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The Role of Tessellation Methods in the Analysis of Three-Dimensional Spatial Point Patterns

This paper concerns the analysis of three-dimensional spatial point patterns using the Dirichlet tessellation. The tessellation method is illustrated by means of a case study in which spatial patterns of chromosomes in cells of a plant hybrid called Leontodon hispidus x L. taraxacoides are investigated. The objective of the analysis is to determine if chromosomes originating from the two species that produce the hybrid occupy separate spatial domains within the cell. The Dirichlet tessellation is used to construct a test for random labelling of chromosome locations by parent species. An alternative test based on the mean nearest-neighbor distance between chromosomes of different types is also illustrated. The tessellation test leads to rejection of the null hypothesis of random labelling whereas the test based on the mean nearest-neighbor distance does not. The reasons for the differences between the results of the two approaches are explored.