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

Area of Science:

• Statistics
• Geospatial Analysis
• Manifold Theory

Background:

• Statistical analysis of point patterns is crucial for understanding spatial distributions.
• Analyzing point patterns on complex spaces like manifolds requires specialized frameworks.
• Existing methods may not fully capture the spatiotemporal dynamics of point processes on curved surfaces.

Purpose of the Study:

• To introduce a novel modeling framework for statistical analysis of point patterns on manifolds.
• To characterize the evolution of microscale point patterns in time across different spatial scales on a manifold.
• To investigate aggregation and regularity phenomena in spherical point process models.

Main Methods:

• Utilized temporal Cox processes driven by a log-intensity function defined on a manifold.
• Implemented aggregation schemes via the time-varying discrete Jacobi polynomial transform of the log-risk process.
• Employed K-function analysis to assess temporal dependence and spatial patterns.

Main Results:

• Demonstrated the construction of spherical point process models.
• Observed aggregation at low Legendre polynomial transform frequencies (large spatial scales).
• Identified regularity at high frequencies (small spatial scales), supported by K-function analysis across various temporal dependencies.

Conclusions:

• The proposed framework effectively models spatiotemporal point patterns on manifolds.
• The study highlights the scale-dependent nature of aggregation and regularity in point processes.
• The methodology provides a robust tool for analyzing complex spatial data on curved surfaces.