Symmetric Discrepancy And Its Improvement

As a kind of robust experimental design,uniform design is one of the most commonly used space-filling designs.In the absence of sufficient prior information,each design point is considered equally important in the experiment region.There-fore,uniform design advocates that the experiment points are uniformly distributed in the target area.In order to measure the uniformity of the design points in the experiment region,many discrepancies are defined and developed.In the literature,we often use the following discrepancies:the star discrepancy,the centered discrep-ancy(CD),the wrap-around discrepancy(WD),the symmetric discrepancy(SD),and the mixture discrepancy(MD)etc.In order to compare different discrepancies,the definition of the cover frequency has been proposed recently.In this paper,according to the definition of cover frequency and the definition of the kernel function of the reproducing kernel Hilbert space,we derive the cover frequency formula of various discrepancies.In addition,we study the advantages of various kinds of discrepancies under the cover frequency formula.At the same time,it is found that the symmetric discrepancy has the constant value in the cover frequency.It has outstanding advantage.Based on this advantage,we construct a new mixed discrepancy drawing on the idea of the mixed discrepancy.Finally,we study the symmetric discrepancy and the new mixed discrepancy from different perspectives,such as the cover frequency,GMA criterion and so on.