Sampling And Design On Simplex With Convex Constraints

Many authors studied the random number generation method on simplex due to its widespread applications,such as experiments with mixtures,multiple crite-ria decision analysis,portfolio optimization and so on.However,the dimension of practical problem is higher and higher and constraints become more and more com-plicated,for which the existing methods are not appropriate in efficiency or complex-ity.In terms of sampling,methods based on acceptance-rejection or transformation by vertices are not efficient,conditional distribution method becomes complex or even invalid.Hit-and-Run algorithm is one of Markov Chain Monte Carlo samplers,which has low mixing rate in the case that experiment region is long and narrow.As for design,classical designs of experiments with mixtures put too much points on the border and the number of experiments is not flexible.With the emergence of some new methods and techniques in these years,uniform design for experiments with mixtures has effectively solved these two problems in some degree.While constraint is complicated,most papers adopted the strategy of acceptance-rejection algorithm and stochastic search algorithm under a certain discrepancy.Similar to the sam-pling problem,these methods don't work in high-dimensional case because of the inefficiency.In this paper,we develop a new sampler based on invariance of projected distri-bution of affine constraints and Gibbs algorithm to generate uniform random number on simplex with convex constraints.Simultaneously,some details are discussed,in-cluding starting point,calculation of upper and lower bounds,acceleration methods about rates of sampling and mixing and diagnosis of convergence.Further,to con-struct uniform representative points on experiment region,we present clustering algorithms based on large sample generated by Gibbs algorithm.In practical exper-iment or application,to make full use of prior knowledge and meet the requirements,we put forward the construction method of non-uniform representative points co-incident with prior information,combining with Sampling Importance Resampling(SIR)and clustering algorithms.From the numerical simulation results,we have three conclusions as following.First,the sampler proposed in this paper can generate sample with desired unifor-mity and is suitable for complex situations with high dimension and complicated constraint.Second,under common criteria,designs derived from clustering method performs better than those generated by stochastic search algorithm.Third,the method of generating non-uniform representative points based on SIR and cluster-ing algorithms,which further improves the value of experiment points.