A Note on Causation versus Correlation in an Extreme Situation

Recently, it has been shown that the information flow and causality between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event A taking a harmonic form (sine/cosine), and it generates through some process another event B so that B always lags A by a phase of [Formula: see text]. Here the causality is obviously seen, while by computation the correlation is, however, zero. This apparent contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincaré section; it does not add information. That is to say, though the absolute information flow from A to B is zero, i.e., [Formula: see text] , the total information increase of B is also zero, so the normalized [Formula: see text] , denoted as [Formula: see text] , takes the form of [Formula: see text]. By slightly perturbing the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that [Formula: see text] approaches 100%, just as one would have expected.