Voronoi-based entropy and Markov chains improve spatial pattern analysis
Voronoi-Markov chain and spatial entropy for point pattern analysis
Information Theory
Summary
Analyzing how points are spread out in space can tell us a lot about natural patterns, such as cells in tissue. This paper develops new ways to measure randomness and clustering of points by looking at shapes called Voronoi basins and building a simplified random walk (Markov chain) on these shapes. These new measures help distinguish different types of point arrangements more clearly. The authors test these ideas on simulated data and real tissue samples and provide code for others to use.
What this means in practice
- •For histopathology data teams: Use new entropy metrics based on Voronoi structures to improve detection of clustering in tissue point patterns like cells.
- •For spatial data analysts: Implement Markov chain based metrics to more sharply discriminate spatial point patterns for environmental or geographic data.
Authors
James C. Mathews, Francisco Couzo, Aleksandr Petrov, Avijit Chatterjee, Saad Nadeem
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.