Figure 4

Granger causality and Bayesian network inference approaches applied on a simple non-linear toy model. (A) Five time series are simultaneously generated, and the length of each time series is 1000. They are assumed to be stationary. (B) The five histogram graphs show the probability distribution for these five time series. (C) Assuming no knowledge of MVAR toy model we fitted, we calculated Granger causality. Bootstrapping approach is used to construct the confidence intervals. The fitted MVAR model is simulated to generate a data set of 100 realizations of 1000 time points each. (a) For visualization purpose, all directed edges (causalities) are sorted and enumerated into the table. The total number of edges is 20. 95% confidence interval is chosen. (b) The network structure inferred from Granger causality method correctly recovers the pattern of connectivity in our MVAR toy model. (D) Assuming no knowledge of MVAR toy model we fitted, we approach Bayesian network inference. (a) The causal network structure learned from Bayesian network inference for one realization of 1000 time points. (b) Each variable is represented by two nodes; each node represents different time statuses, so we have 10 nodes in total. They are numbered and enumerated into the table. (c) The simplified network structure: since we only care about the causality to the current time status, we can remove all the other edges and nodes that have no connection to the node 6 to node 10 (five variables with current time status). (d) A further simplified network structure: in order to compare with Granger causality approach, we hid the information of time status, and we obtained the same structure as Granger causality method had.