TCD-Arena: Assessing Robustness of Time Series Causal Discovery Methods Against Assumption Violations
Authors: Gideon Stein
Niklas Penzel
Tristan Piater
Joachim Denzler
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Discovery
Benchmark
Robustness
Time-Series
Causality
Primary area: datasets and benchmarks
TL;DR: large scale study on the robustness of causal discovery algorithms for time series data against violations of their assumptions

Abstract: Causal Discovery (CD) is a powerful framework for scientific inquiry. 
Yet, its practical adoption is hindered by a reliance on strong, often unverifiable assumptions and a lack of robust performance assessment.
To address these limitations and advance empirical CD evaluation, we present **TCD-Arena**, a modularized, highly customizable, and extendable testing kit to assess the robustness of time series CD algorithms against stepwise more severe assumption violations. 
For demonstration, we conduct an extensive empirical study comprising around 30 million individual CD attempts and reveal nuanced robustness profiles for 33 distinct assumption violations. 
Further, we investigate CD ensembles and find that they have the potential to improve general robustness, which has implications for real-world applications. 
With this, we strive to ultimately facilitate the development of CD methods that are reliable for a diverse range of synthetic and potentially real-world data conditions.

Published as a conference paper at ICLR 2026
TCD-ARENA: ASSESSING ROBUSTNESS OF TIME SE- RIES CAUSAL DISCOVERY METHODS AGAINST As- SUMPTION VIOLATIONS
Gideon Stein, Niklas Penzel, Tristan Piater, Joachim Denzler Computer Vision Group Jena Friedrich Schiller University Jena Jena, Thuringia, 07743, DE g deon. stei -etteinnnnj-eeeeeee
ABSTRACT
Causal Discovery (CD) is powerful framework for scientific inquiry. Yet. its prac- tical adoption is hindered by a reliance on strong, often unverifiable assumptions and lack of robust performance assessment. To address these limitations and advance empirical CD evaluation, we present TCD-Arena, modularized, highly customizable, and extendable testing kit to assess the robustness of time series CD algorithms against stepwise more severe assumption violations. For demonstration, we conduct an extensive empirical study comprising around 30 million individual CD attempts and reveal nuanced robustness profiles for 33 distinct assumption vio- lations. Further, we investigate CD ensembles and find that they have the potential to improve general robustness, which has implications for real-world applications. With this, we strive to ultimately facilitate the development of CD methods that are reliable for diverse range of synthetic and potentially real- world data conditions.
INTRODUCTION
Causal Discovery (CD) holds great potential for addressing scientific hypotheses in fields where randomized control trials difficult impossible Glymour al. (2019). Despite this promise, the widespread adoption of CD methods by practitioners remains limited. Recent works (Brouillard et al.)
2024 Yi al. 2025 Faller et 2024) attribute this mainly two key factors: First, existing CD methods often rely on strong, idealized assumptions (e.g., no hidden confounders stationarity) that are difficult validate or, real-wa world scenarios, simply unverifiable, even if they underpin theoretical guarantees. Second, empirical evaluations of CD methods predominantly use idealized synthetic data, which can overestimate performance and offer limited insight into robustness under imperfect but realistic conditions. Consequently, practitioners hesitate to adopt CD) methods where their output reliability is limited (Kaiser Sipos, 2021 Nastl & Hardt 2024 Poinsot et al. (025).. To overcome this issue, there has been recent push towards more benchmarking as it is the de facto golden standard Mogensen et al., 2024 Herdeanu al.) 2025). However, the scarcity of real-world datasets with known causal ground truth continues to hinder a full reliance on empirical validation of CD methods.
Machine Learning (Neal et al. 2023 Stein et al.) 2024a Wang 2024
As possible alternative real-world benchmarks, recent studies investigate CD performance when specific assumptions are violated (Montagna er al.) 2023a; Yi et al., 2025 Ferdous et al., 2025). Furthermore, the robustness of CD methods to hyperparameter selection has recently been highlighted Machlanski et al. 2024). Building on these emerging efforts in empirical evaluation and aiming to unity them, we present TCD-Arena] modularized, fully customizable testing kit to assess CD robustness against assumption violations. Next to an unprecedented scale, TCD-Arens focuses on three so far sporadically addressed aspects: (1) temporal data, that introduces additional challenges and opportunities; (2) stepwise violation intensities, crucial for capturing nuanced performance degradation rather than binary pass/fail outcomes; and (3) a focus on violations often encountered in real-world settings. In this paper, demonstrate the benefits of TCD-Arena by conducting an extensive empirical study on the robustness of CD algorithms. Specifically, we evaluate ten CD algorithms that cover all four common CD archetypes Assaad al. (2022b). We evaluate these

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algorithms across 33 weal-world-insppred, partly semi-synthetic assumption violations, each scaled in intensity. By performing around 30 million individual CD attempts, we find that different methods differ in their ability to handle assumption violations. Additionally, we investigate hyperparameter sensitivities with respect to robustness and model misspecification, two aspects we believe are critical for applications to novel real-world data. Further, we investigate ensembles of CD methods, which have received little attention in the literature, and conclude that they can improve general robustness. Standing with Poinsot et al. (2025) and recognizing the pressing need for more nuanced CD evaluation, we attempt to further establish robustness analysis as an alternative to traditional benchmarking and theoretical analysis. With this, we hope to ultimately facilitate the development of CD methods that are reliably applicable to real-world data. In summary, this paper makes the following contributions:
1. The introduction of TCD-Arena, open-source, customizable toolkit for quantifying the robustness of CD across diverse time series data that fosters long-term comparability.
2. A large-scale empirical study that evaluates robustness of ten time series CD methods against 33 stepwise intensified, partly semi-synthetic, assumption violations.
3. An investigation into ensembling CD methods with respect to violation robustness.
BACKGROUND AND THEORETICAL PRELIMINARIES
To ground our empirical investigation, we begin by selectively revisiting the relevant theoretical background. Let X E DXT be a D-variate time series comprising T samples from D interacting variables, generated by an unknown underlying causal process. The objective of time series Causal Discovery (CD) is infer the causal relationships among D variables from the observed data X. These relationships are commonly represented as a Structural Causal Model (SCM) (Peters et al. 2017). For each variable Xi., the SCM contains assignments of the form:
X Sc (Pa(X.,)), (2)),
(1)
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where Pa(X,.t) is the set of direct causal parents of Xs,, fü is causal mechanism, and 6,,, is independent innovation noise. The set of assignments within an SCM defines a directed graph G = G=(VEE) EV. In this work, we distinguish between contemporaneous (Xj,, for j # i) and lagged effects 5, for k > 0) by evaluating the recovery of the following three distinct graph structures: First, the lagged window causal graph (GLWCG) provides lag- lag-specific causal dependencies up to a maximum lag L. Here V includes each variable at time step and at all past lags: V {x {L..... L}}. A directed edge X3.-- + X,, exists in G if XX, is in
EE ()) Pa(X,,)) Note that in this representation, edges only connect past variables to variables at step I. Second, the lagged summary graph (GLSG) summarizes time-lagged relationships. Its vertices are defined as V {X1,.... )D). A directed edge X, - X, exists in GLS5 if X3 E Pa(X,..) for at least one / 0. Third, the instantaneous graph (G"NST) captures only contemporaneous relationships. is a directed graph with vertices V = ( Xp)_ A directed edge X, - X; exists in GINST iff x,,, is in Pa( .e). While this framework formalizes causal interactions, the identifiability of any G from X requires a number of assumptions about Eq. (I). For time series, the direction of time Bauer et al. 2016) (effects cannot precede causes) aids with identifying lagged relationships (GLWCG and GLSC), ), generally requiring fewer restrictive assumptions. However, recovering GINST is more challenging (and is not addressed by all CD methods), resembling causal discovery from i.i.d. data. We refer to (Pearl] 2009 Peters et al., (017)) for comprehensive introduction as well as to (Spirtes) 2001 concerning constraint-basee algorithms.
ct al.
Despite the fact that many specific assumptions underpinning CD methods can be relaxed individually, a core set of strong. partly implicit, assumptions generally remains necessary to guarantee the identifiability of any SCM, as the causal hierarchy levels almost never collapse |Bareinboim et al. (2022). Further, even these assumptions can be perfectly met in synthetic data, real-world data will often violate many of them, which can degrade the performance of CD algorithms Kaiser & Sipos (2021): Nastl & ( (2024). On top, many assumptions are not verifiable without having access to the full SCM. c.g., the appropriate conditional-indes independenceee test (Shah & Peters (2020).
For widespread practical adoption, it is therefore essential to assess method performance under suboptimal conditions (Ponsoot al. (2025). In response to these challenges, and mirroring trends in other machine learning domains, there is a growing emphasis on developing real-world Steein al. (2024a): MMgensen et al. (2024) as well as semi-synthetic Cheng et al. (2023); Herdeanu et al. (2025) benchmarks, or kits such as Muñoz-Mari et al.] (220) or Zhou et al. (2024) for CD. Notably, while wellwwordd and semi-synthetic datasets are essential, they come with tradeoff of often having an at least partly unknown data generation process (outside of the causal ground-truth), which leaves room for extensive synthetic benchmarks Poinsot et al. (2025) Furthermore, as aggregating extensive real-world causal ground truth is notoriously challenging. alternative approaches have been introduced to enable the empirical evaluation of CD method performance. For instance, [Schkodal et al. (2024) proposes lcave-one-out cross- cross-validation to assess the predictive performance of CD algorithms. Faltenbacher et al. (2025) introduces an internal consistency score, specifically for PC- style algorithms algorithmms/Spirtss al. (200T). to help validate the inferred graph. Moreover, Machlanski et al. (2024) advocates evaluating hyperparameter sensitivity, which has implications for method selection in practical applications. Closely related to our work, et al. (002)) and Montagna et al. (2023a) test the performance of i.i.d. sample-based CD methods for fully violated assumptions. Further, [erdous al. (2025) provides an insightful study that investigates the impact of five different real-world complications on the performance of CD methods. Notably, robustness has also been explored in a general statistical context as well Rasch & Guiard (2004): y: Zimmerman (2014). Nevertheless, the impact of such violations remains under-investigated, particularly for time series data and across differences in violation severity. Finally, ensembling strategies, as a practical tool to improve robustness in other machine learning domains Arpit et al. (2022); Mienye & Sun (2022). remain largely unexplored in the CD literature. While recent work has investigated ensembling over variable subsets recover large graphs Wu et al. (2024), the potential to improve resilience against assumption violations has not yet been studied.
3 STEPWISE INCREASING ASSUMPTION VIOLATIONS
While the CD literature explores relaxing certain assumptions, identifying causal structure from X alone typically relies on a core set of assumptions to guarantee identifiability. Although prior work partially analyzes resilience to binary assumption violations Yi et al. (2025): MMnntgnna et al.)
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(a) In experiments, the sources of randomness for the observational noise variables Çi. are standard normally distributed (A 1)) ran- dom variables Th, and T,s which are consequently influenced by various factors, e.g. the signal strength (Vobsmul). Both a and @ denote hyperpa- rameters (details in Apx. [.I). For Vahuyeat is entirely dependent extemal (possibly processed) time series.
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(2023a); Ferdous et (2025) a pertinent question arises: How robust are certain CD methods against different severities of assumption violation? This question is critical in applied settings. For instance, the mere existence of observational noise is less informative than understanding its robustness to its presence. Addressing this requires framework for varying the severity of these violations. In this study, we introduce TCD-Arena for this purpose. In total, we implement 33 distinct assumption violations, each parameterized allow for stepwise increase of its severity. We individually describe these in the sections to come. Generally, we focus on covering commonly made assumptions Runge (2018) along with real-world complications that practically relevant. We also focus on providing multiple alternative implementations of a specific violation if applicable. Further, we restrict our exploration the following restriction for Eq. (I):
X
Ar,d, fid,((aa + ce,s
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where A specifies coefficient matrix and fi,d, an edge-specific univariate function and ct,i inde- pendent innovation noise. Crucially, any non-zero element in A denotes a corresponding edge G. Further, for violations that do not concern the causal mechanisms, Si is the identity function, and all interactions are linear. To help with clarity, we mark individual violations as Viype-
Notably, we keep the following violation descriptions brief and include a summary table, graphical depictions, specific violation step sizes, and detailed design choices for each violation Apx. A and Apx.[B Finally, while the experiments in this paper focus assessing the effect of each implemented violation individually, TCD-Arena allows for freely combining all implemented violations to create highly customized data scenarios. We further discuss this in Apx. D.6
Observational Noise (Vars) Many theoretical guarantees in causal discovery assume noise-free measurements, despite the fact that measurement errors are practically unavoidable and introduce discrepancies that distort true causal relationships [Scheines & Ramsey (2016). In an additive form, observation noise can be defined as: X,, = X., + Çi,e. where Ç,,e denotes observational noise. While standard independent additive noise (V (obsaadd is prevalent, other noise types can occur depending on the measurement process. To this end. we investigate the impact of five additional types of observational noise structures on CD. Fig. 2a contains a concrete list. Additionally, details, hyperparameters, and discussions can be found in Apx. B.1 In particular, we include multiplicative, signal-dependent noise (V Vobs.mul), with real- worldworld examples such as temperature sensors that exhibit lower precision at higher values Bentley (1984) and speckle noise in image processing Liu al. (2014). We include time-depent deme-deenddent noise (Vob,,ume) to simulate cycles or linear sensor drift. Further, model autoregressive noise structures (V ). i.c., disturbances in the measurements that persist for multiple time steps. Similarly, we include common observational noise (V (Vot.comm, which affects multiple variables simultaneously, e.g., due weather events. Next, include shock noise
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(Vobs, shock) ) to model infrequent events such as measurement failures. Finally, we include observational noise, directly consisting of processed exogenous real-world time series that can be chosen freely (V r. To systematically vary the level of intensity for any of these observational noise structures, we adjust the signal power of Ç to control the corresponding Signal-to-Noise Ratio (SNR) with respect to the data X. To isolate the influence of the noise structure, we use the same decreasing SNR levels for all observational noise violations in Fig. 2a
Causal Sufficiency (Veon) Causal sufficiency posits that for any pair of observed variables X, and X,. there are no unobserved common causes (hidden confounders). That is, there is no unmeasured variable U such that U + X; and U + Xy. Such latent confounders can induce spurious correlations among observed variables, potentially leading to incorrect or misleading causal inferences. While some advanced methods aim to address specific types of confounding Trifunov et al. (2019); Chen ett al. (2024); & Liu (2024), the presence of unmeasured confounders remains a major practical challenge, as it is rarely feasible measure all relevant variables in complex systems. To simulate varying degrees of confounding and assess its impact, we employ two distinct strategies targeting lagged and contemporaneous confounding: First, concerning instantancous confounding V conf.instt we introduce a set of N exogenous variables 2 = Zı..... ZN. where each Zn, ~ W(, 1). These exogenous variables are not causally influenced by any variable in X but can act as common causes to multiple variables in step t. The severity of this type of confounding is controlled by progressively increasing the probability that an observed variable X.,, becomes dependent on any of the exogenous variables Zn. This, in turn, increases the probability that two variables in X share hidden parent at t. Second, for lagged confounding, we introduce an additional variable, Xc, designated as the potential confounder (V lo This variable X is allowed to causally influence, and can be influenced by, other observed variables X; with lagged effects up to a specified maximum lag J.. The severity of confounding is controlled by stepwise increasing the probability that Xe is in the parent set of any other variable in X. After sampling, the time series, Xc is removed from the observed data X, rendering a hidden confounder. We include an in-depth description and the concrete stepwise probabilities for both types in Apx. B.2
Faithfulness (Vtath)) Faithfulness asserts that all conditional independencies observed in the data are precisely those implied by d-separation of the DAG G [ccheines 1997). Formally. for any disjoint sets of variables H,, 3/ 2, 2 IL | holds in the observed distribution iff 2 and y are d-separated by 22' in G. Violations of faithfulness can lead to indistinguishable causal structures they generate no dependencies in X. Some works that examine this assumption and offer alternatives are (Zhang & Spirtes) 2008 Andersen] 2013: Lin & Zhang 2020 Ng et al. (02T). Typically, unfaithfulness is implemented through causal structures like X,, X1, XN.t + X where effects from X5.t to Xú, cancel out through appropriate parameter configurations in A. We implement this case for instantaneous effects (Viith.iss) as well as lagged structure of the form X,.-2 + X1, + X + X (Viant.agg) Furthermore, we implement an alternative approach to generate unfaithful connections by scaling down the effect sizes of causal relationships to near zero (V zei) making effects harder to detect statistically. Notably, this case is related to the notion of X strong faithfulness Zhang & Spirtes (0012). For the first two cases, we stepwise scale the intensities of both violations by updating the parameter configurations in A to approach full path cancellation. For the third case, we stepwise reduce the range of the parameter distribution for values in A. Further, as this is only one case of violating faithfulness Montagna et al. (2023b), include a discussion of this design choice, along with visual examples, in Apx. B.3
Functional Assumptions (Va)) While the general SCM in Eq. is agnostic to the functional forms fe,, between variables Xa.--1 + Xi,,, many discovery algorithms assume specific interactions, e.g., linear-additive relationships X, = ='= A xX, + Hyvärinen et al. (2010); Pamfil et al. (2020). In real-world systems, such assumptions are often violated or are only approximations. Thus, is crucial to study the consequences of corresponding violations, besides attempting to relax them Runge et al. (2019); Monti et al. (202)): Wu et al. (0022). To better emulate the variety found in practical scenarios and simulate data diversity, we employ a range of sunction-generation techniques with distinct characteristics. In particular, we sample individual univariate functions ft.d, from four distinct distributions: (1) Monotonic nonlinear functions (V al mono), (2) Non-monotonic functions with a linear trend (Vn.trema) (3) A Gaussian process with an RBF kernel (V (Vt following related robustness studies Montagna et al. (2023a): Yi et al. (2025). and (4) Random combinations of set
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of base functions, e.g., sin(.) or el) (VaLcomp). Example functions depicted in Fig. 2b and describe the exact distributions from which we sample in Apx. B.4 To stepwise increase the violations, we rely two distinct procedures. First, for V ml.mom and V nl,trend we stepwise adapt the functional distributions such that sampled functions become on average increasingly nonlinear Emaacciptoor & Kroll (1993) (Apx. [..4.. We sample all interactions fi in Eq. (2) from the corresponding distributions. Second, for V Vll.tof and V pllcoop, we stepwise increase the probability of any fi,d, to be drawn from the nonlinear distribution instead of being equal to the identity ft,,,(-) id(.).
Independent Innovation Noise (V innoo) Independent additive innovation noise (Ec,e in Eq. 2 is cru- cial for CD, as it ensures dependencies are attributed to causal links, not shared noise. However, this assumption is often violated in practice, as noise can incorporate unmeasured, dependent effects, and its true distribution is typically unknown or is fully deterministic Li et al. (2024). To evaluate how al- ternative innovation noise distributions might affect the performance of CD algorithms, we deploy the same six noise structures that we use for observational noise, i.e., V Vinno.mul, V inno,amto. V Vinno.com: V Vinoottime V iinnoshhock V inmooreall- However, for innovation noise scaling, the SNR (compared to the observation noise) is nontrivial as it is part of the signal itself. Therefore, we control the violations by blending standard normal noise with each of the five noise terms to stepwise move away from independent additive conditions (details in Apx. B.5). Notably, autoregressive V Vimocauto> common V inno.com and likely real-world innovation noise (V inno.com) fundamentally violate the Markov condition Peters et al. (2017) Additionally, since some identifiability guarantees assume non non-Gaussian noise (Shimizu ee al.) 2006), we test the effect of stepwise moving towards a non-Gaussian distribution. Specifically, simulate this by starting from Gaussian distribution (see Apx. (B.5) and progressively shifting towards either a uniform (V innno.umi) or a Weibull distribution (V weib). Finally, as some works assume equal noise variances, e.g., PPeters Bühlmann (2014). we implement a strategy to move away from this assumption (V (n For this, we draw c, from (0, of) ). where the variance of is individually sampled for each X,, We then increase the corresponding ranges stepwise.
