﻿ 主成分分析在教学质量分析中的应用

# 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

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 . 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.

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。

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

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

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