Voronoi-Markov chain and spatial entropy for point pattern analysis
Abstract
In this paper, we revisit the concept of entropy in the spatial context with the aim of deriving computable and interpretable metrics for point pattern analysis in domains such as histopathology. We discuss stochastic point process assumptions like Poisson homogeneity and Simple Sequential Inhibition (SSI) and review established results for Voronoi and Delaunay tilings of point sets for their potential to provide null distributions for rigorous empirical hypothesis testing. We present (1) a novel "dense basin entropy" defined in terms of the Ord distribution supported on Voronoi basins, which is shown to be sensitive to clustering, and (2) a related Markov chain designed as a simplification or approximation of Brownian motion in the underlying planar domain. We investigate bounds on the dense basin entropy in the SSI regime and in empirical data, and show that the entropy rate and spectral gap of the Markov chain lead to sharp discrimination metrics. Finally, we demonstrate a generalization to multiple-set analysis via aggregation of one set over the Markov invariant measure for another. Simulations and experiments in histopathology show that our proposed metrics offer insights complementary to other instantiations of entropy in spatial analysis. The code can be found on our SMProfiler GitHub: https://github.com/nadeemlab/SMProfiler.
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BibTeX
@misc{jr2026voronoimarkov,
title = {{Voronoi-Markov chain and spatial entropy for point pattern analysis}},
author = {James C. Mathews Jr. and Francisco Couzo and Aleksandr Petrov and Avijit Chatterjee and Saad Nadeem},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36158},
eprint = {2609.36158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36158},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Voronoi-Markov chain and spatial entropy for point pattern analysis
AU - James C. Mathews Jr.
AU - Francisco Couzo
AU - Aleksandr Petrov
AU - Avijit Chatterjee
AU - Saad Nadeem
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36158
UR - https://arxiv.org/abs/2609.36158
AB - In this paper, we revisit the concept of entropy in the spatial context with the aim of deriving computable and interpretable metrics for point pattern analysis in domains such as histopathology. We discuss stochastic point process assumptions like Poisson homogeneity and Simple Sequential Inhibition (SSI) and review established results for Voronoi and Delaunay tilings of point sets for their potential to provide null distributions for rigorous empirical hypothesis testing. We present (1) a novel "dense basin entropy" defined in terms of the Ord distribution supported on Voronoi basins, which is shown to be sensitive to clustering, and (2) a related Markov chain designed as a simplification or approximation of Brownian motion in the underlying planar domain. We investigate bounds on the dense basin entropy in the SSI regime and in empirical data, and show that the entropy rate and spectral gap of the Markov chain lead to sharp discrimination metrics. Finally, we demonstrate a generalization to multiple-set analysis via aggregation of one set over the Markov invariant measure for another. Simulations and experiments in histopathology show that our proposed metrics offer insights complementary to other instantiations of entropy in spatial analysis. The code can be found on our SMProfiler GitHub: https://github.com/nadeemlab/SMProfiler.
ER -