Stationarity (Vewe,,Vaann CD methods typically aim to uncover a single from observations X. Hence, common assumption is that the SCM remains unchanged, i.e, stationary, over time or across regions. However, in many real-world scenarios, causal relationships can be heterogeneous across different populations or evolve over time Nastl & Hardt (2024). Here, works such as Huang et (0200); Günther et al. (2024); Ahmad et al. (2024) attempt to identify causal relationships in systems where parts of the SCM are changing. We include two violation cases to simulate violation of stationarity. First, we keep the causal skeleton (the nonzero elements in A) fixed and redraw the coefficients multiple times. We scale the violation by increasing the maximum allowed distance from the previous value. Second, we fully resample A multiple times throughout the sampling process, violating the principle of causal consistency. Here, any element in A that is non-zero at any is treated as a causal effect. To stepwise scale the violation, we increase the number of times we resample A when generating X (see Apx. B.6 for more details).
Sufficient Sample Sizes (Viength) Causal discovery algorithms necessitate sufficient sample size to reliably detect patterns and estimate relationships Shen et al. (2020); Castelletti & Consonni (2024). For example, statistical tests used to identify conditional independencies may lack power with limited data Spirtes & Zhang (2016). To the best of our knowledge, no work has yet conducted an extensive study on the relationship between CD performance and sample size. To remedy this, we test the effect of stepwise reduction of the length of the sampled time series X (Viength. Apx. B.7).
Data Quality (V (Vmears V m'rrs V mnar> V ) To simulate measurement disturbances beyond obser- vational noise, we introduce four types of quality degradations for X. First, for missing data (Vmcar Vmar. V V,mar), we remove number of samples "completely at random", "at random" (depend- ing on an external real-world time series), or "not at random" (depending of the values of the time series itself) Heitjan & Basu (199)) and fill the resulting NaNs via linear interpolation, common approach for practitioners. To increase the severity of the violation, we stepwise increase the total amount of missing values. Second, model sensor failures (V Vemmpy) by setting all parent sets to 2 for periods, simulating false, cero-information measurements. We then stepwise increase the length of these periods to scale the effect (see Apx. B.8 for more details).

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Data Scaling (V scate Recent works show that synthetically generated data can introduce artifacts in the causal order, which can be exploited by CD methods Reisach et al. (202I); Kaiser & Sipos) (202T); Ormamiec et al. (025)) By rescaling X, these artifacts can be partly removed. To investigate how robust methods are against scaling, we allow for a stepwise scaling of the generated time series. In partieular, we blend the original time series with its standardized version, transformation that is reported to affect CD performance in Reisach et al. (2021); Kaiser & Sipos (2021) (Apx. (.9).
Acyclicity and Sampling Rate Finally, we comment on the assumption of acyclicity, which we deliberately do not address in this work. While central to many algorithms, this assumption can be violated in two primary ways: by genuine feedback loops inherent to the system (c.g., in differential equations) or by apparent cycles that emerge artifacts of temporal aggregation. The latter occurs when a coarse measurement resolution makes a lagged effect appear as a contemporaneous, bidirec- tional relationship Runge (0118). While the first case is connected generation process, the second creates a fundamental ambiguity: A system may be acyclic at one temporal scale but cyclic at another, making single ground truth non non-trivial to define. Additionally, as we find it problematic to define a meaningful stepwise increase in acyclicity, we refrain from investigating this assumption in this work.
fundamentally different data
4 EXPERIMENTS
To evaluate the robustness of CD methods and to showcase the functionality of TCD-Arena, we conducted large empirical study using synthetic data across all previously described violations. For each violation, we systematically increase its intensity across five discrete levels and test the ability of CD methods recover G GLWOG, G and GLAG The severity levels for each violation were individually calibrated to span from negligible impact to a level at which the baseline method's performance (see below) degrades to chance (if the violation type allows it). A complete list of the configurations and further discussion of this methodology are provided in Apx. Our experiments covered a range of data conditions to ensure the generalizability of our findings. We vary the number of time steps (T E {250, 1000)) and the number of variables D (we call the two configurations small and big), together with the maximum causal lag L in the true SCM ((D, L) E {(5,3), (7, 4)}). For each setting, we generated datasets with both sparse and dense causal graphs, and both with and without instantaneous effects. This resulted in 16 distinct data aatta-eneraatng conditions, which we call "data regimes". Generally, we use standard normal innovation noise (c,,2 (0, 1). eq. ( For each violation type, severity level, and data regime, we generated 100 independent structural causal models (SCMs) and corresponding time series. In total, the evaluation for each violation type comprises 8,000 unique time series instances. Further details on the data-generating process are available in Apx. C.1 Notably, while our experiments focus on assessing each violation individually, TCD-Arena allows for free combination of all violations to generate multi-violation scenarios. We demonstrate this capability shortly in Apx. D.6, Finally, to encourage practitioners and authors of novel methods to build and extend on this protocol, we include guide on this topie in Apx. C.7
Regarding CD methods, we conduct experiments on 10 strategics, including the direct Cross Correla- tion matrix, which serves baseline for predicting causal relationships. Further, we include the following nine approaches: We leverage Granger-causal ideas and deploy a vector autoregressive model Granger (1699) where we either rely on p-values or absolute model coefficients to predict causal relationships (GVAR), Varlingam Hyvärinen et al. (2010). PCMCI and PCMCI+ Runge et al. (2019) Dynotears (Pamfil et al. 2020), NTS-NOTears (Sun et al. 2023 FPCMCI et al. (2023), SVARRFCI Gerhardus
Gerhardiss & Runge (2021) and CausalPretraining (Stein et al.) 2024b). With this, we cover all common CD paradigms| Assaad et al.) (20226). Note, we discuss possible reasons for method exclusion in Apx. C.4 Under ideal linear conditions with no assumption violations, all included methods are capable of recovering GLWCG from Fq. 2). Additional details and a list of assumptions for each method are provided in Apx. C.2 Notably, because all methods were evaluated on the same datasets, any potential theoretical non- non-identifiability issues affect all algorithms equally. This ensures a fair comparison of their relative robustness. Details on reproducibility can be found in Apx. C.6
To ensure a fair performance comparison, we adopt an evaluation protocol with three key components. First, to mitigate bias from suboptimal hyperparameters, perform hyperparameter search for each method (Apx. C.3). Second, we selected the minimum normalized Structured- Hauucture-Hammmin Distance as our primary, threshold-agnostic performance metric:
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SHD G.) AAG|
SHD min TET
Here, specifies an arbitrary decision boundary. This allows us to evaluate how well a method distinguishes causal links from non-dependence, without the need select a specific decision threshold. Notably, depending on the graph structure. G and A correspond to either instantaneous, lagged, or summary effects. For completeness, we also report several alternative metrics (AUROC, F1, Accuracy) in Apx. D.I] Third, to measure the robustness of method's specific hyperparameter configuration with respect to a violation Viyne. we average the SHD scores across all data regimes and violation levels. This aggregation accounts for potential variations in the optimal decision boundary across different experimental conditions. Also, compared with Ferdous et al. (2025), this protocol, although computationally heavier, enables more reliable estimation of robustness, as the considered data distributions exhibit greater variability. We include a visual overview of this experimental protocol in Apx.
General Robustness First, we illustrate the robustness of each method against individual violation types for each graph structure in Fig. [ Furthermore, Fig. 33 summarizes the average robustmess scores across all assessed violations. Here, we identify single hyperparameter configuration for each CD method that maximizes average robustness across all violations. We believe this better reflects a practical scenario than optimizing for each violation individually, a protocol used by MMontagna et al.) (2023a): Yi et al. (2025). For complementary purposes, we also report individually optimized results and worst-case performance in apx. D.1 Given the discovery of lagged effects (GIWCG), we find that Varlingam and Dynotears generally show the highest robustness, with Varlingam achieving the highest total robustness across all violations. Furthermore, we observe that constraint- based approaches (Especially PCMCI) tend to be less robust. Concerning uncovering GINST (Fig. D). we find that continuous optimization-bann based methods (Dynotears and Nts-Notears) display the highest robustness. Notably, as PCMCI+ and SVARRFCI predict undirected graphs, their normalized SHD cannot be smaller than as long as A is constant. Therefore, additional metrics that punish False Positives less harshly should be considered (Apx. D.I) Furthermore, as our data generation process does not strictly enforce non-Gaussian, independent, additive noise, Varlingam's 's performance on GINST aligns with theoretical expectations. Its superior performance on GLNCG is consistent with this finding, as the non-Gaussian assumption is primarily required to identify instantaneous links rather than lagged effects. We, however, find it interesting that explicitly modeling instantaneous links facilitates the identification of lagged effects, outperforming GVAR despite it estimating the same model for those lagged components. Further, even in scenarios where we explicitly introduced non- non-Gaussian noise components (i.e.. Vweih and V Vumi), VaMLM did not exhibit improved robustness. This suggests that the innovation noise must diverge sufficiently from a Gaussian distribution to enable effective identification." Concerning uncovering GLSG we find that GVAR depicts the highest average robustness. Notably, this is consistent with our previously published results reported in /teiin et al. (2024a), large-scale real-world CD benchmark. Here, GVAR is also shown to be the most robust CD method for uncovering summary causal graphs, suggesting that TCD-Arena can capture the complexities of real- world applications.
Model Misspecification Second, as we previously assumed known maximum lag L, we further investigate the performance of all tested algorithms under two additional scenarios: (i) The model is allowed to search for causes up lag greater than the true maximum lag (L E {3. 4} while Lmodel E {5, 6). (ii) The model's search space is restricted to lags shorter than the true maximum lag (L {3, 4) while Lmodel {1. 2)). We denote these cases with 1 L and L. and report the effect of these misspecifications in Table (additional visualizations in Apx. D.2). In the 1 L condition, we observe a sharp decline in performance across all methods. We also find that GVAR becomes the most robust method in this regime. Conversely, performance in the t I. regime is much more stable. However, it is notable that GVAR's performance on GLSG degrades more significantly than Varlingam's, s, resulting in a reversal of their rankings. While synthetic results warrant caution, these observations suggest that, in practice, overestimating the maximum lag Lmodel may be beneficial. Finally, CausalPretraining offers distinct advantage: It does not require specifying Lmodet. it is fixed to the maximum lag during training (3). This explains its superior robustness in the + L regime but has limited implications for real-world applications.
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(a) By ensembling different CD methods, general robustness he improved. marks required knowledge about the underlying SCM. f marks methods that do not discover GINST We highlight superior performance with green
Varlingam
SHD
0.5
Method
GLWCG
GINST
GLSG
POMCI+
CrossCorrelation
.582.1.3
1..2
CausalPretraining
.530.:25
.440..23
SHD [T]
GVAR
.244
.330+.23
0.5
Varlingam
.408
.6922.20 20
.344.22
PCMCI
:601
4
4474.22
PCMCI+
:5394:26
.988+.00
.405.233
Nts-Notears
SVAR-RFCI
.5174.28
.997..01
4477.25
SHD (
F-PCMCI
1499..30
.657..24
4341.26
0.5
Dynotears
445..332
15.5
.3651.27
Nis-Notears
.445+228
16
.388.22
Ensemble
.387..30
.550
3300..4
5
EnsembleLinca
.362.1.20
.527
.2.11..16
Violation Step
EnsembleMLP
.445+.06
.523+.20
.343..15
(b) Each curve depicts SHD changes of particular hyperparameter configuration and a particular data regime for Vimo..omm
TnsembleTransfommr
.415+t20
.587
.3244.16
Ensembleparee .376" 11.29
.470 1.27
.296"
1.24
Figure 3: Left: Average robustness of the best hyperparameter configuration per CD method and per ensemble for GLWCG GINST and GLSS Right: Hyperparameter variations with respect to Vscale.
Hyperparameter Sensitivity Third, in line with Machlanski al. (2024) and acknowledging that optimal hyperparameters are unknown in real-world applications, we report the average robustness across all hyperparameters in Table and examine fine-grained pattern of hyperparameter and data regime influence in Fig. 3b While robustness decreases for all Causal Discovery (CD) methods when averaged across hyperparameters, methods with larger hyperparameter sets (c.g., Dynotears and Nts-Notears) exhibit notably steeper decline and higher standard deviation. Although these results depend in part on our search space, they underscore the practical challenges applying these models to real-world data. Additionally, Fig, 3b shows that different CD methods exhibit distinct sensitivity patterns. Relying on inno.commmnn as an example, we observe distinctive behaviors methods (PCMCI+) show monotonic decrease in performance
when uncovering CLSG While across all combinations of data regime and hyperparameter configuration, others (Varlingam) begin with higher performance that eventually diverges. Again, other (Nts-Notears) exhibit high variance (instability) across hyperparameter settings, particularly at low violation levels, even though their performance extreme violation steps is not poor and relatively consistent. We provide full list of these patterns for each violation and method in Apx. D.3 While we estimate that the ideal sensitivity profile depends on the specific application, we argue that such comparisons that extend beyond simple optimal performance metrics are essential for rigorous evaluation of CD methods. To conclude, we find several empirical differences across CD methods, raising the question of whether combining multiple CD methods improve overall robustness. We investigate this question in the next section.
4.1 ENSEMBLING CD TO IMPROVE ROBUSTNESS
Given the variability in robustness across different CD methods, we investigate the potential of ensembling techniques to achieve improved general robustness. While ensembling is a cornerstone of modern machine learning Arpit et al. (2022); Mienye & Sun (2022), its potential to enhance the robustness of time series CD methods remains underexplored in the literature. To remedy this, we learn meta model that predicts the CLW or GINST based on the collection of predicted graphs (GI,..., .....,G,r from M individual base CD methods. Specifically, we investigate linear combination (Ensemblelinew)) a MLP (EnsembleMLP)) and a Transformer Vaswani et al.] (2017) (Ensemble tranfomeee. To train these meta-learners, generate additional, independent training dataset containing time series samples for all violations and data regimes. The exact training procedure is contained in Apx. C.5 Additionally, report the performance of simple unweighted

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Published as a conference paper at ICLR 2026
Table 1: Average robustness under wrongly specified L (4 L denotes too low, denotes too high) for GLWCG and GLSG. In parentheses, we include the change from a correctly specified L. Further, we report the average hyperparameter performance. As CausalPretraining (*) does not require the specification of max lag, its performance for the & L regime is superior. As Cross Corr. has no hyperparameters, has no standard deviation (+). We mark superior performance with green
G
G.S
HP
Method
+
1 L
Avg.
CrossCorrelation
.81((.23)
.622(-00)
(679(-23)
(472(-02)
.582
CausalPretraining
.530
.530
.440i-000*
.440(-00))
.532..00
GVAR
.782(-.36)
(-67(-004
.636(.331)
.358(-03)
.155.13
Varlingam
.784(-38)
.429(-002)
.640(-31)
.346(-01)
.420+.02
PCMCI
(86((-26)
.680(-08)
.710(-226
.472(-002)
.610#.01
PCMCI+
.836(-30)
.592(-05)
.690(-299
.420(-01)
.555#.01
SVAR-RFCI
(862(..3)
.562(-04)
.706(-30)
.425c.02)
.5251.01
F-PCMCI
.826(.33)
.528(003)
.742(-31)
.453(.02)
.504
Dynotears
(-89()334
(468(-02)
.664(-30)
.378(-01)
.507
Nis-Notears
.804(-36)
.483(-04)
.6736-31)
.3866-03)
.634 18
averaging of all Gm (Ensemblewwg) and an oracle strategy that comprises the Pareto front, which call Ensemblepante- For any given assumption violation, Ensemblepaet selects the output from the CD method that achieves the highest measured robustness on that specific violation. While not practically attainable, it serves
baseline, indicating the maximum potential performance gain achievable by perfectly selecting among the outputs of the base methods. All ensembling strategies are evaluated on the datasets used for all other experiments in this paper. We report the average performance of these ensembling approaches in Fig, 3a and provide additional analysis of performance gains in Apx. D.4 We find that simpler ensembling approaches (Ensemble and Ensemblemt.)) notably improve robustness over any individual method for GLWCG and GLSG while the -ententterr-ttesivee approaches show no extrapolation capabilities from the training set (EnsembleMLLP and Ensemblerranstmmel Specifically, EnsembleLinear leads to notable increases and outperforms Ensemblepucto- suggesting that the learned strategy is truly recombining predictions instead of selecting from them. Surprisingly, while the oracle Ensembleparte achieves a higher performance for G NN we find no superior Ensembles in this case. We present these results as a theoretical proof of concept, suggesting that simple ensembling strategies are promising for enhancing the robustness and reliability of CD methods for complex data. Especially when considering that the here-presented ensembles have direct access X. As are aware that applying these approaches to real-world scenarios will require addressing challenges such as domain adaptation and distributional shifts, we provide additional experiments on real-world data in Apx. D.5
5 CONCLUSION
This study presents the first extensive empirical investigation into the robustness of Causal Discovery (CD) methods to assumption violations in time series data. We implemented 33 distinct assumption- violation scenarios inspired by real-world data complexities and evaluated 10 CD algorithms. In particular, we first quantify general robustness over all violations, then analyze how model misspecifi- cation affects performance, and finally investigate general hyperparameter sensitivities. Motivated by observed differences between CD methods, we investigate ensembles of CD methods and conclude that they improve general robustness. Our study is supported by TCD-Arena, framework and testing kit for time series CD that we developed to conduct all our experiments. Given the vast landscape of potential data-generating processes, we releasing TCD-Areng as an open- source, modular package to facilitate future extensions and foster long-term comparability. We also provide a reproducibility statement Apx. C.6 With this, we aim foster deeper understanding of causal discovery methods, including their strengths and weaknesses, across diverse synthetic and semi-synthetic conditions, thereby paving the way for more robust weal-world applications.
empirical
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Contents
High Level Overview
A. Violation List
A.2 Violation Steps
A.3 Experimental Protocol Depiction
A.4 Violation Depictions
5 Discussion on Metric Failure Cases
B
Violation Details
B.
Additional Details Vobs
B.2
Additional Details V coon
B.3
Additional Details V faihh
B.4
Additional Details V
B.5
Additional Details
imoo
B.6 Additional Details Suel
B.7
Additional Details length
B.8 Additional Details V
mear.mar.mnar.comyy
B.9 Additional Details V sscale
Experimental Setup
C.1 Sampling Details
C.2 Causal Discovery Methods Details
C.3 Hyperparameter Search Spaces
C.4 CD Method Selection
C.5 Ensemble Training
C.6 Reproducibility and Computational Resources
C.7 Extensions of the Experimental Protocol
Additional Results
D Additional Metries
D.2 Visualizations of Misspecified Models
D.3 Hyperparameter Sensitivity
D.4 Robustness Gains of Ensembles
D.5 Ensemble Performance on CiverlRivers
D.6 Multi- Violation Robustness)
D.7 Nonlinear Conditional Independence Tests
D.8 Larger Graphs
LLM usage
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A
APPENDIX - HIGH LEVEL OVERVIEW
Violation
Short Description
Yobaadd
Additive measurement noise.
Yohsmmll
Signal dependent measurement noise.
obs,time
Time-varying measurement noise.
obs,auto
Autoregressive measurement noise.
obs.com
Common source measurement noise,
obs,shock
Spike measurement noise.
V Yobureal
Measurement noise extracted from real world TS.
confiinst
Unseen internal common cause.
confflag
Unseen external common causes,
faith.inst
Instantancous effects cancel out.
laithlag
Lagged effects cancel out.
faith,zero
Effects become extremely small.
nl,mono
Monotonic functions.
allrredd
B-spline functions with a linear trend.
GP-RBF functions.
'al.oopp
Composite funetions.
mnno.mul
Signal dependent innovation noise.
innoctime
Time-dependent innovation noise.
inno,auto
Autoregressive innovation noise.
inno.com
Common source innovation noise.
ihno,shock
Spike innovation noise.
innoorcal
Using a real-world ts as innovation noise.
innouuni
Uniform additive innovation noise.
ienowweb
Weibull additive innovation noise.
innoyarr
Unequal variances innovation noise.
coef
Causal link strengths change over time.
