Voronoi tessellation and voronoi cell density

Autoimmune related genes, ranked by disease association (Pletscher-Frankild et al., 2015), and the variants for each gene were plotted in the x-y coordinates of the Voronoi diagram. Normally distributed mock data were used to test the Voronoi tessellation and clustering method (Figure S2, Table S4, Supplemental Information). Built-in Voronoi functions in Matlab⁷ were used to create tessellation of data, which returned the indices of the Voronoi cells and vertices.

A Voronoi cell represents an area of influence of the data point it contains, and thus the local density in the proximity of a given point can be determined as the inverse of the cell area. This provides a direct precise measurement of the local density. Clusters were identified based on the neighboring Voronoi cells with densities above a certain threshold. Polyarea, a built-in function in Matlab⁷, was used to calculate the area for each Voronoi cell, except for the cells on the boundary of the map with infinite areas. For Voronoi tessellation with n (finite number of) cells, the normalized Voronoi cell density (f~) was calculated as the ratio of the cell density (inverse of cell area) over the inverse of the average cell area (Ebeling and Wiedenmann, 1993)

Parameters were obtained by fitting the Chi-square distribution to 80% of the normalized Voronoi cell density distribution (Ebeling and Wiedenmann, 1993). We obtained the threshold for clustering at the significance level of 90% from the fitted Chi-square distribution (Ramella et al., 2001).