数学建模中主成分分析法的应用
为前k个主成分的累计信息贡献率。
（5）求因子载荷ai=姨iai，计算因子载荷矩阵，再计算各因子得分Fi=aix，i=1，2，…，k.（6）按因子得分Fi及贡献率的大小，计算综
再根据综合得分进合得分F=β1F1+β2F2+…+βkFk，行排序。
）。的特征值，确定主成分个数（表2
表2方差分解主成分提取分析表
Component
Total
12345678910111213
6.0141.6241.3601.0340.8700.5980.4640.3250.3000.2100.1085.114E-024.250E-02
InitialEigenvalues46.26212.49010.4627.9566.6954.5973.5712.4992.3061.6130.8290.3930.327
46.26258.75269.21477.17083.86588.46292.03294.53296.83898.45199.28099.673100.000
ExtractionSumsofSquaredLoadings6.0141.6241.3601.034
46.26212.49010.4627.956
46.26258.75269.21477.170
%ofVarianceCumulative%Total%ofVarianceCumulative%
二、实例分析
以伊犁师范学院（生命与资源环境系06汉旅
游管理班）在2006-2007学年的考试课成绩为例，运用主成分分析法对学生综合能力进行评价。班上共有23名同学，将这23名同学作为总体，把2006-2007学年的13科考试课：大学英语1、道德法律、高等数学、旅游心理学、导游基础、旅游学概论、体育、导游业务、旅游美学与交际礼仪、大学英语2、自然地理、西域民俗风情、大学语文作为变量，分别用x1，x2，x3，x4，x5，x6，x7，x8，x9，x10，x11，x12，x13来表示，用xij表示第i个同学在第j门课上的得分，则x=（xij）23×l3，这样就得到了一个23×13的原始数据矩阵。见表1。
表1原始数据矩阵
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10X11X12X13
ExtractionMethod:PrincipalComponentAnalysis.
从表2可以看出，前4个公共因子的累积贡献率达到77.17%，即前4个公共因子可以反映原指标77.17%的信息量。因此，上述13项指标可以综合成公共因子F1，F2，F3和F4并可得到初始因子截荷矩阵（表3）。
表3初始因子截荷矩证
Component
1
X1X2X3X4X5X6X7X8X9X10X11X12X13
0.7050.3150.8000.9080.9250.666-0.4570.8600.4920.7850.7600.518-2.986E-02
2-0.2990.747-0.230-0.140-0.180-0.281-0.443-1.571E-020.280-0.1380.1510.6120.319
30.225-0.107-0.1410.108-7.937E-02-0.1550.4950.111-0.3440.3117.900E-030.2710.831
40.3930.1900.135-6.683E-03-0.158-0.3708.919E-02-0.1030.5960.270-0.343-0.3086.774E-02
183.382.064.583.184.085.090.070.378.366.077.678.084.0274.489.052.961.970.077.081.967.979.356.762.974.084.6379.094.070.682.079.086.585.975.278.666.574.678.086.1475.591.061.683.577.073.086.073.572.771.373.689.083.9581.290.069.794.392.087.576.086.188.270.578.885.085.0653.888.043.961.871.085.084.866.069.939.872.077.083.0781.584.060.481.778.079.079.270.072.873.066.778.080.0885.290.061.591.587.089.083.288.284.582.575.578.083.0983.295.080.485.088.090.081.876.581.268.275.574.080.11089.397.087.589.788.088.582.083.188.089.582.085.085.31177.386.074.594.485.090.081.285.387.573.175.583.081.01284.190.063.779.384.083.684.176.282.171.274.077.083.21354.887.042.164.162.069.090.663.174.258.761.578.085.01480.785.062.785.885.090.085.081.670.671.465.682.086.11582.095.068.292.289.093.580.484.969.877.382.689.085.01677.378.068.487.085.091.590.078.462.469.876.278.083.11770.396.053.976.781.091.087.868.577.465.174.678.082.01856.187.065.060.077.085.084.169.773.458.171.475.083.11984.193.070.486.582.088.087.472.187.964.368.878.083.12071.385.071.589.689.085.478.073.280.966.877.382.081.02166.584.079.284.080.091.886.078.883.369.571.278.082.02285.078.066.883.081.085.093.270.674.777.971.270.084.12367.995.063.989.083.088.085.472.887.377.281.786.084.0
从表3可以看出：x1，x3，x4，x5，x6，x8，x10，x11，在第一主因子Fl上有较大载荷，Fl反映的是学生在大学英语1、高等数学、旅游心理学、导游基础、旅游学概论、导游业务、大学英语2、自然地理方面的信息；x2，x12在第二主因子F2上有较大载荷，F2反映的是学生在道德法律与西域民俗风情方面的
x13，x7在第三主因子F3上有较大载荷，F3反信息；
映的是学生在大学语文与体育方面的信息；x9在
将原始数据标准化，得标准化矩阵表；用计算
机求出标准化矩阵的相关系数矩阵；求相关矩阵