matlab主成分分析法

1、§10.利用Matlab編程實(shí)現(xiàn)主成分分析1. 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rrr11121prrrR=2122.2p1）rrrp1p2pp在（，rij.（i,j=

2、l,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工(x-x)(x-x)kiikjjk=1:丫(x-x).ki1k=12為(x-x)2ikjjk=12)因?yàn)镽是實(shí)對(duì)稱(chēng)矩陣（即riz=rz.）,所以只需計(jì)算上三角元素或下三角元素即可。ijji 計(jì)算特征值與特征向量首先解特征方程卜I-R=0,通常用雅可比法(Jacobi)求出特征值九£=1,2,p)，并使其按大小順序排列，即九以>,>X>0；然后分別求12p出對(duì)應(yīng)于特征值九的特征向量e(i=1,2,p)。這里要求el|=1，即才e2二1，其iiiijj=1中e表示向量e的第j個(gè)分量。iji 計(jì)算主成分

3、貢獻(xiàn)率及累計(jì)貢獻(xiàn)率主成分z的貢獻(xiàn)率為i累計(jì)貢獻(xiàn)率為一般取累計(jì)貢獻(xiàn)率達(dá)8595%的特征值九，九，,九所對(duì)應(yīng)的第一、第12m二,，第m(mWp)個(gè)主成分。 計(jì)算主成分載荷其計(jì)算公式為l=p(z,x)=宀e(i,j=1,2,，p)(3)ijijij(3丿得到各主成分的載荷以后，還可以按照(，得到各主成分的得分zzz11121mzzzZ=21222mzzn1n2znm4)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(guò)編程來(lái)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)子函數(shù)Cwpr

4、int.m主函數(shù)丄Cwfac.mCwscore.mCwstd.m2.2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.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3.1cwstd.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functionstd=cwstd(vector)cwsum=sum(vector,1);%對(duì)列求和a,b=size(ve

5、ctor);%矩陣大小,a為行數(shù),b為列數(shù)fori=1:aforj=1:bstd(i,j)=vector(i,j)/cwsum(j);endend3.2cwfac.m計(jì)算相關(guān)系數(shù)矩陣%cwfac.mfunctionresult=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

6、排序結(jié)果，i為索引fprintf('特征根排序：n')forz=1:length(y)newy(z)=y(length(y)+1-z);endfprintf('%gn',newy)rate=y/sum(y);fprintf('n貢獻(xiàn)率：n')newrate=newy/sum(newy)sumrate=0;newi=;fork=length(y):-1:1sumrate=sumrate+rate(k);newi(length(y)+1-k)=i(k);ifsumrate>0.85break;endend%記下累積貢獻(xiàn)率大85%的特征值的序號(hào)放

7、入newi中fprintf('主成分?jǐn)?shù)：%gnn',length(newi);fprintf('主成分載荷：n')forp=1:length(newi)forq=1:length(y)result(q,p)=sqrt(newval(newi(p)*vec(q,newi(p);endend%計(jì)算載荷disp(result)3.3cwscore.m%cwscore.m，計(jì)算得分functionscore=cwscore(vector1,vector2);sco=vector1*vector2;csum=sum(sco,2);newcsum,i=sort(-1*csu

8、m);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為排序結(jié)果3.4cwprint.m%cwprint.mfunctiOnprint=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',ab);fprintf(標(biāo)準(zhǔn)化結(jié)果如下：n')v1=cwstd(vectOr)re

9、sult=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)指標(biāo)數(shù)據(jù)見(jiàn)下表。北京天津石家莊太原呼和浩特沈陽(yáng)大連長(zhǎng)春哈爾濱上海南京杭州年底總?cè)丝冢ㄈf(wàn)人）非農(nóng)業(yè)人口比（%）農(nóng)業(yè)總產(chǎn)值（萬(wàn)元）工業(yè)總產(chǎn)值（萬(wàn)元）客運(yùn)總量（萬(wàn)人）貨運(yùn)總量（萬(wàn)噸）地方財(cái)政預(yù)算內(nèi)收入（萬(wàn)元）城鄉(xiāng)居民年底儲(chǔ)蓄余額（萬(wàn)元）在崗職工人數(shù)（萬(wàn)人）在崗職工工資總額（萬(wàn)元）1249.900.59781843427199997062032345562279086326806646410.805773301910.170.5809150113622645502

10、325926317112807311301931202.682254343875.400.23322918680688576829291911352348709587595.60758877299.920.65632360382737750193711895203277394310088.65654023207.780.441236534381645223512623105783139658842.11309337677.080.6299129541858267337782154125679199016998135.451152811545.310.4946187973984263851078

11、019187709227755679694230.406818532105966343481095323570964803744102.63884447927.090.462726638554186123672075204814436450020172.7913091511313.120.7384206901954529098640644485431850025971200336.845605445537.440.53419891991307273714269111936642995680472113.811357861616.050.3556141473712000

12、7961788311684449593742596796.901180947寧波538.410.25471428235106228662221510298501723524635062.15合肥429.950.3184628764251412548931517233628162293147.27福州583.130.27332152288655535188517190467524503022069.59廈門(mén)128.990.4865333374575112437282570418758210833146.93南昌424.200.39886882892305881367431891677142640

13、46062.08濟(jì)南557.630.408514863026285882591511775460690412697083.31青島702.970.369323823201149203613408170386584354978045103.52鄭州615.360.34246774255287601104336768387252513533884.66武漢740.200.5869121129175060859793154426046585748055149.20長(zhǎng)沙582.470.31071146367309817987065718323660346124469.57廣州685.000.62141

14、600738233481392200723854176149920401811182.81深圳119.850.793129966220368295875442741847908951990091.26南寧285.870.4064720486114969151303293149700219091845.09?？?4.380.83544481571746153452356115174162680019.01重慶3072.340.2067416878085855255244125124898,9129090969223.73成都1003.560.335193559058942894014019632

15、5611897479684132.89貴陽(yáng)321.500.45573620612247934157034143197908178774855.28昆明473.390.38657933563605729560412042524216412790088.11西安674.500.409473990536659421031197664088965863980114.01蘭州287.590.5445259444294088418324749169540264156865.83西寧133.950.522765848711310174614694913485505127.21銀川95.380.5709171

16、603661226210611937475881410323.72烏魯木齊158.920.824478513184724126689041254870236550855.27824034369577680607657484479,555756696961704696848131476659698630475941890338371809198138160680412006714196818423218851695508902192511786215176224.2運(yùn)行結(jié)果>>cwprint('cwbook.txt',35,10)fid=6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下v1=0.

17、05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.01110.00750.

18、03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.18480.11480.08880.13520.

19、02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.01830.01640.

20、00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.02230.01680.

21、03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.

22、00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.02030.

23、03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.01460.0125相關(guān)

24、系數(shù)矩陣:std=1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390-0.35390.58910.15071.00000.42940.19710.3

25、1820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46890.30900.37420.86140.25950.90270.95270.9

26、7310.93961.0000特征向量(vec)：vec=-0.13670.2282-0.26280.19390.6371-0.21630.3176-0.1312-0.41910.2758-0.0329-0.02170.00090.0446-0.1447-0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.20650.34030.04040.14

27、080.08960.0380-0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.09870.35500.3280-0.01390.00710

28、.3832-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.16680.3799特征值(val)val=0.003900000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.530000000000002.35140000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=0000006.26020.62600.23510.05300.03210.02570.01230.00990.00310.00240.0004第一、二主成分的載荷：0.6901-0.64270.14830.84140.6007-0.68050.85150.31670.4656-0.67540.94630.04260.91