Gaussian Process Modulated Cox Processes Under Linear Inequality Constraints
Gaussian process (GP) modulated Cox processes are widely used to model point patterns in a great variety of applications. Existing approaches require a mapping (link function) between the real valued GP and the (positive) intensity function. This commonly yields solutions that do not have a closed form or that are restricted to specific covariance functions. We introduce a novel finite approximation of GP-modulated Cox processes where positiveness conditions can be imposed directly on the GP, with no restrictions on the covariance function. Furthermore, our approach can ensure other types of inequality constraints (e.g. monotonicity, convexity), resulting in more versatile models that can be used for other classes of point processes (e.g. renewal processes). We demonstrate on both synthetic and real-world data that our framework accurately infers the intensity functions. Where monotonicity is a feature of the process, our ability to include this in the inference improves results.