In many multivariate time series, the correlation structure is nonstationary, that is, it changes over time. The correlation structure may also change as a function of other cofactors, for example, the identity of the subject in biomedical data. A fundamental approach for the analysis of such data is to estimate the correlation structure (connectivities) separately in short time windows or for different subjects and use existing machine learning methods, such as principal component analysis (PCA), to summarize or visualize the changes in connectivity. However, the visualization of such a straightforward PCA is problematic because the ensuing connectivity patterns are much more complex objects than, say, spatial patterns. Here, we develop a new framework for analyzing variability in connectivities using the PCA approach as the starting point. First, we show how to analyze and visualize the principal components of connectivity matrices by a tailor-made rank-two matrix approximation in which we use the outer product of two orthogonal vectors. This leads to a new kind of transformation of eigenvectors that is particularly suited for this purpose and often enables interpretation of the principal component as connectivity between two groups of variables. Second, we show how to incorporate the orthogonality and the rank-two constraint in the estimation of PCA itself to improve the results. We further provide an interpretation of these methods in terms of estimation of a probabilistic generative model related to blind separation of dependent sources. Experiments on brain imaging data give very promising results.

Unsupervised analysis of the dynamics (nonstationarity) of functional brain connectivity during rest has recently received a lot of attention in the neuroimaging and neuroengineering communities. Most studies have used functional magnetic resonance imaging, but electroencephalography (EEG) and magnetoencephalography (MEG) also hold great promise for analyzing nonstationary functional connectivity with high temporal resolution. Previous EEG/MEG analyses divided the problem into two consecutive stages: the separation of neural sources and then the connectivity analysis of the separated sources. Such nonoptimal division into two stages may bias the result because of the different prior assumptions made about the data in the two stages. We propose a unified method for separating EEG/MEG sources and learning their functional connectivity (coactivation) patterns. We combine blind source separation (BSS) with unsupervised clustering of the activity levels of the sources in a single probabilistic model. A BSS is performed on the Hilbert transforms of band-limited EEG/MEG signals, and coactivation patterns are learned by a mixture model of source envelopes. Simulation studies show that the unified approach often outperforms conventional two-stage methods, indicating further the benefit of using Hilbert transforms to deal with oscillatory sources. Experiments on resting-state EEG data, acquired in conjunction with a cued motor imagery or nonimagery task, also show that the states (clusters) obtained by the proposed method often correlate better with physiologically meaningful quantities than those obtained by a two-stage method.