Causal Influence Range (CIR)

• CIR is a framework that quantitatively defines the temporal window over which a cause influences an effect, distinguishing forward and backward causal impacts.
• The methodology integrates Bayesian data assimilation with information-theoretic metrics to objectively assess when causal effects begin and persist in nonstationary systems.
• Practical applications include climate tipping events and atmospheric processes, where CIR enhances predictive early-warning systems and supports detailed attribution analyses.

Causal Influence Range (CIR) delineates the temporal or structural extent over which a cause exerts a discernible effect on an outcome variable within a dynamical system. Recent advances have established mathematically rigorous methodologies to define and estimate both forward and backward CIRs, leveraging assimilative causal inference (ACI) grounded in Bayesian data assimilation and information-theoretic metrics. This framework provides objective and computationally efficient tools to identify when causal effects initiate, how long they persist, and when observed outcomes can be attributed to past drivers, thus unifying predictive and retrospective aspects of causality in nonstationary and complex systems.

1. Conceptual Foundations of CIR

CIR quantifies the time window during which a given state or variable influences (forward CIR) or is responsible for (backward CIR) another variable’s evolution in a dynamical system. The forward CIR measures the prospective reach of a cause, characterizing for how long an initiated causal effect continues to influence the state of the effect variable. Conversely, the backward CIR identifies, for a realized effect, the temporal interval in the past wherein causal events contributed significantly to the observed outcome. This duality is particularly crucial in nonstationary or intermittent systems, where causal roles can shift and influence durations are variable.

Classical approaches, such as Granger causality or cross mapping, detect time-local causality but do not systematically quantify the temporal support or onset/persistence of causal effects. CIR, as formulated via ACI, provides this missing information and thus advances both the predictive (prognostic) and attributional (diagnostic) dimensions of causal inference.

2. Mathematical Formulation of Forward and Backward CIR

The ACI framework uses Bayesian data assimilation to integrate observational time series with dynamical system models, propagating and updating uncertainties. For state variable pairs (y(t), x(T)), ACI computes two conditional probability distributions at each time:

• Filtering distribution: $p_t^{(f)}(y | x(s \leq t))$ (using data up to $t$)
• Smoothing distribution: $p_t^{(s)}(y | x(s \leq T))$ (using all data up to terminal time $T$).

The discrepancy between these is quantified via the Kullback–Leibler (KL) divergence,

$Q(p, q) = \int p(z) \log \frac{p(z)}{q(z)}\,dz.$

A positive $Q(p_t^{(s)}, p_t^{(f)})$ indicates that future observations contribute significant information about the state at time $t$, signaling a causal link.

Forward CIR

For a given cause variable $y(t)$ and effect variable $x$, the forward CIR is formulated by considering, for $t$ fixed, the KL divergence $\delta(t, T')$ as $T' > t$ varies:

$$\delta^f(\tau; t) := Q(p_t^{(s)}(y|x(s \leq t + \tau(T-t))),\, p_t^{(f)}(y|x(s \leq t)) )$$

where $T' = t + \tau(T-t)$ and $\tau \in [0,1]$ normalizes the interval $[t, T]$.

Subjective Forward CIR

Given a threshold $\epsilon > 0$, the subjective forward CIR at time $t$ is the maximal $\tau_{(\epsilon)}(t)$ such that $\delta^f(\tau; t) > \epsilon$.

Objective Forward CIR

The objective forward CIR is obtained by integrating over all possible thresholds—analogous to the integrated autocorrelation time:

$$\tau^f(t) = \frac{1}{M} \int_0^M \tau_{(\epsilon)}(t)\, d\epsilon$$

where $M = \max_{\tau} \delta^f(\tau; t)$. Closed-form approximations are provided for practical computation, involving $L^1$ and $L^\infty$ norms of $\delta^f$ over $[0,1]$.

