  Crime has both varying patterns in space, related to features of the
environment, economy, and policing, and patterns in time arising from criminal
behavior, such as retaliation. Serious crimes may also be presaged by minor
crimes of disorder. We demonstrate that these spatial and temporal patterns are
generally confounded, requiring analyses to take both into account, and propose
a spatio-temporal self-exciting point process model which incorporates spatial
features, near-repeat and retaliation effects, and triggering. We develop
inference methods and diagnostic tools, such as residual maps, for this model,
and through extensive simulation and crime data obtained from Pittsburgh,
Pennsylvania, demonstrate its properties and usefulness.
