EM (Expectation Maximization)算法是统计学中的核心算法,也是本校近代数理统计课程教学过程中的一个重难点。论文采用案例式、启发式、研讨式教学方法,以基于高斯混合模型(GMM)的轴承退化阶段划分问题为例,引导学生发现隐变量模型极...
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EM (Expectation Maximization)算法是统计学中的核心算法,也是本校近代数理统计课程教学过程中的一个重难点。论文采用案例式、启发式、研讨式教学方法,以基于高斯混合模型(GMM)的轴承退化阶段划分问题为例,引导学生发现隐变量模型极大似然估计(MLE)存在的困难,设计问题链启发学生探寻参数估计的数值方法,并总结出EM算法的一般过程。基于matlab编程可视化EM算法下的GMM模型参数更新过程,对比MLE目标函数和EM迭代目标函数,分析EM算法的内涵思想并结合图形进行直观展示,并且挖掘其中蕴含的思政元素,在知识传授的同时实现价值塑造。Expectation maximization (EM) algorithm is a core algorithm in statistics and also a key and difficult point in the teaching process of modern mathematical statistics courses in our school. The paper adopts a case-based and heuristic teaching method, taking the Gaussian Mixture Model (GMM) based bearing degradation stage division problem as an example, guiding students to discover the difficulties of maximum likelihood estimation (MLE) in the latent variable model, designing a problem chain to inspire students to explore numerical methods for parameter estimation, and summarizing the general process of EM algorithm. Based on Matlab programming, the parameter update process of GMM based on EM algorithm is visualized. Comparing the MLE objective function and EM iteration objective function, the intrinsic thought of EM algorithm is analyzed and visually displayed with graphics. The ideological and political elements are also explored, so as to achieve value shaping while knowledge transmission.
图的染色理论在模式识别、生物信息、社交网络和电力网络上有重要的应用。对于图G的一个点染色φ:V(G)→{ 1,2,⋯,k },若满足对任意非孤立点v∈V(G),都存在c∈{ 1,2,⋯,k }使得| φ−1(c)∩N(v) |是一个奇数,则称φ是图G的一个奇k-染色。特别地,若| φ−1(c)∩N(v) |=1,则称φ是图G的一个正常无冲突k-染色。图G的奇(正常无冲突)色数是使图G有一个奇(正常无冲突) k-染色的k的最小值,记作χo(G)(χpcf(G))。本文研究笛卡尔乘积图的奇染色和正常无冲突染色,确定了Pm和Pn的笛卡尔乘积图的奇色数和正常无冲突色数,确定了奇色数的上界,丰富了图的染色理论,为实践应用提供了理论指导。The coloring theory of graphs has important applications in pattern recognition, biological information, social networks and power networks. For a vertex coloring φ:V(G)→{ 1,2,⋯,k }of a graph G, it is called an odd k-coloring of G if for each non-isolated vertex v∈V(G), there exist c∈{ 1,2,⋯,k }such that | φ−1(c)∩N(v) |is odd. Especially, it is called a proper conflict-free k-coloring of G when | φ−1(c)∩N(v) |=1. The odd (proper conflict-free) chromatic number of a graph G, denoted by χo(G)(χpcf(G)), is the minimum k such that G has an odd (proper conflict-free) k-coloring. In this paper, we study the odd coloring and proper conflict-free coloring of cartesian product graph, determine the odd chromatic number and PCF chromatic number of cartesian product graph of Pmand Pn, and determine the upper bound of odd chromatic number, which enriches the coloring theory of graphs and provides theoretical guidance for practical application.
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