Causal relationships are completely resampled over time.
length
Reduced time series length.
V mear
TS with values missing data completely at random
V mar
TS with values missing data at random (depending real-world TS)
V
TS with values missing not data at random (depending on TS itself)
mmat
cmpty
Temporary complete loss of causal signal.
scale
Data standardization.
Table 2: List of all 33 violations contained in TCD-Arena with corresponding short descriptions.
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A.1 VIOLATION LIST
Table 2 contains brief overview of all 33 violations contained in TCD-Arena and investigated empirically in this work. Further details on the implementation and evaluated severity levels can be found in the following chapters.
A.2 VIOLATION STEPS
Vinlation
Increasing Intemmill via
Parameter Values
Vanau
Reducing the SNR
(1.1,0 0.312,,
Vacral
Reducing SNR
(1.1,08355, 0.376,03115,00
Yaxses
Reducing SNR
(1.1, 0.575, 0.3125, 0.05)
Yacan
Reducing the SNR
(1.1, 0.8.8.8 0.55...55, 0.05)
Vansn
Reducing the SNR
(1,1,08375, 0.6.66666.1111111 0.3116, 0.06)
Vanaba
Reduucing the SNR
(1.1,083755, 0.576,, 0.315.0.31500000 0.05)
Vacoal
Reducing the SNR
(1.1,0.8355 0.325, 0.05)
Youtia
probability
confounders
(0.135, 0.27025, 0.4176,0.88871, 0.7)
Yeutha
Incressed link probability toffrmm hiöden confounder X
20.135, 0.27625, 0:4175, 0.550.5555,00 0.7)
Lagged effecis increasingly cancel out.
(0.2, 0.15, 0.1, 0.05, 0.0)
Instretaneous effects incressingly cancel our.
(0.2, 0:15, 0.1, 0.06, 0.0)
Vanaam
Decrease the range which elements are drewn.
Mar (0.24, 0.1925, 0.145, 0.0875, 0.06]
Yamam
Increasing nonlincarity sampled monotonic Reducing number interpolation points for B-spline functions.
See Ey. Aps.
Vannd
(12, 8,
Vauu
Higher probability of nonlinear links in the SCM (GP-RRP functions).
(0-425, 0.56886, 0.7125, 0.85625, 1)
Vayse
Higher probability of nonlinear links the SCM (compasite finerions).
(0.05, 0.2875, 0.525, 0.7625, 13
Signal dependent innovation
portion.
(0.91, 0.93, 0.96. 0.99)
Time- eependepeneent
portion.
(0.8,
1)
Yamama
Autoregresivve innovation noise portion.
20
0.8, 0.955
Common source innovation nosse portion.
(0.475, 0.60625, 0. 0.7375. 0.86875,
Vinaa
Spike innovation noise portion.
(0.8,0.8 0.9, 0.95, 1)
Veaama
Noise comsisting of Real-worid lime series.
(0.5,0 0.85,, 0.9, 0.9,0.95,
Yumm
Unilomm additive
20.05, 0.25, 0.25,0.5, 0.75
Yaeanch
Weibull innovation noise stale.
(0.2, 0:4, 0.8,
Outer boundaries the interval from which of are sampled. Increase the maximum allowed distances for changes in A.
[[0.55149] (04125, 1.0875), [0. (0.275, 1,75,,7777 1.725), 10.1375, 1, 1.866), [o.,
Vost
(0.275, 0.331 25, 0.3876, 0.44377,
Van
Incresse the number resamples of A. stops T Increasing probubility missing data points.
2.
Yeun
Reducing
(70, 23,0)
Year
(0.2, 0.3375, 0.475, 0.6115666677
Vew
Increasing prohability of missing data points.
(0.2, 0 033555, 0.476 0.6.2266665
Vewr
Increasing probability of missing data points.
(0.2, 0.3375, 01.70.076800000 0.8125,
Vawu
Lengthening of timesteps with temporary causal signal.
(0.32, 0.6, 0.6,0.006.... 0.824, 0.928)
Viak
Mixing
standanlization.
(0.2, 0,8
Table 3: A short description of how we scale all 33 violations contained in TCD-Arena. We also include list of the specific parameter values to reproduce our empirical study.
Table 3 contains a list of parameter values used to intensify all 33 violations contained in TCD- Arena and included in our study. Additionally, we note short description of how each violation is scaled. We refer to Apx. B for more in-depth descriptions. The experimental violations were individually configured establish range that challenges the performance of Causal Discovery (CD) methods. Recognizing that the disruptive impact of each violation type varies considerably, a standardized upproach was employed. Instead, for each violation, the parameters were calibrated according to three-step procedure. First, a parameter configuration was identified that induced minor performance degradation for the baseline Cross-Correlan violation type allowed, the maximum intensity was set to reduce the CC's performance to an Area Under the Receiver Operating Characteristic (AUROC) of approximately 0.5 (random performance). Third, discrete levels of the violation were established by equally spacing them between minimal- effect level and the determined maximum. This methodology ensures that various CD methods can be effectively compared within challenging, relevant operational range for each specific violation. Notably, though, it precludes direct performance comparisons across different violation types. We fully document this process in TCD-Arena for maximum clarity. Finally, we only evaluate five
(CC) method. Second, where the
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violation levels in this study, it is worth discussing when robustness metric might fail. We discuss this in Apx. A.5
A.3 EXPERIMENTAL PROTOCOL DEPICTION
To clarify the protocol we used assess the robustness of various CD methods against assumption violations, we depict the process in Fig. 4 The process can be divided into three aspects. Data generation, CD method evaluation, and the extraction robustness profiles.
Visladion data
Nertie ol villatiis Sinis
setiig vit du highen seeegg
Dokaree
pertemance
grarser
Craii Mittel
Velaiee levi
celigaation
Kander saggls (N]
Figure The experimental protocol was used to create robustness profiles for various Causal Discovery methods. The process can be divided into three steps: 1. Data generation, 2. CD method evaluation and 3. Extraction of results.
A.4 VIOLATION DEPICTIONS
We provide graphical illustration for each violation type. Although the visualization methods vary. each figure isolates specific data property that evolves with the severity of the violation. This is intended enhance the intuitive understanding the specific mechanism (Fig. 5 Fig. 10
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obs.add
0.75
0.5
0.25
0.05
Count
2.5
Noise value
Noise value
Noise value
Noise value
Noise value
(a) Additive innovation noise distributions for all violation levels
V oos.
0.75
0.5
0.25
0.05
Notse valuue
10
-10
Signal
Signal
Signal
Signal
Signal
(b) Relationship between multiplicative innovation the signal itself.
V obssttmee
0.75
0.5
0.25
0.05
10
Notse
500
1000
500
1000
500
1000
500 Jodex
1000
500
1000
TS Index
TS Index
TS Index
Indrex
(c) Development time-dependen observational noise
V Vote
0.75
0.5
0.25
0.05
Noise
2
-25
0.0 Noise [t-1)
-2.5
0.0
2.5
-10
Noise (1-1)
Noise (I-1)
Noise (1-1)
Noiee (1-1)
(d) Correlation of autoregressive observational noise between and t
V Voha.com
0.75
0.5
0.25
0.05
Naise X()
110
0.0
2.3
-2.5
0.0
2.5
Noise X(i)
Noise
X(i)
Noise X(i)
Noise X())
Noise X(i)
(c) Relationship between two observational noise variables in X₁
1
0.75
0.5
0.25
0.05
Count
500
Viloe
Value
Vialo
Valoe
Value
(f) Distributions over noise values when sampled from a shock distribution.
Figure 5: Various depictions of different violations of observational noise. We depict the severity of the violation from left to right and denote the SNR above the figure.
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V obsureal
0.75
0.25
0.05
Value
Aww.ee
10
000
1000
0
600
1000
500
1000
DXX
1000
0
500
1000
Indes
TS Index
Index
TS Index
TS Index
(a) Distributions over noise values when sampled from real-world data.
V conff
0.2
0.4
0.6
0.8
1.0
VAn MAB
0.5
2,04
0.0
with
with
with
with
with
(b) Mean Absolute error between a VAR model that has or has no access to the exogenous variables
V comm..
0.1
0..2
0.5
0.7
0.9
VAR MAB
0.75
1.00
1.25
0.75
1.00
1.25
0.75
1.000
1.25
0.75
1.00
1.25
0.75
1.00
(c) Mean Absolute emror when modeling with VAR process with the confounder.
V fiitha
Count
Peanson Correlation
Peason Coereiation
Penson Corredation
Peansn Convelation
Pearson Correlation
(d) Pearson correlation between start and end node of unfaithful connection A.
V Inith,
4
Count
50
50
50
0
0.0
0.5
0.0
0.5
0.0
0.5
0.0 Pearson Corzelation
0.5
0.0 Pearson Correlation
0.5
Peocron Correlation
Pearson Cocrelation
Peassnn Correlation
(e) Total sum of all parameters the A.
V Gaithu
(0.12, 0.24]
00.0975. 0.1925)
[0.075, 0.145)
[0.0525, 0.0975)
00.03, 0.05)
Count
20
20
20-
Abbolule Coeflicion
Absolate Coellicien
Absolute Coeflicint.
Absolute Coefficient
Absolute Cueficiont
(I) Total sum of all parameters in the A.
Figure 6: Graphical depictions different violations and their intensities.
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almmoo 10.125, 0.25, 4,
[0.5, 1, 1,
[0.25, 0.5, 2, 4]
10,0833, 0.126, 8, 12)
[0.05, 0.0833, 12, 20)
153
0-
-1
(a) Distributions from which we draw nonlinear functions the monotonic family.
V alltrend
64
32
300
-1
0
0
(b) Distributions from which we draw the nonlinear functions of the trend family.
V nnrtif
0.25
0.5
0.75
0.0
-2.5
0
(c) Distributions from which we draw the nonlinear functions of the RBF family.
Val.oopp
0.25
0.5
0.75
150
(d) Distributions from which we draw nonlinear functions the composite family.
Vinnommll
0.4
0.8
Nojoe
-2.5
Signal
Signal
Signal
Signal
Signal
(c) Relationship between signal strength and innovation noise.
inao,timee
0.2
0.4
0.6
0.8
Noise
-10
5001
1000
50O
1000
500
1000
1000
500
1000
Index
TS Index
Ladex
Index
TS Index:
(f) Innovation noise time.
Figure 7: Graphical depictions of different violations and their intensities.
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inmoauto
0.2
0.4
0.6
0.8
0.95
2.5
Nosse (t)
8.0
-2.5
-2.5
2.5
0.25 0.00 0.25
(I-1)
Noiee (1-1)
Noize (I-1)
Noize (t-1)
(I-1)
(a) Relationship between innovation noise of X and X..
V immo.com
0.2
0.4
0.6
0.8
Noiso N.D
0.0
-2.3
-25
8.0
225
-2.0
0.0
2.5
-2
-2.5
0.0
2.5
-2.5
0.0
2.5
X(()
X()
X
X(I)
X(i)
(h) Relationship hetween different innovation noise terms in X,.
imnozreal
0.2
0.4
0.6
0.8
2.5
Noio:
0..0
2.5
500
1000
500
1000
500
1000
1000
500
1000
TS Indlx
TS Index
TS Index
TS Tndkx
TS [nclex
(c) innovation noise impact.
V Innoslock
0.6
0.8
Dausty
0-
-25
0.01 Vilue
2.5
-2.5
0.0
2.5
01.0
2.5
0.0
2.5
0
Value
Value
Viduee
Value
(d) Distributions from which draw the shock innovation noise.
V iumooumi
0.4
0.6
0.8
Count
500
2.5
Value
Value
Viluue
Value
Value
(e) Distributions from which we draw non-Gaussian innovation noise.
Winooweib
0.2
0.4
0.6
0.5
1.0
Count
500
0)
Value
Value
Vilue
Value
Value
Distributions from which draw non-Gaussian innovation noise.
Figure 8: Graphical depictions of different violations and their intensities.
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V
iuno, vaz
1.1
0.75,1.25
0.5,1.5
0,25,1.75
Comr..
250
250
-2.5
0.0
-2.5
2.5
-2.5
0,0
2.5
-0.5
0.0
0.5
(a) Distributions from which draw innovation noise with unequal variance for each variable in X.
wtat
Change:
Change 0.125
Change 0.25
Change 0.375
Change 0.5
Couns
5
Total change
Total A change
Total chango
Total change
Total A change
(b) Causal Coefficient absolute changes (elements in between first last section of X.
V
Change points:
Change points:
Change points:
Change points:
Change points:
10
10
10
10
Count
50
25
Sum Nonzero coeffeientsSam of Nomzero cooffeicursSum Nonzero codffeeenntSum of Nonzero cocffeiinntSum Nonsero coeffieien
(c) Number of temporarily non-zero elements in A.
length
250
100
50
25
12
Value
0
-5-
100
200
100
200
100
200
0
100
200
100
200
TSS Indexx
TS Index
TS Index
Index
Index
Length of generated time series.
V moaa
0.25
Mil Mo mappy
Value
100
200
100
100
100
200
200
TS Index
TS Inddex
TS Incles
TS Index
Index
(c) Time series with inereasingly missing (mcar) and afterwards interpolated values.
mar
0
0.25
0.5
0.75
0.95
Numa
Value
-5-
100
200
100
2301
100
100 Index
200
200
TS Index
TS Index
TS Indes
Index
(1) Time series with increasingly missing (mar) and afterwards interpolated values.
Figure 9: Graphical depictions of different violations and their intensities.
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V mana
0.25
Value
-3-
100
200
100
100
100
200
200
TS Index
TS Index
TS Indes
TS Index
TS Indie
(a) Time series with increasingly missing (mnar) and afterwards interpolated values.
cmppy
Value
-5-
100
200
100
2001
100
200
100
200
100
200
TS Index:
TS Index
T3 Index
Index
Indeex
Timeseries with partially empty A. Sections where empty are marked as yellow areas.
Vacale 0.5
0.25
0.75
0-
Valne
TS Value
TS Volue
TS Valuo
TS Value
(c) Mean and standard deviations of individual variables in X.
Figure 10: Graphical depictions of different violations and their intensities.
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A.5 DISCUSSION ON METRIC FAILURE CASES
In this study, we fundamentally quantify the robustness of a method to an assumption violation using a limited number of samples (five violation intensities). As the underlying robustness is often continuous, we deem it reasonable to discuss under what conditions our methodological approach may yield potentially misleading results. To this end, we depict three simplified scenarios in Fig.
In the first scenario (green box), we observe a consistent separation in robustness across methods. In this ideal scenario, the green curve consistently outperforms the blue curve across the entire range of violations. The discrete measurements (marked by stars) accurately capture this relationship, yielding robust, faithful score.
In the second scenario (blue box), the curves cross. The discrete sampling points suggest that the performance of both curves roughly equal across the measured violation levels, resulting very similar robustness score. However, this discrete evaluation fails to account for the precise dynamics of the robustness curves. Notably, depending on the application, either the blue or the green curve could be preferable.
the third scenario, non-monotonic and highly volatile curve highlights the most critical risk. At the discrete evaluation points, the blue curve is preferred even though its robustness is disputable.
Performance
Perfomance
Performence
Elstatte violetion
Viotation
Evallatt visustn Awee'
Viokation
Figure 11: Depiction of problematic relationships between violation property and robustness measured any performance score for single data regime and single method configuration. Left: Optimal case in which all performance curves are monotonically decreasing, and one curve is Pareto superior. Middle: While both curves monotonically decreasing, our metric does not directly distinguish between them. Right: any performance curve highly non-monotonic, the comparison can be misleading.
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B APPENDIX - VIOLATION DETAILS
Fundamentally, the base SCM that we use for each violation is linear causal process with additive Gaussian noise:
x ΣΣ A (ia,((a ce
(4)
where f is the identify function, A a coefficient tensor, X the time series and additive gaussian noise. To bring this into more compact form, omitting f:
D
L
x Σ
Ata - X C,
(5)
d=1 1=0
Further, we can bring this into matrix notation:
X, = A
(6)
where & is vector of independent innovation noise variables. Crucially, the non-zen entries in A correspond to links in the causal graph G. While sampling in the base linear process, control the density of links using corresponding probability that determines whether entries Ai,d, are equal zero.
Finally, we can separate instantaneous effects as they are not always present and are implemented in a different manner:
X, BX; + AX
where A is coefficient matrix and B is coefficient vector.
To implement all our violations, we alter this basic linear additive process.
B.1 ADDITIONAL DETAILS VOBS
To briefly recap, when we violate the no observational noise assumption, we not directly observe the measurements X.,.- Instead, we measure noisy versions X., X.,; + Si,t. We define a concrete list of observational noise variables and structures in Fig. 2a In this section, we provide additional details for the specific design choices of the various implemented Çr,e- Afterward, we present the concrete formula use to control the signal-to-noise ratio we increase the respective observational noise violations.
First, consider independent additive noise (V (Vobs,dd), where model the noise as standard normal G, N, 1).
Second, consider multiplicative noise (Vobs.mul,, which in the signal-depende noise model is an
additive noise scaled by function of the signal strength Torricelli et al. (2002); Liu et al. (2014). Here,
Gi.t N (0, (X,,)))), )), i.e., multiplication of standard normal noise with the signal X5, (see Fig. 2ap. Real-world examples include temperature sensors whose precision degrades at high signal values Bentley (1984) and speckle noise in image processing Liu et al. (2014).
Third, V Vobs.time. specifies noise with distribution characteristics changing over time. Real-world examples of such noise sources include sensor drift and interference from periodic environmental factors. We model this by scaling the variance by periodic signal, i.e., Si,e ~ N/(0, ((1 + at).
sin )2) where a and are hyperparameters. In experiments, we fix them to simulate an annual cycle and a small linear trend to simulate sensor degradation. Specifically, we use a = 0.01 and B = 365 730.
Fourth, Vots.auto indicates an autoregressive noise structure (Ga, JL G.,-1) that can be found when disturbances of the measurement process persist over multiple timesteps. Here, could imagine a sensor that is overshadowed by cloud for multiple consecutive time steps. We model this term as Ci.t N' (aGat- (1 oo)A(O,1) where is the weighting coefficient which is
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hyperparameter. Intuitively, the mean of the distribution depends the last sampled noise, similarly to random walk. In our implementation, we equally mix the previous step with the random source, Le., a 1/2. Notably, keeping below 1 ensures that the overall process, while dependent prior noise, remains nondeterministic.
Fifth, noise sources across different variables can be dependent (Vob..om): i.e., Çe.t 1d. G3.t for i Such a scenario can occur when multiple sensors are affected by a shared, unmeasured environmental factor (e.g., temperature or power fluctuations). We model this by sampling from a single noise source Se ~ (0, 1) that is shared for all variables in X, i.e. Vi E {1...., D3 Si,e Ge for a timestep L.
Sixth, observed data might be subject infrequent, large disturbances measurement failures with probability Pshock where else
s
(V V. Using a shock probability Pshocks we model Ça,e 0
a fixed scalar. In our experiments, we set 5 and Pshock 0.05.
Seventh, we directly extract noise from real-world time series to generate semi-synthetie measurement noise (V real). We deploy a High-Pass Butterworth filter [Butterwotth (1930) extract high- frequency components from real-world stock price time series. We then randomly sample appropriate- length sequences as observational noise components.
As specified in our main paper, we isolate the influence of the noise structure by using five discrete, decreasing SNR levels. Inparticular, we are using {10, 1, 1/2, 1/10] for all observational noise
violations in Fig. 2a
To rescale the noise vector, to achieve desired Signal-to-Noe to-sss Ratio aagga with respect to X, we first compute the average power of the signal and the unscaled base noise Chase.
The signal power, Px, is defined as:
D
T
Px
T D d=l
x2
d.t
t=1
The power of Stase: is:
D T
Ps. base
Shase
d=1
Given target SNRarget the desired power for the final noise, PN Marget is calculated as:
Px
PN targer
SNR
We find a scaling factor, a, that transforms the base noise power to the target noise power.
