The Construction And Properties Of Uniform Projection Designs

The experimental design is one of the important branches of statistics,and it is a discipline formed by the intersection of scientific experiments and statistical analy-sis methods.In scientific experiments,a well-designed experiment is an effective way to learn about the world.In statistical models,choosing a well-selected training set can effectively improve the accuracy and efficiency of statistical inferences.In mod-ern scientific experiments,the experiment process can be realized through computer codes.The computer experiments that emerged as the times require are widely used to simulate complex physical systems,and have extremely important applications in many fields such as modern industry,aerospace,and medicine,etc.In the modeling of computer experiments,the choice of design is particularly critical,and space-filling design is one of the most popular computer experiment methods at present.A space-filling design refers to a design in which the experimental points are dis-tributed as evenly as possible within the experimental area.There are many criteria for measuring space-filling properties and corresponding optimal designs,such as:Latin hypercube designs based on 1-dimensional hierarchical properties;maximin distance designs and minimax distance designs based on distance criterion;uniform designs based on discrepancy criterion;and maximum projection designs and uniform projec-tion designs focusing on low-dimensional projections,etc.In practice,only a few factors are active.In order to measure the effectiveness of the selected design,space-filling designs that can achieve low-dimensional projection uniformity are particularly needed.The maximum projection designs assume that all subspaces are equally important,while the uniform projection designs assume that low-order effects are more important than high-order effects,so the uniform projection designs pay more attention to the low-dimensional projections.Representative studies on uniform projection designs include:Sun et al.（2019）^[82]proposed the uniform projection criterion for the first time.They proved that the uniform projection designs can evenly distribute points in all dimensions,and they are space-filling under differen criteria.Chen et al.（2021）^[10]provided a method to construct uniform nested Latin hypercube designs combining with the uniform projection criterion.Sun and Tang（2021）^[77]established the connec-tions between uniform projection designs and strong orthogonal arrays,gave the de-composition of the criterion,and established some optimality results of certain strong orthogonal arrays.Wang et al.（2022）^[93]established some theoretical connections be-tween the maximin distance criterion,column orthogonality,and uniform projection criterion,providing new theoretical justifications for each criterion.However,the cur-rent researches on constructing uniform projection designs and the higher-dimensional properties of such designs are still blank.To this end,we have studied related con-tents of constructing uniform projection designs and the high-dimensional properties of them.Let’s briefly introduce our research work.Firstly,we provide the construction methods of uniform projection designs with flexible parameter settings（numbers of runs and factors）.We systematically study the construction methods of uniform projection designs through level permuta-tion and/or level expansion.Specifically,we explore the effects of level permutation and level expansion on designs’projection uniformity.For each approach,we estab-lish theoretical results connecting the uniform projection properties of the generated designs with the properties of the corresponding initial designs.Based on the theoret-ical findings,efficient algorithms are developed to construct UPDs with flexible sizes,which leads to many practically useful designs.Numerical simulation results illustrate the effectiveness of the generated designs.Secondly,we propose the 3-dimensional uniform projection design,and s-tudy its theoretical properties.2-dimensional projection uniformity cares the unifor-mity of designs corresponding to the main effects model.In more complex models,it is not enough to consider the models including linear main effects only,and the second-order interaction effect also often exists.In order to reduce the impact of the second-order interaction effect on the main effect,the 3-dimensional projection uniformity of a design is needed.In addition,the 2-dimensional uniform projection designs are not unique.In order to select the better design,it is also rational to compare their higher-dimensional projection uniformity.In this article,we propose the 3-dimensional uni-form projection designs,which can realize the 3-dimensional projection uniformity in all projection spaces.In addition,we establish some theoretical connections between the 3-dimensional uniform projection criterion and the criteria of maximin distance and 3-orthogonality,which help in finding better space-filling designs under multiple criteria.Numerical simulations confirm the rationality of our proposed designs.Finally,we propose a generalized uniform projection criterion,and provide the search algorithm finding the optimal design:minimum uniform projection de-sign.Theoretically,combined the idea of”sequential”,we want to improve the higher-dimensional projection uniformity of a design step by step,under the assumption of ensuring low-dimensional projection uniformity.However,the values of each crite-rion may not be consistent,sometimes even contrary,so the calculation of the direct search is too large to achieve it.For this reason,according to the principle of sorting effects,we comprehensively consider the 2-dimensional and 3-dimensional projection uniformity of a design,and propose the generalized uniform projection criterion.Its essence is the weighted average of these two criteria.Because the values of these two criteria are in different range,we correct them and use relative efficiencies to replace the corresponding uniform projection criteria in the goal function.The optimal de-sign is called minimum uniform projection design,which can simultaneously realize the 2-dimensional and 3-dimensional projection uniformity on all projection spaces.Based on the theoretical results,we provide the optimization search algorithm of the minimum uniform projection designs.Real simulations also confirms the superiority of proposed designs.