Event-Level Temporal Causality

• Event-level temporal causality is a framework that defines and analyzes causal links between discrete, often extreme, events in dynamic systems.
• The method uses state space embedding and nearest neighbor analysis to detect causality, emphasizing the timing and intensity of individual events.
• It offers improved detection of nonlinear, multiscale causal relations in systems such as climate, neuroscience, and finance by focusing on event-to-event interactions.

Event-level temporal causality concerns the identification, modeling, and quantification of causal relationships that are inherently bound to, and structured by, the sequence and timing of discrete events. Unlike causality defined solely over variables or time-averaged patterns, event-level temporal causality focuses on understanding how individual events—particularly extreme occurrences—initiate, mediate, or propagate subsequent events in dynamic systems. This topic integrates conceptual, mathematical, and computational advances from dynamical systems, information theory, neuroscience, geophysical modeling, and nonlinear time series analysis. Contemporary research extends causal inference from average or aggregate behaviors to the granularity of event-to-event interactions, with an emphasis on addressing causality among extremes, multi-scale temporal dependencies, and nonlinear dynamical processes.

1. Conceptual Foundation: Event-level Causality and Dynamical Systems

Event-level temporal causality departs significantly from traditional, time-averaged causal inference by focusing on the causal connections between individually identifiable events—often extremes—rather than steady-state or mean behaviors. This approach is motivated by domains where system behavior is dominated by transient, high-impact phenomena, such as extreme weather events, environmental shocks, or episodic disruptions in biological or social systems. Under this perspective, events are defined as particular states or occurrences within a dynamical system—e.g., a marine heatwave, drought, or financial crash—that are of sufficient rarity or intensity to warrant special causal analysis (Yu et al., 5 Sep 2025).

This methodology is distinct in that it targets:

• Causal relationships linked to the timing and co-occurrence of discrete events rather than generalized cross-variable dependencies.
• Directionality and temporal precedence evaluated at the granularity of single event occurrences.
• The capacity for one extreme event in a subsystem to trigger, mediate, or be necessary for the emergence of a subsequent extreme in a coupled subsystem.

A pivotal distinction is drawn between global or mean-field causal analysis—which may mask transient or rare causal pathways—and instantaneous, event-triggered causality where precise timing and state-dependence are critical.

2. Methodological Advances: Dynamic System-based Detection of Event-to-Event Causality

The core methodology advanced by recent work extends dynamic system–based causality detection techniques to target event-to-event causality, adopting state space embedding and nearest-neighbor statistics to quantify whether a given extreme event in one system is causally responsible for another event in a coupled system (Yu et al., 5 Sep 2025).

The approach consists of the following steps:

1. State Space Embedding. Construct the phase space representations of two potentially coupled systems, X and Y. Each point in time is mapped to a state vector in this space, either using all system variables or through time-delay embedding if only partial measurements are available.
2. Event Characterization. Identify extreme events by their exceedance over chosen quantile thresholds (e.g., above the 95th percentile).
3. Nearest Neighbor Analysis. For each reference state corresponding to an event in one system, locate K nearest neighbors in its phase space.
4. Cross-mapping. Transfer the time indices of these nearest neighbors to the other system (e.g., mapping events in X to corresponding states in Y).
5. Causality Index Calculation. Compute the rank-based causality index:

$$P(X \rightarrow Y) = \frac{1}{T-2}\sum_k \frac{\text{rank}(d_x(t, t_k))}{K}$$

where $T$ is the time series length and $d_x(t, t_k)$ denotes the Euclidean distance in X's phase space.

Statistical significance is established via surrogate testing (e.g., shuffling time indices to estimate the null distribution), and nonlinearity is accounted for naturally through phase space geometry.

Critically, this method evaluates causal asymmetry by comparing $P(X \rightarrow Y)$ and $P(Y \rightarrow X)$, where a significant excess in one direction suggests a causal influence specific to the timing and configuration of extreme events.