Backward CIR

For attribution, the backward CIR is formulated by fixing the terminal time $T$ (when the effect is realized) and considering $\delta(t, T)$ as a function of past $t < T$:

$$\delta^b(\tau; T) := Q(p_t^{(s)}(y|x(s \leq T)),\, p_t^{(s)}(y|x(s \leq t + \tau(T-t))) )$$

where again $\tau \in [0,1]$ parameterizes how much of the past trajectory is assimilated.

Subjective and Objective Backward CIR

The subjective backward CIR corresponding to threshold $\epsilon$ is the minimal $\tau_{(\epsilon)}(T)$ such that $\delta^b(\tau; T) < \epsilon$. The objective backward CIR is its mean over all $\epsilon\in[0, M]$, with $M = \max_\tau \delta^b(\tau; T)$. An upper-bound approximation is provided (Theorem 2 in the paper) by exchanging the order of integration.

3. Computational Algorithms and Analytical Solutions

Efficient numerical routines are developed for both CIR directions, circumventing direct quadrature via semi-analytical bounds. For certain classes of systems, specifically conditional Gaussian nonlinear systems (CGNS), closed-form filtering and smoothing formulas are available, enabling real-time CIR computation. An adaptive-lag online smoother is proposed for these cases, leveraging computational tractability to extend the applicability to high-dimensional and nonlinear systems.

4. Application Examples

The CIR framework is validated and demonstrated on several prototypical and complex systems:

• Earth system tipping models: The methods quantify both the forward CIR—providing an early-warning timescale for the persistence of a tipping event—and the backward CIR—tracing attributive causes such as shifts in bifurcation control variables or internal variability responsible for regime shifts.
• Multiscale atmospheric models: The decomposition of overlapping causal paths is achieved via conditional ACI and CIR, enabling separate tracking of influence ranges for multiple interacting processes.
• Equatorial atmospheric blocking: Application of CIR illuminates the onset and duration of blocking events, including regime transitions where causal relationships reverse.

These examples underscore that CIRs derived via ACI reveal both the emergence and dissipation of causal effects, as well as the temporal structure underlying critical system transitions.

5. Interpretation and Impact

The dual CIR formalism bridges the historic gap between causal prediction (forward reasoning) and attribution (retrospective analysis). It allows not only detection of when a cause becomes active but also quantifies the duration and timing of its effect, crucial for both operational forecasting and forensics in disciplines such as climate science, neuroscience, and economics.

Key advantages include (i) the removal of arbitrary thresholds via objective integration, (ii) grounding in dynamical models via data assimilation, and (iii) computational feasibility even in nonlinear, high-dimensional, and partial observation settings. The approach enables practitioners to assess and contrast predictive and attributive confidence intervals for causal claims, advancing the design of early warning systems and post-event diagnostics.

6. Limitations and Extensions

While the introduced algorithms have analytical and computational advantages, their efficacy depends on the quality of the underlying dynamical model and suitability of Bayesian assimilation (typically under Gaussian assumptions). Approximate solutions and smoothing may have limited fidelity in strongly non-Gaussian regimes. The CIR framework is formulated for systems with accessible process models; in data-only settings or in the absence of an adequate mechanistic understanding, application requires additional methodological developments.

Further, while forward and backward CIRs capture distinct but complementary perspectives, their equivalence or divergence can themselves be indicators of irreversible dynamics or nonreciprocal causal roles, which constitute an ongoing topic of research.

7. Broader Implications

By rigorously quantifying both forward and backward CIRs, the framework establishes a systematic method for temporal delimitation of causal impact. This has significant ramifications for operational risk assessment, tipping point detection, and the attribution of complex systems behaviors to underlying drivers. The ability to handle overlapping, time-reversing, and multiscale causal interactions promotes a more nuanced understanding—pivotal for scientific, policy, and strategic decision-making in complex arenas ranging from Earth system sciences to engineered networks.

---

This synthesis is drawn from the methodology, mathematical derivations, and case studies of “Bridging Prediction and Attribution: Identifying Forward and Backward Causal Influence Ranges Using Assimilative Causal Inference” (Andreou et al., 24 Oct 2025), supplemented by foundational elements from the original ACI work (Andreou et al., 20 May 2025).