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.
工程例证式教学法(EETM)是一种以工程案例进行组织教学的教学方法,也是一种更侧重工程实践的过程式教学,以侧重问题牵引和思维启发为基本原则组织教学。与传统的“以理论支撑理论”的教学模式相比,工程例证式教学是一种“以实例支撑理论”的教学模式,贴近工程实际,更符合工程类数学课程“以数学为工具并且学以致用”的根本目的。The Engineering Example-based Teaching Method (EETM) is an instructional approach that organizes teaching through engineering case studies. It is a process-oriented teaching method that emphasizes engineering practice, with a fundamental principle of problem-driven and thought-provoking instruction. Compared to the traditional “theory-supported-by-theory” teaching model, EETM adopts a “theory-supported-by-examples” approach, which is closer to real-world engineering scenarios and better aligns with the fundamental purpose of engineering mathematics courses: to use mathematics as a tool and apply it in practical contexts.
在概率论与数理统计中,贝叶斯公式是一个非常重要的工具,它用于计算关于复杂事件的条件概率。该研究主要分析了贝叶斯公式在军事案件、生活中以及教学质量评估中的应用,并创新性地融入了丰富的课程思政元素。让抽象的概率概念变得生动可感,使学生更好地理解贝叶斯的先验概率和后验概率,极大地激发了学习者的兴趣与好奇心。Bayesian formula is a very important tool in probability theory and mathematical statistics, which is used to calculate the conditional probability of complex events. This study mainly analyzes the application of Bayesian formula in military cases, life and teaching quality evaluation. Let the abstract concept of probability become vivid and sensible, make students better understand Bayesian prior probability and posterior probability, and greatly stimulate learners’ interest and curiosity.
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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