreasing over time, rather than trivially applying results from nonstationary causal discovery.

1. Without an explicit statement in the paper, this algorithm appears to be an extension of CD-NOD. The main paper describes phase 1 (Algorithm 1) in detail; however, I believe Algorithm 1 and Theorem 1 have already been introduced in Huang et al., 2020. The citation in phase 1 gave the impression that only step 4 in Algorithm 1 and the complete proof are originally presented in Huang et al., 2020. However, after reviewing the previous work by Huang et al., 2020, it seems the entire Algorithm 1 is from that paper. Therefore, I think the citation in this section is problematic.

2. There is no theorem guaranteeing phase 2, whereas Theorem 1 in phase 1 has been proved in previous work.

3. Considering that measurement trends are a type of measurement error, could the models introduced in the related work aimed at causal discovery in the presence of measurement errors be used as baselines? I understand that there are no previous methods capable of identifying time trend variables, but for the causal discovery part, these models could at least be treated as baselines in Fig. 6 (b). The effectiveness of the proposed algorithm for estimating the correct causal graph is difficult to evaluate without any baselines.

4. Algorithm 1 and Theorem 1 are demonstrated in a setting that assumes there are no measurement errors, correct? If there are errors in the dataset, how can it be guaranteed that the results from Algorithm 1 are correct, given that phase 2 heavily relies on the correctness of Algorithm 1?

• The current model assumes that all causal relationships are instantaneous, an oversimplification for domains requiring lagged causality (e.g., health impacts of pollution). To broaden its relevance, the algorithm could integrate methods for time-lagged effects allowing it to account for delayed causation in complex time-series data.
• A crucial element of the TrendDiff algorithm is Algorithm 2, which represents the main novelty of this paper. However, this algorithm is placed in the appendix rather than the main text, obscuring its importance. Upon closer examination, it appears that Algorithm 2, along with other components, primarily serves as a data preprocessing step rather than a standalone causal discovery technique. This distinction is not clearly stated in the paper, which may lead to confusion about the intended scope and purpose of the method. Meanwhile, Algorithm 1, which is nearly identical to the CD-NOD algorithm, could be moved to the appendix, as it offers limited novel contribution and primarily replicates an established process for detecting time-influenced variables. Additionally, a key part of this preprocessing involves the Savitzky-Golay filter to detrend measurement-influenced variables and reduce noise. However, the filter’s role, configuration, and the rationale for its selection are not discussed in the main paper, leaving a gap in transparency regarding this essential step. A clear justification for using the Savitzky-Golay filter, including details on parameterization and potential alternatives, would improve understanding of the algorithm’s robustness and clarify its intended application as a preprocessing technique.
• The paper does not provide a complete description of how the final causal graph is constructed, leaving out important details on whether additional steps were applied beyond the initial graph derived in Phase 1. While the initial causal graph is generated through conditional independence tests and orientation based on dependency relationships, it is unclear if this graph is directly used as the final causal graph or if further refinement steps were applied. The absence of a full algorithm or detailed methodology for finalizing the causal graph limits the reproducibility and clarity of the approach. Including this information would clarify whether the initial skeleton suffices or if other techniques (such as additional orientation rules or refinement methods) are necessary to achieve the final, fully directed causal graph.
• The proposed TrendDiff algorithm is based on CD-NOD, which theoretically identifies time-influenced variables as trends. However, an examination of the CD-NOD GitHub repository and example datasets (https://github.com/Biwei-Huang/Causal-Discovery-from-Nonstationary-Heterogeneous-Data/tree/master) indicates that the synthetic data relies on sinusoidal functions to represent time dependencies. Sinusoidal patterns more accurately model seasonality rather than long-term trends, introducing ambiguity in Phase 1 of TrendDiff. This means that Phase 1 of the algorithm may be detecting seasonality along with trends, which undermines the core objective of differentiating intrinsic trends from measurement errors. Without a clear separation between seasonality and true trends, the validity of the paper's approach is in question, as the methodology may not be isolating trends as intended. A more refined approach to differentiate between seasonality and trends would strengthen the algorithm’s theoretical foundation and improve its practical applicability.
• In Section 5.1, the authors state that "all trends were modeled as sinusoidal functions with periods w chosen randomly from a uniform distribution Unif([5, 25])". However, this approach models seasonality rather than a true trend, as sinusoidal functions are inherently cyclical and represent recurring patterns over time. True trends, in contrast, are typically non-repeating, directional changes that occur over a longer timeframe. By modeling trends with sinusoidal functions, the paper conflates seasonality with trend, which may lead to inaccuracies in trend differentiation and limit the algorithm’s effectiveness in real-world applications, where identifying genuine trends—rather than cyclic seasonal effects—is crucial.
• Phase 1 of TrendDiff relies on CD-NOD to generate the initial causal graph, using a surrogate variable to represent time. However, CD-NOD’s approach is nearly identical to the PC algorithm and does not account for time lags in causal relationships. Given that the proposed approach uses the causal skeleton and time-influenced variables as foundational components, employing algorithms like JPCMCI+ (https://proceedings.mlr.press/v216/gunther23a/gunther23a.pdf) and CDANS (https://proceedings.mlr.press/v219/ferdous23a/ferdous23a.pdf) would likely produce a more accurate causal skeleton with time lag considerations. Additionally, these algorithms are more efficient in handling high-dimensional data, which would improve the scalability and robustness of the method.
• The experimental section lacks a comprehensive evaluation of TrendDiff's performance relative to similar algorithms, such as JPCMCI+ and CDANS, both of which incorporate the time index as a surrogate variable and are better suited for time-series data. Benchmarking TrendDiff against these methods would provide a more robust assessment of its efficiency and accuracy, offering insights into its relative strengths and weaknesses. While the paper focuses solely on TrendDiff, the real data analysis only presents results from PCMCI+ with TrendDiff, without evaluating the effectiveness of TrendDiff or comparing it with other methods. Given that TrendDiff primarily functions as a data preprocessing step rather than a standalone causal discovery approach, a comparative study using different causal discovery methods on synthetic datasets, both before and after applying TrendDiff, would yield more meaningful insights. Such an evaluation would clarify the preprocessing step's impact on causal discovery accuracy and enhance understanding of its role in improving downstream analyses. Additionally, Figure 6(b), titled "Performance of PC algorithm using data pre- and post-elimination of detected measurement trends," compares the PC algorithm's performance before and after removing measurement trends. However, the PC algorithm is not suitable for time-series data as it assumes an i.i.d. structure and does not consider temporal lags, which prevents it from capturing the dependencies and causal lags inherent in time-series data. This limitation raises concerns about the validity of using the PC algorithm in this evaluation, as the lack of time-series-specific adaptations undermines the reliability of the results presented in Figure 6(b) and calls into question the overall evaluation method's accuracy in a time-dependent context.