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One Thing Leads to Another: The Essentials of Time Series Causal Discovery

Kurt Butler, Damian Machlanski, Michael P. J. Camilleri, Zhihua Liu, Panagiotis Dimitrakopoulos, Feng Chen, Ricardo Silva, Sotirios A. Tsaftaris

arXiv:2610.13430v1eess.SPcs.LGstat.ML

TL;DR

TSCD methods rely on differing principles and assumptions, and the tutorial surveys representative approaches and their trade-offs. It concludes that no general-purpose TSCD method exists, so selecting an approach for the data remains necessary.

  • Problem

    TSCD methods use differing principles and assumptions, and the tutorial concludes that no general-purpose method exists for every data problem.

    Source: §VII · p. 14

  • Method

    The tutorial presents exemplary TSCD methods and unifies their assumptions, philosophies, connections, and trade-offs.

    Source: §VII · p. 14

  • Results

    The tutorial concludes that there is still no general-purpose TSCD method, so the approach must be selected for the data.

    Source: §VII · p. 14

  • Takeaways & Limitations

    Studies in agriculture and Earth system science show TSCD can support causal analysis where interventions are impossible.

    Source: A. Agriculture and Earth system science · p. 13

    The tutorial is selective, excluding many methods and specialized approaches for missing data, irregular sampling, and non-stationarity.

    Source: A. Limitations and scope · p. 2

Abstract

from arXiv · show

Time series causal discovery --- the process of detecting causation between time series --- has its roots in signal processing research from over a half century ago. It has since received active and widespread interest, evolving into different forms across domains such as statistics, dynamical systems and machine learning. Entering this fragmented and constantly evolving research landscape can be daunting. This tutorial provides a unified overview of time series causal discovery across research communities. We introduce the key principles invoked by modern discovery methods, and we discuss their limitations and assumptions. We also discuss how discovery methods are evaluated, benchmarked and fine-tuned. We further discuss how causal discovery for time series is becoming crucial for downstream tasks in agriculture and Earth science, in healthcare, and in artificial intelligence systems. We believe that this tutorial will serve as a valuable resource in navigating the deluge of recent causality research, as well as an invitation to signal processing audiences to contribute to causal problems. To further help readers, we host a companion repository to link to useful code packages and resources at https://github.com/CHAI-UK/Awesome-Time-Series-Causal-Discovery.

I. INTRODUCTION

Time-series causal discovery grew from Granger’s work on detecting causation between signals, but later intervention-based definitions and distinct research traditions have made the field difficult to navigate. This tutorial offers a unified introduction while remaining selective in its coverage.

I. INTRODUCTION

  • Granger’s work popularized detecting causation between time series, using the principle that causes precede their effects.That temporal-precedence principle is tailored to time series, but it does not directly apply to tabular data or images.Source: §I · p. 1
  • Pearl’s intervention-based definition asks whether changing a cause would change its effect, distinguishing it from Granger’s notion.The intervention need only be possible in principle, not necessarily in real life.Source: §I · p. 1
  • The tutorial unifies a fragmented field so readers can navigate its literature, identify methods’ underlying principles, and evaluate methods for their problems.It directs readers seeking implementation-level detail to the companion repository and primary references.Source: §I · p. 2

A. Limitations and scope

  • This tutorial is selective: it covers representative principles and methods, excluding many alternatives and specialized methods for missing data, irregular sampling, or non-stationarity.The companion repository links to additional methods, software, and papers and is intended to be updated.Source: A. Limitations and scope · p. 2

B. How do I get started with TSCD?

  • The paper proceeds from notation and problem formulation to methods, evaluation and tuning, applications, and remaining challenges.A companion quick-start guide provides a programming-oriented entry to TSCD.Source: B. How do I get started with TSCD? · p. 2, §II · p. 2

A. Basic notation

The tutorial represents time-series causal relationships with time-indexed graphs and formalizes TSCD as inferring graph structure from observed signals. What an edge means depends on the method’s definition of causality.

B. Graphs and graph notation

  • The tutorial’s default window graph represents the minimal time points needed to show causal effects and delays; under causal stationarity, it characterizes the process.Full and window graphs are acyclic, while summary graphs can contain cycles; summary graphs omit lag information.Source: B. Graphs and graph notation · p. 3, Figure 1 · p. 3

A. Basic notation

  • A causal graph’s nodes represent variables and its directed edges represent causal relationships; Pa(Xj,t) denotes the nodes with edges into Xj,t.The tutorial assumes a single realization of the system; methods exploiting multiple realizations are outside its scope.Source: A. Basic notation · p. 2, B. Graphs and graph notation · p. 3

