主成分分析法MATLAB的實(shí)現(xiàn)

1、MATLAB 結(jié) 課 作 業(yè)指導(dǎo)老師：張肅 班級(jí)：信管 121 姓名：桂亞?wèn)| 利用 Matlab 編程實(shí)現(xiàn)主成分分析概述Matlab 語(yǔ)言是當(dāng)今國(guó)際上科學(xué)界 （ 尤其是自動(dòng)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(yùn)算，并已經(jīng)發(fā)展成一種高度集成的計(jì)算機(jī)語(yǔ)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(yǔ)言的便捷接口的功能。 Matlab 語(yǔ)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(lái)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(lái)看，這 是一種降維處理技術(shù)。1.1 主成分分析計(jì)算步驟 計(jì)算相關(guān)系數(shù)矩陣1

2、112r1 pR21222prp1rp2rpp在（3.5.3 ）式中,rj （i , j=1 , 2,，p）為原變量的xi與xj之間的相關(guān)系數(shù),其計(jì)算公式為rij(Xkik 1Xi)(XkjXj)nXi)2(Xkjk 1-2Xj)(2)因?yàn)镽是實(shí)對(duì)稱矩陣（即rj=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 ;然后分別求出對(duì)應(yīng)于特征值i的特征p向量q（i 1,2, , p）。這里要求ei =1,即e21，其中eij表示向量e的第j個(gè)分量

3、。j 1 計(jì)算主成分貢獻(xiàn)率及累計(jì)貢獻(xiàn)率主成分乙的貢獻(xiàn)率為累計(jì)貢獻(xiàn)率為般取累計(jì)貢獻(xiàn)率達(dá)8595%勺特征值1, 2, , m所對(duì)應(yīng)的第一、第二，第 m（me p）個(gè)主成分。 計(jì)算主成分載荷其計(jì)算公式為(4)得到各主成分的載荷以后，還可以按照(3.5.2 )式進(jìn)一步計(jì)算，得到各主成分的得 分Z11Z12Z1mZZ21Z22Z2mZn1Zn2Znm2.函數(shù)作用Cwstd.用總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣Cwfac.計(jì)算相關(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)算各特征值貢獻(xiàn)率；挑選主成分(累計(jì)貢獻(xiàn)率大于 85%，輸出主成分個(gè)數(shù)；計(jì)算主成分載 荷Cwscore.計(jì)算各主成分得分、綜合得分并排序Cw

4、pri nt.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fun cti on std二cwstd(vector)cwsum=sum(vector,1); %對(duì)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:bstd(i,j)= vector(i,j)/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

5、') 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:length(y)newy(z)=y(length(y)+1-z);end fprintf('%gn',newy) rate=y/s

6、um(y);fprintf('n 貢獻(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;endnewi 中end % 記下累積貢獻(xiàn)率大 85%的特征值的序號(hào)放入 fprintf(' 主成分?jǐn)?shù)： %gnn',length(newi);fprintf(' 主成分載荷： n')for p=1:length(newi)for q=

7、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') score=sco,csum,j%得分矩陣：SCO為各主成分得分；csum為綜合得分；j為

8、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,reSult);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)

9、指標(biāo)數(shù)據(jù)見(jiàn)下表。年底非農(nóng)農(nóng)業(yè)工業(yè)客運(yùn)貨運(yùn)地方財(cái)政城鄉(xiāng)居民在崗職在崗職工城市業(yè)總量總量年底儲(chǔ)蓄總?cè)丝诳偖a(chǎn)值總產(chǎn)值預(yù)算內(nèi)收余額工人數(shù)丄資總額名稱人口比（萬(wàn)（萬(wàn)（萬(wàn)人）(%)（萬(wàn)元）（萬(wàn)元）人）噸）入（萬(wàn)元）（萬(wàn)元）（萬(wàn)人）（萬(wàn)元）0.597北京1 249.9081 843 42719 999 70620 32345 5622 790 86326 806 646410.805 773 3010.580天津910.1791 501 13622 645 5023 25926 3171 128 07311 301 931202.682 254 3430.233石家莊875.4022 918 680 688

10、5 768 2 929 1 911352 3487 095 87595.60758 8770.656太原299.923236 0382 737 750 193711 895203 2773943 10088.65654 0230.441呼和浩特207.782365 343816 4522 351 :> 623105 7831 396 58842.11309 3370.629沈陽(yáng)677.0891 295 418 5826 733 7 73215 412567 9199 016 998135.451 152 8110.494大連545.3161 879 739 8426 38510 7801

11、9 187709 2277 556 79694.15965 9220.406長(zhǎng)春691.2381 853 210 5966 343 4 810 9 532357 0964 803 744102.63884 4470.462哈爾濱927.0972 663 855 4186 123 6 720 7 520481 4436 450 020172.791 309 1510.738上海1 313.1242 069 01954 529 0986 40644 4854 318 50025 971 200336.845 605 4450.534南京537.441989 19913 072 73714 2691

