Measure Of Uniformity Of Points Distribution In A General Convex Polyhedron

Measure of uniformity is essential for constructing uniform design.However,existing measurement criteria are defined in the unit cube.Based on the distance concept in this paper,potential function and balance function are introduced and used to measure the uniformity of points distributed in a general convex polyhedron.In the second chapter,we give their definition mode in detail.In the third chapter,we explain the computational process of two functions through special examples.At the same time, we compare their results of measuring several special points distribution on the unit square with those obtained by centralized L₂-discrepancy and wrap-around L₂-discrepancy. And we obtain the consistent conclusion.In the following chapters,we calculate the values of potential function,balance function,centralized L₂-discrepancy and wrap-around L₂-discrepancy for 50 pairs of points randomly generated on the unit square and trapezoid.Have arranging their ranks in the order of increasing, we carry on hypothesis tests based on the rank correlation analysis of variables. Numerical examples and multivariate Kendall concordance coefficient of multivariate test indicate when the experiment region is limited to the unit cube,potential function and balance function have consistent conclusion with centralized L₂-discrepancy and wrap-around L₂-discrepancy which are often used to measure uniformity in existing literatures.