C. Interventions and structural causality

  • The do-operator represents an intervention that clamps Xt to xt, changes the data-generating process, and deletes edges entering Xt.An adjustment set can equate an interventional distribution with a conditional distribution, enabling estimation without interventional data.Source: C. Interventions and structural… · p. 3

D. Problem statement

  • In window-graph TSCD, data are used to infer time-indexed edges, but an edge may encode predictive causality, synchronized dynamics, or structural causality.These definitions can yield the same graph in common cases, but not always.Source: D. Problem statement · p. 4
  • Choosing a method begins with identifying its assumed definition of causality; structural notions may suit applications where interventions influence data generation.Granger causality is used in computational neuroscience because it can provide sufficient information to study brain activity.Source: D. Problem statement · p. 4

E. Typical assumptions

TSCD methods often rely on assumptions about observed causes, statistical dependencies, or causal mechanisms, and predictive causality tests whether past signals improve future predictions. The tutorial compares these principles and representative methods.

E. Typical assumptions

  • Causal sufficiency assumes no unobserved process causes two variables, while faithfulness links conditional independence to separation in the causal graph.Faithfulness is the converse of the Markov property, which links graphical separation to statistical independence.Source: E. Typical assumptions · p. 4
  • Causal stationarity means the causal model is invariant over time, a property distinct from statistical stationarity of the signal.A random walk is given as an example of a causally stationary process whose variance grows over time.Source: E. Typical assumptions · p. 4

III. METHODS FOR CAUSAL DISCOVERY WITH TIME SERIES

  • The tutorial organizes TSCD methods by inference principles and compares representative methods’ assumptions.Its taxonomy is conceptual rather than an exhaustive listing of principles or methods.Source: §III · p. 5, Table I · p. 5

A. Predictive causality

  • Granger causality tests whether a signal’s past improves prediction of another signal’s future, including after conditioning on other available signals.The original linear detector compares two order-Q autoregressive models and uses an F-statistic or likelihood-ratio test; allowing future values in conditioning can model instantaneous effects, often a sign of insufficient sampling resolution.Source: A. Predictive causality · p. 5

1) Extensions of GC:

Granger-causality extensions adapt the approach to different signal properties and modeling challenges.

1) Extensions of GC:

  • Fourier-analysis spectral detectors extend Granger’s approach.Source: 1) Extensions of GC · p. 6
  • Kernel methods and deep neural networks generalize Granger causality to nonlinear systems.Source: 1) Extensions of GC · p. 6
  • Other variants address latent confounding or use sparse and regularized formulations for high-dimensional settings.These high-dimensional settings have variable counts approaching or exceeding the number of observations.Source: 1) Extensions of GC · p. 6

2) Transfer entropy:

Transfer entropy measures predictive causality as a reduction in uncertainty and can capture arbitrary statistical dependencies nonparametrically. Predictive causality alone, however, is insufficient to infer structural causality without additional assumptions.

2) Transfer entropy:

  • Transfer entropy measures how much another signal’s past reduces uncertainty about a target’s future, allowing arbitrary statistical dependencies in a nonparametric setting.Variants address multivariate, nonlinear, non-stationary, and high-dimensional time series.Source: 2) Transfer entropy · p. 6

3) Limitations of GC:

  • Different data-generating processes can produce the same probability distributions, so predictive causality alone cannot identify structural causality without additional assumptions.Under certain conditions, predictive causality does imply structural causality.Source: 3) Limitations of GC · p. 6

B. Dynamical systems

  • In deterministic dynamics, a unidirectional cause can still make its own past predictable from the effect, leading Granger causality to report a false positive in the reverse direction.The passage describes a system in which X_t causes Y_t, yet Y_t’s history predicts X_t.Source: B. Dynamical systems · p. 6
  • Takens’ embedding theorem motivates cross mapping: sufficiently many lags of Y can represent the system state and estimate X, while CCM tests for this reverse-direction map.The procedure forms a delay-embedding, maps it to estimates, and checks those estimates using Pearson correlation.Source: B. Dynamical systems · pp. 6–7

1) Limitations of cross mapping:

Cross mapping can address a blind spot of Granger causality, but depends on strong conditions. Structural-causality approaches instead infer graph structure using assumptions that make it identifiable from observational data.

1) Limitations of cross mapping:

  • CCM can be sensitive to measurement noise, stochastic dynamics, non-stationarity, and embedding choices; strong forcing can make unidirectional causation appear bidirectional.Its conclusions also depend on Takens’ theorem conditions, although some conditions can be tested with care.Source: 1) Limitations of cross mapping · p. 7

C. Structural causality

  • Structural-causality methods frame discovery as graph inference or structural-equation modelling, using assumptions to identify causal structure from observational data.The time-series graph is a window graph, and the framework starts from Pearl’s structural equation model.Source: C. Structural causality · p. 7
  • Structural methods include constraint-based, function-based, and score-based approaches, which differ in how they orient edges, restrict model classes, or rank candidate causal models.These categories also apply to structural-causality approaches for time series.Source: C. Structural causality · p. 7

1) Constraint-based methods:

The tutorial groups structural time-series discovery into constraint-based, function-based, and score-based methods. Their strategies trade on conditional-independence testing, assumptions about functional form and noise, or differentiable optimization with graph constraints.

