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REVIEW 4 major objections 5 minor 16 references

Temporal windows let PCMCI+ recover causal graphs from irregularly sampled event streams

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T0 review · deepseek-v4-flash

2026-08-01 15:36 UTC pith:XUSMEJCA

load-bearing objection A plausible, well-posed incremental extension of PCMCI+ to irregular time series, but the empirical support is weaker than the abstract claims—window alignment is user-specified and thresholds are oracle-selected. the 4 major comments →

arxiv 2607.18226 v1 pith:XUSMEJCA submitted 2026-07-20 cs.LG stat.ME

Causal Discovery on Irregular Time Series

Martim Penim , Ricardo Ribeiro Pereira , Jacopo Bono , Hugo Ferreira , M\'ario A.T. Figueiredo , Pedro Bizarro This is my paper
classification cs.LG stat.ME
keywords causal discoveryirregular time seriesPCMCI+temporal windowsGaussian weightingevent streamsconditional independencestructural Hamming distance
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that standard PCMCI+, which aligns observations by fixed discrete lags, fails when events arrive at irregular times. Its proposed replacement aggregates causal influence over temporal windows: rectangular windows treat every event in an interval as a valid lagged realization, and Gaussian windows weight events by their distance from a target delay. On synthetic irregular event streams with known causal structure and added noise events, both variants recover the underlying graph with lower structural Hamming distance than standard PCMCI+, with the Gaussian variant the strongest overall. The paper's message is that conditional-independence causal discovery can be carried to irregular data by changing only how temporal neighbourhoods are built.

Core claim

The central claim is that aligning past observations by event index is the wrong operation for irregularly sampled data, because equal index differences do not correspond to equal time delays. The paper replaces index-based lags with time-aware windows: an event at time t is paired with observations in [t-kΔ, t-(k-1)Δ), or assigned Gaussian weights exp(-(δ-kμ)^2/(2σ^2)) centered on a target delay kμ. Using partial-correlation conditional independence tests, the windowed variants achieve lower structural Hamming distance than standard PCMCI+ across causal-to-noise ratios from 1:0 to 1:4; the Gaussian version reaches maximum recall in aligned, overlapping, and high-resolution settings. The cor

What carries the argument

The load-bearing object is the temporal-window triplet constructor. It converts the irregular-sampling problem into a neighbourhood-construction problem: each current event is paired with all observations whose timestamps fall inside a predefined window around the target delay, or weighted by a Gaussian kernel centered on that delay. This leaves the conditional independence tests and the overall PCMCI+ pipeline untouched. Window centre, width, and overlap are the free parameters that determine whether true delays fall near window centres or boundaries, which is exactly what drives performance in the four experiments.

Load-bearing premise

The method assumes the user can supply temporal windows whose centres and widths align with the true causal delay; when the true delay falls on a window boundary, performance drops sharply, as Experiment 2 shows.

What would settle it

Run the Gaussian-window variant on a ground-truth irregular stream while sweeping window configurations whose centres are shifted from the true delay by a fraction of the interarrival time; the claim of consistent recovery would be falsified if every misaligned configuration—and no configuration, absent privileged knowledge—recovers the graph.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • On irregular event streams with unrelated noise events, the windowed variants keep recovering the causal graph while standard PCMCI+ deteriorates rapidly, so users can expect substantially better structure recovery in low-signal settings.
  • Overlapping windows nearly restore aligned-setting performance even when the true delay sits on a boundary, giving practitioners a robust hedge when they are unsure of the causal delay.
  • Shorter, denser windows improve precision and SHD, making temporal resolution a practical tuning knob for localizing causal dependencies.
  • The Gaussian-window variant consistently achieves the best SHD and maximum recall whenever windows are not adversarially misaligned, making it the recommended default among the two variants.
  • Because the conditional-independence machinery is unchanged, existing extensions of PCMCI+ such as nonlinear conditional independence tests can likely be ported to irregular data using the same windowed triplet construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A data-driven procedure for choosing window centres and widths—for instance, from the empirical interarrival distribution or from a preliminary cross-correlation scan—would likely remove the main practical obstacle left by the paper's dependence on user-supplied windows.
  • In settings where event rates vary strongly over time, a kernel whose width adapts to local event density would probably outperform the fixed-width Gaussian used in the paper.
  • The synthetic setup isolates sampling irregularity from the causal mechanism, so the results transfer most directly to processes with one characteristic causal delay; chains with widely varying delays would need additional study.
  • The experiments report the best SHD over p-value thresholds, which represents an upper bound on achievable performance; deployment would require a threshold-selection rule, since the optimal threshold is unknown in practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes an extension of PCMCI+ for causal discovery on irregularly sampled event streams. Instead of indexing lags by discrete event steps, the method defines temporal windows (rectangular or Gaussian-weighted) and pairs each event with observations falling within those windows, then applies the standard PCMCI+ conditional-independence machinery. The method is evaluated on synthetic data with known causal structures, varying the ratio of causal to noise events, and is compared with standard PCMCI+ in terms of SHD, recall, and precision. The authors report that both variants outperform standard PCMCI+, with the Gaussian variant generally achieving the lowest SHD, and conclude that time-aware window construction improves causal discovery under irregular sampling.

