Objective Causal Influence Range (threshold-free)

Reduces a divergence-versus-lag profile to a single, threshold-free causal influence range (CIR) – the effective horizon over which a cause keeps informing the estimate of its effect (Andreou et al. 2026, eqs. 8-9). Given the relative entropy divergence between the lagged smoother (future of the effect included only up to a lag L) and the complete smoother (all future included), the divergence falls from $M$ at lag 0 (the filter, no future) towards 0 as L grows. The subjective CIR at tolerance $\varepsilon$ is $\tau_\varepsilon = \inf\{L : D(L) \le \varepsilon\}$; integrating it out, $$\tau = \frac{1}{M}\int_0^M \tau_\varepsilon \, d\varepsilon = \frac{1}{M}\int_0^{L_{\max}} D(L)\, dL,$$ the second equality by parts. The result is a decorrelation-time analogue: a lead time in the units of lag, free of any cut-off threshold.

Usage

cir_objective(lag, divergence)

Arguments

lag: Numeric vector of non-negative, increasing lags L (units of time). Should start at (or include) 0.
divergence: Numeric vector, same length as lag: the relative entropy $D(L)$ of the lag-L smoother from the complete smoother. The value at the smallest lag is taken as $M$ (the normalising maximum).

Value

A single non-negative numeric: the objective CIR (a lead time in the units of lag). Zero when there is no recoverable future information ($M = 0$).

Details

The producer of the divergence profile (the lagged/complete smoother passes) is the upstream filtering engine; this function owns only the range integral.

References

Andreou, M., Chen, N. & Bollt, E. (2026). Assimilative causal inference. Nature Communications 17, 1854.

Examples

lag <- seq(0, 10, by = 0.5)
D <- 2 * exp(-0.6 * lag)            # divergence decays with lag
cir_objective(lag, D)
#> [1] 1.674986