伞形约束统计推断问题的优化方法

发布时间：2018-05-29 10:20

本文选题：伞形约束 + 优化理论　；参考：《北京交通大学》2017年硕士论文

【摘要】：统计推断是统计学中的一类重要课题.在统计推断问题中,通常被估计的量或待检验的假设会有一些先验条件,这些条件作为约束条件放在统计推断模型中,形成有约束条件的统计推断问题.带约束的统计推断应用背景非常广泛,在医药,经济,环境科学,基因研究等领域起着很重要的作用,所以至今仍然是统计界研究的一个热门.剂量—反应模型是一类经典的带约束的统计推断问题,研究的是如何设置药物的剂量使药物的使用即安全又有效,副作用最小.处理好这类问题,对药品的制造和使用有很大作用.在药物的剂量—反应研究中,药物的反应—剂量比率并不总是随着剂量的增加而一直增加,当剂量达到一定程度后,药物的毒副作用也会增加,其反应—剂量比率反而会下降.这样,反应—剂量比率形成一种先增后减的伞形顺序,其作为约束条件我们称之为伞形约束.从数学实质上看,带伞形约束的剂量—反应模型就是一类带约束的最优化问题.不同于统计中的保序回归方法,本文通过优化理论对带伞形约束的剂量—反应模型进行分析,用优化的思想解决统计学中的经典模型,为求解此类统计问题提供了一种新思路.首先,本文研究了带伞形约束的剂量—反应模型的性质,证明了其解的唯一存在性,给出了其一阶最优性条件—KKT条件,二阶最优性条件,并研究了该模型的对偶性理论.其次,本文给出了该模型的三种优化算法—序列二次规划法,Rosen梯度投影法和乘子法,也给出了确定该模型伞形顶点的方法.最后,本文对一组真实数据进行了数值实验,并对三种算法进行了比较.
[Abstract]:Statistical inference is a kind of important subject in statistics. In statistical inference, there are usually some prior conditions for the estimated quantity or the hypothesis to be tested. These conditions are placed in the statistical inference model as constraints, and form a statistical inference problem with constraints. Medicine, economics, environmental science, gene research and other fields play an important role. So far, it is still a hot topic in the field of statistics. The dose response model is a class of classic statistical inference problems with constraints. It is studied how to set the dosage of drugs to make the use of drugs safe and effective and the side effects are minimal. The problem has a great effect on the manufacture and use of drugs. In the dose response study of the drug, the reaction dose ratio of the drug does not always increase as the dose increases. When the dose reaches a certain extent, the drug side effects will increase, and the reaction dose ratio will decrease. In this way, the reaction dose ratio is also reduced. In fact, the dose response model with umbrella shaped constraints is a class of constrained optimization problems. Different from the sequential regression method in statistics, the dose response model with umbrella constraints is introduced in this paper. This paper provides a new idea for solving such statistical problems. First, this paper studies the properties of the dose response model with umbrella shaped constraints, proves the only existence of the solution, and gives the first order optimality conditions, the KKT condition, the two order optimality condition, and the study. The duality theory of the model. Secondly, three optimization algorithms of the model - sequence two times programming, Rosen gradient projection method and multiplier method are given in this paper. The method of determining the apex of the model umbrella is also given. Finally, this paper carries out numerical experiments on a group of real data and compares the three algorithms.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

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