Gaussian Process Modulated Cox Processes under Linear Inequality Constraints
This work addresses a limitation in point process modeling for researchers and practitioners by enabling more versatile and constrained inference, though it is incremental in extending existing methods.
The paper tackles the problem of modeling point patterns with Gaussian process modulated Cox processes by introducing a finite approximation that directly imposes positivity and other inequality constraints on the GP, without restrictions on the covariance function, and demonstrates accurate inference of intensity functions on synthetic and real-world data, with improved results when monotonicity is included.
Gaussian process (GP) modulated Cox processes are widely used to model point patterns. Existing approaches require a mapping (link function) between the unconstrained 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. Our approach can also 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.