Px/SN Px/SNRagee
Pc.baae
The final noise vector ( is then obtained by scaling the base noise:
< = a Chase-
B.2 ADDITIONAL DETAILS V VOONT
In our main paper, we specify two possible methods for introducing confounding into a sampled time series. Specifically, we separate instantaneous effects (V (Vconfinna) and and lagged confounding (V (Veunflug). We model the former by generating a set of N independent potential parent variables Zi...., ZN. In all time steps f, the Zn are standard normally distributed and can act as common causes for any variable in X. Hence, the causal assignment (Eq. (I) for an observed variable X,, becomes: X = fs (Pax (X,,,) U Pag (X..), G. We scale V Veonfinst by increasing the probability of links from Z to variables in X. In particular, Finally, to make the more pronounced, links from Z to X are parameterized slighly higher ((0.8.0.9) than elements in A ((.3,
use the probabilities {0.2, 0.4, 0.6. 0.8. 1.0).
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To model Voonflag. generate the time series with additional confounding variable Xc. This variable part of the normal SCM and can influence any variable X,. Further, it can also be influenced by all other variables. After generating the complete time series, create the observed data X by removing Cc, rendering it a hidden confounder. To scale V Vevnflay. we again increase the probability of links from and Xc. Specifically, links from Z to X are parameterized slighly higher ((0.8,0.9) than elements in A ((0.3,0.5)) to render the effect more pronounced.
the probabilities 10.1, 0.2, 0.5, 0.7, 0.9). As before,
B.3 ADDITIONAL. DETAILS V FAITH
To violate faithfulness via path cancellation (Viithu Vtain.)), have to ensure that there variables with a connection in the causal graph G, which have no measurable dependency, i.e,, cancel each other out. We separate instantaneous effects (iihtimm)) and lagged effects (Xyait.lag) and visualize the structures we implement in Fig. [2
Xj,02
Xz.0
X
X
XR.T11
X
(a) Structure for Viihi
(b) Structure for Vinhhl
Figure 12: The two structures, we enforce to violate faithfulness. Note that in Fig. 12b some of the variables are lagged. In both cases, the connection from X, to X; is partially canceled by the connection over XR. so X; and Xj become closer to independent during the violations.
Now, to scale the violations of the faithfulness assumption, we adapt the parameters of the causal graph to cancel information of X, from X,, using the path over X₃ (see Fig. 12 Specifically, we implement the following assignments for V Vatthintt and Vamnaag respectively:
Un3,orm(0.300.33
Xx.1 = 20Xx, ELe X², == (-v d)X 0.5. 61,
Uniform X1.-1 = 2n:X, + G4
Xu. = d) 5, 0.5. 64,1
Here, d denotes the distortion parameter, which we decrease as the violation severity increases. We choose the levels {0.2, 0.15, 0.1, 0.05, 0.0) for the violation severity in both cases. Further, we note that this implementation only one way, arguably the most straightforward, to generate Unfaithful- introduce V Viaiheee where stepwise reduce the range of elements in A
ness. As alternative, to be closer to 0, making it harder to detect statistical dependencies. During our experiments, the elements of A are sampled uniformly from the intervals 10, 0.24], [0.0975, 0.1555), (0.075, 0.145).
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[0.0525, 0.0975], and [0.03, 0.05], respectively. Finally, as other ways of generating Unfaithfulness, such as deterministic relationships, conditional links (XOR), or specific mixtures of causal models, possible, we plan extending TCD-Arena in these directions in the future to gain additional insights.
B.4 ADDITIONAL DETAILS VNL
To introduce nonlinearities, studies on CD robustness sample functions from Gaussian Processes (GPs) with Radial Basis Function (RBF) kernels Yi al. (2025); Montagna et al. (2023a). The smooth and oscillatory nature of these functions provides difficutt test case for algorithms that assume linear relationships, motivating the development of more robust nonlinear methods.
However, they may not fully represent the diverse spectrum of nonlinearities encountered in real- wolldwrldd applications. In many domains, such as those governed by physical constraints, nonlinearities often adhere to specific characteristics like monotonicity or saturation, rather than arbitrary nonlinearity (e.g., SIR-models Kermack et al. (1997) or the Michaelis-Menteee Kinetics Ainsworth (1977)).
In this section, we detail our four specific choices in nonlinear function distributions and the respective design paradigms. Further, a critical consideration in generating synthetic time series from such structural equations is ensuring the stability of the process (i.e., preventing divergence) because the functions are applied iteratively. This often requires constraining the output range or characteristics of the sampled functions fi,t,-- As the specific constraints depend on the functional class, first detail the normalization or bounding procedures possible during the generation process. Then we note the respective choices for the four families of functions we investigate. We also formalize what we mean by increasing the nonlincarity of the structural causal model (Eq.
B.4.1 DETAILS ON BOUNDARY CONDITIONS
For sampling nonlinear functions used to iteratively generate observations X using Eq. 7 it is important to consider the boundary behavior when inputs are either very large x > 0 or very small & as it is possible for any of the /) time series to diverge towards positive or negative infinity. This behavior could lead to numerically unstable values even in our finite simulated time steps T, While we will discuss specific checks test for such conditions in Apx. C.I we consider it here explicitly for the set of violations Vni concerning the functional relationships Ji,da
In our implementation, we focus on the input interval x [-1,1]. Again, this specific setup is motivated by the time series sampling process, in which we apply these functions iteratively as described in our main paper. To enforce saturation for x - =o,, we wrap sampled univariate functions f using hyperbolic tangents to roughly enforce value ranges of [-1, 1]. Consequently, values contained in the generated time series X stay close [-1,1] 1].
Specifically, we either wrap a function f with respect to the input or the output f(x) to ensure saturation. In particular, we employ
ff(z)
if e [-a,a]
sx(((x)
(8)
(tanh())
else
and
[((x) (fan)((=))
if]s(>) <=
sy (f(z))
else
respectively. In Eq. (8) and Eq. 9 a defines the symmetric intervals around zero, which in our experiments set to one. When detailing the specific functional families used for the violations
Val.onn- Vul.uennd V Vul.bb and V Vnl.comp= we will specify which concrete wrapper sy or Sy, we use to ensure saturation of the sampled ft.d
B.4.2 QUANTIFYING AND INCREASING NONLINEARITY
To formalize the concept of nonlinearity of univariate function f. multiple scores proposed in the literature. For instance, roughness penalties for spline smoothing, e.g., (Ramsay & Silverman 2005 Sec. 5.2.2), quantify it using the squared curvature of function over specified interval. In
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particular, they calculate the deviation from a linear function
Powr(() = J (f(
(10)
where f" denotes the second derivative. If and only if the second derivative is zero over the complete interval [-a, al, then f is linear in said interval. Further, given the squared integrand, Deurv is strictly nonnegative with minima exactly when is linear function. Intuitively, this score quantifies changes the derivative of f. which constant only linear functions.
In contrast, in Emancipator & Kroll 1993), the authors measure the minimum possible mean squared error of f to any linear function in the interval interest. Specifically, for a given f. we follow their approach and define the nonlinearity MSE in an interval [-a, a]] as
2x [
DM
min a,
(f(x) - (ar + b)) de
(11)
Intuitively, îMSE measures the minimum possible mean squared error to any linear function in [-a, ai] and is greater than or equal to zero for any f. Further, f can be expressed as linear function, then bMSE is exactly zero. To compute ПMSE, we have to consider the optimal a", b' E R. which are necessary to minimize the mean squared error. Emancipator & Kroll (1993), the authors give general solutions for arbitrary interval boundaries. In case of boundary symmetric around r 0 of [-a, a), the optimal solutions that minimize the error for a function f are given by
3
a
I.f (x)dd, and
2a
-a
(12)
a
6a 2a
((x)dx.
Hence, the measure for nonlinearity becomes
PASE (ʃ) [ (s(x) (a'x (J(0) (a)x
(13)
Both Deure and LMSE behave differently and are not always aligned. Specifically, MMSE considers the absolute distance to a line, meaning it can change if we multiply f with constant factor, while Peury would not change. However, Denry necessitates that the function twice differentiable, i.e., in C², in the interval of interest [-a, a). Hence, it cannot distinguish nonlinearity between step functions or absolute values, even if they closely follow a linear function in absolute deviation. In contrast, MMSE is finite in such cases but includes optimization process. However, using linear regression, empirically estimate MSR for any given function.
In our work, generate time series at random. Hence, we interested in the approximate behavior of the resulting processes. We formalize this by describing families of distributions F for a function with ttewise-varryng nonlinearity. Specifically, ensure that sampled functions F have controllable expected nonlinearity
Ey-r(?(()),
(14)
where 9 is measure of nonlinearity. Thus, to increase the nonlinearity of sampled processes, we stepwise change J from which we sample the functions ft,,, in Eq.
Lastly, consider that in our formulation of the general structural causal model (Eq. (2)) the functions in the causal graph G are univariate and connect two, possibly lagged variables. Hence, another approach to increase nonlinearity in stepwise manner is to sample only subset of the fi.d, from a distribution of nonlinear functions, while keeping the rest linear. This leaves us with third possibility to increase the overall SCM's nonlinearity.
In the following, we specify four families of distributions of nonlinear functions and describe how we can adjust the nonlinearity of the resulting sampled time series processes.
B.4.3 1. MONOTONIC FAMILY:
For the first family, we sample uniformly from three univariate functions, where a parameter > determines the nonlinearity. Before formally analyze the influence of this parameter, detail
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specific functions and motivate design choices. We use
(x;;3) = sgn(x) |x]
f2( (x
1)
1,
and
(15)
c
[1r 1) 8
fs(x;8)
1
where e is hyperparameter which we 2 in experiments. The specific choice of the added and subtracted in f2 and fa depends on our interval of interest [-a, a), where we want to change the non-linearity using |3
Our intention in designing this family of functions is to ensure monotonicity in the specified interval, hypothesizing that this property is important for CD methods. We investigate this hypothesis empirically in our main paper. As first step, we now show that all three of our functions are monotonically increasing in [-1, and for 3 >
To prove this statement, it is enough to show that the first derivative is greater than equal to zero for all x E 1.. Note that given Eq. (2 in our main paper, monotonically decreasing functions are possible because the coefficient matrix A can have negative entries. Hence, without loss of generality, we focus in the following on the monotonically increasing nature. Specifically, consider our three
functions (Eq. [5 only in the interval of interest [-1, 1]
f1 B) = sgn(x) ))) |
8 f2(2; (3) (xr 1)
-,
(16)
[s(;; -c (
(x 1)
8 1,
where the absolute value can be removed from f2 and fa because + is always positive or negative, respectively.
We start our analysis with the derivative of f1, which has three cases, i.e., x 0, and
We start with the first two:
Case 1, - -1 < 0: In this case, sgn(x) = -1 and |x]| -r apply, leading to fi f1(x; B)
(-1)(-2) =2 Using the chain rule, we find
f1 3) = (-1)3(-x) 8-1 (-1) -((-))-1
(17)
3-
(18)
Case 2, 0 < < 1: Here, the absolute value becomes the identity and sgn(x) = 1. Thus, we have [1 B) =, and
fi B) 3. x B|2c|
(19)
In both cases, we can see that the derivative is equal to Jí (x:; 33 B ||00| For all x E [-1,1]] {0} and / this is strictly nonnegative. Lastly, to prove that fi monotonically increasing in the interval of interest, it is left to show that the derivative is also larger than or equal to zero for & = 0. Here, the value depends on the specific setting of B. For 3 > 0, we have three cases and we study the limits of fi from both directions
Case 3.1, 3 = 1: In the linear case, we find
lim f(; = lim 1- 1a| = lim
= 1, and
x->01
x->0
lim 1(x11) 2--
lim
1- |x/0
lim 1.
a--0-
2-0-
In other words, the derivative is constant and larger than zero
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Case 3.2, 3 > 1: Here we find the exponent 3 - leading the following two limits
lim fi(;; 1) = lim 8 |x| 8- = - --10 80 = 0, and
x-
m-01
lim (1; = lim 3 |c|$-1 = B. 10-10/8-1_ 0. z=-
2-0
Hence, we find a saddle point, where the rate of change is exactly zero when x = 0. Case 3.3, 0 < B < 1: In this case, the exponent 8 1 becomes negative meaning |z] 1/101- Consequently, limits from both sides diverge towards
1 +00, and
lim fi::; x -
lim S
|x|1-8
x z-0++
1
lim [(;;1) = lim B
|=|1-8
= +00.
Crucially, in all three cases, the limits of the derivative from both sides are equal and strictly nonnegative. Hence, fi is monotonically increasing in [-1, for all B >> 0.
Next, we investigate the derivatives of fz and fs. Following the observation that for æ E [-1,
1] the we calculate f and f' using the chain rule
absolute values can be rewritten as in Eq. (16
8-1
x +
f2::; B) 3
and
c
(20)
1
f1 () =
For both functions, we have a strictly positive number (3 > 0) which is multiplied by a base raised to a real power. Remember that in our experiments, we set c = 2, meaning both (x-1)/2 and (1-#)/2 vary in [0, 1], i.e., are strictly nonnegative. Therefore, raising it the real power (3 1) leads, for both f and f3. to positive factor times a nonnegative factor. Hence, for all x E [-1, 1] and / > 0, we find (2; B) and [a( B) 2 0. Note that in both cases, when 0 < B 1, again find limits for both derivatives, where they become infinite. Specifically, for f2, we observe a vertical tangent when (compare case 3.3 of fi). Nevertheless,
-1 and for fa, we similarly observe one for the derivatives of all three functions f1, f2. and fa are strictly nonnegative in the specified interval. Hence, the functions themselves are monotonically increasing in [-1, 1] for all 8 > 0. Next, we discuss how we can increase the nonlinearity of all three functions.
Monotonic Family Increasing Nonlinearity As specified above, for given distribution of func- tions, we can quantify the linearity by considering the corresponding expectation. For the monotonic family of functions, the distribution we consider is a uniform choice of {ffs, f2. fa]. Hence, for fixed B, we are interested in
Uni-- MMs :))]]
(21)
In this uniform distribution, all three cases are equally likely. Thus, the expectation for fixed is equal to the average of MSE for the three functions. We specifically choose PMSE because the integral over the squared second derivative (Seurv) of fi diverges for < 3 1.5. Further, we are interested in measuring the squared deviation from any possible line in 1-,
Consider that the parameter 3 directly controls the nonlinearity of Si, /2, and fa. In particular, all three functions are equal and linear in [-1, if =
f1 1) sgn(ær) |x|| = x,
r + 1
fz 1)
1 =
c
(--11
fa(z; 1) -c
+ ==
Hence, the expectation in Eq.
21 becomes zero for 3 1.
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Now, by changing away from all three functions become nonlinear in the sense of MMSB Specifically, we construct five discrete levels E (1,2,3,4.5) to scale VnLmono and sample B with an equal chance from either of the following intervals
f = 1 + 8 E 1/2,, 1] or 8 E [1, -= - 8 e [14, or e (2,4]
f f= + 8 [4] o [4,8).
(22)
4 + 8 E 1/12,
{ 5 + 8 E 1/20, 1/12] or E [12,20].
For concrete level we denote the lower and upper boundaries of the two intervals with [8 0
and [sL. ) respectively. Fig. 13 visualizes examples functions drawn from the five levels of the resulting distributions. Crucially, the intervals of the distinct levels only overlap at maximum of two concrete boundary points with any of the other intervals.
To analyze the non-linearity of our functions f (5) in the interval [-1, consider second expectation over 3 distributed uniformly from either of the two intervals given by a level E, i.e.
E , MSE ( :1)))...
(23)
where we omit the specific distributions for brevity.
Here, both the function fj and 39 are sampled independently. Hence, Eq.
23 is equal to
1
MMSE (.
203 - ((())) A/0
(24)
1
Σ
M (3)(33
(/ 2(0(11) -((
where both of the integrals describe one the two equally likely and symmetrical intervals from which J is sampled, respectively. Further, for fixed level e, the factors contained in the intervals a fixed normalization given by the probability density of the corresponding uniform distributions over the intervals [o(44) ((41 and [3((t) (
By linearity of expectation, we can reorder Eq.
24 into
AMS (58)) 2(3 - -1((2)) d3
1=1
(25)
M ;19) 2(3 - g(tti) d3
As stated above, the only point in all intervals we consider where the fj are linear is for 3 = In any other case, ( ;;)) > 0 applies.
To now show that the expectation increases with the level E, consider that the integrals Eq. (5) calculate averages over all values of MMSE for in the corresponding intervals. Hence, it is enough to show that for 3 0, MSE is smooth and increases when moving away from the global minimum at B 1 in specified intervals. In all cases, focus our analysis the set of functions {f1,f, fs} we defined above. Consider that for the interval [-1, 1]. the optimal parameter for the MSE minimizing line (Eq. (12) become
a" S xf(x)d and
(26)
6" ["] f(x)dr.
We visualize examples for f1, f2, and fa and the corresponding optimal lines in Fig. 13
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0.8581, 6' 0.0000
1,0117,
-0.1987
1.0317, 0.1987
fi(;;3 1.490)
faja: 1.60)
fatx: 1.490)
t=
0.55
-0.5-
-0.5-
-100
-20-
-1.0-
1.0
0.5
0.0
0.5
1.0
0.5
0.0
0.5
1.0
-6.6
0.0
1,8003, 6
0.3560,
0.7481
0.3560,
0.7481
(1(11
0.144)
8"
0.0
-0.5
-0.5
-0.5
1.00
-1.0-
-1.0-
1.0
-0.5
0.00
0.5
1.0
-1.0
0155
0.0
0.5
-1.0
-0.5
0.0
0.5
1.0
1.4462, 6
0.2050)
0.8600
.0.8609
fi(e: (0044)07)))
falm: 0.074)
8"
0.0
0.0-
0.0
0.5
-0.5
-0.5
0.044)
-100
-1.0-
o'x
-1.0-
1.0
-0.5
0.00
0.5
1.0
-1.0
-0.5
0.0
0.5
100
-10
-0.5
0.0
0.5
1.0
Figure 13: Examples of f1. f2, and f3 with randomly sampled from the respective level We also visualize the optimal line and denote the corresponding parameters.
Now to determine whether MmE is smooth with respect changes in 3, we have consider the three terms in Eq. (13) that are functions of 3: fj. a", and b*, where the last two also depend on the specific function fy (Eq. [2 Eq. (26)).
For all three of our functions fy. the critical part is of the form [g(x)) where g(-) is defined in Eq. [5] Hence, the following equality holds
1x(x)) (l(ss())
(27)
In particular, the function has exponential form, which is infinitely differentiable (Co) with respect to 33 for any fixed x (where g(x) + 0). In other words, f1, f2, and fs are smooth with respect direct consequence of the Leipnitz integral rule, e.g., (Protter Morrey 1985 Chap. 8), integrals of the form / f, (z; 3) and =f, (x; 3) are also smooth functions with respect to 3. Finally, consider that MME is again an integral with respect to square of the sum of three functions that smooth with respect to B. Hence, using the Leipnitz integral rule and the chain rule for differentiation, we can conclude that MMS is also smooth for the functions fi, fz, and fa-
to 3 in [-1, 1]. Further,
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To show that the nonlinearity increases if shift 3 away from the linear case of B 1, we now study 8g MSE. In particular, using the Leibniz rule and the chain rule, we have
0 (
S (15(1; - (a'x 1a'++01 b²))
03
(28)
(0f)(r;))
da
ōb
dx.
03
0
03
Expanding the product in the integral leaves us with three separate terms
= S (1z(1; 8) (a xx0)xb)))))) 0fz(2: (x
dx
03
05
da
[ x[J,(1;33 axxx b²) )x
(29)
03
[[ (J)(1; - a' x--x-10 bd)dx. 0ß
Crucially, note that is possible move the partial derivatives On and outside of the integral 03 03
because they are independent of z. This is important because the integrals in the second and third terms are exactly the first-order optimality conditions of a" and 6*, respectively Emanniiptoorr & Kroll (1993). Hence, both these integrals vanish, and we are left with
MSMSE (fa) 03
+ S" (1)(z; (a (a(
b*) (z; 03
dr.
