主成分分析法MATLAB的实现

1、MATLAB结 课 作 业指导老师：张肃班级：信管121姓名：桂亚东学号：201200654118利用Matlab编程实现主成分分析概述Matlab语言是当今国际上科学界（尤其是自动控制领域）最具影响力、也是 最有活力的软件。它起源于矩阵运算，并已经发展成一种高度集成的计算机语言。 它提供了强大的科学运算、灵活的程序设计流程、高质量的图形可视化与界面设 计、与其他程序和语言的便捷接口的功能。Matlab语言在各国高校与研究单位 起着重大的作用。主成分分析是把原来多个变量划为少数几个综合指标的一种统 计分析方法，从数学角度来看，这是一种降维处理技术。1.1主成分分析计算步骤 计算相关系数矩阵r

2、pi在（3.5.3 ）式中,rij(i，1222rp2j=1ri p2prPP,2,，p）为原变量的xi与xj之间的相关系数，其计算公式为rijn(Xkik 1nn(XkiXi)2 (Xkj Xj)2k 1k 1Xi)(Xkj Xj)(2)因为R是实对称矩阵（即rij=rji），所以只需计算上三角元素或下三角元素即可。 计算特征值与特征向量首先解特征方程| I R 0 ,通常用雅可比法（Jacobi ）求出特征值 i（i 1,2, , p），并使其按大小顺序排列，即12, p 0;然后分别求p出对应于特征值i的特征向量e（i 1,2, p）。这里要求le =1,即ei2 1，其j 1中e表示向

3、量ei的第j个分量。 计算主成分贡献率及累计贡献率主成分乙的贡献率为累计贡献率为第m (m< p)个主成分 计算主成分载荷其计算公式为lj P(z,Xj).iej(i,j得到各主成分的载荷以后，还可以按照成分的得分z11Z12Z1 mZz21Z22Z2mZn1Zn2Znm1,2, p)(3)(3.5.2 )式进一步计算，得到各主(4)2.函数作用Cwstd.m用总和标准化法标准化矩阵Cwfac.m计算相关系数矩阵；计算特征值和特征向量；对主成分进行排序；计算各特征值贡献率；挑选主成分(累计贡献率大于85%，输出主成分个数；计算主成分载荷Cwscore.m计算各主成分得分、综合得分并排序C

4、wpri nt.m读入数据文件；调用以上三个函数并输出结果3.源程序3.1 cwstd.m 总和标准化法标准化矩阵%cwstd.m,用总和标准化法标准化矩阵fun ctio n std=cwstd(vector)cwsum=sum(vector,1); %a,b=size(vector); %for i=1:a对列求和矩阵大小,a为行数,b为列数for j=1:bstd(i,j)= vector(i,j)/cwsum(j); endend3.2 cwfac.m计算相关系数矩阵%cwfac.mfun ctio n result=cwfac(vector);fprintf('相关系数矩阵:

5、n')std=CORRCOEF(vector) %计算相关系数矩阵 fprintf('特征向量(vec)及特征值(val) : n')vec,val=eig(std) %求特征值(val)及特征向量(vec)n ewval=diag(val);y,i=sort(newval) ;%对特征根进行排序，y为排序结果，i为索引fprintf('特征根排序：n')for z=1:le ngth(y)n ewy(z)=y(le ngth(y)+1_z);endfprin tf('%gn',n ewy)rate=y/sum(y);fprin tf(&

6、#39;n贡献率：n')n ewrate=n ewy/s um(n ewy)sumrate=0;n ewi=;for k=le ngth(y):-1:1sumrate=sumrate+rate(k);n ewi(le ngth(y)+1_k)=i(k);if sumrate>0.85 break;endend%记下累积贡献率大 85%的特征值的序号放入newi中fprintf('主成分数：%gnn',le ngth( newi);fprintf('主成分载荷：n')for p=1:le ngth( newi)for q=1:le ngth(y)re

7、sult(q,p)=sqrt (n ewval( newi(p)*vec( q,n ewi(p);计算载荷end enddisp(result)3.3 cwscore.m%cwscore.m,计算得分function score=cwscore(vector1,vector2);sco=vector1*vector2;csum=sum(sco,2);n ewcsum,i=sort(-1*csum);n ewi,j=sort(i);fprintf('计算得分：n')score=sco,csum,j9得分矩阵：sco为各主成分得分；csum为综合得分；j为排序结果3.4 cwpri

8、 nt.m%cwpri nt.mfunction prin t=cwpri nt(file name,a,b);%filename为文本文件文件名，a为矩阵行数(样本数)，b为矩阵列数(变量指标数)fid=fope n( file name,'r')vector=fsca nf(fid,'%g',a b);fprintf('标准化结果如下：n')v1=cwstd(vector)result=cwfac(v1);cwscore(v1,result);4.程序测试4.1原始数据中国大陆35个大城市某年的10项社会经济统计指标数据见下表。城名市称年底总

