完整版主成分分析法matlab實(shí)現(xiàn),實(shí)例演示

1、利用Matlab編程實(shí)現(xiàn)主成分分析1 .概述Matlab語言是當(dāng)今國(guó)際上科學(xué)界尤其是自動(dòng)限制領(lǐng)域最具影響力、也是 最有活力的軟件.它起源于矩陣運(yùn)算,并已經(jīng)開展成一種高度集成的計(jì)算機(jī)語言. 它提供了強(qiáng)大的科學(xué)運(yùn)算、靈活的程序設(shè)計(jì)流程、高質(zhì)量的圖形可視化與界面設(shè) 計(jì)、與其他程序和語言的便捷接口的功能.Matlab語言在各國(guó)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(gè)變量劃為少數(shù)幾個(gè)綜合指標(biāo)的一種統(tǒng) 計(jì)分析方法,從數(shù)學(xué)角度來看,這是一種降維處理技術(shù).1.1 主成分分析計(jì)算步驟門j (i, j=1 , 2,(D在3.5.3式中, 數(shù),其計(jì)算公式為,p為原變量的xi與xj之間的相關(guān)系rij(Xk

2、iXi)(Xkj Xj)k 1nn(XkiXi )2 (Xkj Xj)k 1k 1(2)由于R是實(shí)對(duì)稱矩陣即rij=rji,所以只需計(jì)算上三角元素或下三角元素即可. 計(jì)算特征值與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),并使其按大小順序排列,即12, p 0；然后分別求P出對(duì)應(yīng)于特征值i的特征向量ei(i 1,2, , p).這里要求同=1,即ej 1 ,其j i中e表示向量ei的第j個(gè)分量.計(jì)算主成分奉獻(xiàn)率及累計(jì)奉獻(xiàn)率主成分zi的奉獻(xiàn)率為(i 1,2, ,p)k k 1累計(jì)奉獻(xiàn)率為k(i 1,2, ,p)k k 1m所對(duì)應(yīng)的第一、第一般取

3、累計(jì)奉獻(xiàn)率達(dá)85 95%的特征值1, 2,第m (m&p)個(gè)主成分.計(jì)算主成分載荷其計(jì)算公式為lijp(Z,Xj). iej(i, j 1,2,p)(3)得到各主成分的載荷以后,還可以根據(jù)3.5.2式進(jìn)一步計(jì)算,得到各主成 分的得分Z11 42Z1mZZ21 Z22Z2 mZnl Zn2Znm韋2 .程序結(jié)構(gòu)及函數(shù)作用在軟件Matlab中實(shí)現(xiàn)主成分分析可以采取兩種方式實(shí)現(xiàn)：一是通過編程來 實(shí)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(jì)算功能編程實(shí)現(xiàn)主成分分析.2.1 程序結(jié)構(gòu)2.2 函數(shù)作用Cwstd.m用總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣Cwfac.m計(jì)算

4、相關(guān)系數(shù)矩陣；計(jì)算特征值和特征向量；對(duì)主成分進(jìn)行排序；計(jì)算各特征值奉獻(xiàn)率；挑選主成分累計(jì)奉獻(xiàn)率大于85%,輸出主成分個(gè)數(shù)；計(jì)算主成分載荷Cwscore.m計(jì)算各主成分得分、綜合得分并排序Cwprint.m讀入數(shù)據(jù)文件；調(diào)用以上三個(gè)函數(shù)并輸出結(jié)果3 .源程序3.1 cwstd.m 總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣%cwstd.m,用總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣function std=cwstd(vector)cwsum=sum(vector,1);% 對(duì)歹 U 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:b std(i,j)= vector(i,j)/

5、cwsum(j);endend3.2 cwfac.m計(jì)算相關(guān)系數(shù)矩陣%cwfac.mfunction result=cwfac(vector);fprintf('相關(guān)系數(shù)矩陣：n')std=CORRCOEF(vector) %計(jì)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);y,i=sort(newval) ;%對(duì)特征根進(jìn)行排序,y為排序結(jié)果,i為索引fprintf('特征根排序：n')for z=1:len

6、gth(y)newy(z)=y(length(y)+1-z);endfprintf('%gn',newy)rate=y/sum(y);fprintf('n 奉獻(xiàn)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 sumrate>0.85 break;endend%記下累積奉獻(xiàn)率大85%的特征值的序號(hào)放入newi中fprintf('主成分?jǐn)?shù)：gnn',length(n

7、ewi);fprintf('主成分載荷：n')for p=1:length(newi)for q=1:length(y)result(q,p)=sqrt(newval(newi(p)*vec(q,newi(p);endend%計(jì)算載荷disp(result)3.3 cwscore.m%cwscore.m,計(jì)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(jì)算得分：n&

