Optimal Experimental Design For Quantile Regression Based On Particle Swarm Algorithm

With the continuous development of scientific technology,optimal experimental design has gradually become a hot topic in research over the past decade.One main reason is the rising cost of experiments,and has led increasingly more fields of application to realize that optimization design ideas can significantly save costs without sacrificing statistical efficiency.Quantile regression has advantages over least squares regression,such as robustness,applicability to heteroscedastic data,and estimation of the overall distribution.It has widespread applications in fields such as biomedical,ecological,and economic research.However,due to the non-convexity of its objective function,finding the optimal experimental design for quantile regression is both theoretically and computationally challenging.Currently,there are no effective algorithms to solve the optimization design problem for quantile regression,which has prevented in-depth research on optimization experimental design for this method.Therefore,it is necessary to develop numerical algorithms for constructing the optimal experimental design for quantile regression.Particle swarm optimization technology has been widely used to solve challenging optimization problems,but so far it has not had an impact on mainstream statistical applications.This thesis constructs a numerical algorithm for solving the optimal experimental design for quantile regression based on the idea of particle swarm optimization.The main research work is divided into the following parts:(1)Since the objective function of the optimal experimental design for quantile regression is not a convex function,there is no general equivalence theorem to support the optimal design for quantile regression.In this thesis,we derive the relevant theorems of A-and c-optimality for quantile regression experimental design,and provide proofs.The above theorems provide the necessary conditions for the A-and c-optimal experimental design for the quantile regression model,which fill the gap in the theoretical foundation of the optimal experimental design for quantile regression.(2)Traditional experimental design algorithms are not feasible for solving the optimal design of quantile regression and can easily lead to results falling into local optima.Even for simple linear models,there is currently no effective algorithm to find the optimal experimental design for quantile regression.Inspired by the idea of particle swarm optimization,we construct a numerical algorithm for solving the optimal experimental design problem of quantile regression,which has good optimization ability for complex problems such as nonlinearity,non-convexity,and high dimensionality.We prove that this algorithm has strong global search capability and can be applied to various optimality criteria and models with merely any assumptions,finding the optimal solution.(3)For dose-response models widely used in pharmacology and toxicology,such as the Michaelis-Menten,Emax,and Exponential models,the research of optimal designs for quantile regression is still lacking.In this thesis,we use numerical algorithms to obtain the local optimal experimental design and Bayesian optimal experimental design for the above models in quantile regression,and verify the accuracy of the optimal results with the derived theorem.We demonstrate that our numerical algorithm is easy to implement and can effectively find different types of optimization designs for models with multiple factors.