## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 2 (2017), 5081-5104.

### Poisson intensity estimation with reproducing kernels

Seth Flaxman, Yee Whye Teh, and Dino Sejdinovic

#### Abstract

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.

#### Article information

**Source**

Electron. J. Statist. Volume 11, Number 2 (2017), 5081-5104.

**Dates**

Received: June 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1513306868

**Digital Object Identifier**

doi:10.1214/17-EJS1339SI

**Subjects**

Primary: 62G05: Estimation 60G55: Point processes 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

**Keywords**

Nonparametric statistics computational statistics spatial statistics intensity estimation reproducing kernel Hilbert space inhomogeneous Poisson processes

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Flaxman, Seth; Teh, Yee Whye; Sejdinovic, Dino. Poisson intensity estimation with reproducing kernels. Electron. J. Statist. 11 (2017), no. 2, 5081--5104. doi:10.1214/17-EJS1339SI. https://projecteuclid.org/euclid.ejs/1513306868