Nanjing Center for Ocean-Atmosphere Dynamical Studies

Another important application of Liang-Kleeman information flow is the establishment of a quantitative and rigorous causality analysis. Given two time series X_{1 and}  X_2,  (Liang 2014) proved that the maximum likelihood estimator of the information flowing from X_{2 to} X₁ is：

where C_i,j  is the covariance between X_{i and} X_j, and C_{i,dj that between} X_{i and} the difference approximation of dXj /dt with the Euler forward scheme.

Ideally, when T_2→1 = 0, X₂ is not the cause of X₁, and vice versa. It is easy to see that, if C₁₂ = 0, then T_2→1 = 0, but when T_2→1 = 0, C₁₂ does not need to vanish. That is, contrapositively, causation implies correlation, but correlation does not imply causation. In an explicitly quantitative way, this resolves the long-standing debate over causation versus correlation.

The above remarkably concise yet rigorous formula has been validated with many touchstone problems, and applied successfully to many real world problems. See, for example, the stories on "Seven Dwarfs and a Giant" (Liang, 2015) and "CO₂  vs. global warming" (Stips et al., 2016).

Click to download related articles：

X. San Liang, 2014: Unraveling the cause-effect relation between time series. Phys. Rev. E. 90, 052150.

X. San Liang,2016: Information flow and causality as rigorous notions ab initio, Phys. Rev. E 94, 052201.

X. San Liang,2014 Entropy Evolution and Uncertainty Estimation with Dynamical Systems. Entropy 16, 3605-3634

X. San Liang, 2015: Normalizing the causality between time series. Phys. Rev. E. 92, 022126.

Stips A, Macias D, Coughlan C, et al., 2016: On the causal structure between CO2 and global temperature[J]. Scientific Reports, 6.