Modeling Occurrences of Residential Burglary via Bayesian Network Regularized Regression
Elizabeth Upton, Luis CarvalhoAnalyses of occurrences of residential burglary in urban areas have shown that crime rates are not spatially homogenous: rates can vary sharply across regions in the city resulting in some neighborhoods or streets being considered far more dangerous than others. Motivated by the importance of understanding these spatial patterns, we consider a statistical model of burglary occurrences defined on the network of city streets in Boston, Massachusetts. More broadly speaking, we study the problem of statistical inference for a process defined on a network.
Our network of interest, the street network of Boston, contains residential street intersections (vertices) connected by street paths (edges), forming an undirected simple graph. We pooled occurrences of residential burglary over time and mapped each occurrence to its closest intersection. 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. Using the graph Laplacian, 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. Furthermore, we include a latent vertex-indexed indicator to identify residential burglary hot spots in the city. This hierarchical structure allows the model to capture both abrupt and smooth changes in the process rate.
To fit the model, we examine efficient expectation-maximization fitting algorithms and provide computationally-friendly methods for eliciting hyper-prior parameters and sampling from posterior distributions. The resulting model and interpretations provide insight into the spatial patterns and dynamics of residential burglary in Boston. Specifically, the model allows an interpretation of how a covariate’s influence on the attribute process changes across the network. Furthermore, a simulation study demonstrates that our proposed model outperforms competing regression methods. While the focus of this talk is on modeling occurrences of burglary in Boston, the introduced methodology is applicable to modeling vertex attributes in a wide variety of applications.