In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.
In the context of fMRI, the work of @cite_8 used a similar kernel-weighted estimator for the spatial covariance, however they modeled the temporal covariance with a simple AR-1 model which they did not estimate, and their estimator did not attempt to remove. Similarly, @cite_5 used a smoothed kernel estimator for @math with a penalty to further promote smoothness, but did not model the temporal correlations. Our additive model allows the direct estimation of the temporal behavior, revealing a richer structure than a simple AR-1, and allowing for effective denoising of the data, and hence better estimation of the spatial graph structures.
@cite_5 @cite_8
2069908974 1754350508
Recent functional magnetic resonance imaging studies have shown that the brain is remarkably active even in the absence of overt behavior, and this activity occurs in spatial patterns that are reproducible across subjects and follow the brain’s established functional subdivision. Investigating the distribution of these spatial patterns is an active area of research with the goal of obtaining a better understanding of the neural networks underlying brain function. One intriguing aspect of spontaneous activity is an apparent nonstationarity, or variability of interaction between brain regions. It was recently proposed that spontaneous brain activity may be dominated by brief traces of activity, possibly originating from a neuronal avalanching phenomenon. Such traces may involve different subregions in a network at different times, potentially reflecting functionally relevant relationships that are not captured with conventional data analysis. To investigate this, we examined publicly available functional magnetic resonance imaging data with a dedicated analysis method and found indications that functional networks inferred from conventional correlation analysis may indeed be driven by activity at only a few critical time points. Subsequent analysis of the activity at these critical time points revealed multiple spatial patterns, each distinctly different from the established functional networks. The spatial distribution of these patterns suggests a potential functional relevance. In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T, n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.