12、1 193664 2995 680 472113.811 357 8610.355杭州616.0561 414 73712 000 79617 88311 684449 5937 425 96796.901 180 9470.254寧波538.4171 428 23510 622 86622 21510 298501 7235 246 35062.15824 0340.318合肥429.954628 7642 514 125 4893 15；17233 6281 622 93147.27369 5770.273福州583.1332 152 288 6555 351 8 851 7 190467

13、 5245 030 22069.59680 6070.486廈門(mén)128.995333 3745 751 124 3728 2 J；70418 7582 108 33146.93657 4840.398南昌424.208688 2892 305 881 3674 3 189167 7142 640 46062.08479 ,5550.408濟(jì)南557.6351 486 302 6285 882 5 91511 775460 6904 126 97083.31756 696青島702.970.36932 382 32011 492 03613 40817 038658 4354 978 04510

14、3.52961 704鄭州615.360.3424677 4255 287 60110 4336 768387 2525 135 33884.66696 848武漢740.200.58691 211 291 7506 085 9 79315 442604 6585 748 055149.201 314 766長(zhǎng)沙582.470.31071 146 367 3098 179 8 706 5 718323 6603 461 24469.57596 986廣州685.000.62141 600 73823 348 13922 00723 8541 761 49920 401 811182.813 0

15、47 594深圳119.850.7931299 66220 368 2958 754 41 274 1847 908 9 5'9 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 :2 356115 1741 626 80019.01198 138重慶3 072.340.20674 168 780 8585 52552 44125 124898,9129 090 969223.731 606 80

16、4成都1 003.560.3351 935 590 5894 28940 14019 632561 1897 479 684132.891 200 671貴陽(yáng)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287.590.544525

17、9 4442 940 884 1832 4 :49169 5402 641 56865.83550 890西133.950.522765 848711 3101 746 46949 134855 05127.21219 251銀川95.380.5709171 603661 2262 106 19374 758814 10323.72178 621烏魯木齊158.920.824478 5131 847 241 2668 9 ()41254 8702 365 50855.27517 6224.2運(yùn)行結(jié)果 >> cwpri nt('cwbook.txt',35,10) f

18、id =6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下：v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.004

19、50.00620.01110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.184

20、80.11480.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.018

21、30.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.022

22、30.01680.03440.03490.02860.02550.02680.03770.02590.02540.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.002

23、50.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.02030.031

24、30.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.0146 0.0125 相關(guān)系數(shù)矩陣 :std =1.

25、0000-0.34440.84250.36030.73900.62150.40390.49670.6761 0.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.1570 0.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.5575 0.37420.36030.30960.33581.00000.15070.76640.94120.84800.7320 0.86140.7390-0.35390.58910.15071.00000.42940.19710.31820.38

26、93 0.25950.62150.19710.50560.76640.42941.00000.83160.89660.9302 0.90270.40390.35710.32360.94120.19710.83161.00000.92330.8376 0.95270.49670.26000.44560.84800.31820.89660.92331.0000 0.9201 0.97310.67610.1570 0.55750.73200.38930.93020.83760.92011.0000 0.93960.46890.3090 0.37420.86140.25950.90270.95270.

27、97310.9396 1.0000特征向量 (vec) ：vec =-0.13670.2282 -0.26280.19390.6371-0.21630.3176-0.1312-0.4191 0.2758-0.0329-0.0217 0.00090.0446-0.1447-0.44370.4058-0.55620.5487 0.0593-0.0522-0.0280 0.2040-0.0492-0.5472-0.42250.34400.3188-0.4438 0.24010.0067-0.4176-0.2856-0.23890.1926-0.4915 -0.4189 0.27260.2065 0.

28、3403 0.04040.14080.08960.0380 -0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.2360 0.0640-0.82940.03770.26620.1356-0.12900.0278 0.37820.29810.4739 0.56850.23580.1465-0.1502-0.26310.12450.2152 0.36440.15670.3464 -0.64850.2489-0.40430.2058-0.07040.04620.1214 0.38120.4879 -0.5707 0.1217 0.1761 0.0987

29、 0.3550 0.3280 -0.01390.0071 0.3832 -0.7894 -0.1628 0.1925 0.2510 -0.0422 0.2694 0.0396 0.04560.1668 0.3799 特征值 (val)val = 0.0039 0 0 0 0 0 0 000.0240 0 000000 000 0.0307 000000 000 0 0.099100000 0 0 0 0.1232 0 0 00 00 0 0 00 00 0 0 00 00 0 0 00 00 0 0 00 0.2566 0 00 0 0.3207 00 0 0 0.53002.3514 00 6.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分貢獻(xiàn)率：newrate =0.00990.6260 0.2351 0.0530 0.0321 0.0257 0.01230.0031 0.0024 0.0004 第一、二主成分的載荷：0.690 1 -0.64270.148 3 0.84140.600 7 -0.68050.851 5 0.31670.465 6 -0.67