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$K$ -functions for point processes on complex surfaces

This paper proposes and evaluates several methods for extending the classical $K$ -function to point processes on complex surfaces, ultimately identifying a surface area $K$ -function based on geodesic ball areas as the most effective approach for accurately assessing clustering or regularity without the distortions caused by 2D projections.

Original authors: Francisco Cuevas-Pacheco, Scott Ward, Ed Cohen, Rasmus Waagepetersen

Published 2026-08-25

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CC0 1.0

Original authors: Francisco Cuevas-Pacheco, Scott Ward, Ed Cohen, Rasmus Waagepetersen

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand how trees grow in a dense forest by looking only at a flat map. You would see where the trunks touch the ground, but you would miss the hills, valleys, and steep slopes that shape their world. In the study of point patterns—scientific terms for how objects like trees, stars, or bacteria are scattered across a space—researchers have long relied on tools designed for flat, two-dimensional maps. These tools help scientists decide if objects are randomly scattered, clustered together in groups, or spaced out evenly. However, the real world is rarely flat. When objects exist on a curved surface, such as the uneven terrain of a mountain or the rounded body of a bacterium, flattening that surface onto a map distorts the distances between them. This distortion can trick researchers into seeing patterns that do not actually exist, leading to incorrect conclusions about how nature organizes itself.

A team of researchers has addressed this problem by developing a new way to measure these patterns that respects the true shape of the surface. Instead of forcing the data onto a flat plane, they created a method that measures the space between objects along the surface itself. Their work focuses on a specific statistical tool known as the K-function, which counts how many neighbors an object has within a certain distance. The researchers realized that on a curved surface, the distance between two points is not a straight line through the air, but the path one would have to walk along the ground to get from one to the other. They found that the traditional way of measuring this distance fails on complex shapes, often making random groups of objects look like tight clusters or making actual clusters look scattered.

To solve this, the team proposed and tested several different approaches, eventually settling on a method they call the surface area K-function. This approach changes the fundamental question asked by the statistic. Rather than asking, "How many neighbors are within a specific walking distance?" it asks, "How many neighbors are within a specific amount of surface area?" This distinction is crucial because on a curved surface, a circle of a fixed walking distance can cover very different amounts of ground depending on where it is located. By measuring the actual area covered, the new method provides a consistent and accurate baseline for comparison. The researchers tested this idea using computer simulations of points on various curved shapes, including surfaces with bumps, saddles, and hills. They also applied the method to real-world data, examining the locations of thousands of trees in a Sri Lankan rainforest and the positions of molecules on the surface of a bacterium.

The results showed that the old, flat-map methods often produced misleading answers. In the case of the rainforest trees, the traditional method suggested a much stronger level of clustering than actually existed, simply because the steep terrain compressed the distances when viewed from above. The new surface area method corrected this, revealing a more accurate picture of how the trees are distributed. Similarly, when applied to the bacterial molecules, the new tool clearly identified a clustered pattern that previous methods had struggled to interpret in a meaningful way. The researchers found that their new approach is not only more accurate but also easier to interpret, as it directly relates the count of neighbors to the actual surface area they occupy. This work demonstrates that to truly understand how objects are arranged in our three-dimensional world, we must stop treating the ground as a flat sheet and start measuring the space exactly as it exists.

Technical Summary: K-functions for Point Processes on Complex Surfaces

Problem Statement
The statistical analysis of point processes is predominantly confined to Euclidean spaces ( $\mathbb{R}^d$ ). However, many practical applications involve point patterns generated on complex surfaces in three-dimensional space (e.g., topographic landscapes or biological membranes) that are often projected onto a two-dimensional plane for analysis. This projection ignores the third dimension (elevation or surface curvature), which can lead to erroneous conclusions regarding the clustering or regularity of the underlying point process. Specifically, the application of the classical K-function to projected planar data without accounting for surface geometry can introduce spurious second-order inferences and bias intensity estimation. While recent work has addressed point processes on spheres and linear networks, there is a lack of unified methodology for general smooth surfaces, particularly those representing geographical topography.

Methodology
The authors propose and evaluate three distinct extensions of the classical K-function to point processes defined on a parametric surface $S \subset \mathbb{R}^3$ . The surface is defined as the image of a mapping $\phi: D \to \mathbb{R}^3$ , where $D \subseteq \mathbb{R}^2$ . The methodology relies on geodesic distances ( $d_S$ ) and surface area measures ( $\nu$ ) rather than Euclidean distances.

