[Paper Review] A comparison of functional summary statistics to detect anisotropy of three-dimensional point patterns

Farzaneh Safavimanesh, Claudia Redenbach|arXiv (Cornell University)|Apr 14, 2016

Point processes and geometric inequalities1 references3 citations

TL;DR

This paper compares the conical and cylindrical K-functions—two directional summary statistics for detecting anisotropy in 3D point patterns. It provides parameter recommendations based on simulation-based isotropy testing and demonstrates that the conical K-function is more effective for compressed regular patterns (e.g., ice cores), while the cylindrical K-function excels in detecting columnar structures (e.g., pyramidal cells in the brain), with performance peaking when parameters align with structural features like cluster diameter or hardcore radius.

ABSTRACT

The growing availability of three-dimensional point process data asks for a development of suitable analysis techniques. In this paper, we focus on two recently developed summary statistics, the conical and the cylindrical $K$-function, which may be used to detect anisotropies in 3D point patterns. We give some recommendations on choosing their arguments and investigate their ability to detect two special types of anisotropy. Finally, both functions are compared on some real data sets from neuroscience and glaciology.

Motivation & Objective

• To evaluate and compare the effectiveness of the conical and cylindrical K-functions in detecting anisotropy in three-dimensional point patterns.
• To provide data-driven recommendations for choosing key parameters (e.g., r2, rcl, h, rcn, θ) in these functions to maximize detection power.
• To assess the performance of both statistics on real data from neuroscience (pyramidal cells) and glaciology (ice core bubbles), representing distinct anisotropy types.
• To investigate how parameter choices affect the power of isotropy tests based on directional K-functions.
• To guide practitioners in selecting the most appropriate directional K-function depending on the expected geometric structure of the point pattern.

Proposed method

• The conical K-function estimates the expected number of points within a cone of fixed height h and radius rcl, centered at a typical point and oriented along a specified direction.
• The cylindrical K-function estimates the expected number of points within a cylinder of radius rcn and height h, similarly oriented and centered at a typical point.
• A nonparametric isotropy test is applied to simulated data to evaluate detection power across different parameter combinations.
• Parameter recommendations are derived by identifying settings where the isotropy test achieves maximum power, particularly aligning r2 with structural features like cluster diameter or hardcore radius.
• The methods are applied to real datasets: pyramidal cell locations (columnar anisotropy) and ice core bubble centers (compressed regular pattern).
• Directional K-functions are estimated along the x, y, and z axes and compared visually and statistically to detect anisotropy.

Experimental results

Research questions

• RQ1 Which directional K-function—conical or cylindrical—performs better in detecting anisotropy in 3D point patterns with columnar structure?
• RQ2 How does the choice of parameters (e.g., r2, h, rcl, rcn, θ) affect the power of isotropy detection using these functions?
• RQ3 In what parameter ranges is the conical K-function most effective for detecting compression-induced anisotropy in regular point patterns?
• RQ4 For patterns with linear clustering (e.g., pyramidal cells), does the cylindrical K-function outperform the conical K-function in detecting anisotropy?
• RQ5 Can parameter recommendations be derived from simulation studies that maximize detection power while remaining applicable to real-world data?

Key findings

• The conical K-function is more powerful than the cylindrical K-function in detecting anisotropy in compressed regular point patterns (e.g., ice core data), particularly when r2 is set close to the hardcore radius (R = 0.05).
• The cylindrical K-function is more effective than the conical K-function in detecting columnar anisotropy (e.g., pyramidal cell data), especially when r2 is chosen to match the cluster diameter (e.g., 4σ ≈ 0.004 for σ = 0.001).
• The power of the isotropy test peaks when the integration interval captures the full extent of the structural feature—such as the entire column in columnar patterns or the full compressed cluster in regular patterns.
• For the pyramidal cell data, the cylindrical K-function shows clear directional dependence along the z-axis, indicating strong anisotropy aligned with the pial surface normal.
• For the ice core data, the conical K-function reveals stronger directional effects along the z-axis, confirming compression-induced anisotropy.
• The performance of both functions is highly sensitive to parameter choice, with suboptimal settings leading to poor detection power, even when anisotropy is visually apparent.