Matlab編程實現(xiàn)主成分研究

1、10.利用Matlab編程實現(xiàn)主成分分析1. 概述Matlab語言是當今國際上科學界（尤其是自動控制領(lǐng)域）最具影響力、也是 最有活力的軟件。它起源于矩陣運算，并已經(jīng)發(fā)展成一種高度集成的計算機語言。 它提供了強大的科學運算、靈活的程序設(shè)計流程、高質(zhì)量的圖形可視化與界面設(shè) 計、與其他程序和語言的便捷接口的功能。Matlab語言在各國高校與研究單位起著重大的作用。主成分分析是把原來多個變量劃為少數(shù)幾個綜合指標的一種統(tǒng) 計分析方法，從數(shù)學角度來看，這是一種降維處理技術(shù)。1.1主成分分析計算步驟11r21計算相關(guān)系數(shù)矩陣r1 pr2prp2rpp(1)在（3.5.3）式中,數(shù)，其計算公式為rij (i,

2、 j=1 , 2,，p）為原變量的xi與xj之間的相關(guān)系6 / 10nrij(XkiXJ(Xkj XJk ：!nn(Xki -兇)5 (Xkj Xj)2(2)kmk d因為R是實對稱矩陣（即rij=rji），所以只需計算上三角元素或下三角元素即可。 計算特征值與特征向量首先解特征方程卩J-R=0,通常用雅可比法（Jacobi）求出特征值 i（i 2，p），并使其按大小順序排列，即 匕一，一p0 ;然后分別求p出對應(yīng)于特征值i的特征向量（i =1,2,p）。這里要求e =1，即a ei2 =1，其j 二中q表示向量e的第j個分量計算主成分貢獻率及累計貢獻率主成分zi的貢獻率為十(i ,2, ,p

3、)- kk 1累計貢獻率為iV kk生pkk=1(i =1,2/ , p)般取累計貢獻率達85 95%的特征值1,2/ ,m所對應(yīng)的第一、第，第m （mW p）個主成分計算主成分載荷其計算公式為lj = p(z,Xj)=屮憐億 j =1,2，,p)(3)得到各主成分的載荷以后，還可以按照（3.5.2）式進一步計算，得到各主成 分的得分z1 mZ2mznm(4)2. 程序結(jié)構(gòu)及函數(shù)作用在軟件Matlab中實現(xiàn)主成分分析可以采取兩種方式實現(xiàn)：一是通過編程來 實現(xiàn)；二是直接調(diào)用Matlab種自帶程序?qū)崿F(xiàn)。下面主要主要介紹利用 Matlab的 矩陣計算功能編程實現(xiàn)主成分分析。2.1程序結(jié)構(gòu)2.2函數(shù)作

4、用Cwstd.m用總和標準化法標準化矩陣Cwfac.m計算相關(guān)系數(shù)矩陣；計算特征值和特征向量；對主成分進行排序；計算各特征值貢獻率；挑選主成分（累計貢獻率大于85%）,輸出主成分個數(shù)；計算主成分載荷Cwscore.計算各主成分得分、綜合得分并排序Cwpri nt.m讀入數(shù)據(jù)文件；調(diào)用以上三個函數(shù)并輸出結(jié)果3. 源程序3.1 cwstd.m%cwstd.m, 用總和標準化法標準化矩陣 function std=cwstd(vector) cwsum=sum(vector,1);%對列求和a,b=size(vector);%矩陣大小 ,a 為行數(shù) ,b 為列數(shù)for i=1:afor j=1:bs

5、td(i,j)= vector(i,j)/cwsum(j);end end3.2 cwfac.m%cwfac.m function result=cwfac(vector); fprintf( 相關(guān)系數(shù)矩陣 :n) std=CORRCOEF(vector) % 計算相關(guān)系數(shù)矩陣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)newval=diag(val) ;i 為索引y,i=sort(newval) ;%對特征根進行排序， y 為排序結(jié)果，fprintf( 特征根排序: n)for z=1:lengt

6、h(y)newy(z)=y(length(y)+1-z);end fprintf(%gn,newy) rate=y/sum(y);fprintf(n 貢獻率: n) newrate=newy/sum(newy) sumrate=0;newi=;for k=length(y):-1:1sumrate=sumrate+rate(k); newi(length(y)+1-k)=i(k);if sumrate0.85 break;endend% 記下累積貢獻率大 85%的特征值的序號放入 newi 中fprintf( 主成分數(shù)： %gnn,length(newi);fprintf( 主成分載荷： n)

7、for p=1:length(newi)for q=1:length(y)result(q,p)=sqrt(newval(newi(p)*vec(q,newi(p);endend%計算載荷disp(result)3.3 cwscore.m%cwscore.m, 計算得分 function score=cwscore(vector1,vector2); sco=vector1*vector2; csum=sum(sco,2);newcsum,i=sort(-1*csum); newi,j=sort(i);fprintf( 計算得分： n) score=sco,csum,j%得分矩陣： sco 為

8、各主成分得分； csum 為綜合得分； j 為排序結(jié)果3.4 cwprint.m%cwprint.m function print=cwprint(filename,a,b);%filename 為文本文件文件名， a 為矩陣行數(shù) (樣本數(shù) )， b 為矩陣列數(shù) (變量指標數(shù) ) fid=fopen(filename,r) vector=fscanf(fid,%g,a b);fprintf( 標準化結(jié)果如下： n) v1=cwstd(vector) result=cwfac(v1);cwscore(v1,result);4. 程序測試4.1 原始數(shù)據(jù)中國大陸 35 個大城市某年的10 項社會經(jīng)

