Fig. 3

Illustration of causal asymmetry between two variables with linear relations. The data were generated according to equation 3 with , i.e., the causal relation is \(X\rightarrow Y\). From top to bottom: X and \(\varepsilon\) both follow the Gaussian distribution (case 1), uniform distribution (case 2), and a certain type of super-Gaussian distribution (case 3). The two columns on the left show the scatter plot of X and Y and that of X and the regression residual for regression of Y given X, and the two columns on the right correspond to regression of X given Y. Here we used 1000 data points. One can see that for regression of X given Y, in cases 2 and 3 the residual is not independent from the predictor, although they are uncorrelated by construction