﻿ 主成分分析在教学质量分析中的应用 The Application of Principal Component Analysis in the Analysis of Teaching Quality

Statistics and Application
Vol.07 No.02(2018), Article ID:24738,8 pages
10.12677/SA.2018.72031

The Application of Principal Component Analysis in the Analysis of Teaching Quality

Rong Hu1,2, Zeyu Li3, Weiyan Mu1,2

1School of Science, Beijing University of Civil Engineering and Architecture, Beijing

2Beijing Key Laboratory of Functional Materials for Building Structure and Environment Remediation, Beijing University of Civil Engineering and Architecture, Beijing

3Canvard College, Beijing Technology and Business University, Beijing

Received: Apr. 5th, 2018; accepted: Apr. 23rd, 2018; published: Apr. 30th, 2018

ABSTRACT

The quality of the teaching of a school is often closely related to the students’ academic performance. In order to study the teaching quality of several primary schools of Shaanxi, we made a principal component analysis of the average scores of 12 subjects in order to know the achievement of each school student [1] . The research results show that the quality of a school’s teaching can be analyzed by student’s academic record. In fact, only a few linear combinations consisting of the average scores of each subject can be considered, which can simplify the problem and improve the efficiency of analysis.

Keywords:Principal Component Analysis, Score of the Principal Component, Quality of Teaching

1北京建筑大学理学院，北京

2北京建筑大学，建筑结构与环境修复功能材料北京市重点实验室，北京

3北京工商大学嘉华学院，北京

1. 引言

2. 数据收集

3. 主成分分析

3.1. 主成分分析的基本思想

3.2. 主成分分析的数学模型

$X=\left[\begin{array}{cccc}{x}_{11}& {x}_{12}& \cdots & {x}_{1p}\\ {x}_{21}& {x}_{22}& \cdots & {x}_{2p}\\ ⋮& ⋮& \ddots & ⋮\\ {x}_{n1}& {x}_{n2}& \cdots & {x}_{np}\end{array}\right]=\left[{X}_{1},{X}_{2},\cdots ,{X}_{p}\right],\text{}{x}_{j}=\left[\begin{array}{c}{x}_{1j}\\ {x}_{2j}\\ ⋮\\ {x}_{nj}\end{array}\right],j=1,2,\cdots ,p$

$\left\{\begin{array}{l}{F}_{1}={\gamma }_{11}{X}_{1}+{\gamma }_{12}{X}_{2}+\cdots +{\gamma }_{1p}{X}_{p}\\ {F}_{2}={\gamma }_{21}{X}_{1}+{\gamma }_{22}{X}_{2}+\cdots +{\gamma }_{2p}{X}_{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {F}_{p}={\gamma }_{p1}{X}_{1}+{\gamma }_{p2}{X}_{2}+\cdots +{\gamma }_{pp}{X}_{p}\end{array}$

${F}_{j}={\gamma }_{j1}{X}_{1}+{\gamma }_{j2}{X}_{2}+\cdots +{\gamma }_{jp}{X}_{p}$

$Var\left({F}_{j}\right)={\lambda }_{j}$

$Cov\left({F}_{i},{F}_{j}\right)=0$

$F=\gamma X$

$F=\left[\begin{array}{c}{F}_{1}\\ {F}_{2}\\ ⋮\\ {F}_{p}\end{array}\right],\text{\hspace{0.17em}}\gamma =\left({\gamma }_{1},{\gamma }_{2},\cdots ,{\gamma }_{p}\right)$

γ为主成分系数矩阵。

$\frac{\sum _{j=1}^{k}{\lambda }_{j}}{\sum _{j=1}^{p}{\lambda }_{j}}\ge 85%$

3.3. 主成分分析模型的应用

$\begin{array}{l}{F}_{1}=0.411417{x}_{1}+0.412534{x}_{2}+0.160213{x}_{3}+0.434863{x}_{4}+0.309819{x}_{5}-0.16245{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-0.30982{x}_{7}-0.18868{x}_{8}+0.334381{x}_{9}-0.24227{x}_{10}-0.12337{x}_{11}+0.080385{x}_{12}\end{array}$

$\begin{array}{l}{F}_{2}=0.235393{x}_{1}+0.313423{x}_{2}+0.05007{x}_{3}+0.25425{x}_{4}+0.103391{x}_{5}-0.45648{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.446076{x}_{7}+0.269206{x}_{8}-0.14761{x}_{9}+0.122248{x}_{10}+0.439573{x}_{11}+0.242546{x}_{12}\end{array}$

$\begin{array}{l}{F}_{3}=-0.07099{x}_{1}+0.140462{x}_{2}+0.415279{x}_{3}+0.000763{x}_{4}-0.31833{x}_{5}-0.247335{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-0.08779{x}_{7}+0.558795{x}_{8}+0.176341{x}_{9}-0.30459{x}_{10}-0.17787{x}_{11}-0.40536{x}_{12}\end{array}$

$\begin{array}{l}{F}_{4}=0.111978{x}_{1}+0.177146{x}_{2}+0.610374{x}_{3}-0.12483{x}_{4}-0.12391{x}_{5}-0.16246{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.238643{x}_{7}-0.15695{x}_{8}-0.31758{x}_{9}+0.301975{x}_{10}-0.44791{x}_{11}+0.228546{x}_{12}\end{array}$

$\begin{array}{l}{F}_{5}=-0.21108{x}_{1}-0.04242{x}_{2}+0.220397{x}_{3}+0.191425{x}_{4}+0.548406{x}_{5}-0.18004{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.169696{x}_{7}+0.023799{x}_{8}-0.03208{x}_{9}+0.339391{x}_{10}+0.112785{x}_{11}-0.61359{x}_{12}\end{array}$

$\begin{array}{l}{F}_{6}=0.139163{x}_{1}+0.143512{x}_{2}-0.10655{x}_{3}-0.09459{x}_{4}-0.28159{x}_{5}+0.278327{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-0.06849{x}_{7}-0.19026{x}_{8}+0.541432{x}_{9}+0.646892{x}_{10}-0.05219{x}_{11}-0.16743{x}_{12}\end{array}$

Table 1. Interpretation of the total variance

Figure 1. Scree plot

Table 2. Matrix of components

Table 3. The coefficient matrix of the principal component

4. 结果分析

Table 4. The score of the principal component

Figure 2. A scatter plot of the principal component score

5. 结论与建议

The Application of Principal Component Analysis in the Analysis of Teaching Quality[J]. 统计学与应用, 2018, 07(02): 263-270. https://doi.org/10.12677/SA.2018.72031

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