基于约束选择下概率密度函数的最大熵法估计

发布时间：2018-04-05 17:28

本文选题：概率分布　切入点：约束选择　出处：《西南交通大学》2017年硕士论文

【摘要】：概率密度函数包含了随机变量几乎所有的信息,根据已经得到的样本数据去估计随机变量的概率密度函数,即概率密度函数估计,它是概率与数理统计中的一个基本问题。与此同时,在许多与实际问题相关的应用研究当中,也都以此为基础,从而开展对本领域知识及问题的研究和探讨。由此可见,概率密度函数估计在理论研究以及实际的工程应用中都扮演着十分重要的角色。按照传统对概率密度函数估计方法的分类标准,可将这一问题的研究分为以下三类:参数化方法、非参方法以及半参方法。由于在大多数与现实问题相关的应用研究当中,对于概率密度函数的具体模型所对应的信息往往无从得知,因此,类似这类问题的求解通常不大会使用参数化方法和半参方法。正是基于这样的考虑,使得非参方法成为人们在研究概率密度函数估计问题时应用最为普遍的一种方法。而在非参方法中,由于核方法最终能给到概率密度函数的具体的显示解,因而被人们广泛地研究和使用。尽管如此,对概率密度函数使用核方法进行估计时,依然存在核函数及窗口宽度较难确定的缺点。基于此,本文在较深入理解最大熵原理的情况下,针对其如何对常见分布开展参数估计进行了较详细论述,并总结出了基于最大熵原理对常见分布开展参数估计的一般步骤。最大熵方法的思想大致如下:在给定某些约束条件的情形下,从符合这些约束的分布当中,选择熵值最大的分布作为理想的分布才是合情合理的。而针对实际问题而言,要想使得所推导出来的分布与所要研究的系统的已知信息相一致,找出确定分布的约束便成了使用最大熵原理估计概率密度函数的关键。也正是基于这样的考虑,本文提出了一种有效的选择约束条件的方法,在此基础上利用最大熵原理对概率密度函数进行估计。通过仿真数据表明,该方法能较合理地选择出数据服从真实分布下基于最大熵原理所需要满足的约束,并得出结论:基于本文提出的选择约束的方法利用最大熵原理去估计概率密度函数确实是一种无论从理论还是从实现上来讲都是较为容易、且行之有效的方法。
[Abstract]:The probability density function contains almost all the information of random variables. It is a basic problem in probability and mathematical statistics to estimate the probability density function of random variables according to the obtained sample data.At the same time, in many practical problems related to the application of research, also on the basis of this, so as to carry out the field of knowledge and problems of research and discussion.Thus, probability density function estimation plays an important role in both theoretical research and practical engineering applications.According to the traditional classification criteria of probability density function estimation, the research on this problem can be divided into the following three categories: parameterization method, non-parametric method and semi-parametric method.Because the information of the specific model of probability density function is often unknown in most practical application studies, parameterized methods and semi-parametric methods are not usually used to solve such problems.Because of this consideration, nonparametric method has become the most popular method in the study of probability density function estimation.In the nonparametric method, the kernel method is widely studied and used because it can finally give the concrete explicit solution of the probability density function.However, when the probability density function is estimated by kernel method, the kernel function and the width of the window are still difficult to determine.Based on this, this paper discusses in detail how to carry out parameter estimation of common distribution, and summarizes the general steps of parameter estimation for common distribution based on maximum entropy principle.The idea of the maximum entropy method is as follows: it is reasonable to choose the distribution with the largest entropy value as the ideal distribution from the distribution according to these constraints given some constraint conditions.In order to make the distribution consistent with the known information of the system, finding out the constraints of the distribution is the key to estimate the probability density function using the maximum entropy principle.Based on this consideration, this paper proposes an effective method to select constraint conditions, and then estimates the probability density function by using the maximum entropy principle.The simulation data show that the method can reasonably select the constraints that must be satisfied by the principle of maximum entropy in the real distribution.It is concluded that the method of choosing constraints proposed in this paper to estimate the probability density function by using the maximum entropy principle is an easy and effective method both theoretically and practically.
【学位授予单位】：西南交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

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