(30)
when our ʃz become In Eq. (30p, we have two factors: the residual error to the MSF optimal line and the sensitivity of
with respect changes in Given that the residual is constant
linear, we again confirm that this is a minimum of ASE. Hence, 35hu = if 3 = 1. Consider d3
now that for all values B > 0 which are not 3 = 1, all our functions f1, fz, and fs are nonlinear, we know that MSE has to be strictly larger than zero. This implies that 5 is a unique global minimum. Given this observation and the previous insight that 9MSE is smooth with respect to B, we conclude that the averages over IMSE have to increase locally the neighborhood minimum at 8 = 1. However, this does not necessarily imply that the only critical point is at 3 1. To test whether our claim that the expected nonlinearity increases for an increase level we use Eq. (12) and Eq. 13 to simulate the nonlinearity.
the global
We visualize MSE in Fig. 14 and confirm that the nonlinearity increases when we move away from the global minimum B 1. However, observe local maxima in the interval (0, 20). Hence, is unclear how the expected nonlinearity for randomly sampled f, and 3 according to the defined levels é behaves.
f1
fz
fa
E (MMSEE
0,005178
0.006049
0.005783
0.005670
2
0.032103
0.030878
0.029993
0.030991
3
0.066364
0.046946
0.048134
0.053815
0.087101
0.046346
0.044872
0.059440
5
0. 003752
0.039548
0.037819
0.060373
Table 4: Approximated nonlinearity scores for the three functions fi, f2, and fa and different levels é. The last column contains the accumulated DMSE over all fy for all / sampled in the respective level.
Thus, we estimate the expected nonlinearity (Eq. (25) per level f. Specifically, we sample 1000 values for each level and use the theoretically optimal line parameters a" and 6*. We list the approximated expected nonlinearity in Table 4 We find that the expected 2MSE does stepwise increase for fi while it decreases slightly for fu and fs again after = 3. However, the accumulated expectation over all functions (Eq. (23)) does grow for = 1,..,,.. 5. Therefore, we conclude that the empirical nonlinearity does increase stepwise for our defined violation levels.
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fi(;
0.20
f2(:,
f3(,,
0.15
ØMSEU
0.10
0.05
0.00
20
40
60
80
100
B
Figure 14: Nonlinearity measured with 2MSE for the three functions Si, ʃ2, and fa for increasing
values of B > 0.
B.4.4 2. B-SPLINES FOLLOWING A TREND:
Next, we investigate univariate functions that exhibit an overall increasing trend but are not neces- sarily monotonic. To do this, rely on B-spline interpolations, e.g. de Boor (200T). Specifically, we sample sample Np scalar values (interpolation points) {vi, U2 ...: UNp from a uniform distribution Uniform(-D1 11, Next, sort the values vy and set them as targets for [(x) at equidistant abscissae in the range x 1-1, Consequently, B-spline f(x) Σee
و eą By, (,)((())) of degree k is constructed to smoothly interpolate these points. The corresponding B-spline basis elements are given by
(1 if Sa x < Ttt,, B5,(x)) = {0 else
Bax(x)
x Tj
Bj,k- (xx)
Tjth Tj
+ Tjth+1 - 2
B B)+118-1(2))
J Ti+1
where we determine the entries the knot vector as described de Boor (200T) using the implementation provided in Virtanen et al. (2020). Given that the basis functions are piecewise cubic, the fitted curve is not monotonic, as visualized in Fig. 7b To ensure saturation, we use the procedure described in Eq. (8). For V Valuendd sample different fi,d, for all non-zero interactions in Eq. 2 by sampling different Uz. Next, we discuss how we stepwise increase in nonlincarity of the sampled functions.
B-Splines Increasing Nonlinearity To stepwise scale the nonlinearity of the sampled functions ft..t we decrease the number Np of interpolation points. Specifically, we use the following values: {25, 15, 10, 6, 4). Intuitively, higher number interpolation points indicates more values v, that have to be interpolated and which are strictly increasing. To empirically show that larger Np leads to more linear functions in the sense of 9MSE (Eq. (1 ). we use Eq. (12 to approximate the average nonlinearity in Fig. 15 Further, following the uniform distribution of independently sampled vy approximate the expected 2ME of our concrete Vnl.rend in Table Similar to the monotonic family of functions, observe that the nonlinearity of sampled fi,d,1 increases average.
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0.04
Mean Spline Nonlincarity
0.03
MMEE
0.02
0.01
0.00
10
20
30
40
50
Number of Interpolation Points
Figure 15: Nonlinearity measured with MSR for the spline functions for increasing number of interpolation points. We report the average and the standard deviations.
Number of Inter- polation Points
Bfgiu [MMS (fspline)
25
0,003792
15
0.006517
0.010198
0.017728
0.019475
Table 5: Estimated expected AMS for the spline functions sampled for the five decreasing number of interpolation points.
B.4.5 3. GAUSSIAN PROCESSES WITH RBF KERNELS:
Related studies focusing the robustness of CD methods for i.i.d. sample data Montagna et al. (203a) Yi et al. (225)) primarily replace standard linear relationships with interactions modeled by Gaussian processes using Radial Basis Function (RBF) kernels. In our third family of functions for the violation V VuLabe, we follow the same approach. Specifically, we use Gaussian process prior
(z-x')²
f() CP(O, KRBF(Z, x²)) with the RBF kernel KRBF(I, x') exp
where is the
2x
length scale which we set to one. In particular, we use the implementation of Pedregosa et al. (2011).
Further, do not employ wrapping in this case, as the Gaussian process's 's mean zero ensures that the sampled functions do not diverge. Next, we describe how we stepwise increase the nonlinearity of the resulting SCM.
Stepwise Increasing Nonlinearity To stepwise increase the nonlinearity, we use a different ap- proach for VaLate Specifically, we sample nonlinear links ft.d, with increasing probability from the Gaussian process. Here, we use the following probabilities: {0.2, 0.4, 0.6, 0.8, 1.03. All remaining links in the causal graph G are linear and use the identity function in Eq. (2). Hence, we increase the overall SCM's nonlinearity by increasing the likelihood of sampling nonlinear links. To be specific, for two empirical scenarios (D, L) (5, and LD. (7, 4), have maximum of 5 x 5 x (3 + 1) and 7 x (4 + 1) links, respectively. where is the number of variables and T. the number of lags (+1 for instantaneous). For lagged links, employ the link probabilities
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Plag
0.075 Plag 0.15, while for instantaneous links, we use Pinst 0.0 and Pinst 0.1 (see Eq.
We list the number of expected links for the eight combinations in Table 9 For each of these eight combinations, we can calculate the number of expected nonlinear links for the increasing intensities {0.2, 0.4, 0.6, 0.8, 1.0} via standard multiplication because they are sampled independently. This leads to the following increasing numbers of expected nonlinear links for the smallest and sparsest scenario (D 5, L 3, Plag 0.075, Pinst 0.0): {1.125, 2.25, 3.375, 4.5, 5.625J. Conversely, we get the following expected nonlinear links in the largest scenario (D = 7, L = 4, Plag 0.15, Pins 0.1): {6.86, 13. 13.72, 20.58, 27.44. 34.3). Hence, we conclude that sampling the SCM in this stepwise manner progressively increases the amount of nonlinear interactions.
B.4.6 4. COMPOSITE FUNCTIONS
Lastly, and inspired by symbolic regression, we sample the Si,. through a random hierar- chical composition. First, we define a set of base functions B. Specifically, we implement Then, independent
(>/²/, tanh(2)), sinh- max(x,, J), 2 |æ=], cosh(r))) sin(x)),,os()) chains h0) (x)) are formed, each by randomly selecting and sequentially composing Ng functions Finally, the results of the independent chains get multiplied by -1 with probability of 1/2, i... Cii up to
from B, i.c., (x) bN₂( (.. (b,,
( Uniform (-1, 1}, before all chains get summed
f(x)
U up
11000
This construction allows for wide range of possible, potentially highly nonlinear functions. Hence, we apply Eq. (9) to enforce stable behavior. In our empirical evaluation, we use two chains, each composed of two base functions uniformly sampled from B model V Vnl.oop.. Next, we describe how we stepwise increase the nonlinearity of the resulting SCM.
Stepwise Increasing Nonlinearity For V vllcoopp strictly follow the procedure also employed for Valzer- Specifically, we increase the probability of sampling nonlinear interactions when generating the SCM. Again. we use the following probabilities: {0.2, 0.4, 0.6, 0.8, 1.0) to scale the violation intensity. Table |ummmarizes the number of links for the various scenarios in our experiments, and the same caleulations as for V nlabrr apply to estimate the expected number of nonlinear links. Hence, we conclude that sampling the SCM in this stepwise manner progressively increases the amount of nonlinear interactions.
B.5 ADDITIONAL DETAILS INNO
The standard assumption of independent additive innovation noise often violated in practice. Here, we detail our setup for evaluating the impact of such violations. Specifically, we consider three different paradigms:
First, we discuss direct changes to the noise structures by introducing dependencies. Here, we also deploy real-world time series as direct innovation noise components, relying on "Rivers flood," set of river discharge time series from (Seein et al. (2024a). Second, we shift the distribution from Gaussian to non-Gaussian variations. Third, we consider widely different variances, leading to stronger variation among the variables X,. which can be problematic Peters & Bühlmann) (2014). In the first set of violations, we test robustness to six alternative noise structures, as discussed in Apx. B. To be specific, implement V Vinoo.mul> V Vinno.auto- V Vinno.com- Vinootime- V Shonk and V Vinouccal
All of these structures change the distributions of the independent additive noise Ei,t in Eq. 2 We list the specific distributions in Table 6 Regarding the corresponding hyperparameters, i.e., O, S. and Patack. we use the same setting as for the observational variants (see Apx. B.I.
In contrast to scaling the SNR, blend the different 6,, distributions with decreasing amount of standard normal noise to intensify the five violations. This is important because innovation noise is part of the signal (Eq. (2) and relying solely on the specified noise can lead arbitrarily low signal
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Violation
Definition of 6i,t
Depends On
innozmul
E Xi,t : Thit
the signal Xx, time step t autoregressive
inno,time
Gitt Tiit (1 + at) sin sin(22t/a) G = a. 10.Gi--11 + 0.) niit
inno,auto
inno.com
Vi: Ei, 1/+
S
with prob. Pshock: else.
fixed scalar S shock prob, Pshock exog. real-world TS Z
inno,shock
G 10
innocreal
Z
Table 6: First set of innovation noise violations. In our experiments, the sources of randomness for the variables X; are routed in the innovation noise Ci,, which are typically additive and standard normal. Here, we include various dependencies by using random variables 7/, and 1/. (standard normal (0, 1)), which are subsequently influenced by various factors, c.g., the signal strength (V m.m.mmmmi) Both a and / denote hyperparameters.
variances. In particular, use alpha blending, i.e.,
a + (1 ai)Ei,
(31)
where Eitt NO0 1). We specify the values that use for all innovation noise violation types in Table 3
For the second set of innovation noise violations, i... V Vinooani and V weib. progressively blend non-Gaussian distributed noise with standard normal noise. Let w ~ Sang be a non-Gaussian random variable and let di NO 1) be standard normal noise. Then, we define C2
(1 (w B(w]) anp var(w) - 2a) + a (var(ww +
G,
(32)
where is a blending parameter. To stepwise scale the intensity, we use the following blending values for a: {0.2, 0.4, 0.6, 0.8, 1.0].
The denominator in Eq. (32) and substracting B[w] in the numerator ensure that E[er,-] 0 and var(cu.) = To verify this, note that the expectation is linear and the denominator is constant factor for fixed w. Hence, is enough to analyze the numerator. Here, B|w] - [[[[[]] = [[w] - [[]] 0 and E[p) which directly implies E] 0. Further, to show that the denominator scales the mixture to unit variance, we have show that it equivalent to the standard deviation of the numerator. Given that the standard deviation is the square root of the variance, is enough to show that the variance of the numerator is equal to the squared denominator. Crucially, both w and e are independent random variables, which means their covariance is zero. Hence, var( - (w B(w]) + anb)
(1 var(w) + a var(()
(1 2a + a² var((w) ) a²1
var(w:) 2a) a² (var(w) + +1,
i.., the squared denominator in Eq. 332 A direct consequence of this is that var(G..) = 1.
In our experiments, we use two different non-Gaussian distributions to model the violations V Vino,uni and V Winnh In the first case, for V Vinno,ui,, we use uniform distribution over the interval [-2, 2] with the corresponding density
1
PUnifomm -r)=1(- 11-22(-2 4'
(33)
where 1|-2,2 is a unit function that is equal to one iff x E [-2,2]. we employ a Weibull distribution Weibull (1939). which described by two parameters: scale which we set to one, and shape a, where we use 1.5. The corresponding density is defined as
2). In the second case, for V weih,
a-1 e (4/A)*
Pwesbull (I) A
(34)
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0.2
0.6
PUniform T
PWelbill
0.4
0.1
0.2
0.0
0
1 X
2
-2.5
0.0
2.5
X
Figure 16: The densities of the two non-Gaussian distributions, we employ to violate standard normal Gaussian innovation noise. Specifically, we employ Weibull distribution Weibull (I939) with scale and shape = 1.5 and a uniform distribution over the interval [-2,
We visualize the densities of both non-Gaussian distributions in Fig.
Lastly, as a final violation concerning innovation noise, which is known to be potentially problematic Peters Bühlmann (2014), we introduce the equal variance violation (V Vmno.var). We implement this violation by sampling variance of for each variable X; at the initialization of the sampling process. This value used every time step to draw innovations εitt ~ (O,,o²) In particular, we sample of uniformly from the union of two disjoint intervals. To stepwise intensify V inno.var we increase the separation distance between these intervals. Specifically, we use the following order of interval unions to model the widening distributions:
[0.55, 0.75) U [1255, 1.45),
|0.4125, 0.6125 U [1.3875, 1.5875).
[0.275, 0.475) U [1.525, 1.725).
[0.1375, 0.3375 U [1.6625, 1.8625), and
[0. 0.2) U [1.8,2].
B.6 ADDITIONAL DETAILS VCOEF, VSIAT
Stationarity assumes that the structural assignments Eq. 7 do not change during the genera- tion/measurement process. First, for V Vocet we keep the nonzero entries in A fixed, but resample the coefficients during the generation of the time series. To increase the severity of the violation, we stepwise allow larger changes (8) to the current coefficients and sample the actual changes uniformly from (-5, 5). In total, we resample the coefficients times at Sehunge E {50, 100, 150, 200} for T 250 and Schange E {200, 400, 600, 800) for T 1000. Second, for V stats fully resample A at certain change points in the generation process. To scale this violation, we introduce more change points during the time series generation. Notably, we treat the union of all existing A the ground truth for the process (If there is an effect for some limited amount of time, there is a causal effect), As we set the number of time steps T to either 250 or 1000 our experiments, we introduce Nehange {1. 2, 3, 4, 5) change points, using the time steps denoted in Table 7
B.7 ADDITIONAL. DETAILS
V VLENGTH
To reliably estimate relationships and identify patterns, CD algorithms need sufficient number of samples. To violate this, often implicit, assumption (Viength), reduce the number of time
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Selected Time Steps for change points
N Nehange
T 250
T = 1000
125
500
83, 166
333,666
3
63, 63.126,187
250,500, 750
4 50, 500,000,15 10,
200,400,6000 600,800
5
41,82.122.163.2203
166,333,500,666,88
Table 7: The specific time steps we use for the respective number of change points to violate stationarity (Vst.). We separate the time steps for the two settings T 250 and 1000 in our experiments.
steps 7 we sample from Eq. 77. Specifically, we employ the following five discrete levels T {76, 58, 41, 23,61
B.8 ADDITIONAL. DETAILS V MCAR. V VMAR. V MANAR. V VEMPTY
Empty Periods
250
T 1000
2x 0.24
(40, 100) and (150, 210)
(160, 400) and (600, 840)
2x 0.30
(31, 106) and (144, 219)
(124, 424) and (576, 876)
2x 0.352
(23, 111) and (139, 227)
(92, 444) and (556, 908)
2x 0,412
(14, 117) and (133, 236)
(56, 468) and (532, 944)
2x 0.464
(6, 122) and (128, 244)
(24, 488) and (512, 976)
Table The specific time periods, denoted by (start, end), where we the parent sets Ø during generation of X, Vempty- We separate the time steps for the two settings 7 = 250 and T 1000 in our experiments. In all cases, we introduce two periods with no causal signal and scale the length to increase intensity. We denote the average ratio of empty periods for sampled time series in the first column. Note that this ratio, i.e., the violation intensity, increases with each row.
In practical applications, data quality cannot always be controlled, leading to various forms of degradation beyond observational noise. We investigate four quality violations that are common across domains, namely missing measurements (V Wenar Vmar V mmnr) and sensor failures V Vempty. Concerning missing measurements, we define binary mask matrix M E {0, 1|NxT, where m,, - indicates a value is missing, and m,, = 0 indicates it is observed. Further, we set a target missingness rate For Vinarr each value in the time series has the same probability of missing m4, ~ Bernoulli(p). For V Vmar: we let M depend on an auxiliary observed, in our case, real-world (Stein al. (2024a)), time series. Let Z R N be dataset. The missingness probability is determined by P(m specifies the sensitivity to Z. To ensure the empirical missingness rate matches the target the intercept a is numerically determined. For V Vmmars we let M depend on the time series itself. Here, the probability of value missing is determined by P(mi, where a and 3 fulfill the same roles as described beforehand. In our experiments, we rely on 3= 3.5. To increase the severity of these violations, we stepwise increase the target missingness rate p E {0.2, 0.3375, 0.475, 0.6125, 0.75). Regarding sensor failure (Vempty), we periodically set the parent set of variables Ø during the generation of X, simulating temporary unnoticed sensor failures Izero-information measurements). Specifically, we introduce two such periods for each sampled time series and list the corresponding intervals in Table & Crucially, the average ratio of each of the empty periods during the T 250 T 1000 time steps increases as follows: (0.25, 0.345, 0.4, 0.420,4255.. 0.455).
of external, real-world time series sampled from reference
2i,4) = o(a + 3 =2,a), where
.x,,) a(a + x.,))
B.9 ADDITIONAL DETAILS VSCALE
Related work suggests that synthetically generated data introduces artifacts that are beneficial for identifying causal order, e.g., Ormaniec al. (2025). This phenomenon is problematic because
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it can lead to overestimation of CD method's efficacy. Because it can be partly remedied using standardization to mean zero and variance one Reisach et al. (2021); Kaiser & Sipos (2021). we violate this condition by mixing the original observations X with standardized version X after generating the time series. In particular, alpha-blend both versions of the time series: X = + (1 - X, where determines the intensity of V scale- Specifically, in our investigation, we use a E {0.2, 0.6, 0.8, 1}.
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C
APPENDIX - EXPERIMENTAL SETUP
C.I SAMPLING DETAILS
(D,L)
(5.3)
(7,4)
Plag Pinst = (0.075, 0.0)
5.625
14.7
Plag Pinst (0.075, 0.1)
8.125
19.6
(lag Pinsa] = (0.15, 0.0)
11.25
29.4
Plag Pinsa) (0.15, 0.1)
13.75
34.3
Table 9: Expected number of links in sampled SCMs. Here, D denotes the number of variables in X and L denotes the number of lags. The probabilities (Plag: Plnst), correspond to the likelihood of lagged and instantancous connections in the causal graphs G (i.c., nonzero elements in A and B, Eq. 77))
In this section, describe the data-gen aata-generation process use throughout experiments and for all violations. Generally, we base all of our experiments on Eq. (7) and alter according to the violations described in Apx. B We employ two combinations of the number of variables D and the maximum lags L, resulting in small and a big scenario. Specifically, we set (D, L) = (5, 3) or (D, L) (7, 4). respectively. Then, with a probability Plag {0.075, 0.15) and probability Pinsa E {0.0, 0.13 links in A and B are being selected to be nonzero. Finally we generate either 250 or 1000 samples for each time series. This leads 16 "data regime" combinations, and we list the expected number of links in Table Next, each nonzero element in A and B receives a value that is uniformly sampled from the joint interval [-0.5, -0.3] U [0.3, 0.5). Notably, we explicitly exclude coefficients close to render causal relationships detectable. If f is not the identity function (i.e., for Vai), a univariate function is drawn from the corresponding distribution. We then generate iteratively using Eq. 7 To initialize, we sample every variable from (N(0, 1)).
