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Abstract:
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Multi-dimensional data constituted by measurements along multiple axes have emerged across many scientific areas such as genomics and cancer surveillance. A common objective is to investigate the conditional dependencies among the variables along each axes taking into account multi-dimensional structure of the data. Traditional multivariate approaches are unsuitable for such highly structured data due to inefficiency, loss of power and lack of interpretability. In this paper, we propose a novel class of multi-dimensional graphical models based on matrix decompositions of the precision matrices along each dimension. Our approach is a unified framework applicable to both directed and undirected graphs as well as arbitrary combinations of these. Exploiting the marginalization of the likelihood, we develop efficient posterior sampling schemes based on partially collapsed Gibbs samplers. We illustrate our approaches through extensive simulations and an application in cancer surveillance.
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