1) Constraint-based methods:

  • PCMCI uses a two-stage conditional-independence procedure: PC1 selects candidate parents, then MCI filters edges to address false positives and bias from autocorrelation.Temporal filtering and correlation prioritization help narrow the search in high-dimensional time series.Source: 1) Constraint-based methods · p. 7

2) Function-based methods:

  • Function-based methods infer direction from asymmetry: the chosen function and noise assumptions may permit an effect model while ruling out the reverse model.VARLiNGAM assumes linear links and non-Gaussian errors; TiMINo tests additive-noise models and can return “do not know.”Source: 2) Function-based methods · p. 8
  • Function-based methods suit settings with partially known physical mechanisms, but non-additive error structure can prevent them from finding valid causal links.Their reliability depends on the model assumptions.Source: 2) Function-based methods · p. 8

3) Score-based methods:

  • DYNOTEARS turns graph search into continuous optimization, minimizing reconstruction loss while enforcing differentiable acyclicity for instantaneous relationships.It models time series as a VAR process with instantaneous and lagged relationships.Source: 3) Score-based methods · pp. 8–9
  • DYNOTEARS can converge faster than combinatorial approaches, but is not invariant to rescaling and may require careful penalty tuning.Its documented caveats include varying signal scales and avoiding overfitting or cycles.Source: 3) Score-based methods · p. 9

D. Methods for latent confounding

Methods for latent confounding relax causal sufficiency by representing uncertain ancestral relationships in partial ancestral graphs. This accommodates unobserved processes but weakens what the graph identifies and increases computational demands.

D. Methods for latent confounding

  • LPCMCI relaxes causal sufficiency by accommodating latent processes and outputs a partial ancestral graph rather than PCMCI’s DAG.Its iterative, order-independent search refines candidate parent sets and ancestral relationships across repeated passes.Source: D. Methods for latent confounding · p. 9
  • A PAG’s bidirected edge indicates latent confounding and its approximate temporal footprint, but does not identify the confounder itself, making effect estimation less direct.SVAR-FCI and tsFCI also output PAGs.Source: D. Methods for latent confounding · p. 9
  • Confounder-robust methods are more computationally demanding and typically suit modest systems, especially when observational data leave unmeasured drivers plausible.Their search tracks ancestral and latent-confounding possibilities jointly.Source: D. Methods for latent confounding · p. 9

IV. BEST PRACTICES FOR EVALUATION AND TUNING

Hyperparameter choices shape which temporal relationships TSCD can detect, while evaluation requires measures of both edge-level accuracy and global graph structure. No single metric captures every relevant error, especially when intervention reliability matters.

A. Hyperparameter selection

  • Window length Q balances capturing slow-propagating influences against the memory, processing, and false-discovery costs of including too many lags.Ideally, Q matches the system’s longest physical delay, which is rarely known a priori.Source: A. Hyperparameter selection · p. 10
  • Aggressive downsampling can collapse delayed effects into zero-lag links, obscure directionality, or leave the resulting data causally underdetermined.Modest downsampling may improve detection power by smoothing stochastic fluctuations that do not contribute to the causal mechanism.Source: A. Hyperparameter selection · p. 10

B. Evaluation metrics

  • Precision, recall, and F1 evaluate predicted causal edges, respectively reflecting link reliability, recovery of true links, and their harmonic-mean balance.Each ranges from 0 to 1, with 1 representing perfect reconstruction of the graph’s skeleton and orientation.Source: B. Evaluation metrics · p. 10, Table II · p. 11
  • Structural Hamming Distance (SHD) measures graph-edit errors, but reversed edges may be penalized twice under one definition and location-specific costs are ignored.An alternative SHD definition counts edge reversals as a single operation.Source: B. Evaluation metrics · p. 11
  • Structural Intervention Distance (SID) counts variable pairs whose predicted and true interventional distributions differ, addressing errors with unequal consequences for causal effects.SID is asymmetric, and a low value suggests interventional reliability even when SHD is nonzero.Source: B. Evaluation metrics · p. 11

C. No free lunch

TSCD methods depend on assumptions, so their performance varies with the data-generating process and no universal benchmark winner is the goal. Benchmarking instead maps methods’ applicability boundaries using complementary data types.

