Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

TL;DR

This paper proposes a novel statistical framework for analyzing evolving point patterns on compact two-point homogeneous spaces, utilizing temporal Cox processes and Jacobi polynomial transforms to characterize spatial scales and dependencies.

Contribution

It introduces a new modeling approach for spatiotemporal point patterns on manifolds, incorporating polynomial transforms and scale-specific analysis, including simulation validation.

Findings

Models exhibit aggregation at large scales and regularity at small scales.

K-function analysis confirms scale-dependent dependence patterns.

Simulation demonstrates effectiveness of the proposed framework.

Abstract

This paper introduces a new modeling framework for the statistical analysis of point patterns on a manifold M_{d}, defined by a connected and compact two-point homogeneous space, including the special case of the sphere. The presented approach is based on temporal Cox processes driven by a L^{2}(\mathbb{M}_{d})-valued log-intensity. Different aggregation schemes on the manifold of the spatiotemporal point-referenced data are implemented in terms of the time-varying discrete Jacobi polynomial transform of the log-risk process. The n-dimensional microscale point pattern evolution in time at different manifold spatial scales is then characterized from such a transform. The simulation study undertaken illustrates the construction of spherical point process models displaying aggregation at low Legendre polynomial transform frequencies (large scale), while regularity is observed at high frequencies (small scale). K-function analysis supports these results under temporal short-, intermediate- and long-range dependence of the log-risk process.