Causal Discovery of Feedback Networks with Functional Magnetic Resonance Imaging
We test the adequacies of several proposed and two new statistical methods for recovering the causal structure of systems with feedback that generate noisy time series closely matching real BOLD time series. We compare: an adaptation for time series of the first correct method for recovering the structure of cyclic linear systems; multivariate Granger causal regression; the GIMME algorithm; the Ramsey et al. non-Gaussian methods; two non-Gaussian methods proposed by Hyv{diaeresis}arinen and Smith; a method due to Patel, et al.; and the GlobalMIT algorithm. We introduce and also compare two new methods, the Fast Adjacency Skewness (FASK) and Two-Step, which exploit non-Gaussian features of the BOLD signal in different ways. We give theoretical justifications for the latter two algorithms. Our test models include feedback structures with and without direct feedback (2-cycles), excitatory and inhibitory feedback, models using experimentally determined structural connectivities of macaques, and empirical resting state and task data. We find that averaged over all of our simulations, including those with 2-cycles, several of these methods have a better than 80% orientation precision (i.e., the probability a directed edge is in the true generating structure given that a procedure estimates it to be so) and the two new methods also have better than 80% recall (probability of recovering an orientation in the data generating model). Recovering inhibitory direct feedback loops between two regions is especially challenging.