Testing for complete spatial randomness on three dimensional bounded convex shapes.

Area of Science:

• Statistics
• Computational Geometry
• Spatial Data Analysis

Background:

• Existing methods for point pattern analysis are limited to Euclidean spaces (planar, spatial).
• Analyzing point patterns on non-Euclidean surfaces, like 3D objects, presents theoretical challenges, particularly in defining stationarity.
• Previous extensions focused on spherical surfaces, leaving other convex shapes unexplored.

Purpose of the Study:

• To develop functional summary statistics for point processes on 3D convex shapes.
• To create a statistical test for complete spatial randomness versus spatial preference on these surfaces.
• To address the gap in analyzing point patterns in non-Euclidean spaces.

Main Methods:

• Utilizing the Mapping Theorem to transform point processes from convex shapes to a unit sphere.
• Constructing first and second-order properties of functional summary statistics for Poisson processes.
• Developing a test statistic based on these summary statistics and comparing it with an analogue K-function.
• Conducting simulations on ellipsoids to evaluate Type I and II errors.

Main Results:

• Functional summary statistics were successfully constructed for Poisson processes on 3D convex shapes.
• A novel test statistic was developed to differentiate between complete spatial randomness and spatial preference.
• The methodology allows for the analysis of point patterns on surfaces beyond spheres and Euclidean domains.
• Simulations provided insights into the performance and error rates of the proposed test statistic.

Conclusions:

• The Mapping Theorem provides a viable pathway to analyze point processes on complex surfaces.
• The developed functional summary statistics and test statistic offer a robust tool for spatial data analysis in non-Euclidean settings.
• This work expands the theoretical framework for point pattern analysis to a broader range of object surfaces.