Further, before we start to generate X, we need to evaluate the following two conditions: First, concerning instantaneous coefficients B, we guarantee the sample graph to be acyclic by checking the following necessary and sufficient condition for acyclicity:
tr(c 1 D,
(35)
where fr is the trace operator. If this condition is not met, we resample until it passes. See Zheng ett al. (2018) for details of this condition. To account for potential divergence of the SCM, we test the VAR stability of A. In particular, we investigate whether it is stationary by evaluating the eigenvalues of its companion matrix F:
A
A
A
A
Ip
0
0
0
F
0
D
(36)
ID
where Tn is a D X D identity matrix. To guarantee the stationarity of the corresponding process, all eigenvalues of F have to lie in the complex unit circle. This condition applies if
max (F)(F)) <.
(37)
If it does not hold for the sampled A, we resample the coefficients.
However, if the Sa, nonlinear, then the VAR stability test does not apply. Hence, we additionally check for divergence with the following two tests: First, while generating X we continuously check whether any variable in X is monotonically increasing over the last T time steps by testing:
3i (1,..., dd, 36
Vk E (O...., T),
(38)
[X 1- < [Xee-kl
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If this condition is met, we halt the generation process and resample SCM. In experiments, we set T = 10.
Second, test whether any time series in X exceed maximum value (likely indicating divergent processes). In our experiments, this value is set to =25, Again, if this condition is met, we halt the generation process and resample new SCM.
For each violation intensity and data regime, we sample 100 random SCMs along with a corresponding X. As discussed before, "data regime" is combination of D, L, Ping- and Pinst (compare Table 9 well as the length of the time series T E {250, 1.00]. In summary, this results in 2 x 22x x x 100 1600 SCMs per violation intensity level. Considering that evaluate 5 stepwise violation intensities, we typically sample 8000 SCMs for each violation contained in Table 2 Notably, for violations that are concerned with instantanous links (V faith.i V conff.). we do not generate samples with Pinst effectively halving the amount of SCMs to 4000. For V Vengthh. the time series length becomes the violation property, also halving the data-regime space. total, the results on the robustness of time series causal discovery in this paper are based on: 30 x 8000 + 3 x 4000 252000 pairs of SCM and time series.
as
C.2 CAUSAL DISCOVERY METHODS DETAILS
CausaPPretraiiing CoosCurrdation
GYAR
Vartingau
POMCI
POMCH+
SFAR
F-PCMCI
Dynoteus
NIS-NOTears
Temporal Dynomics
Lagged
/
/
Ohrervational nober
Hiddon confonnding
x
x
x
x
x
x
x
x
Unfail//sless
x
x
x
x
1
1
Noelivarrty
/-
Jnsovation
Non Non-Addinve
Gassian aliiivve moike
7
7
7
7
NoeNaaiionnll
8
Table 10: Comparison of core assumptions and capabilities of selected causal discovery algorithms. Note, constraint- based approaches can handle both linearity and nonlinearity. We, however, only test these methods with linear conditional independence test in this study (partial correlation and robust partial correlation). Therefore, we mark their ability to handle nonlinearity with a We include a small study on this matter in apx. D.6. We found no information on the faithfulness assumption for CausalPretraining and therefore mark it with t. Finally, we omit data quality issues such as Veegth. Vg. and V Vecale from this table, as they are typically not explicitly mentioned as assumptions.
We include details the method assumptions of all causal discovery methods involved in Table TO Note, many of the data quality assumptions that we test, such as V Vienghh. Vq. or V Vscates are not explicitly assumed by most methods, however they nonetheless often implicitly modeled in the synthetic data that is used for testing algorithm performance.
C.3 HYPERPARAMETER SEARCH SPACES
In Table II we include list of the full hyperparameter space that we evaluated for each causal discovery method used throughout this paper. Note, as we not consider nonlinear conditional independence tests in our experiments due to computational overhead, we include small study on this matter in apx. D.6 In total, we evaluate 143 hyperparameter configurations. For our main experiment, therefore perform 252000 x 143 36.036000 causal discovery attempts.
C.4 CD METHOD SELECTION
As a wide variety of time series causal discovery algorithms exist, many of which are partially integrated into TCD-Arena, we had to narrow the set of included CD algorithms. Our method selection for this study was guided by threefold rationale:
1. Established Baselines: We prioritized the most well-established methods, typically focusing original method formulations without domain-specific extensions.
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Method (Combos)
Parameters
Values
Cross Correlation (3)
mmndel
L 2,
CausalPretraining (2)
Architecture
TRF. GRU
Imodel
L - 2, I,, +
Varlingam (6)
Prune
Truc, False
Imodel
L 2, I,, L +
GVAR (6)
Use
coeff, p-val
Imodel
L L- L,
PCMCI (6)
test
ParC, RParC
Lmodel
L- L, L +
PCMCI+ (12)
CI test
ParC, RParC
RLL
True, False
Imodel
L- 2, L,
SVAR-RFCI (6)
CI test
ParC, RParC
Lmodel
L 2, L,
F-PCMCI (6)
CI test
ParC, RParC
Lmodet
L 2, L +
Lambda-w
0.1, 0.3
Dynotears (48)
Lambda-a
0.1, 0.3
Max iter
100, 40
H-tol
le-8, 1e-5
Lmodel
L --2, L +
h-tol
le-60, le-10
NTS-NOTears (48)
Rho-max
lc+16, lc+18
Lambdal
0.005, 0.001
Lambda2
0.01, 0.001
Table 11: Hyperparameter space and number of combinations in the hyperparameter grid. For NTS-NOTcars and Dynotears, we use default parameters (first value) and an alternative value per HP. ParC and RParC denote the Partial Correlation conditional independence test and the Robust Partial Correlation conditional independence test, respectively. RLL denotes the resetting of the lagged links before calculating instantaneous effects. Importantly, for the main robustness results (Fig. 3b we only consider Lmodtt = L. We refer to the implementations of all methods in TCD-Arena for further details.
2. Implementation Reliability: We selected algorithms with reliable, publicly available implementations ensure reproducibility.
Scope Analysis: We focused on the fundamental form of causal discovery: analyzing single time series to produce single window causal graph.
Consequently, we purposefully excluded specialized algorithms developed for specific data properties or extended settings that fall outside this scope. For instance, methods targeting extended summary graphs (e.g., PCGCE Assaad et al. (2022a)), irregular or gappy data (e.g.. Cuts Cuts et al. (2022)), latent context (e.g., J-PCMCI Günther et al. (202)). multiple datasets (e.g., Space'T Time Mameche ctt al. (2025)), or interventional data (c.g., CANDOIT et al. (2024)) were not included. While analysis of these specialized methods would be valuable, this study aims first establish
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comprehensive benchmark for the most common causal discovery setting. Despite this, committed to continually expanding the methods available in TCD-Arena to increase the range of experiments that can be conducted. We refer to our TCD-Arena for continually growing list.
C.5 ENSEMBLE TRAINING
Category
Hyperparameter
Evaluated Values
Common to all architectures
General
Batch Size
[32, 128,256, 1024, 2048} (BCE (1,2.5,3), MSE, Focal) (BCE (0.01,0.1), MSE, Focal)
Loss Function
Weight Decay
AdamW
Learning Rate
(Ie-4, le-2)
Base Model: Linear
Base Model: MLP
Architecture
Hidden Layers
(3528+1764)
Regularization
Dropout
(0.0.1)
Base Model: Transformer
Regularization
Dropout Rate
(0.0, 0.1)
Architecture
Model Dim
[256,512} {2)
Num Layers
Table 12: Summary of hyperparameters that were evaluated for CD ensembling.
To train our examples, we generate a separate training dataset that holds SCMs and corresponding X from all violations, respective intensities, and data regimes. These samples are combined into single joint training dataset that we use to train ensembles with trainable parameters. In the most general sense, all our ensembles take tensor of the shape B x M D D Lmodel, where M denotes the number of individual CD methods, D the number of variables, Lmodel the model order, and B the batch size. This tensor then, necessary, reshaped to match the first layer of the respective network architectures (Ensembler.inem) Ensemblewı.. and EnsembleTransoomer) All network architectures return D x D Lmndel tensor that is directly used as the final predicted graph G. Notably, for the EnsembleMaan and Ensemblepucto we directly recombine elements in the input tensor by either taking the average over the model dimension M or selecting the optimal element. EnsembleLineat is implemented single fully-connected layer without activation function. For EnsembleMLP, use a 2-layer MLP with RELU activation functions and batch norm. For Ensemblernasiiomer we use a standard Transformer architecture Vaswani al.) (2017), with learnable embeddings and a learnable CLS token, from which we use the embedding vector as the final prediction. Further, we evaluate the hyperparameters specified in Table 12 to select the best model that we report in Fig. 3a
C.6 REPRODUCABILITY AND COMPUTATIONAL RESOURCES
To facilitate reproducing our results, we have made all code and required resources available in the TCD-Arena repository. This repository includes seeded functionality to generate the datasets used this paper, along with hashing functions verify their integrity. The code for all evaluated Causal Discovery methods and ensembling approaches, as well as the scripts to generate the figures presented, is also included. Experiments were conducted on 7-node Slurm cluster using 14-24 CPU cores per node, with the exception of ensemble training, which was performed on single Nvidia RTX 3090 GPU. While most individual experiments are not resource-intensive, reproducing the complete set of approximately 36 million causal discovery attempts will require multi-day runtime on a comparable setup. However, as many scripts can be executed in parallel on a Slurm cluster, the total runtime may vary depending on the specific hardware and configuration. As the execution of various CD methods requires vastly different computational resources, we provide statistics on the average runtimes per tested hyperparameter configuration and per method in table 13
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Method
Correct Lag
t L
CrossCorrelation
5.2 1
7.7+ f
2.8
CausalPretraining
24.3
24.3
24.3
GVAR
.34.1
.41.1
.31.1
Varlingam
9.2+664
11,5+9.0
7.2+4.3
PCMCI
44.00
444.9=173.8
86.2++77.
PCMCI+
357.0
47.7
194.3
SVAR-RFU
275.3:255..0
1417.7:1626.2
221.8
F-PCMCI
801. 444.72444
1155.3425.3
413.5
Dynotears
18.3488
22.2
16.5
Nts-Notears
441.6-112.2
505.3
402.8
Table 13: Average computational efforts (in seconds) per 100 samples and for a single hyperparameter configuration for each of the 10 CD strategies. The standard deviations denote differences between hyperparameter configurations. As Cross Corr has no Method hyperparameters, no standard deviation provided. Note that we run 8,000 samples per HP combination and violation (See apx. C.1
C.7 EXTENSIONS OF THE EXPERIMENTAL PROTOCO1.
While the experimental protocol for this study was fixed to ensure consistency, we designed TCD- Arena as an extendable dynamic benchmark (Shiraii et al. (2022). The framework highly config- urable: users can define novel combinations of violations, data regimes, and generation parameters via single YAML configuration file or command line arguments. Furthermore, the codebase features modular architecture explicitly intended facilitate community contributions. We encourage researchers to extend the benchmark by integrating additional assumption violations and Causal Discovery (CD) methods.
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D APPENDIX - ADDITIONAL RESULTS
LSG
Neter
Saftemnc
Nealaneity
Statasurity
Quality
1.0
0.8
0.66
530
0.4
0.22
Credd Eaecovtry
Vinds
Vinki
Visum
Vista
Viraa
Vianataa
Figure 17: Robustness profile for GLSG and of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured as average normalized SHD over various data regimes.
D.1 ADDITIONAL METRICS
To extend our empirical evaluation, include alternative metries and additional depictions. First, we include additional depictions of the main metric (normalized minimum SHD) in Fig. 18 and Fig. T9 Second, we include the same depiction for three alternative metrics (AUROC, Maximum F1, and Maximum Accuracy) in Fig. 20 Fig. [2 Generally, we find that most metrics show a similar picture (e.g., the ordering of methods when looking at performance metries for GISTT or the generally low performance of Cross Correlation and CausalPretraining), A notable exception to this is the difference between the normalized SHD and other metrics for GINST Since the normalized SHD cannot become smaller than 1 for undirected graphs (PCMCI+ and SVARRFCI) by definition, other metries provide more informative picture. Third, we include the same depictions of normalized minimum SHD when selecting the optimal hyperparameters individually (contrary to our protocol; see Fig. 4 Fig. Fig. 27 Notably, differences between these protocols typically quite small, suggesting that a general, well-performing hyperparameter configuration exists. Finally, we also report the worst-case performance per violation for the selected hyperparameter configuration in the main results in Fig. 28 Fig. 20 Again, find that method ordering is generally consistent with the results that were presented in the main section of the paper.
method for each violation
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G
Elonratioal
Crasd Salullacyy
Tearalke Xale
Tulier
Quuaitty
SUDD L
Chaan Pre
81
94
35
10
180
35
15
43
190
17
Varka
G INST
Clonrationl
Grad
Tepealke
Sulker
Quality
15
PEWOL
1.00
SWAR AFCI
LIU
LO0
110
100
1X0
1.00
300
LIU
1102
110
L.91
LOI
110
64
71
44
52
35
11
41
75
10
47
29
LSSC
Nian
Salleweyy
Quility
70
0YM
TOUCL
71
71
40
55
16
11
10
Vancali
Va
Figure 18: Alternative depiction of robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured average SHD over various data regimes. From top to bottom: results for CLWC (NNT and LLSG.
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GUING
QINET
GL
Figure 19: Alternative depiction robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured average SHD over various data regimes. From top to bottom: results for LCCG GIST and GLS
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D.1.1 ALTERNATIVE METRIC AUROC
G LWWG
Chacrstooal Nouse
Careal Buftrknny
Incorationn
Date Quality
Hirafiitoe
Noniicority
Statocaiily
0.5
0.8
3.6
Diaonny Auttoca
CYVE
0,5
Vincoa
Lands
Vanu
pabs
Vigua
Yenal
G INST
Inrocation
Fillafduce
Notlicensite
Stutoxaaile
Consol Diecower Stitvek
1.5
0.5
0.3
Ana.
0.8
V
V
Valama
V
V Valernl
V
1: Vani
G 188
Choerwtsonn
Caraal
Incovation
Nosse
Sultikency
Hiralidtese
Notiicearity
Nocle
Stutocaiityy
Quality
Cacaal Donoeyy
Mediais
Cheed
AVOREL
Vituel
V Vidua
LWOG
Salli Salmwery
Puridekmo
Sheknouttr
DVAR
FUMII
1.00
1.00
PCMC1
20
20
700
70
77
23
35
10
55
Natwe
Figure 20: Robustness profiles ten Causal Discovery algorithms against a multitude of stepwise assumption violations measured average AUROC over various data regimes. From top to bottom: results for LWWC GINST and GLSG Notably, we include three depictions for each graph to improve data comprchensibility. First, a scatter plot (1-3), second, heatmap (4-6), and third, a Vinylplot (7-9, see next page) that specifies the ranking of the methods.
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INST
Vala
Sa Saluvwery
Sucifelms
Quart
AUBOC
PCMII-
83
94
90
50
85
80
9
10
48
58
68
80
64
82
65
50
12
18
50
91
Vile
F. Kahadook
LLS
DVAR
PUSCI-
71
71
84
GUIOG
Main
7
G
Figure 21: Additional depictions of the robustness profiles of Causal Discovery algorithms against assumption violations measured as average AUROC. Top: heatmaps for GINST GLSG. Bottom: Method performance rankings. Best viewed in conjunction with the figures on the previous page.
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D.1.2 ALTERNATIVE METRIC F1 MAX
G LLWG
Chocrstaonl
Carcal Suftrknny
Incoration Noce
Date Quality
Nouso
Hitafitoee
Notiicority
Statocaiiyy
3.0
0.8
0.7
94
Vin Sio
V V.
V
aburk
Vinua
pahs
Visua
V
V2
10
G INST
Chvaroat
Falafiduce
Notlicanile
Stutoxaadle
Conoot Dlecower Miitvek
A
e:
P
Volcand
Vilama
V
V
Liah
V
V-
1.
V Sleves
G L88
Choervtooon
Caraal
Incovation
Nosse
Saltikency
Hiraliltess
Notiiceadlyy
Statoraiityy
Quality
3.5
0.4
0.6
Vita
Vituree
V
Vinue
V Vidua
LWCC
Cheqratena
Salleiery
Nedammm
Nasi
DVAR-
14
15
34
50
124
72
30
195
DOUCL
135
PCSKI-
Vinual
V Va
V
Figure 22: Robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured average maximum FI over various data regimes. From top to bottom: results for LWWG INST and GLSG, Notably, we include three depictions for each graph to improve data comprehensibility. First, a scatter plot (1-3), second, a heatmap (4-6), and third, Vinylplot (7-9, see next page) that specifies the ranking of the methods.
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GINST
Note
Ra Rabio
Seltamy
Sutecaac
Quarity
PCSXI
14
47
23
97
4
A3
46
83
28
53
41
59
28
24
80
82
40
P-PEUCE
24
Viluant
Vilo
GLSG
Ellorational
Sellnntyy
OVAR
23
77
25
15
158
25
50
100
81
180
POVEL
PCSCC
(0)
Vimaal
V
QUAWOG
F
f
Figure 23: Additional depictions of the robustness profiles of ten Causal Discovery algorithms against assumption violations measured average maximum F1. Top: heatmaps for CINTT ,LSG Bottom: Method performance rankings. Best viewed in conjunction with the figures on the previous page.
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D.1.3 ALTERNATIVE METRIC MAXIMUM ACCURACY
G GLWOG
Oitervation
Coraa Sutieieary
Innovation
Unto
Note
Nociianoiy
Note
Stalicmeity
Quality
140
0.53
0.90
0.85
3,83
O
0.75
B
Vools
Yanas
Visua
Vishe
Vanan
V Mammans
V
Lann
Hark
GINST
Clecrvaticte
Cuaed
Interalion
Xite
Sullieacy
Rutifiius
Nazliamiis
Nolid
Staticnacit
Quilit
1.0+
A
335-
M
Are
40
A
0.57
8.75-
receo
8.50
Vistta
Vinkal
Vashe
Vini
Vaun Laae
Vanu
Vina
LSG
Cusal
Insoration
Date
Bethfflltew
Noclitnantty
Oscial Quanncy Xariaaiaa
CEANI
0,4
V. Kanaet
Vilam
V Vone
V
Valcm raae
V Vanabon
V Valund
Vanuaa
Madu
Vanta
V
1
V
V
1
1
1
G AWOG
Caral
Seevelen
Thtk Quility
Aus
65
101
63
83
50
20
200
1.00
20
S7
56
2.00
231
5.00
54
803R RPC1-
55
58
56
50
111
31
55
Sta
Figure 24: Robustness profiles of Causal Discovery algorithms against multitude of stepwise assumption violations measured as average maximum accuracy over various data regimes. From top to bottom: results for WCC GINST and GG Notably, we include three depictions for each graph to improve data comprehensibility. First, scatter plot (1-3), second, a heatmap (4-6), and third, a Vinylplot (7-9, see next page) that specifies the ranking of the methods.
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INST
Necibaaity
PEMET
31
21
30
31
5L
30
36
51
50
31
31
50
32
50
J1
50
56
12
24
31
31
20
31
30
Finaali
V
LLS
Keue
Quiiity
Cre
DVAR-
PUSII
GUweS
QASSE
1
Figure 25: Additional depictions of the robustness profiles of ten Causal Discovery algorithms against assumption violations measured average maximum accuracy. Top: heatmaps for GINST .GS Bottom: Method performance rankings. Best viewed in conjunction with the previous page.
