matlab主成分分析法1

第頁(yè)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)開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\*GB3①計(jì)算相關(guān)系數(shù)矩陣〔1〕在〔，rij〔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〔2〕因?yàn)镽是實(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。=2\*GB3②計(jì)算特征值與特征向量首先解特征方程，通常用雅可比法〔Jacobi〕求出特征值，并使其按大小順序排列，即；然后分別求出對(duì)應(yīng)于特征值的特征向量。這里要求=1，即，其中表示向量的第j個(gè)分量。=3\*GB3③計(jì)算主成分奉獻(xiàn)率及累計(jì)奉獻(xiàn)率主成分的奉獻(xiàn)率為累計(jì)奉獻(xiàn)率為一般取累計(jì)奉獻(xiàn)率達(dá)85—95%的特征值所對(duì)應(yīng)的第一、第二，…，第m〔m≤p〕個(gè)主成分。=4\*GB3④計(jì)算主成分載荷其計(jì)算公式為〔3〕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〔4〕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。主函數(shù)子函數(shù)——用總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣——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)展排序；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——計(jì)算各主成分得分、綜合得分并排序——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(vector);%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);endend計(jì)算相關(guān)系數(shù)矩陣functionresult=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)展排序，y為排序結(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('%g\n',newy)rate=y/sum(y);fprintf('\n奉獻(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%記下累積奉獻(xiàn)率大85%的特征值的序號(hào)放入newi中fprintf('主成分?jǐn)?shù)：%g\n\n',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)%cwscore.m,計(jì)算得分functionscore=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為排序結(jié)果functionprint=cwprint();%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(,'r')vector=fscanf(fid,'%g',[ab]);fprintf('標(biāo)準(zhǔn)化結(jié)果如下：\n')v1=cwstd(vector)result=cwfac(v1);cwscore(v1,result);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。城市名稱年底總?cè)丝?萬(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)北京1818434271999970620323455622790863268066465773301天津91501136226455023259263171128073113019312254343石家莊229186806885768292919113523487095875758877太原323603827377501937118952032773943100654023呼和浩特2365343816452235126231057831396588309337沈陽(yáng)91295418582673377821541256791990169981152811大連61879739842638510780191877092277556796965922長(zhǎng)春818532105966343481095323570964803744884447哈爾濱7266385541861236720752048144364500201309151上海142069019545290986406444854318500259712005605445南京198919913072737142691119366429956804721357861杭州6141473712000796178831168444959374259671180947寧波714282351062286622215102985017235246350824034合肥46287642514125489315172336281622931369577福州321522886555351885171904675245030220680607廈門53333745751124372825704187582108331657484南昌86882892305881367431891677142640460479,555濟(jì)南5148630262858825915117754606904126970756696青島323823201149203613408170386584354978045961704鄭州467742552876011043367683872525135338696848武漢91211291750608597931544260465857480551314766長(zhǎng)沙711463673098179870657183236603461244596986廣州416007382334813922007238541761499204018113047594深圳12996622036829587544274184790895199001890338南寧47204861149691513032931497002190918371809?？?44815717461534523561151741626800198138重慶37416878085855255244125124898,91290909691606804成都119355905894289401401963256118974796841200671貴陽(yáng)736206122479341570341431979081787748419681昆明579335636057295604120425242164127900842321西安473990536659421031197664088965863980885169蘭州52594442940884183247491695402641568550890西寧7658487113101746146949134855051219251銀川91716036612262106119374758814103178621烏魯木齊4785131847241266890412548702365508517622>>cwprint('cwbook.txt',35,10)fid=6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下：v1=0.05810.03560.04350.06800.05570.11120.11940.11840.03460.03540.07700.00890.04070.01390.06880.02340.00800.00470.01390.03910.00560.00930.00530.00970.02630.00860.00280.00640.00640.00450.00620.03150.03750.03050.01980.02130.02530.02950.04430.02860.02950.03210.04310.02760.06280.01420.06100.04400.04880.18530.01760.10860.18480.11480.02500.03180.02330.04440.03910.02860.02500.01520.03370.03610.06090.02510.02590.03270.02860.03440.02710.01850.02700.01050.02390.01400.01390.00560.01330.00250.04970.14280.01230.09830.02920.14370.04660.01990.04560.02000.11000.04790.01490.02200.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.00620.0074相關(guān)系數(shù)矩陣:std=1.0000-0.34440.84250.36030.73900.62150.40390.49670.6761-0.3444-0.4750-0.35390.19710.35710.26000.15700.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.36030.30960.33581.00000.15070.76640.94120.84800.73200.7390-0.35390.58910.15071.00000.42940.19710.31820.38930.62150.19710.50560.76640.42941.00000.83160.89660.93020.40390.35710.32360.94120.19710.83161.00000.92330.83760.49670.26000.44560.84800.67610.46890.3090特征向量(vec)：vec=-0.13670.2282-0.26280.19390.6371-0.21630.3176-0.0329-0.02170.00090.0446-0.1447-0.44370.4058-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.20650.34030.04040.14080.08960.0380-0.1969-0.0437-0.03430.23600.29810.47390.56850.23580.1465-0.1502-0.26310.12450.4879-0.57070.12170.17610.09870.35500.3280-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.1668特征值(val)val=0.0039000000