K$^2$IE: Kernel Method-based Kernel Intensity Estimators for Inhomogeneous Poisson Processes
This work addresses a domain-specific problem in statistical estimation for Poisson processes, offering incremental improvements in efficiency and theoretical insights.
The paper tackles the problem of estimating intensity functions for inhomogeneous Poisson processes by proposing K$^2$IE, a kernel method-based kernel intensity estimator that unifies classical and kernel method approaches, achieving comparable predictive performance and significantly improved computational efficiency over state-of-the-art methods.
Kernel method-based intensity estimators, formulated within reproducing kernel Hilbert spaces (RKHSs), and classical kernel intensity estimators (KIEs) have been among the most easy-to-implement and feasible methods for estimating the intensity functions of inhomogeneous Poisson processes. While both approaches share the term "kernel", they are founded on distinct theoretical principles, each with its own strengths and limitations. In this paper, we propose a novel regularized kernel method for Poisson processes based on the least squares loss and show that the resulting intensity estimator involves a specialized variant of the representer theorem: it has the dual coefficient of unity and coincides with classical KIEs. This result provides new theoretical insights into the connection between classical KIEs and kernel method-based intensity estimators, while enabling us to develop an efficient KIE by leveraging advanced techniques from RKHS theory. We refer to the proposed model as the kernel method-based kernel intensity estimator (K$^2$IE). Through experiments on synthetic datasets, we show that K$^2$IE achieves comparable predictive performance while significantly surpassing the state-of-the-art kernel method-based estimator in computational efficiency.