Modeling space-time data using stochastic differential equations

Abstract

This paper demonstrates the use and value of stochastic difierential equations for modeling space-time data in two common settings. The flrst consists of point-referenced or geostatistical data where observations are collected at flxed locations and times. The second considers random point pattern data where the emergence of locations and times is random. For both cases, we employ stochas- tic difierential equations to describe a latent process within a hierarchical model for the data. The intent is to view this latent process mechanistically and endow it with appropriate simple features and interpretable parameters. A motivating problem for the second setting is to model urban development through observed locations and times of new home construction; this gives rise to a space-time point pattern. We show that a spatio-temporal Cox process whose intensity is driven by a stochastic logistic equation is a viable mechanistic model that afiords mean- ingful interpretation for the results of statistical inference. Other applications of stochastic logistic difierential equations with space-time varying parameters in- clude modeling population growth and product difiusion, which motivate our flrst, point-referenced data application. We propose a method to discretize both time and space in order to flt the model. We demonstrate the inference for the geosta- tistical model through a simulated dataset. Then, we flt the Cox process model to a real dataset taken from the greater Dallas metropolitan area.