C. No free lunch

  • TSCD inference is assumption-driven, and assumptions such as causal sufficiency may be untestable, making method use unjustified without careful application-specific scrutiny.CCM’s compact-attracting-manifold assumption is not directly testable from noisy data.Source: C. No free lunch · p. 11

D. All shades of benchmarking

  • Benchmarking should map algorithm applicability boundaries rather than identify a universal winner, because a method may fail when a dataset violates its assumptions.The tutorial groups evaluation data into simulations, semi-synthetic datasets, and real-world data.Source: D. All shades of benchmarking · pp. 11–12
  • Simulations provide controlled data-generating processes and abundant ground truth, but may lack real-world messiness or expose exploitable statistical artifacts.Researchers can vary noise, nonlinearity, and coupling strength to test specific assumption violations.Source: D. All shades of benchmarking · p. 12
  • Semi-synthetic datasets bridge simulation and reality using domain-specific equations or generative models constrained to provide verifiable ground truth.Examples include equations from climate, power-grid, or biological systems.Source: D. All shades of benchmarking · p. 12
  • Real-world data tests practical impact, but definitive causal ground truth is difficult to obtain for complex systems.Community-driven efforts such as CausalRivers provide high-fidelity signals with verified causal links.Source: D. All shades of benchmarking · p. 12
  • Use controlled synthetic experiments to diagnose algorithmic failures and at least one semi-synthetic or real-world case study to demonstrate impact.The benchmark types serve complementary roles in TSCD development and assessment.Source: D. All shades of benchmarking · p. 12

V. IMPACT AND APPLICATION ACROSS DOMAINS

TSCD supports reasoning about mechanisms when experiments are limited, with method choice reflecting intervention feasibility, confounding concerns, and dimensionality. Applications span agriculture and Earth science as well as healthcare and biology.

A. Agriculture and Earth system science

  • Agriculture and Earth science applications choose methods suited to their settings: cross mapping for deterministic, low-noise attractors and constraint-based approaches for continental climate questions.Physical climate knowledge can narrow an otherwise underdetermined search.Source: A. Agriculture and Earth system science · pp. 12–13
  • TSCD in agriculture and Earth science supports causal analysis where interventions are impossible, not only improved prediction.Examples include extracting wheat-yield precursors months ahead and identifying climate drivers from observational records.Source: A. Agriculture and Earth system science · p. 13

B. Health science and biology

  • Healthcare applications include recovering causal antecedents of hypertension and hypoxia despite confounding, and predicting acute kidney injury 16 hours earlier than existing guidance-based pathways.The ICU setting motivates confounder-robust variants over standard PCMCI.Source: B. Health science and biology · p. 13
  • Neuroscience and genomics applications use distinct TSCD approaches, from causal analyses of brain signals to sparse Granger-causality models for large gene networks.BETS recovered nearly 2,800 genes and over 30,000 edges from lung cells exposed to a glucocorticoid over 12 hours.Source: B. Health science and biology · p. 13
  • Temporal causal graphs turn patient and biological trajectories into structured, interpretable knowledge for treatment planning, patient stratification, and precision medicine.The stated aim is to ground these activities in mechanisms rather than correlations alone.Source: B. Health science and biology · p. 14

C. Artificial Intelligence

TSCD can inform AI system design by identifying causally relevant inputs, while the field still lacks a general-purpose method. The tutorial concludes that method selection must account for each approach’s strengths, limitations, and assumptions.

C. Artificial Intelligence

  • PCMCI-based TSCD has been used to identify causally relevant inputs for a neural prediction model, pointing to a role in AI systems robust under intervention.The broader framing treats causal graphs as design constraints on which variables are reliable for decision-making.Source: C. Artificial Intelligence · p. 14

VI. RESEARCH OUTLOOK

  • TSCD still faces scalability, non-stationarity, efficiency, and unobserved-confounding challenges, and methods underperform on benchmarks such as CausalRivers.It remains unknown whether these issues reflect solvable method limitations or fundamental limits.Source: §VI · p. 14
  • New paradigms target scalability and transfer differently: topological ordering and amortized discovery seek efficiency, while foundation models target zero-shot transfer across unseen domains.Amortized models remain bound to the system distributions or shared dynamics seen during training.Source: §VI · p. 14

VII. CONCLUSION

  • No general-purpose TSCD method exists, so practitioners must select an approach for their data; automated problem-to-solution pipelines remain an open problem.The tutorial presents understanding methods’ assumptions, strengths, and limitations as necessary progress toward that mapping.Source: §VII · p. 14
  • The conclusion frames time-dependent causal discovery as a prerequisite for safe and reliable AI as signal processing moves from monitoring systems toward manipulating them.Source: §VII · p. 15
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