Causal effect estimation for robust detection of critical slowing down

A tipping element is a system that may pass a tipping point, that is, a threshold value of an environmental stressing condition at which a small disturbance can cause an abrupt shift of the tipping element from one state to another, accelerated by positive feedbacks. For example, under rising temperatures and increasing deforestation, the Amazon rainforest could tip from a forest state to a savanna state; one feedback involved is that fewer trees means less evapotranspiration, thus less rainfall and, finally, less trees. Therefore, the fewer trees, the harder it is for the remaining forest to adapt and survive. This phenomenon is called critical slowing down: approaching a bifurcation, a tipping system's resilience decreases, resulting in increasing autocorrelation and variance. The latter indicators are thus often measured to detect bifurcation-induced tipping and are called early-warning signals (EWS).

Let us describe the dynamics of a tipping element Y with the following equation: dY/dt = f(Y, r) + η, with r the environmental stressing condition involved in the tipping behavior and η some noise (e.g., climate variability). Deriving EWS directly from Y's time series relies on the assumption that the noise η is not correlated (white-noise), otherwise any trend in η's autocorrelation would be incorporated in Y's autocorrelation, even if not related to the tipping behavior contained in f(Y, r). In the Amazon rainforest example, increasing deforestation due to human activity is a part of r with a long-term effect, while e.g. ENSO also influences the forest but on short time scales and with sometimes opposite effects depending on its phase (El Niño/La Niña/neutral), and would be part of η.  If for example El-Niño's autocorrelation increases with time, the rainforest autocorrelation might also increase regardless of whether the forest is approaching the bifurcation point or not, therefore the autocorrelation would no longer reflect changes in the forest resilience.

To know how far the system is from the bifurcation point, we want to measure Y's internal autocorrelation (excluding noisy influences η, considering only f(Y, r)), and thus to answer the question: "If we intervene in the system and set the value of Y at time t-1, how does Y evolve at time t?" This defines the direct causal effect of Y_t-1 on Y_t and comes under the heading of causal inference: we look at the influence of setting Y_t-1=y_t-1 on Y_t, whatever the values of the other variables causing Y, which is fundamentally different from a direct measure where the value of Y at time t-1 is, in the general case, dependent on the state of the other variables. To measure how the direct causal effect of Y_t-1 on Y_t  evolves with time (with changing r), we use causal effect estimation, which quantifies the causal effect of hypothetical interventions in a system from observational data --the interventional distribution being rarely available in the majority of systems--- and an assumed causal graphical model that allows us to derive an adjustment expression that controls for confounders. We demonstrate the method on an ideally forced simulated system and discuss potential applications.