Causality in nonlinear systems

Sugihara et al. have a really interesting paper in Science, on detection of causality in nonlinear dynamic systems. It’s paywalled, so here’s an excerpt with some comments.

Abstract: Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

Identifying causality in complex systems can be difficult. Contradictions arise in many scientific contexts where variables are positively coupled at some times but at other times appear unrelated or even negatively coupled depending on system state.

Although correlation is neither necessary nor sufficient to establish causation, it remains deeply ingrained in our heuristic thinking. … the use of correlation to infer causation is risky, especially as we come to recognize that nonlinear dynamics are ubiquitous.

An alternative approach, Granger causality (GC), provides a framework that uses predictability as opposed to correlation to identify causation between time-series variables. …

Variable X is said to “Granger cause” Y if the predictability of Y (in some idealized model) declines when X is removed from the universe of all possible causative variables, U. The key requirement of GC is separability, namely that information about a causative factor is independently unique to that variable (e.g., information about predator effects is not contained in time series for the prey) and can be removed by eliminating that variable from the model. Separability is characteristic of purely stochastic and linear systems, and GC can be useful for detecting interactions between strongly coupled (synchronized) variables in nonlinear systems. Separability reflects the view that systems can be understood a piece at a time rather than as a whole.

However, as Granger realized early on, this approach may be problematic in deterministic settings, especially in dynamic systems with weak to moderate coupling. … This is because separability is not satisfied in such systems, which, unlike the tradition in economics and single-species fisheries management, need to be considered as a whole. That is to say, in deterministic dynamic systems (even noisy ones), if X is a cause for Y, information about X will be redundantly present in Y itself and cannot formally be removed from U — a consequence of Takens’ theorem. … When Granger’s definition is violated, GC calculations are no longer valid, leaving the question of detecting causation in such systems unanswered.

GC applies if the world is purely stochastic. However, to the extent that it is deterministic and dynamics are not entirely random, there will be an underlying manifold governing the dynamics (representing coherent trajectories as opposed to a random tangle).

In dynamical systems theory, time-series variables (say, X and Y) are causally linked if they are from the same dynamic system—that is, they share a common attractor manifold M (movies S1 to S3 illustrate this idea). This means that each variable can identify the state of the other (e.g., information about past prey populations can be recovered from the predator time series, and vice versa). Additionally, when one variable X is a stochastic environmental driver of a population variable Y, information about the states of X can be recovered from Y, but not vice versa. For example, fish time series can be used to estimate weather, but not conversely. This runs counter to Granger’s intuitive scheme … .

The implementation is called Convergent Cross Mapping (CCM). It uses delay embedding to reconstruct a manifold from each of the variables to be examined. Then it compares points on the reconstructed manifolds. If neighborhoods on one manifold can be used to predict those on another manifold, then one concludes that the associated variables are causally connected.

… (i) … Bidirectional causality is analogous to the concept of “feedback” between two time series described by Granger and is the primary case covered by Takens. Simply put, if variables are mutually coupled (e.g., predator and prey), they will cross map in both directions. Thus, each variable can be estimated from the other (predator histories can estimate prey states).


(ii) Unidirectional causality. Here, species X influences the dynamics of Y, but Y has no effect on X. This describes an amensal or commensal relationship, or where X represents external environmental forcing.

One of the fundamental ideas in this work is that when causation is unilateral, X -> Y (“X drives Y,” as in case ii), then it is possible to estimate X from Y, but not Y from X. This runs counter to intuition (and GC), and suggests that if the weather drives fish populations, for example, we can use fish to estimate the weather but not conversely.

In this counterintuitive case, X can be predicted from Y, because the behavior of Y partially encodes X, but not the reverse. There’s a caveat, because if the forcing of X on Y is sufficiently strong, it overwhelms the dynamics of Y, but this is not normally the situation of interest.

… External forcing of noncoupled variables. Consider the case where two species, X and Y, do not interact but are both driven by a common environmental variable Z (example 1 schematic in Fig. 4A). This occurs commonly in ecological systems [the Moran effect (23)] and remains problematic in studies of causation. Here we expect no cross mapping between species X and Y because there is no information flow between variables; however, information about the external forcing variable (Z) should still be recoverable from X and Y.

The authors go on to apply the method to several test cases, where it performs as expected.

Competing hypotheses have been advanced to explain the pattern of alternating dominance of sardine and anchovy across global fisheries on multidecadal time scales. …

We address this controversy using the same analytical protocol …. The results … show no significant cross-map signal between sardine and anchovy landings, indicating that sardines and anchovies do not interact. In addition, as expected, there is no detectable signature from either sardine or anchovy in the temperature manifold; obviously, neither sardines nor anchovies affect SST. However, there is clear asymmetric CCM between sardines and SST as well as between anchovies and SST, meaning that temperature information is encoded in both fishery time series. The recoverable temperature signature reveals a weak coupling of temperature to sardines and anchovies. Thus, although sardines and anchovies are not actually interacting, they are weakly forced by a common environmental driver, for which temperature is at least a viable proxy.

This leaves me with one question though. The finding of sardine/anchovy independence seems to ignore an obvious coupling mechanism: fishermen. Could it be that the dynamics of that relationship are so different that CCM misses it? Or, perhaps the series, which show a single boom-bust, are too short to properly reveal manifolds? Something is fishy here.

Hopefully the supplemental materials are not paywalled, because these movies are excellent illustrations of the concepts:

Movie S1
This movie demonstrates the relationship between time series and dynamic attractors (manifolds, M).
Movie S2
This movie illustrates how Takens’ Theorem (19) can be used to reconstruct a shadow manifold MX from a single time series, and illustrates the 1:1 mapping between M and MX.
Movie S3
This movie explains how convergent cross mapping (CCM) estimates states across variables.

The authors are careful to constrain the method to the kind of ecosystem problems that they study, but I think there are obvious applications elsewhere, and this could be an important paradigm shift in many areas.

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