Figure 3

Granger causality and Bayesian network inference applied on a stochastic coefficients toy model. The parameters in polynomial equation are randomly generated in the interval [-1,1]. For each randomly generated coefficient vector, we applied the same approach as example 1: bootstrapping method and 95% confidence interval for Granger causality; 95% high confidence arcs are chosen from Bayesian network inference. (A) We applied both approaches on different sample size (from 20 to 900). For each sample size, we generated 100 different coefficient vectors, so the total number of directed interactions for each sample size is 500. (a) The percentage of detected true positive causalities for both approaches. (b) Time cost for both approaches. (B) For sample size 900, the derived causality (1 represents positive causality and 0 represents negative) is plotted with the absolute value of corresponding coefficients. For visualization purpose, the figure for Granger causality is shifted upward. (C) Linear model fitting comparison for both Granger causality and Bayesian networks. Using a number of training data points to fit both linear models, one can calculate a corresponding predicted mean-square error by applying a set of test data. And we can find that Bayesian networks inference approach works much better than the Granger causality approach when the sample size is significant small (around 100). When the sample size is significant large, both approaches converge to the standard error which exactly fits the noise term in our toy model.