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# Nanjing Center for Ocean-Atmosphere Dynamical Studies

#### Causality analysis

Causality analysis with time series

Another important application of Liang-Kleeman information flow is the establishment of a quantitative and rigorous causality analysis. Given two time series X1 and  X2,  (Liang 2014) proved that the maximum likelihood estimator of the information flowing from X2 to Xis：

$T_{2\rightarrow&space;1}=&space;\frac{C_{11}C_{12}C_{2,d1}-C_{12}^{2}C_{1,d1}}{C_{11}^{2}C_{22}-C_{11}C_{12}^{2}}$

where Ci,j  is the covariance between Xi and Xj, and Ci,dj that between Xi and the difference approximation of dXj /dt with the Euler forward scheme.

Ideally, when T2→1 = 0, X2 is not the cause of X1, and vice versa. It is easy to see that, if C12 = 0, then T2→1 = 0, but when T2→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).