Gaussian Process Modulated Cox Processes under Linear Inequality Constraints

Point processes are used in a variety of real-world problems for modelling temporal or spatiotemporal point patterns in fields such as astronomy, geography, and ecology (Baddeley et al., 2015; Møller and Waagepetersen, 2004). In reliability analysis, they are used as renewal processes to model the lifetime of items or failure (hazard) rates (Cha and Finkelstein, 2018). Poisson processes are the foundation for modelling point patterns (Kingman, 1992). Their extension to stochastic intensity functions, known as doubly stochastic Poisson processes or Cox processes (Cox, 1955), enables nonparametric inference on the intensity function and allows expressing uncertainties (Møller and Waagepetersen, 2004). Moreover, previous studies have shown that other classes of point processes may also be seen as Cox processes. For example, Yannaros (1988) proved that Gamma renewal processes are Cox processes under non-increasing conditions. A similar analysis was made later for Weibull processes (Yannaros, 1994).