Weighted And Sequential Support Points

Monte Carlo method is an important simulation method,which is based on the theories and methods of probability and statistics.It associates the problem with a certain probability model and generates the corresponding random sample,and then carries on the numerical approximation and statistical inference.Monte Carlo method has the ability to solve a wide range of problems,and it is simple and flexible to use.However,its convergence rate is slow,and the practical application efficiency is greatly reduced when the sampling cost is relatively high.In recent years,some scholars have studied deterministic sampling methods.According to the dependence of models,these methods can be divided into two categories:(1)Model-free methods,which usually adopt some distribution similarity measurements as the objective function such as discrepancy and divergence;(2)Based on a certain or a kind of models,such as a variety of optimal design methods in experimental design.Energy distance is a well-known discrepancy criterion,which was originally used to test the goodness of fit for high-dimensional distributions.Some scholars put the random training sample from the target distribution into the energy distance for optimization,and called the sample set obtained as the support point.The energy distance adopts the negative Euclidean distance kernel,which is easier to optimize than other discrepancies,so the support points are generated quickly.When the target distribution is relatively simple,the representativeness of the support points is better.However,when the target.distribution is relatively complex,the properties of the support points will be affected by the random training sample,and if the training samples themselves are significantly different from the target distribution,the support points will absorb its defects and become worse in representativeness.In this paper,the representativeness of the samples obtained by the support point algorithm is further improved based on the quasi-Monte Carlo sequence and the weighting strategy.The support points generated by improved algorithm have better space-filling property,which is more obvious when the target distribution is multimodal.Combined with the idea of sequential sampling and augmented design,a sequential version of the support point algorithm is proposed in this paper,which can be used for the augmented design of arbitrary distribution.When the calculation cost of the target distribution is high,sample of target distribution can be obtained more efficiently with the combination of interpolation model,such as Gaussian process model,and sequential importance sampling method.In addition,the constructed of sliced design is considered in this paper with new support points algorithm,by introducing two hyperparameters to adjust the full design(all points)and subdesigns(part of the design point).The algorithm generates sliced design eficiently,the number of points in each subdesign is flexible,and it is suitable for the sliced design of general distribution.It can be seen from the visual simulation examples that when we improve the representativeness of the approximate large sample itself,the representativeness of support points has a very obvious improvement.In addition,sequential algorithm of support points is beneficial to reduce the number of calculation of the target density.Based on the sequential support points algorithm,the existing points in the experimental area are augmented uniformly which improves the optimization efficiency of sequential number theoretic optimization(SNTO)algorithm.In the application of sliced design,three sliced design processes of uniform distribution in hypercube,normal distribution and constrained experimental design with mixtures are given,and their performance verifies the balancing effect of hyper parameters on the representation of the full design and subdesigns.