K$^2$IE: Kernel Method-based Kernel Intensity Estimators for Inhomogeneous Poisson Processes

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.

Poisson processes are widely used to analyze and forecast event patterns occurring in space and time, from tweets in SNS to disease outbreaks. A key challenge in using them is estimating the intensity function, which tells us how likely events are to occur at different locations. While recent approaches based on kernel methods provide accurate estimates, they are often very slow for large datasets. In this paper, we introduce a new kernel method-based approach that replaces the commonly used likelihood function with the least squares loss, offering a major boost in computational efficiency. We show that the proposed method achieves comparable accuracy while being significantly faster than previous kernel method-based methods. Moreover, we show that it connects closely to the kernel intensity estimator, a classical method known for its simplicity. These results make our approach both scalable and theoretically sound, helping researchers apply Poisson processes to large-scale scientific data.