Spectral Analysis of Spatial Point Process with Applications

Abstract

This presentation consists of three parts. The first part introduces two relevant statistical problems that arise in spatial point processes: (1) goodness-of-fit testing and (2) denoising of point patterns. We will then review some classical spatial domain approaches to solve these problems and discuss their limitations. The second part of the talk provides alternative solutions to tackle these problems. A key component is the use of the discrete Fourier transform (DFT) of the point pattern. We show that the DFT exhibits properties similar to those found in time series and spatial statistics; it can also be used for inference in point processes. Then, we revisit the two problems and hope to convince the audience that frequency domain methods offer clear advantages in addressing these problems compared to the classical approaches. In the last part, we discuss some ongoing and planned research projects on spectral methods for point processes.

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