数理统计 作业四

1.用主成分分析方法探讨城市工业主体结构。表1是某市工业部门13个行业8项指标的数据。表1某市工业部门13个行业8项指标的数据年末固定资产净值（万元）职工人数（人）工业总产值（万元）全员劳动产率（元/人年）百元固定原资产值实现产值（元）资金利税率（％）标准燃料消费量（吨）能源利用效果（万元/吨）1(冶金)90342524551010911927282.00016.1001974350.1722(电力)4903197320351031334.2007.1005920770.0033(煤炭)6735211393767178036.1008.2007263960.0034(化学)4945436241815572250498.10025.9003482260.9855(机械)1391902035052158981060993.20012.6001395720.6286(建材)122151621910351638262.5008.7001458180.0667(森工)23726572810312329184.40022.200209210.1528(食品)11062230785493523804370.40041.000654860.2639(纺织)17111239075210821796221.50021.500638060.27610(缝纫)12063930612615586330.40029.50018400.43711(皮革)21505704620010870184.20012.00089130.27412(造纸)525161551038316875146.40027.500787960.15113(文教艺术用品)1434113203193961469194.60017.80063541.5741）试用主成分分析方法确定8项指标的样本主成分（综合变量）；若要求损失信息不超过15％，应取几个主成分；并对这几个主成分进行解释；2）利用主成分得分对13个行业进行排序和分类。解：先将给出的的数据导入到Spass软件对导入的数据进行因子分析得到KMO與Bartlett檢定Kaiser-Meyer-Olkin測量取樣適當性。.463Bartlett的球形檢定大約卡方96.957df28顯著性.000首先进行KMO检验和巴特利球体检验，KMO检验系数=0.969570.5P值0.05，所以能进行因子分析。說明的變異數總計元件起始特徵值擷取平方和載入循環平方和載入總計變異的%累加%總計變異的%累加%總計變異的%累加%13.10538.81138.8113.10538.81138.8113.03237.90337.90322.89736.21875.0292.89736.21875.0292.97037.12775.02930.93011.62886.6570.93011.62886.6570.93011.62886.65740.6428.02794.68450.3043.80198.48560.0871.08299.56770.0320.40299.96980.0020.031100.000擷取方法：主體元件分析。循环平方和载入，累加达到85%，所以取三个指标，即累加达到86.657%旋轉元件矩陣a元件123年末固定资产净值.975-.084.108职工人数.965-.093.044工业总产值.989.090.093全员劳动产率.121.822.204百元固定原资产值实现产值-.169.906-.181资金利税率-.088.931.021标准燃料消费量-.020-.700-.289能源利用效果.141.139.961擷取方法：主體元件分析。轉軸方法：具有Kaiser正規化的最大變異法。a.在4疊代中收斂循環。由旋转矩阵分析可知，八个指标分为三类第一类：年末固定资产净值，职工人数，工业总产值第二类：全员劳动产率，百元固定原资产值实现产值，资金利税率，标准燃料消费量第三类：能源利用效果（2）对原始数据进行归一化处理,计算相应的得分，结果如下：最后的结果如上表最后一列所示，根据数值的正负号分成两类，利用主成分得分对13个行业进行排序和分类如下：第一类：1(冶金)4(化学)5(机械)8(食品)13(文教)第二类：9(纺织)6(建材)7(森工)10(缝纫)11(皮革)12(造纸)2(电力)3(煤炭)2.下表是某年美国50州每10万人中各种类型犯罪的犯罪率数据，分析找出主要的犯罪类型、列出主成分与原始变量的线性关系式，分析解释主成分及其特征，排序说明每州主要的犯罪类型。州名杀人罪强奸罪抢劫罪斗殴罪偷盗罪汽车犯罪ALABAMA14.225.296.8278.33017.4280.7ALASKA10.851.696.82844701.5753.3ARIZONA9.534.2138.2312.36813.5439.5ARKANSAS8.827.683.2203.42834.7183.4CALIFORNIA11.549.42873585639.2663.5COLORADO6.342170.7292.95838.4477.1CONNECTICUT4.216.8129.5131.83966.7593.2DELAWARE624.9157194.25361467FLORIDA10.239.6187.9449.15700.4351.4GEORGIA11.731.1140.5256.53521.3297.9HAWAII7.225.512864.15831.9489.4IDAHO5.519.439.6172.53650.4237.6ILLINOIS9.921.8211.32093913.5528.6INDIANA7.426.5123.2153.53584.9377.4IOWA2.310.641.289.83497.6219.9KANSAS6.622100.7180.54009.7244.3KENTUCKY10.119.181.1123.32534