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D.1.4 ALTERNATIVE METRIC INDIVIDUAL HYPERPARAMETER SELECTION PER VIOLATION
G LLWG
Chscrstioal Nouse
Carcal Suftrkncy
Incoration Noke
Date Qualit
Notiieority
Suttoaaiy
0.5
044
3.2
Liaoursy Auttoca
CYVE
Valcin
Vista
Yonal
Famorfark
V
G INST
Inrrestion
Natiicanll
Stutoxadil
e
0.8
D
A
Cland Flacsves Meluoh
065
Volcand
V.
Valuma
Vinua
Vish
V
V.
Viunct
V
GL8G
Chserstoonl
Caraal
Incovation
Nosse
Saltikency
Hiralideese
Notilcearlty
No.bc
Statoraiityy
Quality
16
0.6
[
&
510
0.21
Caccal Doowwry
Cheed SOLSIN
0,5
Viduel
Vis
Vidua
GWCG
Cheqrotena
Selleiery
Nedammm
Nai
DVAR-
130
30
133
13
45
355
TOUCL
30
PCSKI-
Vinual
V Ea
V
Figure 26: Robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured as normalized SHD and per-violation hyperparameter selection over various data regimes. From top to bottom: results for LLCG GINSs and GLSG, Notably, we include three depictions for each graph to improve data comprchensibility. First, a scatter plot (1-3), second, heatmap (4-6), and third, Vinylplot (7-9, see next page) that specifies the ranking of the methods.
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GINST
Nete
Sallwwrry
Selaamy
Naio
Sutecaacty
Quartyy
PCSC LIO 100
100
100
1.00
LUM
100
1.30
L00
180
1.30
1A0
100
100
100
LON
LU0
L10
1.91
110
100
LO0
110
110
PEUCE
Vii
GLSG
Elloratiinal
Selinntyy
[-]
OVAR
43
34
180
411
131
33
200
TAL
X2
446
POVEI
PCSCC
101
Vimaal
V
QUAOO
PASSE F
Figure 27: Depictions of robustness profiles of ten CD algorithms against assumption violations measured normalized SHD and per-violation hyperparameter selection. Top: heatmaps for GINTT GLSG Bottom: Method rankings. Best viewed in conjunction with the previous page.
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D.1.5
ALTERNATIVE METRIC WORST CASE PERFORMANCE VIOLATION
G LLWGG
Chocrstiool
Carzal Suftrkacy
Incoration Nobe
Late Quality
Nooso
Notiicority
Statocaiiyy
0.6
0.4
Liaourity Auttica
9.2
CYVE
Vincoa
Yanks
Vinta
Vigus
Labs
Visua
Vinal
V
G INST
Inrrestion
Ruláfiluc
Stutoxcadie
16
0.8
D
0.2
Cland Flacovrs Metuok
06
Volcand
V.
Valsma
Vana
Vish
V
V-
V.
Viune Vanniaa
V
GLSG
Chserstoon
Curaal
Incovation
Nosso
Sultikeny
Hiralileess
Notiicerilyy
Nollc
Stutooaiity
Quality
3.5
0.8
510
332
Caccal Doowery
Cheed SALSUM
AVOREL
Vidue
Vida
Vidta
GWCC
Chequrieul
Salliiery
Nedammm
Nasi
150
1.01
1.00
335
OVAR--.
120
110
10
24
20
200
23
20
200
231
30
180
150
L.00
29
TOTCE
1.00
110
1.91
110
100
110
110
1.31
110
De
15
Vinual
V
V
Figure 28: Robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured as the highest normalized SHD that was achieved for any data regime and violation level. From top to bottom: results for GLWCG GINST and GLSG Notably, we include three depictions for each graph to improve data comprchensibility. First, a scatter plot (1-3), second, heatmap (4-6), and third, Vinylplot (7-9, see next page) that specifies the ranking of the methods.
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GINST
Nome
Sallwrryy
Sedaamy
Naio
Suttcaacty
Quality
PCSK
L10 100
L00
L.00
LU0
100
1.30
LA0
180
1.30
1A0
100
100
1.0
LOW
LUC
LA0
110
L:0
110
L.00
110
110
PEUCE
100
1.90
1.00
1.00
L-01
Viluant
Vil
GLSG
Eloorational
Sellintty
(-]
OVAR
194
95
30
10
155
200
56
POVEE
110
PCUCCE-
Vimaal
V
QUANO
F
PAREEFA
3
Figure 29: Depictions of robustness profiles ten CD algorithms against assumption violations measured the highest normalized SHD for any data regime and violation level. Top: heatmaps for GNST LSS Bottom: Method rankings. Best viewed in conjunction with the previous page.
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D.2 VISUALIZATIONS OF MISSPECIFIED MODELS
In Table D we report robustness profiles for misspecified modelling parameters, i.e., Lmodet + L. To further support these results, we include more fine-grained depictions of this experiment in Fig. 30
Fig. 33
Notably, because Causal Pretraining does not require specifying a max lag, its performance superior in the low-lag regime, Further, concerning GINST Dynotears seems particularly robust.
GLWCG
Choerstinal
Careal Suftikecyy
Incoration
Date
Nooso
Hirtaidtes
Notitraatty
Nille
Statocarity
Quality
3.5
3000
ard
0.3
Do Saluee
Vile anl
Vinla
Vinual
Vaha
Vidua
V
Campp
V Vanneen
INST
Cloeratonni
Rutafdoce
Notiicnnily
Neki
Suttoaail
14
0.8
D
0.4
D
4
0.2
Clanel Flacover Mettock
V. Vilunnt
Valuanta
V.
Vinua
V Vibb
V
V
V
ctock
work
G LSG
Charratand
Canaa
Insoration Necs
Datn Quility
Nazas
Suftiancy
Notiisarity
Statscurity
0.5
500
3.2
Coned Flaccoes Mar wole
bark
Rhack
GWWC
Nian
Nain
Quiiity
36
OYMI
TOUCL
94
20
110
06
85
37
Viluali
Figure 30: Robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured as normalized SHD and under L. From top to bottom: results for LLWC GINST and GLSG Notably, we include three depictions (also see next page) for each graph improve data comprehensibility.
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GINST
Nete
Sellwwrry
Ra Rabio
Selaamy
Naio
Sutecacce
Quariyy
PCSC 90 LIO 100
290
100
100
1.00
LUM
100
1.30
LA0
180
1.30
1A0
100
110
100
1.0
LO0
L10
35
1.01
110
100
LO0
110
110
PEUCE
Vile
GLSG
Ellorational
Sellnntty
OVAR
22
151
71
72
123
750
53
30
125
105
POVEI
PCUCC
72
Vimaal
V
QUANO
F
3
Figure 31: Depictions of robustness profiles ten CD algorithms against assumption violations measured normalized SHD and under J. L. Top: heatmaps for GNST _LSSG Bottom: Method rankings. Best viewed conjunction with the previous page.
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G GLWGG
Charratoosl Nazeo
Caraal Suttrewr
Inrovation
Nonimnuriy
Neker
Statonarity
Quality
0.5
58%
500
0.2
Pacaal Trevory Vottock
Vil shock
Vilu
Vinid
Viatel
Vasha
Vistus
Vinae
Vine Alree
G GNST
Cusal
Incoration
Dase
Sufloiony
Notilceniiy
Quality
0.4
A
0.2
36
V V
V.
V. dauteen
V
V Laals
Codd
Vahha
V
V-
V
V.
Jame
V Darccass
V Bemovaa
G LSG
Oaarotand
Issoration Nicse
Nazae
Suflickns
Ralhfidus
Notlisearit:
Statacarity
Quiliy
0.4
3P+
500
3.4
3.2
Pacoal Thowary
Chead Narsoii
GWCC
Cloversseal
Teueslky
SHD []
DVMR
47
10
55
34
33
45
164
20
POMCE
166
35
25
25
80
100
Do waw
16
30
Vanati
V
V
dioe
Figure 32: Robustness profiles of ten Causal Discovery algorithms against multitude of stepwise assumption violations measured as normalized SHD and under 1 L. From top to bottom: results for L.CC INST and GLSG Notably, include three depictions (also next page) for each graph to improve data comprehensibility.
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GINST
Nete
Sallwwrry
Selaamy
Naio
Suttcaacty
Quartyy
PCSC LIO 100
190
100
100
1.00
LU0
100
1.30
L00
180
1.30
1A0
100
110
100
2000
100
LON
LU0
3
110
1.01
110
100
LO0
110
110
PEUCE
Vil
GLSG
Ellorational
Selinntyy
OVAR
24
24
130
34
30
133
11
770
POVEL
PCSCC
17
Vimaal
V-
QUAOO
3
f
Figure 33: Depictions of robustness profiles ten CD algorithms against assumption violations measured normalized SHD and under L. Top: heatmaps for GNST _GSSG Bottom: Method rankings. Best viewed conjunction with the previous page.
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Blsering qplrii
03
Merbeel
Vicatlea
Figure 34: Robustness improvements in comparison to the performance of Cross Correlation per violation and for GLWCG Interestingly, the highest robustness gains of ensembles are achieved on missingness violations.
D.3 HYPERPARAMETER SENSITIVITY
To provide a comprehensive view of the hyperparameter sensitivity of all CD methods, we include graphics that illustrate the relationship between violation severity and performance for all combi- nations of hyperparameters (excluding Lmodel * Land data regimes per method. Fig. 3 Fig. 39 contain the sensitivities for lagged effects, while Fig. 40 and Fig. 41 depict methods estimating instantaneous links. Notably, NTS-Notears's performance is often highly hyperparameter-deeendeea and a poor parameter selection can result in almost arbitrary performance. Furthermore, we observe that some algorithms occasionally exhibit unintuitive performance curves, where increase in violation does not necessarily correspond to a decrease in performance (e.g., GVAR for Vınn.com). This is, however, typically only the case if the initial performance is already poor. As only report the best-performing hyperparameter configuration per method in our main results, this has no influence this evaluation
D.4 ROBUSTNESS GAINS OF ENSEMBLES
To gain deeper insights into the efficacy of our ensemble strategies, Fig. Fig. 36 illustrate their robustness gains relative to the Cross Correlation baseline and other CD methods across different violations. The most significant improvements occur under the missing data violations Vmears V mar and V Furthermore, we investigated that the ensemble yields the greatest robustness gains in small data regimes. For instance, on small datasets, EnsembleLinea achieves a score of 0,346, compared to the best individual method (Varlingam) with score of 0.396. For big, their scores are 0.411 I and 0.421, respectively. Finally, this strong performance in low-variable environments, combined with the ability to handle missing data better than individual CD methods, provides plausible explanation for the substantial gains observed on the CausalRivers dataset Apx. [D.5] which similarly consists of 5-variable samples.
D.S ENSEMBLE PERFORMANCE ON CAUSALRIVERS
As we train and evaluate our ensemble approaches exclusively on synthetic and semi-synthetic data in the main section of the paper, we were interested in how reliant ensembles are real-world, out-of-domain data distributions. To test this, we evaluate all CD methods and all ensembles on Rivers Stein al. 2024a)). Here, we take the first 500 samples from the setting Random-5, as this matches the number of variables we use during training. Since CausalRivers provides labels only for GSS. evaluate only the corresponding predictions. We find that all ensembles notably improve performance compared to individual CD methods. From this, we conclude that ensembles could be reliable in real-worke applications despite being trained synthetic data.
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Resuction CronCuradatioon Quter
Metiall
E32 E31
6.3
1.0
Trundl 1005
8m
Revalia: Twwelene 8.1
815
47
Visterkes
Figure 35: Robustness improvements in comparison the performance of Cross Correlation per violation and for GINST
Robostross Goies
SHD Reduction (va ChossCorciltiin Basclin)
IMAL
Valkuu
E10
635
5
Medial
612
S
S
KX4
Dosile DETT
4
Vicadle
Figure 36: Robustness improvements in comparison the performance of Cross Correlation per violation and for GLSG Interestingly, the highest robustness gains of ensembles are achieved on missingness violations.
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Method
CausalRivers (SHD)
CausalPretraining
0.851
PCMCI
0.978
Nts-Notears
0.758
PCMCI+
0.835
Varlingam
0.774
Dynotears
0.715
F-PCMCI
0.840
SVAR-RFCI
0.857
GVAR
0.761
CrossCorrelation
0.798
Ensemble (Mean)
0.659
Ensemble (Linear)
0.666
Ensemble (MLP)
0.685
Ensemble (Transformer)
0.666
Table 14: Performance assessment for individual CD methods and ensembles CausalRivers (first 500 samples from Random-5|Stein et al. (2024a). We find that ensembles improve performance across the board, suggesting that ensembles can be reliable in real-world applications despite being trained on synthetic data.
D.6 MUL.I-VIOLLAION ROBUSTNESS
While we always introduce only single violation in main section, want to emphasize that TCD-Arena also allows an arbitrary combination of violations. To showcase this, we report results for two violation sets in which we simultaneously violate two assumptions. We test double common violation where we generate data by combining V Vinuo.com and Vobe.com- and a violation where introduce both types of confounding 1 and V Yoouf.1- We use the same configuration for the violation levels and the experimental protocol as for the single violations. Results for these multi-violation scenarios are presented in Table 15 through Table 17 alongside the corresponding single-violatie baselines. Generally, the method exhibiting the highest robustness in a multi-vvolation setting is also the top performer for at least one of the constituent single violations. However, observe an exception with GLSG under the combined innovation and observation violation, where Nts- Notears emerges as the most robust approach. Given that we evaluated only two specific combinations, we refrain from over wverinterppeeing these results. Nevertheless, real-world data often violates multiple assumptions simultaneously, we aim to extend ACD-Arena include more complex scenarios the future.
WCC (SHD)
inno.com
obs,com
double.com
Yeonf11
conf,i
dionhle.conf
CrossCorrelation
0.668
0.621
0.958
0.815
0.390
0.819
CausalPretraining
0.727
0.550
0.960
0.810
0.272
0.813
GVAR
0.407
0.395
0.985
0.456
0.053
0.530
Varlingam
0.260
0.425
0.987
0.384
0.067
0.469
PCMCI
0.488
0.905
0.989
0.896
0.381
0.904
PCMCI+
0.633
0.714
0.973
0.824
0.346
0.839
SVAR-RFCI
0.536
0.630
0.971
0.823
0.205
0.829
F-PCMCI
0.454
0.672
0.994
0.775
0.133
0.788
Dynotears
0.391
0.413
0.962
0.376
0.029
0.436
Nts-Notears
0.350
0.430
0.931
0.481
0.102
0.519
Table 15: Robustness evaluation for multi-violation datasets (V comble.com and V Vdouble,conf) measured normalized SHD and for the recovery of GLWCG For comparison, also report the correspond- ing individual violation results. Combining violations further reduces the robustness of CD methods across the board.
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GINST (SHD)
com
obs,com
double.com
conf,1
Yoonfii
conf couf,double
Varlingam
0.906
0.917
0.952
0.747
0.770
0,814
PCMCI+
1.000
1.000
1.000
1.000
1.000
1.000
SVAR-RFCI
1,000
1.000
1.000
1.000
1.000
1.000
F-PCMCI 0.992
0,999
1.000
0.843
0.695
0.885
Dynotears 0.757
0.802
0.940
0.568
0.474
0.677
Nts-Notears 0.872
0.832
0.959
0.716
0.701
0.773
Table 16: Robustness evaluation for multi-violation datasets (V double,com and V ddouble.conf) measured normalized SHD and for the recovery of G For comparison, we also report the corresponding individual violation results. Combining violations further reduces the robustness CD methods across the board.
GLSG (SHD)
inno..om
obs.com
dduble.com
Weonf.1
cconf,i
Voonf,double
CrossCorrelation
0.533
0.500
0.853
0.703
0.281
0.708
CausalPretraining
0.625
0.468
0.880
0.725
0,209
0.728
GVAR
0.352
0.315
0.900
0.355
0,040
0.417
Varlingam
0.317
0.296
0.877
0.290
0.067
0.370
PCMCI
0.356
0.719
0.891
0.775
0.262
0.796
PCMCI+
0.480
0.575
0.865
0.672
0.197
0.691
SVAR-RFCI
0.427
0.533
0.875
0.692
0.145
0.700
F-PCMCI
0.404
0.626
0.990
0.662
0.110
0.677
Dynotears
0.309
0,315
0.841
0.301
0.023
0.344
Nis-Notears
0.283
0,346
0.838
0.394
0.084
0.427
Table 17: Robustness evaluation for multi-violation datasets (V douuble.com and V doouble.conf) ) measured normalized SHD and for the recovery of GLSG For comparison, also report the corresponding individual violation results. Combining violations further reduces the robustness CD methods across the board.
D.7 NONLINEAR CONDITIONAL. INDEPENDENCE TESTS
As we rely on linear conditional independence (CI) tests (Partial Correlation and Robust Partial Cor- relation) for constraint-baser algorithms, we conducted a small study to assess potential performance gains from using nonlinear CI tests (in this case, GPDC) instead. For this, we compared performance on a particular violation (V Val.rbf.co small) where we expected the highest impact from this hyperparam- eter extension. We report the results in Table [8] When using GPDC to run PCMCI+, we found that robustness notably improved. However, it still did not exceed that of other methods. Because the use of nonlinear CI tests increases the computational load immensely (in our settings, roughly hundredfold), and light of these results, we refrained from including them in the hyperparameter search space in the main section of the paper.
D.8 LARGER GRAPHS
To explore settings beyond our standard experimental regimes, we evaluated method robustness under V Viuno.com large graph structures (12 variables, 3 lags), comparing it to our small (5, 3) and big (7, 4) setups. We include the results in Table Ta found that system size affects robustness heterogeneously: while Nts-Notears remains relatively stable, GVAR and PCMCI+ suffer notable performance drops as size increases. This indicates that differences between methods likely become more pronounced larger graphs. However, due to the more-thame paae-than-quadratte computational scaling of many methods with graph size, we refrained from a comprehensive evaluation of this effect, as would have forced to slim down empirical protocol.
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V smalll (SHD)
GLNCG
GINST
GLSG
CrossCorrelation
0,441
0.368
CausalPretraining
0.458
0.391
GVAR
0.399
f
0.336
Varlingam
0.410
0.775
0.370
PCMCI
0.510
0.403
PCMCI+
0,473
1.000
0.380
SVAR-RFCI
0.446
1.000
0.366
F-PCMCI
0,463
0.635
0.422
Dynotears
0.535
0.504
0.420
Nts-Notears
0.387
0.649
0.334
PCMCI+ (GPDC)
0.420
1.000
0.336
Table 18: Robustness comparison for (nl.rbf.smal of all Methods and PCMCI+ with a nonlinear conditional independence test. Even tho robustness is visibly increased, other methods still show higher robustness against this particular violation.
GLWCG (SHD)
iuno,com,,smal
inno,com,bie big.
inno,com,large
CrossCorrelation
0.588
0.748
0.895
CausalPretraining
0.727
t
GVAR
0.371
0.443
0.517
Varlingam
0.258
0.262
0.249
PCMCI
0.436
0.541
0.762
PCMCI+
0.552
0.678
0.814
SVAR-RFCI
0.460
0.611
0.811
F-PCMCI
0.377
0.530
0.944
Dynotears
0.322
0.417
0.438
Nts-Notears
0.341
0.358
0.323
Table 19: Impact of system size on method robustness for the V Vinno.com violation, measured as normalized SHD and for GLWOG Evaluated on small (5, 3), big (7, 4), and large (12, 3) settings, we observe heterogeneous scaling effects: Nts-Notears remains stable, whereas GVAR and PCMCI+ show degraded robustness as graph size increases.
GST (SHD)
imno, comllllll
iuno,com,big
Viuno larg
Varlingam
0.905
0.907
0.929
PCMCI+
1.000
1.000
1.000
SVAR-
1.000
1.000
1.000
F-PCMCI
0.987
0.998
1,000
Dynotears
0.743
0.771
0.768
Nts-Notears
0.909
0.837
0.828
Table 20: Impact of system size method robustness for the V Vinno.com violation, measured as normalized SHD and for GINST We found little impact of the size method robustness.
