几种方差缩减技术的应用研究
发布时间：2018-07-17 20:22
【摘要】：蒙特卡罗(MC)方法作为一种统计方法,起初主要作为确定论方法的补充使用.但是随着计算机的快速发展以及对计算机的普及与应用打破了这一局面,使得原本耗时的模拟过程变得更加快捷,这极大的促进了MC方法的发展.MC方法的应用范围也从起初的核领域延伸到其它领域,现今已成为解决许多物理、工程、金融等领域中问题的重要计算工具,在参数估计和可靠性设计投资风险和投标报价生物医学统计物理学等方面也有极为广泛的应用,对于确定性问题,也可用MC方法进行解决,以及求解各类方程(组)、计算多重积分、无穷级数等.但是由于MC方法经过随机模拟得到的近似值难免会与所估计的真值存在偏差,本文针对该问题展开研究.首先,论文对MC方法的收敛性与误差估计进行了分析,总结了常用的六种方差缩减的方法:重要抽样法、控制变量法、对偶变量法、条件期望法、分层抽样法、相关抽样法.其次,论文讨论了常见的方程组、定积分、级数问题的蒙特卡罗对偶变量法,通过建立概率统计模型、抽样产生随机样本进而获得确定性问题的估计值,结果显示该技巧可有效降低模拟精度且缩短计算机运行时间,充分展示了对偶变量法的高效性.最后,讨论线性代数系统的自适应蒙特卡罗求解算法.该方法包括自适应重要性抽样蒙特卡罗和自适应相关抽样蒙特卡罗,并通过求解具体算例比较蒙特卡罗方法和确定性方法的求解时间和算法效率,从而得出自适应蒙特卡罗方法可以获得更快的收敛速度。
[Abstract]:The Monte Carlo (MC) method, as a statistical method, was used primarily as a supplement to the deterministic method. However, with the rapid development of computers and the popularization and application of computers, this situation has been broken, and the time consuming simulation process has become more rapid. This greatly promotes the development of MC methods. The application of MC methods extends from the original nuclear field to other fields. Now, it has become an important computing tool to solve many problems in physics, engineering, finance and so on. It is also widely used in parameter estimation and reliability design investment risk and bid quotation biomedical statistical physics. For deterministic problems, MC method can also be used to solve them. And solve all kinds of equations (groups), calculate multiple integrals, infinite series and so on. But because the approximate value of MC method after random simulation will inevitably deviate from the estimated true value, this paper focuses on this problem. Firstly, the convergence and error estimation of MC method are analyzed, and six commonly used methods of variance reduction are summarized: important sampling method, control variable method, dual variable method, conditional expectation method, stratified sampling method, correlation sampling method. Secondly, the paper discusses the common equations, definite integrals, series of the Monte Carlo dual variable method, through the establishment of probability and statistics model, sampling to generate random samples to obtain the estimated value of the deterministic problem. The results show that this technique can effectively reduce the simulation accuracy and shorten the running time of the computer, which fully demonstrates the efficiency of the dual variable method. Finally, the adaptive Monte Carlo algorithm for linear algebraic systems is discussed. The method consists of adaptive importance sampling Monte Carlo and adaptive correlation sampling Monte Carlo. The solving time and efficiency of Monte Carlo method and deterministic method are compared by solving concrete examples. It is concluded that the adaptive Monte Carlo method can obtain faster convergence rate.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
[Abstract]:The Monte Carlo (MC) method, as a statistical method, was used primarily as a supplement to the deterministic method. However, with the rapid development of computers and the popularization and application of computers, this situation has been broken, and the time consuming simulation process has become more rapid. This greatly promotes the development of MC methods. The application of MC methods extends from the original nuclear field to other fields. Now, it has become an important computing tool to solve many problems in physics, engineering, finance and so on. It is also widely used in parameter estimation and reliability design investment risk and bid quotation biomedical statistical physics. For deterministic problems, MC method can also be used to solve them. And solve all kinds of equations (groups), calculate multiple integrals, infinite series and so on. But because the approximate value of MC method after random simulation will inevitably deviate from the estimated true value, this paper focuses on this problem. Firstly, the convergence and error estimation of MC method are analyzed, and six commonly used methods of variance reduction are summarized: important sampling method, control variable method, dual variable method, conditional expectation method, stratified sampling method, correlation sampling method. Secondly, the paper discusses the common equations, definite integrals, series of the Monte Carlo dual variable method, through the establishment of probability and statistics model, sampling to generate random samples to obtain the estimated value of the deterministic problem. The results show that this technique can effectively reduce the simulation accuracy and shorten the running time of the computer, which fully demonstrates the efficiency of the dual variable method. Finally, the adaptive Monte Carlo algorithm for linear algebraic systems is discussed. The method consists of adaptive importance sampling Monte Carlo and adaptive correlation sampling Monte Carlo. The solving time and efficiency of Monte Carlo method and deterministic method are compared by solving concrete examples. It is concluded that the adaptive Monte Carlo method can obtain faster convergence rate.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
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相关期刊论文 前10条
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