.3245.4LOUISIANA15.530.9142.9335.53635.4337.7MAINE2.413.538.71703603.8246.9MARYLAND834.8292.1358.94577.7428.5MASSACHUSETTS3.120.8169.1231.63843.51140.1MICHIGAN9.338.9261.9274.64681.7545.5MINNESOTA2.719.585.985.83694343.1MISSISSIPPI14.319.665.7189.12155.5144.4MISSOURI9.628.3189233.53742.5378.4MONTANA5.416.739.2156.83578.1309.2NEBRASKA3.918.164.7112.73076.1249.1NEVADA15.849.1323.13556665.7559.2NEWHAMPSHIRE3.210.723.2763385.6293.4NEWJERSEY5.621180.4185.14210.3511.5NEWMEXICO8.839.1109.6343.44427.3259.5NEWYORK10.729.4472.6319.14510745.8NORTHAROLINA10.61761.3318.33191.9192.1NORTHDAKOTA0.9913.343.82289.1144.7OHIO7.827.3190.5181.13912.8400.4OKLAHOMA8.629.273.82053516.3326.8OREGON4.939.9124.1286.95142.5388.9PENNSYLVANIA5.619130.31282501.6333.2RHODEISLAND3.610.586.52014333.6791.4SOUTHCAROLINA11.933105.9485.33956245.1SOUTHDAKOTA213.517.9155.72274.9147.5TENNESSEE10.129.7145.8203.93036.2314TEXAS13.333.8152.4208.24591.8397.6UTAH3.520.368.8147.34176.2334.5VERMONT1.415.930.8101.23549.2265.2VIRGINIA923.392.1165.73507.4226.7WASHINGTON4.339.6106.2224.84992.5360.3WESTVIRGINIA613.242.290.91939.1163.3WISCONSIN2.812.952.263.73461.1220.7WYOMING5.421.939.7173.93583.8282解：将上表中各项数据导入到SPSS，进行因子分析。KMO检验和巴特利球体检验，KMO检验系数=0.7630.5P值0.05，所以能进行因子分析。元件矩陣a元件123杀人罪.666-.619.279强奸罪.885-.156-.261抢劫罪.822.185.369斗殴罪.833-.321-.079偷盗罪.764.344-.471汽车犯罪.580.694.290擷取方法：主體元件分析。a.擷取3個元件。分析元件矩阵可以知道主要的犯罪类型为：强奸罪，抢劫罪，斗殴罪。主成分与原始变量的线性关系式：F1=0.666X1+0.885X2+0.822X3+0.833X4+0.764X5+0.580X6F2=-0.619X1-0.156X2+0.185X3-0.321X4+0.344X5+0.694X6F3=0.279X1-0.261X2+0.369X3-0.079X4-0.471X5+0.290X6循环平方和累加三项后达87.62%达到要求，分析主成分分为三类。旋轉元件矩陣a元件123杀人罪.948.058.060强奸罪.593.699.190抢劫罪.505.235.732斗殴罪.732.494.150偷盗罪.097.888.356汽车犯罪-.033.251.915擷取方法：主體元件分析。轉軸方法：具有Kaiser正規化的最大變異法。a分为三类，第一类：杀人罪，偷盗罪第二类：强奸罪，斗殴罪第三类：抢劫罪，汽车犯罪（3）经过归一化处理，并与相应的系数相乘，得到如下结果：3.采用因子分析法对美国50州的六种犯罪率数据进行分析，找出公共因子（假定2个），列出六种犯罪与公共因子的关系式，推导公共因子与六种犯罪的关系式，计算因子得分与综合得分，并对各州犯罪率按综合得分排序解释。解：对于第二题采用因子分析法得出以下结论：X1=0.805C1+0.036C2X2=0744C1+0.478C2X3=0.472C1+0.633C2X4=0.789C1+0.324C2X5=0.345C1+0.633C2X6=0.026C1+0.803C2C1=0.805X1+0.744X2+0.472X3+0.789X4+0.345X5+0.026X6C2=0.036X1+0.478X2+0.633X3+0.324X4+0.663X5+0.803X64.某汽车组织欲根据一系列指标来预测汽车的销售情况，为了避免有些指标间的相关关系影响预测结果，需首先进行因子分析来简化指标系统。下表是抽查欧洲某汽车市场7个品牌不同型号的汽车的各种指标数据，试用因子分析法找出其简化的指标系统。品牌价格发动机功率轴距宽长轴距燃料容量燃料效率A215001.8140101.267.3172.42.63913.228A284003.2225108.170.3192.93.51717.225A420003.5210114.671.4196.63.85018.022B239901.8150102.668.2178.02.99816.427B339502.8200108.776.1192.03.56118.522B620004.2310113.074.0198.23.90223.721C269902.5170107