Significance. If the central empirical claim were established, this would be a practical and interpretable extension of a widely used causal discovery method, of interest to applied work on event streams in finance, healthcare, and sensor data. The paper preserves the conditional-independence semantics of PCMCI+, which is a strength. It is also transparent in disclosing that the reported SHD is an upper bound from optimizing the p-value threshold on the evaluation data, and the misalignment experiment is a useful robustness probe. However, the current evaluation does not substantiate the abstract's claim that the method "consistently recovers" the causal graph: the reported performance depends on oracle choices for the temporal windows and for the p-value threshold, and the Gaussian weighting mechanism is not fully specified. The significance of the contribution is therefore conditional on a more rigorous and reproducible evaluation.

major comments (4)
  1. [Section 3.2, Section 5] The Gaussian-window variant defines a weighting function w(δ;k,μ,σ) but never specifies how these weights enter the conditional independence test. The text states in Section 3 that the CI tests remain unchanged, while Appendix B says the Gaussian variant "replaces the binary inclusion rule with a continuous weighting function." If weights are not used in the partial-correlation estimator, the variant reduces to a rectangular window over a wider support; if they are used, the weighted estimator is not defined. This ambiguity is load-bearing because the abstract's "substantially outperforms" claim relies on the reported SHD of the Gaussian variant. Please specify the exact computational role of the weights.
  2. [Section 5, Figure 1] The reported SHD is the best over p-value thresholds chosen on the evaluation data itself: "we report the threshold yielding the best SHD for each method and noise level" and describe this as the "upper-bound of the optimal p-value optimization strategy." This is an oracle metric, not the expected performance of a complete pipeline. Because no calibration or cross-validation is used, the absolute SHD values do not support the conclusion that the method "consistently recovers" the causal graph. Please report results at a fixed threshold (e.g., the pre-specified α=0.01) or use an independent calibration set for threshold selection.
  3. [Section 5, Experiments 1 and 2] The method is evaluated with temporal windows centered on the true causal delay of 6 hours in Experiment 1, which is ground-truth information not available in real applications. Experiment 2 shows that shifting the windows by only 2 hours, placing the delay on a boundary, markedly degrades performance, especially for the Gaussian variant. The paper provides no data-driven procedure for choosing window centers, widths, or the number of windows. Consequently, the method is not established as a drop-in replacement for PCMCI+ on irregular data where the causal delay is unknown. Please provide a sensitivity analysis over delay misalignment and a practical selection rule, or temper the central claim accordingly.
  4. [Section 4, Section 5] The abstract uses "consistently recovers" and the results are presented as curves without error bars or a stated number of random seeds. Figure 1 shows single SHD values per configuration. With only one synthetic realization per setting, the reader cannot assess whether the observed improvement is statistically stable. Please provide multiple replicates with confidence intervals or error bars, and clarify how many random seeds were used.
minor comments (5)
  1. [Section 4, Eq. (1)] The data-generating process defines causal dependence over event indices (X^{(j)}_m depends on X^{(i)}_{m-1}), so the actual causal delay is a random variable with mean 6 hours. Calling 6 hours the "true causal delay" is imprecise; clarify that this is the mean interarrival, not a fixed physical delay.
  2. [Section 3.2] The parameter μ in the Gaussian weighting function is used but not defined in the text. Define it as the target delay spacing (or window-center spacing) and state how σ is chosen.
  3. [Section 5] The number of lag windows used in each experiment is not explicitly stated. For example, Experiment 1 uses three intervals, while Experiments 3 and 4 appear to use more. Listing the exact window configurations and the resulting lag count in the text would improve reproducibility.
  4. [Appendix A.2] Typo: "Faculdade de Ciêncidas" should be "Faculdade de Ciências".
  5. [General] No code or data availability statement is provided. Releasing the synthetic data generator and the implementation would materially strengthen the reproducibility of the empirical claims.

Circularity Check

1 steps flagged

Headline recovery claim rests on test-set-optimized thresholds and oracle-aligned windows; the core PCMCI+ reformulation is not itself circular.

specific steps
  1. fitted input called prediction [Section 5, Results (paragraph following Figure 1); used in abstract and conclusion]
    "To avoid relying on a single edge-selection cutoff that could unfairly favour particular methods, we additionally evaluate all detected edges across p-value thresholds below 0.01 and report the threshold yielding the best SHD for each method and noise level. As such, the reported performance for each method represents the upper-bound of the optimal p-value optimization strategy."

