Uniform Design With Mixtures Based On Lattice

In many fields such as food,chemical industry,materials,finance and so on,many experiments are conducted by mixing some ingredients together.And the influence of different proportion of ingredients(factors)on the response variable is considered by the experimenter.Different from factor experiment,the area of each factor in this experiment is not independent,the proportion of each factor is always greater than 0 and sum to 1.Statisticians put forward the experimental design with mixtures for this kind of experiment.When the model is unknown,a robust experimental effect can be achieved by distribute the experimental points in the target area uniform-ly.This kind of method is called Uniform design with mixtures,which overcomes the shortcomings of excessive dependence on model assumptions in the optimal design.Existing mixture experimental design methods either use regular uniform de-sign inverse transformed return to the mixture design area,or discretize the mixing design area based on the idea of distance optimization,and then get the represen-tative points of the design points by clustering method.However,the uniformity of the inverse transformation method depends on the uniformity of the uniform design,and the uniform design is obtained by the threshold acceptance method.When the number of experiments is large,the algorithm takes too long.For clustering meth-ods,when there are too many centers,the results may not be stable and can only converge to local optimum in practical applications.In view of these shortcomings,the sphere filling method is applied to construct the uniform experimental design with mixtures in this paper.By using the optimum lattice points to fill the tar-get area,the good distance properties of lattice points will be retained.Some of comparison based on different uniform criteria and mse based on modeling are al-so conducted.The results of those comparisons shows that when the dimension is not high but the number of experiments is large,The root mean squared error of the design constructed by lattice points is between the inverse transform and the clustering method.and the design point has a larger separation distance.