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GLSG
imno,com.small
Inno,com,,ig
inno,com,
CrossCorrclation
0,476
0.589
0.835
CausalPretraining
0.625
f
GVAR
0.324
0.380
0.482
Varlingam
0.318
0.191
0.205
PCMCI
0.341
0.370
0.641
PCMCI+
0.440
0.519
0.770
SVAR-RFCI
0.368
0.487
0.740
F-PCMCT
0.335
0),433
0.938
Dynotears
0.266
0.318
0.386
Nts-Notears
0.283
0.283
0.285
Table 21: Impact of system size on method robustness for the Vinno.com violation, measured as
normalized SHD and for GLSG Evaluated small (5, 3), big (7, 4), and large (12, 3) settings, observe heterogencous scaling effects: Nts-Notears remains stable, whereas GVAR and PCMCI+ show degraded robustness as graph size increases.
E APPENDIX - LLM USAGE
During the preparation this manuscript, we used Gemini 2.5 Pro to refine sentence structure and correct grammatical errors. We reviewed and edited all Al-generated suggestions and take full responsibility for the publication's 's final content.
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Vinail
DusCCriirie
Evan
TORT
Vide
CraCComiiiie
Gvan
TORT
7033+
STAM-ICC
Vik
DraComirdie
Dva
Vallar
TORT
STAR-ICCT
Dractoor
Vilison
DrosComiiiie
man
Valhare
TORT
STAB-BICT
Dracioor
monn
Walhar
TORT
soan-rrct
Tracioor
ovan
Valrer
TORT
Traciorr
Vin
ove
Vinfs
Dreul conda
OVER
VUMET
V..i
Dreut
VOMES
thwilng
Visui
Dreat
HAOK
Vasial
Figure 37: Performance differences (measured as normalized SHD) when uncovering the LWCG
depending hyperparameter selection and data regime for the violations V Vobs.add, V Vobs.mul, V Vots.auto
Wobs.oom Vos.time. obs.bbock. V Volsreal Weonfinsts Veont.lggeed- V Faith.inst Laith
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Vinko
DraCCaniizie
EvaB
TORT
Vieco
DuaComirdi
Dvan
Vallar
TORT
7031+
STAM-ICC
Vived
DraComirdi
mva
Vallar
TORT
7033
STAMEICT
Dracton
Vix
DroCCmiiiiie
man
Vallare
TORT
Dracioor
Venep
DroaContiiie
mvnn
TORT
Dracioor
Visarad
Creetherritee
ovan
TOME
Draciorr
Vernzato
OVE
TOMET
Dreutt
VONEE
Dreut
VOMES
Dreut
V
Dreath Conatcemdiie
Figure 38: Performance differences when uncovering the LWCG depending on hyperparameter
selection and data regime for the violations V Vaith.eros Val.ooos V Valuwends Valibe Val.comp V Vino.mul-
Vinooauto cono.com Vinootime: V Vinokshocks V Vinnoreal-
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V Vieec
DusComirie
ovab
Valler
TORT
713E1
300331
V Varesii
DuaComirdie
Dva
Vablax
TORT
70K1+
STAMBICT
V Vieecs
DraComirdie
mvan
Vallar
TORT
STAMBICTT
Vixe
DrasContiidie
man
Walhare
TORT
Dracioor
Vau
Dvnn
Walhare
TORT
Vineth
ovan
Vaber
TORT
TORT
Draciorr
OVN
TORTT
Dreaf Coniaee
VUMET
Dreut
OVE
PORKO
NA Sitee
Vengts
Dreua
Vasto
Figure 39; Performance differences (measured as normalized SHD) when uncovering the LWCG depending on hyperparameter selection and data regime for the violations V Vinno,unis Vino.eibb V Vinoovarr Vecet, V Vsaat length mmaas
V mary
V Ymnars Vempty.
scale-
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Vicial
misann
Valaan
Viski
Voldew
Valnas
Inum
Via
Zamue
Figure 40: Performance differences (measured as normalized SHD) when uncovering INST depending
on hyperparameter selection and data regime for the violations V Vohandd Vons,muls V. Volsautos Vobs.com-
Vobs.tmmes Vobs.hock. V Vobs.reat Voonfinnst V coonf.lags Viatthinst- Vtath.agg V "aath.zcroo Vul.mono- Vuluendd V Vll.bf-
Val.comp. V Vinno.mul- V Vinno,auto, V Vinpo.coms and V inmootime. V ibnock V Vinnouccal V Vinnouunis V Vinnowweib- V Vinoovvr- var
Vstat coef
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Viggi
Figure 41: Performance differences (measured as normalized SHD) when uncovering INST depending on hyperparameter selection and data regime for the violations V Vingghh V mears V mars Ymnars V cempyy and V scale-
78


Reviews:
[{"review_id": "xquogXm22B", "reviewer": "Reviewer_qeAM", "summary": "This paper presents TCD-Arena, a modularized and extendable testing kit designed to assess the robustness of time series causal discovery (CD) algorithms against assumption violations. The authors use this toolkit to conduct a large-scale empirical study, evaluating eight distinct CD methods across 27 different assumption violations, each with stepwise increasing severity. The study involves over 50 million individual CD attempts. The paper also provides an initial investigation into ensembling CD methods, which can improve general robustness. The TCD-Arena toolkit is presented as an open-source package to facilitate future research and comparability.", "questions": "1. Could the authors justify the decision to rely on AUROC as the primary metric in the main paper and to exclude common metrics like TPR, FDR, and SHD from the study entirely? These are standard in the field, and their omission is a significant weakness.\n2. What was the rationale for the specific selection of the eight algorithms? Were more recent, state-of-the-art methods for time series causal discovery considered?\n3. Regarding the ensembles: Beyond the \"proof-of-concept,\" what do the authors see as a realistic path to practical application? How could a meta-learner, trained on this synthetic data, be reliably applied to real-world data where the types and severity of violations are unknown?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper's most significant strength is the sheer scale and thoroughness of the empirical study, evaluating 8 CD methods against 27 distinct violations with increasing severity, totaling over 50 million CD attempts.\n2. The paper addresses the critical and practical problem of algorithm robustness to assumption violations, which is a known barrier to the adoption of CD methods.\n3. The paper provides a novel investigation into ensembling CD methods as a strategy to improve general robustness, which is an under-explored area in the literature.", "weaknesses": "1. This level of resource-intensive evaluation is an \"overkill\" and is not a realistic or practical standard for most researchers to adopt. Instead of clarifying robustness, such a convoluted framework may deter researchers, raise the barrier for reproduction, and make the results more difficult to interpret. The paper does not convincingly argue that this massive complexity offers proportionate benefits over simpler, more targeted robustness tests.\n2. The main paper's evaluation relies exclusively on AUROC. This omits other standard and widely-used metrics in the CD literature, such as True Positive Rate (TPR), False Discovery Rate (FDR), and Structural Hamming Distance (SHD). This makes the results difficult to align and compare with many other benchmark studies.\n3. The authors rightly concede that the ensemble results are a \"theoretical proof-of-concept\". The paper does not provide a clear path for how these ensembles could be deployed in a real-world scenario where ground truth is unavailable and domain adaptation is a challenge.\n4. The paper restricts some methods, like PCMCI and PCMCI+, to linear conditional independence tests, even though they can handle nonlinear cases. While this is noted in the appendix, it means the study is not evaluating the full capabilities of these specific algorithms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZDnG5rJ7hs", "reviewer": "Reviewer_K4FN", "summary": "This paper introduces TCD-Arena, a large-scale and modular benchmark framework for evaluating the robustness of time series causal discovery (CD) algorithms under systematically controlled assumption violations. This study evaluates eight representative CD algorithms through over 50 million runs across 27 assumption violations. Additionally, this work explore hyperparameter sensitivity, model misspecification and ensembling strategies that improve robustness. TCD-Arena is released as an open-source toolkit to support reproducible, large-scale robustness evaluation for time series causal discovery research.", "questions": "- In line 654 and line 771, the references to Ormaniec et al. and Yi et al. appear to be incomplete. Both works were published at ICLR 2025. Moreover, the title of Yi et al. is fully capitalized, which is inconsistent with the other citation styles. Providing complete and correctly formatted references would help readers in the community quickly identify related advances.\n- Could the authors provide more details about the GVAR method and the code source used in the experiments? It seems that this corresponds to an extended version of the standard VAR model, as Table 11 indicates that the method has two hyperparameters, coeff and p-val. Including a brief methodological description would make the work more complete. Also, unless there is a specific reason to retain this name, it might be preferable to change it, since another work [1] is also named GVAR, which could lead to confusion.\n- Could the authors elaborate on how AUROC was computed in Figure 1? My understanding is as follows: for each assumption violation scenario, there are five different violation levels, and for each level, 100 datasets are sampled. Each method, under each hyperparameter configuration, computes AUROC across all these datasets. The results are then averaged to obtain an average AUROC per hyperparameter configuration, and the best-performing configuration is reported. Please confirm whether this interpretation is correct.\n- The paper defines robustness (line 363) in a way that differs from prior works such as Montagna et al. and Yi et al., which report results based on the best-performing hyperparameter setting per violation. Could the authors discuss the differences between these two evaluation strategies and their respective advantages or limitations? Although there may not yet be a community-wide consensus, a discussion on this distinction would be valuable.\n\n[1] Marcinkevičs R, Vogt J E. Interpretable models for granger causality using self-explaining neural networks[J]. arXiv preprint arXiv:2101.07600, 2021.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper presents a comprehensive and rigorously executed empirical benchmark that systematically evaluates the robustness of time series causal discovery methods across diverse assumption violations. By encompassing 27 distinct violation scenarios and analyzing eight representative algorithms, the study establishes a new empirical standard for assessing reliability in time series causal discovery.", "weaknesses": "The authors conducted a commendable and well-structured benchmark study. However, there are a few aspects that could improve the paper’s readability and completeness:\n- In line 408 and line 411, the references to Fig. 3b are mistakenly written as Fig. 2b.\n- In line 2420, the right panel of Figure 18 shows missing results for NTS-NOTEARS under certain conditions.\n- Including a runtime comparison across different algorithms would be valuable for practitioners, as it would help guide the selection of time-series causal discovery methods in real-world applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0wkrfJW5R3", "reviewer": "Reviewer_B3QT", "summary": "The paper introduces TCD-Arena, a modular benchmarking toolkit for time-series causal discovery (CD) with a focus on robustness under stepwise assumption violations. It evaluates 8 CD methods spanning major paradigms across 27 violation types (each with graded severity) and multiple data regimes, totaling 50M runs. The study contrasts recovery of three graph notions (GWCG, GSG, GINST), analyzes hyperparameter sensitivity, and explores ensembling of CD methods as a route to improved robustness. Results suggest (i) notable variability in robustness across methods and violations, (ii) simple Granger-style approaches are comparatively robust for lagged effects, (iii) deep methods can trail simpler baselines, and (iv) ensembling provides tangible gains.", "questions": "* Can you consider testing methods on larger graphs?\n\n* Can you include at least one semi-synthetic or interventional real dataset to validate ranking consistency (even if partial ground truth)? \n\n* Do your qualitative conclusions hold under AUPRC, (adj)SHD, and orientation error measures? \n\n* Any cases where AUROC misleads? \n\n* Why that specific cancellation pattern? Have you tried near-unfaithful settings (small but nonzero effects) and do trends persist? \n\n* How were search budgets matched across methods? Can you report per-method best-vs-average gaps in the main text and show robustness profiles over the top-k configs? \n\n* How do you prevent leakage from the meta-learner training to evaluation regimes? Any evidence of transfer to unseen violation types or severities? Stationarity violation. \n\n* You resample coefficients while keeping the skeleton fixed. Have you tested changing skeletons over segments (structural breaks)? \n\n* Can you include or at least discuss clearly recent algorithms like TiMINo[1], varFCI [2], LPCMCI [3], PCGCE [4] J-PCMCI[5], CBNB [6], SpaceTime [7] \n\n\n* Can you discuss clearly the contribution of this paper with respect to other ICLR benchmark papers on causal discovery. Like for example: [8] (which was cited but not thoroughly discussed)\n\nMinor:\n\n* You might consider citing [9] in addition to Pearl (2009) and Peters et al. (2017) when referring to a comprehensive review of causal discovery in i.i.d. settings, as this book provides a more in-depth treatment of constraint-based causal discovery methods (PC and FCI) than the other two references.\n\n\n* In much of the existing literature, the WCG and summary graph are defined differently: both typically include instantaneous relations. It may therefore be helpful to explicitly state that your definitions diverge from the conventional ones. You might also consider using distinct notation (e.g., LWCG for lagged window causal graph and LSG for lagged summary graph) to avoid confusion. Additionally, note that the literature also defines a third type of graph, the extended summary causal graph, which, as I understand it, combines features of the lagged summary graph and the instantaneous graph.\n\n\nReferences:\n\n* [1] Peters et al. Causal Inference on Time Series using Restricted Structural Equation Models. Neurips, 2013\n\n* [2] Malinsky and Spirtes. Causal Structure Learning from Multivariate Time Series in Settings with Unmeasured Confounding. KDD workshop on CD. 2018\n\n* [3] Gerhardus and Runge. High-recall causal discovery for autocorrelated time series with latent confounders. Neurips, 2020\n\n* [4] Assaad et al. Discovery of extended summary causal graphs from time series. UAI. 2022\n\n* [5] Gunther et al. Causal Discovery for time series from multiple datasets with latent contexts. UAI, 2023\n\n* [6] Bystrova et al. Causal Discovery from Time Series with Hybrids of Constraint-Based and Noise-Based Algorithms. TMLR, 2024\n\n* [7] Ameche et al. SpaceTime: Causal Discovery from Non-Stationary Time Series. AAAI, 2025\n\n* [8] Cheng et al. CAUSALTIME: REALISTICALLY GENERATED TIMESERIES FOR BENCHMARKING OF CAUSAL DISCOVERY. ICLR, 2024.\n\n* [9] Spirtes et al. Causation, prediction, and search. MIT press. 2000.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 1, "strengths": "I found the paper interesting and I do think that there is a need more papers like in the causal discovery community.\n\nThis paper study the robustness to violated assumptions is central for CD adoption\n\nEvaluating GWCG, GSG, and GINST clarifies what each method is able to recover.\n\nIt studies hyperparameter sensitivity analysis which is important and often ignored. This it has a   practical value for deployers.\n\nIf the toolkit is released as open-source with configs and seeds, TCD-Arena could become a useful community resource.", "weaknesses": "* It is very difficult to understand the contribution of the paper without going to the Appendix.\n\n* Despite the motivation that real-world ground truth is scarce, the study remains fully synthetic. This limits the external validity; even a small real or semi-synthetic case  would strengthen claims.\n\n* Many violations are reasonable, but several design choices (e.g., faithfulness violations via path cancellation; innovation noise blends; stationarity via coefficient resampling) feel one specific instantiation among many. It’s unclear how sensitive findings are to alternative parameterizations. More ablations/justifications would help.\n\n* Some algorithms do not target GINST, others require max-lag specification, and assumptions differ (e.g., non-Gaussianity). While you note this, the aggregated “robustness” comparisons may conflate target mismatch with method weakness. Clearer tracks per target/assumption would avoid this.\n\n* Meta-learners trained on synthetic violations may not transfer across domains/violations. The practical ensembles’ training protocol need clearer specification.\n\n* Recent CD algorithms for time series causal discovery are not included. \n\n\n* While the topic of causal discovery robustness is important, this paper is primarily a benchmarking and dataset-style contribution based entirely on synthetic data. It does not introduce a new learning algorithm, theoretical insight, modeling principle, or a new real-data benchmark, types of contributions that, to my understanding, are typically expected for ICLR’s main track. Even though ICLR does occasionally accept benchmark-oriented papers, such works usually have a broad and transformative impact (for example, by introducing a new real-world benchmark or establishing a widely adopted evaluation framework). In its current form, the contribution of this paper seems more incremental in scope (even though I think it is very interesting and needed). That said, I may not be fully aware of the current editorial stance on such submissions, and I would defer to the area and meta chairs regarding venue fit. Given its focus, the work might be better suited for a Datasets and Benchmarks track at a major AI or ML conference, where such empirical frameworks are the main focus.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fyKm75CfCz", "reviewer": "Reviewer_9gRA", "summary": "The paper introduces TCD-Arena, a modular, open-source testbed for stress-testing time-series causal discovery (CD) methods under stepwise assumption violations. It defines three target graphs—WCG (lag-window graph), SG (lagged summary graph), and INST (instantaneous graph)—and implements 27 violation families (e.g., observational/innovation noise structures, latent confounding, faithfulness violations, nonlinear mechanisms, non-stationarity, sample length, data quality, scaling). For each violation, five intensity levels are simulated across 8 data regimes (varying T, D, L, sparsity, and presence of contemporaneous effects). The study evaluates eight CD methods spanning major paradigms (cross-correlation, VAR/GVAR, VARLiNGAM, PCMCI/PCMCI+, DYNOTEARS, NTS-NOTEARS, and CausalPretraining), with extensive hyper-parameter searches, producing over 50 million runs. \n\nKey findings: (i) Granger-based approaches are comparatively robust for lagged edges (WCG/SG); (ii) Dynotears and PCMCI+ are strongest on instantaneous edges (INST) while VARLiNGAM lags; (iii) deep models (CausalPretraining, NTS-NOTEARS) underperform simpler ones; (iv) under-specifying the max lag hurts all methods, whereas over-specifying is mostly benign; (v) ensembles (simple average, linear, MLP, ConvMixer) improve average robustness.", "questions": "1. Could TCD-Arena add skeleton changes and context-specific mechanisms (e.g., regime switches) to stress methods designed for heterogeneous data?\n2. What meta-features (if any) about datasets improve selection? Did you try cross-violation training and measure out-of-violation generalization?\n3. Interesting result that ↑𝐿 is safe while ↓𝐿 is harmful. Can you show sample-complexity curves for varying 𝑇 under ↑𝐿 (variance vs bias trade-off)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. First comprehensive time-series-focused robustness arena with graded violation intensities rather than binary toggles; covers 27 violations with clear, reproducible generators.\n2. Separating WCG/SG/INST clarifies evaluation of lagged vs. instantaneous structure—often conflated in prior work.\n3. Massive run budget with systematic hyper-parameter sweeps; sensible aggregation (AUROC across regimes × levels) and ablations on lag mis-specification and hyper-parameter sensitivity.\n4. Ensembling insight shows that meta-learners fed only with method outputs (no raw input) increase robustness and reduce variance across violations.", "weaknesses": "1. AUROC on adjacency recovery is threshold-free but may obscure class-imbalance and orientation errors. SHD, F1/AUPRC, precision, and arrow-orientation accuracy would provide complementary views; some are included in the appendix but not highlighted.\n2. The DGP (Eq. 2) is additive with univariate link functions and a fixed max lag; some violations (e.g., nonlinearity via RBF/mixtures) are still univariate. Methods exploiting multivariate interactions or cycles may not be assessed.\n3. Non-stationarity is simulated by redrawing coefficients A while keeping the skeleton fixed; real systems also change skeletons, contexts, regimes, and sampling rates.\n4. Only one faithfulness-violation pattern is implemented; non-Gaussian noise is introduced for innovation but not matched to LiNGAM’s identifiability conditions (e.g., independent non-Gaussian; instantaneous vs lagged). The result that VARLiNGAM is not advantageous under this violation could reflect misspecification.\n5. Now it seems only MCAR is used for missingness. More missingness mechanisms, such as MAR and MNAR, should be examined.\n6. Meta-learners are trained on simulated violations; transfer to real data under distribution shift is acknowledged but not tested. The Pareto oracle is instructive but unattainable. Guidance on how to pick ensembles in practice is limited.\n7. Given 50M runs, a brief computational cost profile (GPU hours, carbon) and per-method runtime/memory would help users scope experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)