Low Tree-Rank Bayesian Vector Autoregression Model
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Vector autoregressions have been widely used for modeling and analysis of multivariate time series data. In high-dimensional settings, model parameter regularization schemes inducing sparsity have achieved good forecasting performances. However, in many data applications such as those in neuroscience, the graph estimates from existing methods still tend to be quite dense and difficult to interpret, unless we made compromise in the goodness-of-fit. To address this dilemma, in this article we propose to incorporate a commonly used structural assumption – that the ground-truth graph should be largely connected, in the sense that it should only contain at most a few components. We take a Bayesian approach and develop a novel tree-rank prior for the regression coefficients. Specifically, this prior forces the non-zero coefficients to appear only on the union of a few spanning trees. Since each spanning tree connects p nodes with only (p-1) edges, this prior effectively achieves both high connectivity and high sparsity. In analyzing test-retest functional magnetic resonance imaging data, our model produces a much more interpretable graph estimate, compared to popular existing approaches. In addition, we show appealing properties of this new method, such as differentiable computation, mild stability conditions and posterior consistency.
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