主成分分析法MATLAB的實現(xiàn)

1、MATLAB結(jié) 課 作 業(yè)指導(dǎo)老師：張肅班級：信管121姓名：桂亞東學(xué)號：201200654118利用Matlab編程實現(xiàn)主成分分析概述Matlab語言是當(dāng)今國際上科學(xué)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(xué)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(biāo)的一種統(tǒng) 計分析方法，從數(shù)學(xué)角度來看，這是一種降維處理技術(shù)。1.1主成分分析計算步驟 計算相關(guān)系數(shù)矩陣r

2、pi在（3.5.3 ）式中,rij(i，1222rp2j=1ri p2prPP,2,，p）為原變量的xi與xj之間的相關(guān)系數(shù)，其計算公式為rijn(Xkik 1nn(XkiXi)2 (Xkj Xj)2k 1k 1Xi)(Xkj Xj)(2)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| 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出對應(yīng)于特征值i的特征向量e（i 1,2, p）。這里要求le =1,即ei2 1，其j 1中e表示向

3、量ei的第j個分量。 計算主成分貢獻率及累計貢獻率主成分乙的貢獻率為累計貢獻率為第m (m< p)個主成分 計算主成分載荷其計算公式為lj P(z,Xj).iej(i,j得到各主成分的載荷以后，還可以按照成分的得分z11Z12Z1 mZz21Z22Z2mZn1Zn2Znm1,2, p)(3)(3.5.2 )式進一步計算，得到各主(4)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.m計算各主成分得分、綜合得分并排序C

4、wpri 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 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 ctio n std=cwstd(vector)cwsum=sum(vector,1); %a,b=size(vector); %for i=1:a對列求和矩陣大小,a為行數(shù),b為列數(shù)for j=1:bstd(i,j)= vector(i,j)/cwsum(j); endend3.2 cwfac.m計算相關(guān)系數(shù)矩陣%cwfac.mfun ctio n result=cwfac(vector);fprintf('相關(guān)系數(shù)矩陣:

5、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)n ewval=diag(val);y,i=sort(newval) ;%對特征根進行排序，y為排序結(jié)果，i為索引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貢獻率：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%記下累積貢獻率大 85%的特征值的序號放入newi中fprintf('主成分?jǐn)?shù)：%gnn',le ngth( newi);fprintf('主成分載荷：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);計算載荷end enddisp(result)3.3 cwscore.m%cwscore.m,計算得分function score=cwscore(vector1,vector2);sco=vector1*vector2;csum=sum(sco,2);n ewcsum,i=sort(-1*csum);n ewi,j=sort(i);fprintf('計算得分：n')score=sco,csum,j9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 cwpri

8、 nt.m%cwpri nt.mfunction prin t=cwpri nt(file name,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=fope n( file name,'r')vector=fsca nf(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)濟統(tǒng)計指標(biāo)數(shù)據(jù)見下表。城名市稱年底總

9、人口（萬人）非農(nóng)業(yè)人口比（%）農(nóng)業(yè)總產(chǎn)值（萬元）工業(yè)總產(chǎn)值（萬元）客運總量（萬人）貨運總量（萬噸）地方財政預(yù)算內(nèi)收入（萬元）城鄉(xiāng)居民年底儲蓄余額（萬元）在崗職工人數(shù)（萬人）在崗職工工資總額（萬元）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石家莊 太 原 呼和浩特 沈 陽 大 連 長 春 哈爾濱 上 海 南 京 杭 州 寧

10、波 合 肥 福 州 廈 門 南 昌 濟 南 青 島 鄭 州 武 漢長 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廣州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重慶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貴陽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運行結(jié)果>> cwpri nt('cwbook.tx

18、t',35,10) fid =6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下:v1 =0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.

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相關(guān)系數(shù)矩陣：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各主成分貢獻率：n ewrate =0.00240.62600.23510.05300.03210.02570.01230.00990.00310.0004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 6-0.67540