Matlab编程实现主成分分析

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1、10.利用 Matlab 编程实现主成分分析1. 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r11r12r1 pr21r22r2 pRr p1r p 2rpp（1）在（3.5

2、.3）式中， r ij （ i，j=1 ，2， ，p）为原变量的 xi 与 xj 之间的相关系数，其计算公式为n(xkixi )( xkjx j )rijk 1nn(xkixi ) 2(xkjx j ) 2k 1k1（ 2）因为 R 是实对称矩阵（即r ij =r ji ），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 (i1,2, , p) ，并使其按大小顺序排列，即 1 2, p 0 ；然后分别求peij2出对应于特征值 i 的特征向量 ei (i1,2, , p) 。这里要求 ei =1，

3、即1，其j1中 eij 表示向量 ei 的第 j 个分量。 计算主成分贡献率及累计贡献率主成分 zi 的贡献率为i(i 1,2, , p)pkk 1累计贡献率为ikk1(i 1,2, , p)pkk1一般取累计贡献率达85 95% 的特征值1 , 2 ,m 所对应的第一、第二， ，第 m（m p）个主成分。 计算主成分载荷其计算公式为lij p( zi , x j )i eij (i , j 1,2, , p)（3）得到各主成分的载荷以后，还可以按照（3.5.2）式进一步计算，得到各主成分的得分z11z12z1mz21z22z2 mZzn1 zn 2znm（4）2. 程序结构及函数作用在软件

4、Matlab 中实现主成分分析可以采取两种方式实现：一是通过编程来实现；二是直接调用Matlab 种自带程序实现。下面主要主要介绍利用Matlab 的矩阵计算功能编程实现主成分分析。2.1 程序结构Cwprint.m主函数子函数Cwstd.mCwfac.mCwscore.m2.2 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Cwprint.m 读入数据文件；调用以上三个函数并输出结果3.

5、 源程序3.1 cwstd.m%cwstd.m, 用总和标准化法标准化矩阵function std=cwstd(vector)cwsum=sum(vector,1);%对列求和a,b=size(vector);%矩阵大小 ,a 为行数 ,b 为列数for i=1:afor j=1:bstd(i,j)= vector(i,j)/cwsum(j);endend3.2 cwfac.m%cwfac.mfunction result=cwfac(vector);fprintf( 相关系数矩阵:n)std=CORRCOEF(vector)% 计算相关系数矩阵fprintf( 特征向量 (vec) 及特征值

6、 (val) ： n)vec,val=eig(std)% 求特征值 (val) 及特征向量newval=diag(val) ;y,i=sort(newval) ;%对特征根进行排序，(vec)y 为排序结果，i 为索引fprintf( 特征根排序：n)for z=1:length(y)newy(z)=y(length(y)+1-z);endfprintf(%gn,newy)rate=y/sum(y);fprintf(n 贡献率： n)newrate=newy/sum(newy)sumrate=0;newi=;for k=length(y):-1:1sumrate=sumrate+rate(k)

7、;newi(length(y)+1-k)=i(k);if sumrate0.85 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,length(newi);fprintf( 主成分载荷：n)for p=1:length(newi)for q=1:length(y)result(q,p)=sqrt(newval(newi(p)*vec(q,newi(p);endend%计算载荷disp(result)3.3 cwscore.m%cwscore.m, 计算得分function score=cwscore(vector1,vec

8、tor2);sco=vector1*vector2;csum=sum(sco,2);newcsum,i=sort(-1*csum);newi,j=sort(i);fprintf( 计算得分： n)score=sco,csum,j%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 cwprint.m%cwprint.mfunction print=cwprint(filename,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=fopen(filename,r)vector=fscanf(fid

9、,%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。年底非农农业工业客运货运地方财政城乡居民在岗职在岗职工城市业年底储蓄总人口总产值总产值总量总量工人数工资总额名称人口比预算内收余额入(万元 )(万人 )(万人 )(万元 )(万元 )(万人 ) (万吨 )(万元 )(%)(万元 )北京1 249.900.597 81 843 42719 999 706 20 323 45 5622 790 8

10、6326 806 646410.805 773 301天津910.170.580 91 501 13622 645 5023 25926 3171 128 07311 301 931202.682 254 343石家 庄875.400.233 22 918 6806 885 7682 9291 911352 3487 095 87595.60758 877太原299.920.656 3236 0382 737 7501 93711 895203 2773 943 10088.65654 023呼和浩特207.780.441 2365 343816 4522 3512 623105 7831 3

11、96 58842.11309 337沈阳677.080.629 91 295 4185 826 7337 78215 412567 9199 016 998135.451 152 811大连545.310.494 61 879 7398 426 38510 780 19 187709 2277 556 79694.15965 922长春691.230.406 81 853 2105 966 3434 8109 532357 0964 803 744102.63884 447哈尔 滨927.090.462 72 663 8554 186 1236 7207 520481 4436 450 020

