基于贝叶斯B样条的结构方程模型
发布时间：2018-08-31 12:31
【摘要】：本文主要是利用B样条估计SEM中的结构方程,并进一步改进结构方程模型(SEM),使改进后的SEM可以刻画潜变量间更复杂的函数关系;此外,将B样条结点个数视为随机变量,利用贝叶斯平均方法确定模型的最优结点个数,保证了结点选取的客观性;最后,给出估计SEM未知参数的具体MCMC算法。本文主要研究内容为:第一部分是对本文研究所需的理论基础的简要介绍,为后续研究铺垫必要的理论基础。本节主要涉及对结构方程模型简介、贝叶斯理论、样条函数基础以及MCMC算法内容。第二部分是对一般性非参数SEM模型的构建。首先分别构建的测量方程适用于处理两种不同数据类型;然后利用不定结点个数的B样条估计SEM的结构方程;在确定了样条次数和结点位置的基础之上,通过概率测度转化方法保证样条结点区间可以覆盖观察变量的取值区间,保证模型的有效性。第三部分给出模型的限制性条件。第二部分中构建了一般意义的SEM,但在其框架下,新SEM的参数并不是完全确定的。此模块对相关参数取值进行一些合理约束,保证新构建的SEM可被识别。第四部分是对SEM的未知参数进行贝叶斯解释。此部分给出相关参数合理的先验分布,主要是对结点个数先验分布的选取,以及在结点个数给定的情况下,各参数的全条件的分布。第五部分给出估计新SEM未知参数的MCMC算法。由于是将结点个数看成随机变量,导致模型的维数在不断变化。为估计未知参数的具体值,本文使用可逆跳的MCMC算法进行抽样,并给出具体的抽样步骤。行文最后指出此模型还有哪些不足之处,以及后续可以继续研究的方向;并在附录中给出一个MCMC抽样原理的演示程序。
[Abstract]:In this paper, the structural equations in SEM are estimated by B-spline, and the structural equation model (SEM), is further improved so that the modified SEM can depict the more complex functional relations among latent variables, in addition, the number of B-spline nodes is regarded as a random variable. The Bayesian average method is used to determine the optimal number of nodes in the model, which ensures the objectivity of node selection. Finally, a specific MCMC algorithm for estimating the unknown parameters of SEM is given. The main contents of this paper are as follows: the first part is a brief introduction of the theoretical basis needed for the study of this paper. This section mainly deals with the introduction of structural equation model, Bayesian theory, the basis of spline function and the content of MCMC algorithm. The second part is the construction of general non-parametric SEM model. First of all, the measurement equations constructed separately are suitable for dealing with two different data types. Then, the structural equations of SEM are estimated by B-spline with indefinite number of nodes, and the number of splines and the location of nodes are determined on the basis of determining the number of splines and the location of nodes. The probabilistic measure transformation method is used to ensure that the spline node interval can cover the value interval of the observation variable and ensure the validity of the model. In the third part, the restrictive conditions of the model are given. In the second part, the general meaning of SEM, is constructed, but under its framework, the parameters of the new SEM are not completely determined. This module has some reasonable constraints on the value of relevant parameters to ensure that the newly constructed SEM can be recognized. The fourth part is the Bayesian interpretation of the unknown parameters of SEM. In this part, the rational prior distribution of the relative parameters is given, which is mainly about the selection of the prior distribution of the number of nodes and the distribution of all the conditions of each parameter when the number of nodes is given. In the fifth part, the MCMC algorithm for estimating the unknown parameters of new SEM is given. Because the number of nodes is regarded as a random variable, the dimension of the model is constantly changing. In order to estimate the specific values of unknown parameters, the reversible jump MCMC algorithm is used to sample and the sampling steps are given. In the end, the author points out the shortcomings of the model and the direction of further study, and gives a demonstration program of MCMC sampling principle in the appendix.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
本文编号：2214994
[Abstract]:In this paper, the structural equations in SEM are estimated by B-spline, and the structural equation model (SEM), is further improved so that the modified SEM can depict the more complex functional relations among latent variables, in addition, the number of B-spline nodes is regarded as a random variable. The Bayesian average method is used to determine the optimal number of nodes in the model, which ensures the objectivity of node selection. Finally, a specific MCMC algorithm for estimating the unknown parameters of SEM is given. The main contents of this paper are as follows: the first part is a brief introduction of the theoretical basis needed for the study of this paper. This section mainly deals with the introduction of structural equation model, Bayesian theory, the basis of spline function and the content of MCMC algorithm. The second part is the construction of general non-parametric SEM model. First of all, the measurement equations constructed separately are suitable for dealing with two different data types. Then, the structural equations of SEM are estimated by B-spline with indefinite number of nodes, and the number of splines and the location of nodes are determined on the basis of determining the number of splines and the location of nodes. The probabilistic measure transformation method is used to ensure that the spline node interval can cover the value interval of the observation variable and ensure the validity of the model. In the third part, the restrictive conditions of the model are given. In the second part, the general meaning of SEM, is constructed, but under its framework, the parameters of the new SEM are not completely determined. This module has some reasonable constraints on the value of relevant parameters to ensure that the newly constructed SEM can be recognized. The fourth part is the Bayesian interpretation of the unknown parameters of SEM. In this part, the rational prior distribution of the relative parameters is given, which is mainly about the selection of the prior distribution of the number of nodes and the distribution of all the conditions of each parameter when the number of nodes is given. In the fifth part, the MCMC algorithm for estimating the unknown parameters of new SEM is given. Because the number of nodes is regarded as a random variable, the dimension of the model is constantly changing. In order to estimate the specific values of unknown parameters, the reversible jump MCMC algorithm is used to sample and the sampling steps are given. In the end, the author points out the shortcomings of the model and the direction of further study, and gives a demonstration program of MCMC sampling principle in the appendix.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
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