Derived Discrepancy On Hypersphere

In many fields,such as geographical science,satellite space,medical mirroring and so on,the experimenter often need to distribute points uniformly on the hyper-sphere.So how to measure the uniformity of point set effectively is an important problem.We know that in some regular region(such as hypercube)the uniformity measurement of uniform design point set been studied very deeply.Thereinto,the discrepancy based on the number theory integral has been discussed and studied the most.Some researchers further extended the definition of discrepancy to irregular experimental region,such as simplex,sphere.However,there are few researches on the measurement criterion of uniformity on the hypersphere.At present,the quasi F-discrepancy and the spherical cap discrepancy are the more popular methods in practical applications.It's worth noting that the quasi F-discrepancy is not direct-ly defined on the hypersphere,and there is no explicit expression for it,while the definition of the spherical cap discrepancy does not consider the uniformity of the low-dimensional projection.In this paper,we propose the measurement criteria of uniformity of points on the hypersphere with considering the uniformity of low-dimensional projection.They are collectively called derived discrepancy on sphere(SDD).Different with the quasi F-discrepancy,our discrepancy is directly defined on the hypersphere and the computational expression of it is given by a explicit function.This discrepancy takes the uniformity of low-dimensional projection into account compared to the spherical cap discrepancy.To compare the properties of different derived discrepan-cies,we have done a lot of numerical simulation and example analysis,these analysis results show that the measurement of design uniformity by other derived deviations is consistent with intuition except for the derived full discrepancy on hypersphere(FD).Compared with other derived discrepancy,the derived wrap-around discrep-ancy on hypersphere(SDWD)has rotational invariance making it more suitable for measuring the uniformity of point sets on hypersphere.The example of numerical integration shows that the variation of the derived discrepancy on hypersphere(S-DD)is consistent with the variation of the integral error,which is more sensitive than the spherical cap discrepancy and more fully depicts the uniformity of the hypersphere point set.