3. Empirical Validation and Case Studies

The method’s performance is validated through canonical and real-world systems:

• Coupled Lorenz-Lorenz Systems: Using numerical integration of coupled Lorenz equations, with event-driven coupling (e.g., active only upon exceeding a state threshold), the technique demonstrates that only a subset of extreme events in the driving system X elicit a corresponding extreme in the response system Y. This selective triggering unambiguously distinguishes causal from coincidental event pairs (Yu et al., 5 Sep 2025).
• Walker Circulation Phenomenon: In analyzing climate data, such as sea-level pressure (SLP) over the Western Pacific and surface air temperature over the Central/Eastern Pacific, the approach detects an asymmetric event-level causal influence—whereby extreme temperature anomalies in CEPAC more strongly predict subsequent SLP extremes in WPAC than vice versa. This aligns with physical mechanisms underpinning atmospheric circulation.

Measurement strategies are systematically compared:

• Full system measurements maximize detection efficacy.
• State space reconstruction using delay embedding compensates for incomplete observations, improving event-to-event causality detection over naive partial measurement.

4. Comparison with and Extension Beyond Traditional Causal Frameworks

The dynamic system event-to-event causality approach contrasts with conventional techniques such as Granger causality, transfer entropy, and convergent cross mapping that typically:

• Detect global (time-averaged) causal relations, insufficient for stochastic or nonlinear event triggers.
• Underperform in high noise or high-dimensional, nonstationary systems where causality may only emerge in the tail behaviors.

In direct comparison, this event-level approach:

• Demonstrates improved detection in concurrent or rare-event scenarios, particularly with nonlinear or threshold-based couplings.
• Is robust against non-Gaussian noise and tolerates measurement limitations, provided sufficient consideration is given to embedding dimension and delay parameters.

The method recognizes that not all extreme events are causally effective; coupling coefficients and delay structure (i.e., synthetic lag) are systematically evaluated, ensuring that only temporally consistent, dynamically plausible pathways are identified.

5. Theoretical and Practical Implications

The shift to event-level temporal causality analysis yields several significant implications:

• Multiscale and Nonlinear Dynamics: By targeting causality among discrete, often rare, extremes, the method addresses the nonlinear, multiscale nature of many real-world systems—advancing understanding in fields such as climate, neuroscience, and complex engineered systems (Yu et al., 5 Sep 2025).
• Clarification of True Drivers: Event-level analysis allows for the discrimination of true causal drivers from correlative or coincidental events, a crucial advancement for causal inference in systems where averages obscure critical dependencies.
• Applicability Across Domains: While validated in geophysical and climate settings (e.g., Walker circulation), the approach is adaptable to epidemiology, finance, and other domains where high-impact, non-repetitive events dominate.
• Guide for Data Collection: The finding that richer measurements (full system states) facilitate causality detection motivates investment in comprehensive monitoring and the use of state space embedding when only partial data are available.

6. Limitations and Future Directions

Current limitations and open research avenues include:

• Parameter Selection: Optimization of hyperparameters, such as detection window width and embedding dimensions, remains nontrivial, impacting detection sensitivity and specificity.
• Spatial Heterogeneity: Regional averaging may obscure fine-grained spatial dynamics. Integrating spatially resolved data to construct event-driven causal networks among sub-regions represents a promising direction.
• Generalization Across Domains: Extending techniques to high-dimensional, shorter-length datasets and hybridizing with machine learning or Bayesian methods may further improve robustness.
• Real-world Implementation: Coupling with process-based models and observational networks is needed to evaluate event-level causal inference in operational decision-making and policy contexts.

7. Summary Table: Key Features of the Event-to-Event Dynamic System Method

Aspect	Description	Note
Causal Granularity	Instantaneous, event-to-event, with explicit timing and state mapping	Enables asymmetric detection
System Requirement	Works with multivariate and/or reconstructed state spaces (delay embedding possible for partial measurements)	Embedding dimension must be carefully chosen
Dynamical Regimes	Nonlinear, multi-scale, extreme-event-driven dynamics are explicitly accommodated	Outperforms time-averaged methods
Statistical Inference	Surrogate data, shuffling for significance; causality direction by comparing indices in each direction	Robust in presence of noise
Real-world Applicability	Successfully applied to coupled atmospheric phenomena (Walker circulation), validated in synthetic nonlinear models	Supports operational and research applications

In summary, event-level temporal causality, as approached through the dynamic system method, establishes a rigorous, state-space-based framework for detecting and quantifying causal influences among discrete, temporally localized extreme events. This development provides significant insight into the operation of complex, high-impact systems, with broad theoretical and practical consequences across the sciences (Yu et al., 5 Sep 2025).