    The reported SHD is not the performance of any fixed procedure: it is defined as the minimum over p-value thresholds on the same test data, optimized separately per method and noise level. The abstract's claim that the method 'consistently recovers the underlying causal graph and substantially outperforms the standard PCMCI+' is therefore supported only by in-sample upper bounds, not by a prediction from a complete pipeline. The comparison is constructed by the evaluation rule, so the headline advantage is partly an artifact of selecting the best threshold after observing the target graph.

full rationale

The method itself is not derived from its target: Section 3 simply replaces fixed-lag triplet construction with rectangular or Gaussian temporal windows while leaving the PCMCI+ conditional-independence machinery unchanged; no equation defining the causal output is reused as an input. There is no load-bearing self-citation (PCMCI+ is cited from the original authors, who are not the present authors), no imported uniqueness theorem, and no ansatz smuggled via citation. The main circularity-like element is the evaluation protocol: Section 5 reports the p-value threshold that minimizes SHD on the same test set per method and noise level, explicitly calling it an upper bound, and the abstract/conclusion use these oracle-tuned numbers as evidence of consistent recovery. Additionally, Experiment 1 builds the window configuration around the known 6-hour causal delay, so the favorable results require privileged alignment information; Experiment 2 shows that a two-hour shift markedly degrades performance. This second issue is a generalizability limitation rather than a construction-circularity, because the method does not claim to estimate the windows. On balance, one evaluation choice partially reduces the headline empirical claim to an in-sample optimum, but the underlying causal-discovery pipeline is not circular by construction, giving a partial evaluation-level circularity score of 5.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim depends on user-chosen window geometry and a test-set-optimized threshold; no invented entities are introduced. The key axioms are that time-windowed aggregation preserves CI-test validity and that event-index AR causality is a suitable proxy for irregular time-series causality.

free parameters (3)
  • Temporal window geometry (number of lags, centers, width Δ) = Exp1: Δ=4h, intervals [−4,0), [−8,−4), [−12,−8)h; Exp3/4: overlapping/finer variants
    Set manually; Experiment 1 centers windows on the true 6h mean interarrival of the causal stream, using ground-truth knowledge rather than a data-driven procedure.
  • Gaussian kernel standard deviation σ = Not reported
    The Gaussian weighting function w(δ;k,µ,σ) requires σ to control decay, but no value or tuning procedure is given anywhere in the paper.
  • Significance threshold p for edge selection = Best SHD per method and noise level
    Section 5 reports the threshold yielding the lowest SHD on the evaluation data, i.e., an oracle choice on the test set rather than a pre-specified rule.
axioms (3)
  • domain assumption Temporal windows, when centered appropriately, capture the causal delay underlying the irregular stream.
    The method defines lag k via temporal intervals in Section 3.1. Experiment 2 shows that misaligned windows degrade performance, so this assumption is load-bearing for the claimed 'consistent recovery'.
  • domain assumption Replacing each discrete lag with a set of observations inside a temporal window preserves the validity of PCMCI+'s conditional independence tests.
    Section 3.1 states that 'only the mechanism used to construct aligned triplets is adapted', but no argument is given that multiple observations per window yield valid inputs for partial correlation or other CI tests. This is a structural assumption about statistical validity.
  • domain assumption The synthetic construction in Eq. (1), where causality runs over event indices, is representative of real irregular time-series causal processes.
    Data generation defines X_m^(j) in terms of X_{m-1}^(i), so time itself is not in the causal equation; the 6h delay is an emergent property of the sampling process. This makes the benchmark only indirectly about continuous-time causality.

pith-pipeline@v1.3.0-alltime-deepseek · 6451 in / 12405 out tokens · 98691 ms · 2026-08-01T15:36:41.097528+00:00 · methodology

0 comments
read the original abstract

Causal discovery methods have shown strong performance in temporal systems, but they typically rely on regular and discrete lag structures, limiting their applicability to regularly sampled data. However, many real-world tasks require dealing with irregularly sampled streams of events, such as sensor streams, healthcare data, and financial transactions. In this work, we propose an extension of PCMCI+, a state-of-the-art method for causal discovery on regular multivariate time series, to allow for handling irregular time series. Instead of modelling causal relations through fixed-lag dependencies, our method aggregates causal influence over predefined temporal windows. We evaluate our method on synthetic irregular event streams with known causal structures under different signal-to-noise ratios, showing that it consistently recovers the underlying causal graph and substantially outperforms the standard PCMCI+ on irregularly sampled data.

Figures

Figures reproduced from arXiv: 2607.18226 by Hugo Ferreira, Jacopo Bono, M\'ario A.T. Figueiredo, Martim Penim, Pedro Bizarro, Ricardo Ribeiro Pereira.

Figure 1
Figure 1. Figure 1: SHD for experiments 1 (top-left), 2 (top-right), 3 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Histograms of interarrival times across different causal-to-noise ratios. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time series of features, grouped according to their underlying causal structures, for the first 100 events, under [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between rectangular-window (left column) and Gaussian-window (right column) temporal weights [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the window configurations utilized in Experiment 1. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the window configurations utilized in Experiment 2. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the window configurations utilized in Experiment 3. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the window configurations utilized in Experiment 4. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Recall and precision for Experiments 1-4 (first through fourth columns, respectively) across varying causal-to-noise [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: SHD, recall and precision for Experiments 1-4 (first through fourth columns, respectively) across varying [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

discussion (0)

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