9、人口（万人）非农业人口比（%）农业总产值（万元）工业总产值（万元）客运总量（万人）货运总量（万吨）地方财政预算内收入（万元）城乡居民年底储蓄余额（万元）在岗职工人数（万人）在岗职工工资总额（万元）北京1 249.900.59781 843 42719 999 70620 32345 5622 790 86326 806 646410.805 773 301天津910.170.58091 501 13622 645 5023 25926 3171 128 07311 301 931202.682 254 343石家庄 太 原 呼和浩特 沈 阳 大 连 长 春 哈尔滨 上 海 南 京 杭 州 宁

10、波 合 肥 福 州 厦 门 南 昌 济 南 青 岛 郑 州 武 汉长 沙875.40299.92207.78677.08545.31691.23927.091 313.12537.44616.05538.41429.95583.13128.99424.20557.63702.97615.36740.20582.470.23320.65630.44120.62990.49460.40680.46270.73840.53410.35560.25470.31840.27330.48650.39880.40850.36930.34240.58690.3102 918 680 6885 768 2 92

11、9 1 911352 3487 095 87595.60236 0382 737 750 193711 895203 2773943 10088.65365 343816 4522 351 :! 623105 7831 396 58842.111 295 418 5826 733 7 78215 412567 9199 016 998135.451 879 739 8426 38510 78019 187709 2277 556 79694.151 853 210 5966 343 4 810 9 532357 0964 803 744102.632 663 855 4186 123 6 72

12、0 7 520481 4436 450 020172.792 069 01954 529 0986 40644 4854 318 50025 971 200336.84989 19913 072 73714 26911 193664 2995 680 472113.811 414 73712 000 79617 88311 684449 5937 425 96796.901 428 23510 622 86622 21510 298501 7235 246 35062.15628 7642 514 125 4893 15)17233 6281 622 93147.272 152 288 655

13、5 351 8 851 7 190467 5245 030 22069.59333 3745 751 124 3728 2 J)70418 7582 108 33146.93688 2892 305 881 3674 3 -89167 7142 640 46062.081 486 302 6285 882 5 91511 775460 6904 126 97083.312 382 32011 492 03613 40817 038658 4354 978 045103.52677 4255 287 60110 4336 768387 2525 135 33884.661 211 291 750

14、6 085 9 79315 442604 6585 748 055149.201 146 367 3098 179 8 706 5 718323 6603 461 24469.57758 877654 023309 3371 152 811965 922884 4471 309 1515 605 4451 357 8611 180 947824 034369 577680 607657 484479 ,555756 696961 704696 8481 314 766596 986广州685.0070.62141 600 73823 348 13922 00723 8541 761 49920

15、 401 811182.813 047 594深圳119.850.7931299 66220 368 2958 754 4274 1847 908 9 519 90091.261 890 338南285.870.4064720 4861 149 691 5130 3 :>93149 7002 190 91845.09371 809海口54.380.835444 815717 4615 345 :356115 1741 626 80019.01198 138重庆3 072.340.20674 168 780 8585 52552 44125 124898,9129 090 969223.7

16、31 606 804成都1 003.560.3351 935 590 5894 28940 14019 632561 1897 479 684132.891 200 671贵阳321.500.4557362 0612 247 93415 7034 143197 9081 787 74855.28419 681昆明473.390.3865793 3563 605 729 560412 042524 2164 127 90088.11842 321西安674.500.4094739 9053 665 94210 3119 766408 8965 863 980114.01885 169州287.5

17、90.5445259 4442 940 884 1832 4 '749169 5402 641 56865.83550 890西133.950.522765 848711 3101 746 -46949 134855 05127.21219 251银川95.380.5709171 603661 2262 106 -19374 758814 10323.72178 621乌鲁木齐158.920.824478 5131 847 241 2668 9 ()41254 8702 365 50855.27517 6224.2运行结果>> cwpri nt('cwbook.tx

18、t',35,10) fid =6数据标准化结果如下:v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.

19、00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.

20、01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.

21、02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.

22、02860.01650.01660.02270.02230.01680.03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.

23、01410.00800.00640.00970.01190.00900.00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.

24、01540.02940.02240.01820.02320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.

25、00730.02210.01090.01050.01460.0125相关系数矩阵：std =1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000 -0.4750 0.3096-0.35390.1971 0.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390

26、-0.35390.58910.15071.00000.42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.468

27、90.30900.37420.86140.25950.90270.95270.97310.93961.0000特征向量(vec)：vec =-0.13670.2282 -0.26280.19390.6371 -0.2163 0.3176 -0.1312 -0.41910.2758-0.0329-0.02170.00090.0446-0.1447-0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.2389)0

28、.1926-0.4915-0.41890.27260.20650.3403 0.0404 0.14080.08960.0380-0.1969 -0.0437-0.4888-0.6789 -0.4405 0.1861-0.0343 0.23600.0640-0.82940.03770.2662 0.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4

29、879-0.57070.12170.17610.09870.35500.3280-0.01390.00710.3832-0.7894 -0.1628 0.1925 0.2510 -0.0422 0.2694 0.0396 0.04560.16680.3799特征值(val)val =0.00390 0000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分贡献率：n ewrate =0.00240.62600.23510.05300.03210.02570.01230.00990.00310.0004第一、二主成分的载荷：0.690 1 -0.64270.148 30.84140.600 7-0.68050.851 50.31670.465 6-0.67540