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Abstract:
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We consider the problem of statistical inference and prediction for processes defined on networks. We assume that the network is known and measures similarity, and our goal is to learn about an attribute associated with its vertices. Classical regression methods are not immediately applicable to this setting, as we would like our model to incorporate information from both network structure and pertinent covariates. Our proposed model consists of a generalized linear model with vertex indexed predictors and a basis expansion of their coefficients, allowing the coefficients to vary over the network. We employ a regularization procedure, cast as a prior distribution on the regression coefficients under a Bayesian setup, so that the predicted responses vary smoothly according to the topology of the network. We motivate the need for this model by examining occurrences of residential burglary in Boston, Massachusetts. Noting that crime rates are not spatially homogeneous, and that the rates appear to vary sharply across regions in the city, we construct a hierarchical model that addresses these issues and gives insight into spatial patterns of crime occurrences.
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