Surface Corrected K-function: This approach applies the standard planar K-function to the projected point pattern $X_D = \phi^{-1}(X)$ but incorporates a correction for the surface area element $\|n(u)\|$ (where $n(u)$ is the normal vector) within the intensity function. While computationally efficient, the authors note that the pre-images of Euclidean discs on the surface can become highly distorted (e.g., elongated ellipsoids on steep slopes), potentially leading to neighborhoods that lack physical interpretability for the original process.
Tangent Plane K-function: This method maps points from the surface to the tangent plane at each observation point using a projection (orthogonal or logarithm/exponential map). It defines the K-function based on Euclidean discs in the tangent plane. While conceptually a natural refinement, the authors identify significant computational hurdles: calculating the image of the observation window on the tangent plane ( $A_{T_s}$ ) and the associated Jacobians are computationally expensive, particularly for general surfaces. Consequently, this method is excluded from the simulation and data applications.
Surface Area K-function: This is the primary contribution of the paper. Instead of defining neighborhoods by a fixed geodesic radius $r$ (where the area of the geodesic ball varies with location on a curved surface), this approach defines neighborhoods by a fixed surface area $a$ .
- Definition: For a point $s \in S$ , the neighborhood is the geodesic ball $b_S(s, r^*)$ such that its area $\nu(b_S(s, r^*)) = a$ .
- Estimator: The estimator $\hat{K}_{area}(a)$ sums interactions within these area-defined neighborhoods, normalized by the surface area of the observation window and edge corrections.
- Implementation: Numerical implementation requires solving the Eikonal equation (using the Fast Marching Method) to compute geodesic distances on a grid. This allows for the approximation of geodesic balls and the determination of the radius $r^*$ corresponding to a specific area $a$ .

Key Contributions

Theoretical Formulation: The paper establishes that for general surfaces, the argument of the K-function is more naturally the area of the neighborhood rather than the radius. The classical K-function is shown to be an area-based statistic where radius is merely an equivalent parameterization valid only for flat geometry or constant curvature surfaces (like spheres).
Computational Framework: The authors provide a practical implementation of the Surface Area K-function using the Fast Marching Method to solve the Eikonal equation, enabling the calculation of geodesic distances and areas on arbitrary smooth surfaces.
Comparative Analysis: Through simulation studies and real-world data applications, the paper compares the performance of the proposed methods against the planar approach and the surface-corrected planar approach.

Results

Simulation Experiments: The study utilized three surface types (Monkey Saddle, Gaussian Bump, Polynomial Hills) and three point process models (Homogeneous Poisson, Matérn Cluster, Matérn Type II Inhibitory).
- The planar K-function failed significantly, exhibiting high Type I error rates (rejection of Complete Spatial Randomness, CSR) even for Poisson processes (rejection rates >87%).
- The Surface Corrected K-function showed improved Type I error control but suffered from lower statistical power in detecting clustering or inhibition compared to the Surface Area K-function.
- The Surface Area K-function demonstrated the best performance, maintaining Type I error rates close to the nominal 5% level and achieving the highest power in detecting deviations from CSR across all surface types.
Real Data Applications:
- Sinharaja Tree Data: Analysis of tree locations in a 500m $\times$ 500m plot in Sri Lanka with significant elevation variation (150m difference). Both the Surface Area and Surface Corrected K-functions confirmed clustering for two species, but the planar K-function indicated a much larger deviation from the null hypothesis, attributed to spurious clustering induced by projection.
- E. coli Lipopolysaccharide (LPS) Data: Analysis of LPS molecule locations on a pill-shaped bacterial surface. The Surface Area K-function rejected the CSR hypothesis, indicating clustering. The authors note that the area-based metric provides a more direct biological interpretation (in units of surface area) compared to spherical projection methods that rely on great-circle distances.

Significance and Claims
The paper claims that the Surface Area K-function provides a "natural formulation" for analyzing point processes on complex surfaces. Its primary significance lies in correcting the bias introduced by projecting 3D surface data onto 2D planes, which can lead to misleading inferences about spatial dependence.

The authors modestly frame their work as clarifying which aspects of the Euclidean K-function reflect general principles (dependence on neighborhood area) versus those specific to planar geometry (dependence on radius). They argue that for non-flat surfaces, the Surface Area K-function is the superior analogue because it ensures that the expected number of points in a neighborhood is directly proportional to the area of that neighborhood, a property that holds for Poisson processes regardless of surface curvature. The paper concludes that accounting for the surface domain is crucial for obtaining quantitatively correct assessments of clustering levels in spatial statistics.

Technical Summary: K-functions for Point Processes on Complex Surfaces

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