Partially Observed Two-Phase Point Processes

In this paper, a two-phase spatio-temporal point process (STPP) defined on a countable metric space and characterized by a conditional intensity function is introduced. In the first phase, the process is memoryless, generating completely random point patterns. In the second phase, the location and occurrence time of each event depend on the spatial configuration of previous events, thereby inducing spatio-temporal correlation. Theoretical results that characterize the distributional properties of the process are established, enabling both efficient numerical simulation and Bayesian inference. A statistical inference framework is developed, for the setting in which the STPP is observed at discrete calendar dates while the spatial locations of events are recorded, their exact occurrence times are unobserved, i.e., interval-censored. This partial observation scheme commonly arises in ecological and epidemiological applications, such as the monitoring of plant disease or insect pest spread across a spatial grid over time. The methodology is illustrated through an analysis of the spatio-temporal spread of sugarcane yellow leaf virus (SCYLV) in an initially disease-free sugarcane plot in Guadeloupe, FrenchWest Indies.