Dynamic connectivity factorization: Interpretable decompositions of non-stationarity

In many multivariate time series, the correlation structure is non-stationary, i.e. it changes over time. Analysis of such non-stationarities is of particular interest in neuroimaging, in which it leads to investigation of the dynamics of connectivity. A fundamental approach for such analysis is to estimate connectivities separately in short time windows, and use existing machine learning methods, such as principal component analysis (PCA), to summarize or visualize the changes in connectivity. Here, we use the PCA approach by Leonardi et al as the starting point and present two new methods. Our goal is to simplify interpretation of the results by finding components in the original data space instead of the connectivity space. First, we show how to further analyse the principal components of connectivity matrices by a tailor-made two-rank matrix approximation, in which the eigenvectors of the conventional low-rank approximation are transformed. Second, we show how to incorporate the two-rank constraint in the estimation of PCA itself to improve the results. We further provide an interpretation of the method in terms of estimation of a probabilistic generative model related to blind source separation methods and ICA. Preliminary experiments on magnetoencephalographic data reveal possibly meaningful non-stationarity patterns in power-to-power coherence of rhythmic sources (i.e. correlation of amplitudes).