Innovations orthogonalization: a solution to the major pitfalls of EEG/MEG "leakage correction"
1.The problem of interest here is the study of brain functional and effective connectivity based non-invasive EEG-MEG inverse solution time series. These signals generally have low spatial resolution, such that an estimated signal at any one site is an instantaneous linear mixture of the true, actual, unobserved signals across all cortical sites. False connectivity can result from analysis of these low-resolution signals. Recent efforts toward \"unmixing\" have been developed, under the name of \"leakage correction\". One recent noteworthy approach is that by Colclough et al (2015 NeuroImage, 117:439-448), which forces the inverse solution signals to have zero cross-correlation at lag zero. One goal is to show that Colcloughs method produces false human connectomes under very broad conditions. The second major goal is to develop a new solution, that appropriately \"unmixes\" the inverse solution signals, based on innovations orthogonalization. The new method first fits a multivariate autoregression to the inverse solution signals, giving the mixed innovations. Second, the mixed innovations are orthogonalized. Third, the mixed and orthogonalized innovations allow the estimation of the \"unmixing\" matrix, which is then finally used to \"unmix\" the inverse solution signals. It is shown that under very broad conditions, the new method produces proper human connectomes, even when the signals are not generated by an autoregressive model.