A Note on Causation versus Correlation in an Extreme Situation.

X San Liang,Xiu-Qun Yang

doi:10.3390/e23030316

Abstract

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 . 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., , the total information increase of B is also zero, so the normalized , denoted as , takes the form of . 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 approaches 100%, just as one would have expected.

Highlights

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 τA→B approaches 100%, just as one would have expected
The causality between two time series can be analyzed in a quantitative sense, and, besides, the resulting formula is very concise in form
In the case with only two time series X1 and X2, under the assumption of a linear model with additive noise, the maximum likelihood estimator (MLE) of the rate of information flowing from X2 to X1 is [2]

Summary

A Review of the Rigorous Information Flow-Based Causality Analysis

Causal inference is a fundamental problem in scientific research. Recently it has been shown that the problem can be recast into the framework of information flow, another fundamental notion in general physics which has wide applications in different disciplines (see [1]), and can be put on a rigorous footing. In the case with only two time series (no dynamical system is given) X1 and X2, under the assumption of a linear model with additive noise, the maximum likelihood estimator (MLE) of the rate of information flowing from X2 to X1 is [2]. A combination of some sample convariances will give a quantiative measure of the causality between the time series. This makes causality analysis, which otherwise would be complicated with the classical empirical/half-empirical methods, very easy. It has been successfully applied to the studies of many real world problems such those in financial economics (e.g., the “Seven Dwarfs vs. a Giant” problem [6]), earth system science (e.g., the Antarctica mass balance problem [7] and the global warming problem [8]), neuroscience (e.g., the concussion problem [9]), to name but a few

The Question

The Solution The problem can be more formally stated with the harmonic system

Discussion