Distinguishing time-delayed causal interactions using convergent cross mapping

Abstract

An important problem across many scientific fields is the identification of causal effects from observational data alone. Recent methods (convergent cross mapping, CCM) have made substantial progress on this problem by applying the idea of nonlinear attractor reconstruction to time series data. Here, we expand upon the technique of CCM by explicitly considering time lags. Applying this extended method to representative examples (model simulations, a laboratory predator-prey experiment, temperature and greenhouse gas reconstructions from the Vostok ice core, and long-term ecological time series collected in the Southern California Bight), we demonstrate the ability to identify different time-delayed interactions, distinguish between synchrony induced by strong unidirectional-forcing and true bidirectional causality, and resolve transitive causal chains.

Figure 1. Model demonstration of causal lags and optimal cross mapping using a 2-species logistic model with bidirectional forcing.

Cross-mapping skill (ρ) is shown as a function of cross-mapping lag for three different time delays, τ_d, in the effect of x on y. Here, “y xmap x” refers to using y and its lags to cross map variable x with time lag l. (A) With τ_d = 0, both variables respond to each other within a single time-step (y(t + 1) is influenced by x(t) and vice-versa), and so the optimal cross map lag occurs at l = −1, falling within the embedding vector (green bar) as expected. (B,C) For τ_d = 2 or 4, the effect of x on y is delayed, and so the optimal lag for y cross mapping x (i.e., red line, measuring the effect of x on y) shifts back by a corresponding amount, while x cross mapping y is unchanged. Plots show mean cross map skill and standard deviation over 100 random libraries (see Materials and Methods).

Figure 2. Generalized synchrony in a 2-species logistic model with unidirectional forcing.

In this system, the dynamics of y becomes enslaved to x, and so y can be predicted from x. Since x affects future values of y, x is best able to cross map y forward in time (l ~ 3 > 0), whereas cross mapping in the true direction shows optimal prediction for negative time lags (l ~ −1 < 0, as in Fig. 1). Thus, even though there is cross mapping in both directions, we can use the positive optimal prediction lag to distinguish the direction of causality. As in Fig. 1, “y xmap x” refers to using y and its lags to cross map variable x with time lag l; plots show mean cross map skill and standard deviation over 100 random libraries (see Materials and Methods).

Figure 3. Direct and indirect causality in a transitive causal chain.

(A) In this system, y₁ causes y₂ causes y₃ causes y₄ such that indirect causation from y₁ to y₃, y₂ to y₄, and y₁ to y₄ occurs. (B) Using extended CCM, the direct links (top row) are strongest with the highest cross map skill and the most immediate effects (l ~ −2), the indirect links separated by one node (middle row) have moderate cross map skill and somewhat delayed effects (l ~ −4), and the indirect link from y₁ to y₄ (bottom row) is the weakest and with the longest time delay (l ~ −6). Here “y_i xmap y_j” refers to using y_i and its lags to cross map to y_j. Plots show mean cross map skill and standard deviation over 100 random libraries (see Materials and Methods).

Figure 4. Applying extended CCM to real world examples.

(A) Extended CCM analysis of time series from Veilleux’s predator-prey experiment with Paramecium aurelia (prey) and Didinium nasutum (predator) reveals bidirectional causality. While the effect of predators on prey (red, “ para. xmap didi.”) is immediate, the effect of prey on predators (blue, “didi. xmap para.”) shows a distinct lag, as prey ingestion does not instantaneously translate into population growth. (B) Analysis of causality between Earth atmospheric CO₂ and temperature using time series data from the Vostok ice core for the previous 412,000 years. As expected CO₂ has a nearly instantaneous effect on temperature (blue, “temp. xmap CO₂”) due to the fast-acting greenhouse gas effect, while the influence of temperature on CO₂ is much slower, with an optimal CCM lag of ~3000 years (red, “CO₂ xmap temp.”). (C) Analysis of weekly averages of sea surface temperature (SST) and chlorophyll-a at SIO pier in La Jolla, CA suggests that the effect of SST occurs with a lag of 1–4 weeks (blue, “chl. xmap SST”). All plots show mean cross map skill and standard deviation over 100 random libraries (see Materials and Methods).

Figure 5. Effect of time delays on cross mapping.

Panel (A) shows causation for two cases: (i) no time delay in the effect of x on y (i.e., y responds instantaneously to x), and (ii) y responds to x with a time delay of 4 (time steps). Panel (B) shows (i) cross mapping with l = 0, equivalent to the original formulation by Sugihara et al. and (ii) cross mapping with l = −4, which may be expected to be better than l = 0 when x acts on y with some time delay.