On Analytical Approximation of Cell Volume Distribution in Three-Dimensional Poisson-Voronoi Tessellation
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The cell volume distribution in the three-dimensional Poisson-Voronoi tessellation is simulated for a data set of 400000000 cells. The probability density of the distribution is estimated using the Gaussian-kernel method. Analytical approximation of the estimated probability density by the generalized gamma distribution with different sets of the three parameters is discussed. We show that the set of parameters proposed by Lazar et al. (2013) does not lead to a satisfactory analytical approximation, whereas the set of Tanemura (2003) ensures a significantly better accuracy of approximation. We propose a method for determining the parameters that provides the required values of the mean volume, mean squared volume, and mean cubed volume of cells. These three values may be obtained theoretically or from simulation. As a result, we obtain a system of three equations for three parameters of the generalized gamma distribution. The solution of this system leads to the approximation whose accuracy is close to that of the Tanemura approximation. In addition, we use the least squares method to determine values of the three parameters and obtain the best analytical approximation for the results of our modeling of the distribution of cell volumes in the spatial Poisson-Voronoi tessellation.
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