Velocities for spatio-temporal point patterns

Abstract Point patterns gathered over space and time are receiving increasing attention in the literature. Examples include incidence of disease events, incidence of insurgent activity events, or incidence of crime events. Point pattern models can attempt to explain these events. Here, a log Gaussian Cox process specification is used to learn about the behavior of the intensity over space and time. Our contribution is to expand inference by introducing the notion of the velocity of a point pattern. We develop a velocity at any location and time within the region and period of study. These velocities are associated with the evolution of the intensity driving the spatio-temporal point pattern, where this intensity is a realization of a stochastic process. Working with directional derivative processes, we are able to develop derivatives in arbitrary directions in space as well as derivatives in time. The ratio of the latter to the former provides a velocity in that direction at that location and time, i.e., speed of change in intensity in that direction. This velocity can be interpreted in terms of speed of change in chance for an event. The magnitude and direction of the minimum velocity provides the slowest speed and direction of change in chance for an event. We use a sparse Gaussian process model approximation to expedite the demanding computation for model fitting and gradient calculation. We illustrate our methodology with a simulation for proof of concept and with a spatio-temporal point pattern of theft events in San Francisco, California in 2012.

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