Poisson intensity estimation with reproducing kernels
This provides a tractable nonparametric modeling approach for Poisson intensity estimation, addressing a bottleneck in stochastic process theory and applications, especially for high-dimensional data.
The paper tackles the problem of nonparametric intensity estimation for inhomogeneous Poisson processes in high-dimensional spaces by developing a new RKHS formulation that models the square root of the intensity. The result is a computationally tractable method that scales to high dimensions and large datasets, with a proven representer theorem ensuring finite-dimensional optimization.
Despite the fundamental nature of the inhomogeneous Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches of intensity functions exist, especially when observed points lie in a high-dimensional space. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. Whereas RKHS models used in supervised learning rely on the so-called representer theorem, the form of the inhomogeneous Poisson process likelihood means that the representer theorem does not apply. However, we prove that the representer theorem does hold in an appropriately transformed RKHS, guaranteeing that the optimization of the penalized likelihood can be cast as a tractable finite-dimensional problem. The resulting approach is simple to implement, and readily scales to high dimensions and large-scale datasets.