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1、-1-东 北 大 学 秦 皇 岛 分 校课程名称：多元统计分析试卷类型：答案考试形式：开授课专业：数学与应用数学考试日期：2012 年 12 月 12 日 试卷：共 3 页题号一二三总分得分阅卷人一、填空题：（每空 2 分，共 30分）1、设(1)(2)(,)pNXXX(2)p，(1)(2)，11122122，其中(1)X，(1)为1r，11为 rr，则(1)X(1)11(,)rN，(2)X(2)22(,)p rN2、系统聚类分析的方法很多，其中的五种分别为最短距离法、最长距离法、重心法、类平均法、离差平方和法。3、若p维随机向量(,)pXN，(,)pWWn，且X与W相互独立，则1()()n
2、XWX2(,)Tp n，21(,)npTp npn(,1)Fp np。4、iX 与前个主成分的全相关系数的平方和21(,)mkikYX称为12,mY YY 对原始变量iX 的方差贡献率，在因子分析中也称之为共同度。5、Q型因子分析研究样品之间的相关关系，R型因子分析研究变量之间的相关关系。6、Fisher判别法的基本思想是投影，并利用方差分析的思想来导出判别函数。二、判断题（每题2 分，共 10 分）1、（）随机向量12(,)pXXX的协方差阵()D X是对称非负定阵。2、（）因子载荷矩阵A是对称阵。3、（）聚类分析中快速聚类法指的就是模糊聚类法。4、（）设(,)pNX，(,)pWnW，且X与
3、W相互独立，则12()()(,)nTp nXWX-。5、（）主成分分析中，从相关矩阵出发求解的主成分一定会比从协方差矩阵出发求解的主成分更可信。三、计算题（共 60 分）1、设12(,)XXX=的协方差阵为2889，试分别从协方差和相关矩阵出发求解总体主成分1Y，2Y。（10 分）解：特征方程28|089E，得特征根：110，21110的特征方程：1288081xx，得特征向量11383u21的特征方程：1218088xx，得特征向量28313u1121833Yxx，1128133Yxx5 分由2889，得：12 32 31R，装订线装订线内不要答题学 号姓 名班 级-2-特征方程213|02
4、13ER，得特征根：153，213153的特征方程：122 32 302 32 3xx，得特征向量11212u213的特征方程：122 32 302 32 3xx，得特征向量21212u1121122Yxx，2121122Yxx 10分2、设 5个样品两两之间的距离矩阵：0406901710063580，试用最长距离法作系统聚类，并画出谱系聚类图。（15 分）解：样品与样品之间的明氏距离为：1234512(0)3450406901710063580 xxxxxxxDxxx样品最短距离是 1，故把1X与4X 合并为一类3 分计算类与类之间距离（最长距离法），得：1423514(1)235,070
5、10908350 x xxxxx xDxxx类与类的最短距离是3，故把2X与5X 合并为一类，6 分计算类与类之间距离（最长距离法），得：1425314(2)253,0,801090 x xxxxx xDxxx类与类的最短距离是8，故把14,XX与25,XX合并为一类9 分计算类与类间距离（最长距离法），得：12453(3)12453,0100 x x xxxDx xxxx12 分谱系聚类图（略）15 分3、设三维随机向量1233(,)(,)XXXXN，其中120250002。试问下列5对随机变量或随机向量中哪几对是相互独立的，为什么？（1）1X 与22X；（2）2X 与3X；（3）12(,)
6、XX与3X；（4）1X 与23(,)XX。（10 分）解：由120250002知，1X 与22X不独立；2X 与3X独立4 分12(,)XX与3X独立；1X 与23(,)XX不独立10 分4、在 1984 年洛杉矶奥运会上，有55 个国家和地区参与了男子径赛，其中有这样8项比赛指标：x1：100米（秒），x2：200 米（秒），x3：400 米（秒），x4：800 米（秒），x5：1500米（分），x6：5000 米（分），x7：10000米（分）和 x8：马拉松（分）。这 8 个变量的样本相关系数矩阵的个特征根和前三个标准正交特征向量分别为：123456786.6223,0.878,0.15
7、9,0.113,0.105,0.068,0.045,0.008装订线装订线内不要答题学 号姓 名班 级文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J
8、9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编
9、码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10
10、O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3
11、Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3
12、D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T
13、1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9-3-123(0.318,0.337,0.356,0.369,0.
14、373,0.364,0.367,0.342)(0.567,0.462,0.248,0.012,0.140,0.312,0.307,0.439)(0.332,0.361,0.560,0.532,0.153,0.190,0.182,0.263)UUU（1）写出前三个主成分的表达式；（5 分）（2）计算每个主成分的方差贡献率，并按照一般性原则选取主成分的个数。（5 分）（1）*1123456780.3180.3370.3560.3690.3730.3640.3670.342Yxxxxxxxx*2123456780.5670.4620.2480.0120.1400.3120.3070.439Yxxxx
15、xxxx*3123456780.3320.3610.5600.5320.1530.1900.1820.263Yxxxxxxxx5 分（2）方差贡献率分别为182.8%v，211.0%v，32.0%v，1293.7%vv，12395.7%vvv，故主成分应保留2 个。10 分5、已知某实际问题选用了5 个指标作为原始变量，利用了因子分析模型，经Spss软件计算，部分输出如下：试回答以下问题：（1）解释表 1 中的列 Extraction 中各得数；（5 分）（2）结合表 2 和表 3，写出因子分析模型，并简要阐述公因子是如何综合原始指标的信息的。（10分）解：（1）表 1 中的列 Extract
16、ion 中各得数分别是保留的三个公因子从五个原始指标中的各指标所提取信息的百分比；（2）因子分析模型：112310.8330.3060.036XFFF212320.9240.0210.066XFFF312330.9260.1330.05XFFF412340.0260.0030.999XFFF512350.0420.9830.004XFFF从旋转因子载荷矩阵可以看出，公因子1F 主要反映了原始指标1X，2X，3X公因子2F 主要反映了原始指标5X公因子3F 主要反映了原始指标4X表2 Rotated Component Matrixa表1CommunalitiesComponent Initia
17、l Extraction 1 2 3 X1 1 0.856 X1 0.833-0.306 0.036 X2 1 0.726 X2 0.924-0.021 0.066 X3 1 0.909 X3 0.926 0.133-0.05 X4 1 0.978 X4 0.026 0.003 0.999 X5 1 0.981 X5-0.042 0.983 0.004 表3Rotation Sums of Squared Loadings Total%of Variance Cumulative%2.448 49 49 1.078 22 71 1.007 20 91 装订线装订线内不要答题学 号姓 名班 级文
18、档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF
19、10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3
20、Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 H
21、J3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T
22、9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7
23、ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9文档编码:CF10O7F3Z3Z1 HJ3D10T9T1P7 ZL4K7Y6J9M9