8、#39;)score=sco,csum,j%得分矩陣：sco為各主成分得分；csum為綜合得分；j為排序結(jié)果3.4 cwprint.m%cwprint.mfunction 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(biāo)數(shù))fid=fopen(filename,'r')vector=fscanf(fid,'%g',a b);fprintf('標(biāo)準(zhǔn)化結(jié)果如下：n')v1=cwstd(vector)result=cwfac(v1);cwscore(v1,resu

9、lt);4 .程序測(cè)試4.1 原始數(shù)據(jù)中國(guó)大陸35個(gè)大城市某年的10項(xiàng)社會(huì)經(jīng)濟(jì)統(tǒng)計(jì)指標(biāo)數(shù)據(jù)見下表.年底非農(nóng)工業(yè)城鄉(xiāng)居民城市農(nóng)業(yè)客運(yùn)貨運(yùn)地方財(cái)政在崗職在崗職工業(yè)年底儲(chǔ)蓄總?cè)丝诳偖a(chǎn)值總產(chǎn)值總量總量預(yù)算內(nèi)收余額工人數(shù)工資總額名稱萬人人口比%萬元萬元萬人萬噸入萬元萬人萬元萬元北京1 249.900.597 81 843 42719 999 70620 32345 5622 790 86326 806 646410.805 773 301天津910.170.580 91 501 13622 645 5023 25926 3171 128 07311 301 931202.682 254 343石家莊87

10、5.400.233 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 396 58842.11309 337沈陽677.080.629 91 295 4185 826 7337 78215 412567 9199 016 998135.451 152 811大連545.310.494 61 87

11、9 7398 426 38510 78019 187709 2277 556 79694.15965 922長(zhǎng)春691.230.406 81 853 2105 966 3434 8109 532357 0964 803 744102.63884 447哈爾濱927.090.462 72 663 8554186 1236 7207 520481 4436 450 020172.791 309 151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

12、19913 072 73714 26911 193664 2995 680 472113.811 357 861杭州616.050.355 61 414 73712 000 79617 88311 684449 5937 425 96796.901 180 947寧波538.410.254 71 428 23510 622 86622 21510 298501 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

13、 555 3518 8517 190467 5245 030 22069.59680 607廈門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 46062.08479 ,555濟(jì)南557.630.408 51 486 3026 285 8825 91511 775460 6904 126 97083.31756 696青島702.970.369 32 382 32011 492 03613 40817

14、038658 4354 978 045103.52961 704鄭州615.360.342 4677 4255 287 60110 4336 768387 2525 135 33884.66696 848武漢740.200.586 91 211 2917 506 0859 79315 442604 6585 748 055149.201 314 766長(zhǎng)沙582.470.310 71 146 3673 098 1798 7065 718323 6603 461 24469.57596 986廣州685.000.621 41 600 73823 348 13922 00723 8541 761

15、49920 401 811182.813 047 594深圳119.850.793 1299 66220 368 2958 7544 2741 847 9089 519 90091.261 890 338南寧285.870.406 4720 4861 149 6915 1303 293149 7002 190 91845.09371 809?？?4.380.835 444 815717 4615 3452 356115 1741 626 80019.01198 138重慶3 072.340.206 74 168 7808 585 52552 44125 124898,9129 090 9692

16、23.731 606 804成都1 003.560.3351 935 5905 894 28940 14019 632561 1897 479 684132.891 200 671貴陽321.500.455 7362 0612 247 93415 7034 143197 9081 787 74855.28419 681昆明473.390.386 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 16

17、9蘭州287.590.544 5259 4442 940 8841 8324 749169 5402 641 56865.83550 890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(yùn)行結(jié)果>> cwprint('cwbo

18、ok.txt',35,10)fid =6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下：v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00

19、280.00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18

20、530.01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02

21、230.02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01

22、800.02860.01650.01660.02270.02230.01680.03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00

23、390.01410.00800.00640.00970.01190.00900.00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01

24、230.01540.02940.02240.01820.02320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00

25、630.00730.02210.01090.01050.01460.0125相關(guān)系數(shù)矩陣:std =1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.739

26、0-0.35390.58910.15071.00000.42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46

27、890.30900.37420.86140.25950.90270.95270.97310.93961.0000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 -0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492 -0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.2389 0.19

28、26-0.4915-0.41890.27260.20650.34030.04040.14080.08960.0380 -0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23600.0640-0.8294 0.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489 -0.40430.2058-0.07040.04620.12140.38120.4879-0.5

29、7070.12170.1761 0.09870.35500.3280-0.01390.00710.3832-0.7894-0.16280.19250.2510 -0.04220.26940.03960.04560.16680.3799特征值(val)val =0.003900000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分奉獻(xiàn)率:newrate =0.6260 0.2351 0.0530 0.03210.0257 0.0123 0.0099 0.00310.00240.0004第一、二主成分的載荷:0.690 1-0.64270.148 30.84140.600 7-0.68050.851 50.31670.465 6-0.67