12、172.791 309 151上海1 313.120.738 42 069 01954 529 0986 40644 4854 318 50025 971 200336.845 605 445南京537.440.534 1989 19913 072 737 14 269 11 193664 2995 680 472113.811 357 861杭州616.050.355 61 414 73712 000 796 17 883 11 684449 5937 425 96796.901 180 947宁波538.410.254 71 428 23510 622 866 22 215 10 2985

13、01 7235 246 35062.15824 034合肥429.950.318 4628 7642 514 1254 8931 517233 6281 622 93147.27369 577福州583.130.273 32 152 2886 555 3518 8517 190467 5245 030 22069.59680 607厦门128.990.486 5333 3745 751 1243 7282 570418 7582 108 33146.93657 484南昌424.200.398 8688 2892 305 8813 6743 189167 7142 640 46062.0847

14、9 ,555济南557.630.408 51 486 3026 285 8825 91511 775460 6904 126 97083.31756 696青岛702.970.369 32 382 32011 492 036 13 408 17 038658 4354 978 045103.52961 704郑州615.360.342 4677 4255 287 60110 433 6 768387 2525 135 33884.66696 848武汉740.200.586 91 211 2917 506 0859 79315 442604 6585 748 055149.201 314 76

15、6长沙582.470.310 71 146 3673 098 1798 7065 718323 6603 461 24469.57596 986广州685.000.621 41 600 73823 348 139 22 007 23 8541 761 49920 401 811182.813 047 594深圳119.850.793 1299 66220 368 2958 7544 2741 847 9089 519 90091.261 890 338南宁285.870.406 4720 4861 149 6915 1303 293149 7002 190 91845.09371 809海口5

16、4.380.835 444 815717 4615 3452 356115 1741 626 80019.01198 138重庆3 072.340.206 74 168 7808 585 52552 441 25 124898,9129 090 969223.731 606 804成都1 003.560.3351 935 5905 894 28940 140 19 632561 1897 479 684132.891 200 671贵阳321.500.455 7362 0612 247 93415 703 4 143197 9081 787 74855.28419 681昆明473.390.3

17、86 5793 3563 605 7295 60412 042524 2164 127 90088.11842 321西安674.500.409 4739 9053 665 94210 3119 766408 8965 863 980114.01885 169兰州287.590.544 5259 4442 940 8841 8324 749169 5402 641 56865.83550 890西宁133.950.522 765 848711 3101 7461 46949 134855 05127.21219 251银川95.380.570 9171 603661 2262 1061 193

18、74 758814 10323.72178 621乌鲁木齐158.920.824 478 5131 847 2412 6689 041254 8702 365 50855.27517 6224.2 运行结果 cwprint(cwbook.txt,35,10) fid =6数据标准化结果如下：v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.0080

19、0.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.0132

20、0.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.0609

21、0.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.0162

22、0.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.02230.01680.03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.0603

23、0.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.1100

24、0.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.0048

25、0.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.01460.0125相关系数矩阵 :std =1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.4750

26、1.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390-0.35390.58910.15071.00000.42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.260

27、00.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46890.30900.37420.86140.25950.90270.95270.97310.93961.0000特征向量 (vec)：vec =-0.13670.2282-0.26280.19390.6371 -0.21630.3176-0.1312-0.41910.2758-0.0329-0.02170.00090.0446-0.1447 -0.44370.4058-0

28、.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.20650.34030.04040.14080.08960.0380 -0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.235

29、80.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.09870.35500.3280 -0.01390.00710.3832-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.16680.3799特征值 (val)val =0.003900000000000.024000000000000.030700000000000.09910000

30、0000000.123200000000000.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分贡献率：newrate =0.62600.23510.05300.03210.02570.01230.00990.00310.00240.0004第一、二主成分的载荷：0.690 1-0.64270.148 30.84140.60

31、0 7-0.68050.851 50.31670.465 6-0.67540.946 30.04260.911 70.32990.953 70.18620.958 90.01090.950 60.2558第一、二、三、四主成分的得分：score =0.718 50.049 90.768 42.00000.380 60.038 60.419 24.00000.184 8-0.043 30.141 421.00000.118 60.031 10.149 720.00000.054 90.011 50.066 433.00000.228 80.007 00.235 87.000 00.2364-0.

32、00810.228310.00000.1778-0.01670.161116.00000.2292-0.03370.195514.00000.83820.13390.97211.00000.22760.00640.23408.00000.2279-0.02220.205612.00000.1989-0.03820.160718.00000.0789-0.00610.072832.00000.1711-0.03170.139423.00000.09260.02660.119225.00000.0900-0.00000.089928.00000.1692-0.00820.161017.00000.

33、2441-0.03180.212411.00000.1507-0.01080.139922.00000.23160.00120.23289.00000.1294-0.02110.108327.00000.47160.03280.50453.00000.27370.08340.35705.00000.0754-0.00130.074131.00000.04480.03490.079730.00000.4759-0.20280.27316.00000.2907-0.08830.202413.00000.0944-0.01180.082629.00000.15460.00350.158119.00000.1718-0.00920.162615.00000.08650.02300.109526.00000.03490.02160.056635.00000.03430.02280.057234.00000.08890.04220.131024.0000