9、濟統(tǒng)計指標數(shù)據(jù)見下表。在崗職工工資總額（萬元）年底非農(nóng)農(nóng)業(yè)工業(yè)客運貨運地方財政城鄉(xiāng)居民在崗職城市名稱總?cè)丝冢ㄈf人）業(yè)人口比（%）總產(chǎn)值（萬元）總產(chǎn)值（萬元）總量（萬人）總量（萬噸）預算內(nèi)收入（萬元）年底儲蓄余額（萬元）工人數(shù)（萬人）北京1 249.900.597 81 843 42719 999 70620 32345 5622 790 86326 806 646410.80天津910.170.580 91 501 13622 645 5023 25926 3171 128 07311 301 931202.68石家莊875.400.233 22 918 6806 885 7682 9291

10、911352 3487 095 87595.60太原299.920.656 3236 0382 737 7501 93711 895203 2773 943 10088.65呼和浩特207.780.441 2365 343816 4522 3512 623105 7831 396 58842.11沈陽677.080.629 91 295 4185 826 7337 78215 412567 9199 016 998135.45大連545.310.494 61 879 7398 426 38510 78019 187709 2277 556 79694.15長春691.230.406 81 85

11、3 2105 966 3434 8109 532357 0964 803 744102.63哈爾濱927.090.462 72 663 8554186 1236 7207 520481 4436 450 020172.79上海1 313.120.738 42 069 01954 529 0986 40644 4854 318 50025 971 200336.84南京537.440.534 1989 19913 072 73714 26911 193664 2995 680 472113.81杭州616.050.355 61 414 73712 000 79617 88311 684449 5

12、937 425 96796.90寧波538.410.254 71 428 23510 622 86622 21510 298501 7235 246 35062.15合肥429.950.318 4628 7642 514 1254 8931 517233 6281 622 93147.27福州583.130.273 32 152 2886 555 3518 8517 190467 5245 030 22069.59廈門128.990.486 5333 3745 751 1243 7282 570418 7582 108 33146.93南昌424.200.398 8688 2892 305 8

13、813 6743 189167 7142 640 46062.08濟南557.630.408 51 486 3026 285 8825 91511 775460 6904 126 97083.31青島702.970.369 32 382 32011 492 03613 40817 038658 4354 978 045103.52鄭州615.360.342 4677 4255 287 60110 4336 768387 2525 135 33884.66武漢740.200.586 91 211 2917 506 0859 79315 442604 6585 748 055149.20長沙582

14、.470.310 71 146 3673 098 1798 7065 718323 6603 461 24469.57廣州685.000.621 41 600 73823 348 13922 00723 8541 761 49920 401 811182.81深圳119.850.793 1299 66220 368 2958 7544 2741 847 9089 519 90091.26南寧285.870.406 4720 4861 149 6915 1303 293149 7002 190 91845.09?？?4.380.835 444 815717 4615 3452 356115 17

15、41 626 80019.01重慶3 072.340.206 74 168 7808 585 52552 44125 124898,9129 090 969223.73成都1 003.560.3351 935 5905 894 28940 14019 632561 1897 479 684132.89貴陽321.500.455 7362 0612 247 93415 7034 143197 9081 787 74855.28昆明473.390.386 5793 3563 605 7295 60412 042524 2164 127 90088.11西安674.500.409 4739 9053

16、 665 94210 3119 766408 8965 863 980114.01蘭州287.590.544 5259 4442 940 8841 8324 749169 5402 641 56865.835 773 3012 254 343758 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 9863 047 5941 890 338371 8091

17、98 1381 606 8041 200 671419 681842 321885 169550 8908 / 10西寧133.950.522 765 848711 3101 7461 46949 134855 05127.21219 251銀川95.380.570 9171 603661 2262 1061 19374 758814 10323.72178 621烏魯木齊158.920.824 478 5131 847 2412 6689 041254 8702 365 50855.27517 6224.2運行結(jié)果 cwpri nt(cwbook.txt,35,10) fid =6數(shù)據(jù)標準化

18、結(jié)果如下:v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.0

19、1110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.18480.11480.0

20、8880.13520.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.0

21、1830.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.0

22、2230.01680.03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.0

23、1190.00900.00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.0

24、2320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.0

25、1460.0125相關(guān)系數(shù)矩陣 :std =1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390-0.35390.58910.15071.00000.

26、42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46890.30900.37420.86140.25950.

27、90270.95270.97310.93961.0000特征向量 (vec)：vec =-0.13670.2282-0.26280.19390.6371 -0.21630.3176-0.1312-0.41910.2758-0.0329-0.02170.00090.0446-0.1447-0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.2065

28、0.34030.04040.14080.08960.0380-0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.09870.35500.32

29、80-0.01390.00710.3832-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.16680.3799特征值 (val)val =0.003900000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分貢獻率：newrate =0.6260 0.2351 0.0530 0.03210.0257 0.0123 0.00990.00310.00240.000412 / 10第一、二主成分的載荷0.690 10.148 30.600 70.851 50.465 60.946 30.911 70.953 70.958 90.950