Projection Weighted Symmetric Discrepancy

Experimental design is used to arrange experiments economically and scientif-ically,and plays an important role in industrial production and engineering design.Better experimental design will get twice the result with half the effort to achieve the desired purpose.In order to better measure the quality of the experimental de-sign,so many evaluation criteria are proposed.Uniformity is one of the commonly used criteria.Uniformity plays an important role in the construction of space filling design,as well as in the study of model robustness and effective numerical analy-sis.There have been many measurements of uniformity.One of the commonly used measures of uniformity is discrepancy,which is studied and applied in many liter-atures.The centered L2-discrepancy(CD),the wrap-around L2-discrepancy(WD),the full L2-discrepancy(FD),the mixture L2-discrepancy(MD),the symmetrical L2-discrepancy(SD)are commonly used.Each discrepancy has its own advantages and disadvantages in measuring uniformity.From the aspects of the low dimensional projection,we study the properties of various discrepancies,and we find out that the proportion of discrepancy value in lower dimensional is larger than that in high-er cases except SD.The result in SD is not consist with the hierarchical ordering principle.In order to overcome the shortcoming of SD,a new discrepancy,called the weighted symmetric discrepancy,is proposed,and study various excellent properties of it in detail.WSD is consist with the hierarchical ordering principle and Devia-tion,and better than SD.Also WSD overcome the shortcoming that CD neglects the center point and WD lacks translation invariance.Inspired by the definition of the mixture discrepancy,we consider to define a new mixture discrepancy(NMD),just the mixture of the weighted symmetric discrepancy and the wrap-around dis-crepancy.And we also verify the rationality of the new mixture discrepancy and the relationship with other design criteria for experimental design.