We introduce the notion of velocity in an effort to expand inference for stochastic processes defined over space and time. For a realization of a stochastic process defined over a spatio-temporal domain, we can obtain the instantaneous gradient of the surface in time and space for a given location and time. The ratio of these two gradients can be interpreted as a velocity; the change in space in a given direction per unit time. With a Gaussian process realization for the spatio-temporal surface, we can obtain these gradients as directional derivative processes. The direction of the maximum gradient in space and the associated magnitude, which yields the direction and magnitude of minimum velocity, offer practical interpretations. Dimension reduction through predictive processes and sparsity through nearest neighbor Gaussian processes provide computational efficiency. We apply our method to two case studies. First, we specify a geostatistical model for average annual temperature across the eastern United States for the years 1963– 2012. Estimates of the velocity of temperature change are compared across a collection of spatial locations and time points. Then, for a spatio-temporal point pattern of theft events in San Francisco in 2012, we specify a log Gaussian Cox process model to explain the events. We estimate the velocity of the point pattern, where the magnitude and direction of the minimum velocity provides the slowest rate and direction of movement required to maintain constant chance for an event

Dr. Erin Schliep is an Assistant Professor in the Department of Statistics at the University of Missouri. She received her PhD at Colorado State University in 2013. Prior to starting at Missouri, she spent two years at Duke University as a postdoctoral fellow. Dr. Schliep's research focuses on developing statistical methodology to further the understanding of environmental processes. Her work often entails methods for dependent data, with emphasis on spatio-temporal and multivariate data.