主成分分析法實(shí)例

1、利用Matlab編程實(shí)現(xiàn)主成分分析1.概述Matlab語齊是為今國際上科學(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(yùn)算，并已經(jīng)發(fā)展成種高度集成的計(jì)算機(jī)語言。它提供了強(qiáng)人的科學(xué)運(yùn)算、靈活的程序設(shè)計(jì)流程、高質(zhì)顯的圖形可視化與界血設(shè)計(jì)、與其他程序和語言的便捷接口的功能。Matlab語汗在各國高校與研究單位起著車人的作用。主成分分析是把原來多個變量劃為少數(shù)兒個綜合指標(biāo)的-種統(tǒng)計(jì)分析方法，從數(shù)學(xué)和度來看，這是-種降維處理技術(shù)。1.1主成分分析計(jì)算步驟計(jì)算和關(guān)系數(shù)矩陣在(3.5.3)式中，q(ifj=l2,p)為原變量的與寸之間的相關(guān)系數(shù)，其計(jì)算公式為(2)因?yàn)镽是實(shí)對稱矩陣(

2、即0所以只需計(jì)算上二和元素或卜二介兀素即唄計(jì)算特征值與特征向量首先解特征方程|x/-/?|=0,通常用雅可比法(Jacobi)求出特征值入(心1,2,丿)，并使其按人小順序排列，即2,22.-,0；然后分別求出對應(yīng)于特征值人的特征向量勺(山1,2,刃。這里要求|ej|=l,即其7=1中e)表示向量勺的第/個分量。計(jì)算主成分貢獻(xiàn)率及累計(jì)貢獻(xiàn)率工成分玄的貞獻(xiàn)率為(心12,卩)ZaA=l累計(jì)貢獻(xiàn)率為VI(7=1,2,丿)般取累計(jì)貢獻(xiàn)率達(dá)8595%的特征值入A，九所對應(yīng)的第-、第二，第加(mWp)個上成分n計(jì)算主成分載荷其計(jì)算公式為(3)Ijj=P(可宀)=7烏=12，p)得到各主成分的載荷以后,還可

3、以按照(3.5.2)式進(jìn)步計(jì)算,得到各主成分的得分(4)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)：-足通過編程來實(shí)現(xiàn)：是門接調(diào)用Matlab種自帶程丿T；實(shí)現(xiàn)。卜面丄要主要介紹利用Matlab的矩陣計(jì)算功能編程實(shí)現(xiàn)主成分分析。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)算相關(guān)系數(shù)矩陣；計(jì)算特征值和特征向量；對主成分進(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)擁文件：調(diào)用以匕三個函數(shù)并輸出結(jié)杲3.源程序

4、3.1cwstd.m總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣%cwstd.mj|j總和標(biāo)準(zhǔn)化法標(biāo)準(zhǔn)化矩陣functionsid=cwstd(vector)cwsum=sum(vector,I);%對列求和a?bl=size(vector);%矩陣人小,a為行數(shù),b為列數(shù)fori=l:aforj=l:bst(J(i.j)=vcc(or(ij)/cvvsuni(j);endend3.2cwfac.m計(jì)算相關(guān)系數(shù)矩陣%cwfac.mfunctionresu1t=cwfac(vector):fprintf(相關(guān)系數(shù)矩陣:n)std=CORRCOEF(vector)%i|算相關(guān)系數(shù)矩陣fprintff特征向量(vec)及

5、特征fll(val)：n)vec,val=cig(std)%求特征值(val)及特征向吊(vcc)newval=diag(val);yJ=son(newval):%對特征根述行排序，y為排序結(jié)果.i為索引fprintff特征根排序:W)forz=l:length(y)newy(z)=y(length(y)+1z);enclfprintfC%gnnewy)rate=y/sum(y);fprinif(n貢獻(xiàn)率:n*)newrate=newy/sum(newy)suniratc=();ncwi=J;fork=cngih(y):-l:lsumrate=sumraie+ra(e(k);newi(leng

6、th(y)+l-k)=i(k);ifsumraie0.85break:endend%記下累枳貢獻(xiàn)率大85%的特征侑的序號放入newifprimf(主成分?jǐn)?shù)：%gnnlcngth(ncwi);fprintf(1成分載荷:n)forp=l:length(newi)forq=klength(y)resull(q,p)=sqrt(newval(newi(p)*vec(qjiewi(p):endend%計(jì)算戦荷clisp(result)3cwscorem%cwscore.m,計(jì)算得分functionscore=cwscore(vectorI,vector2);sco=veciorI*vector?;cs

7、um=sum(sco2);newcsumi=sort(-Icsum);newij=sort(i);fprintfCil算得分：n)score=fsco,csumj%得分知陣：sc。為各主成分得分：csum綜合得分：j為排序結(jié)果3.4cwprint.m%cwprinl.mfunctionprini=cwprini(filenames,b);%filename文本文H文fl名a為矩陣行數(shù)(樣本數(shù)),b為矩陣列數(shù)(變鈕指標(biāo)數(shù))nd=fopcn(filenameJr)vector=fscanf(fid/%gab);fpriiiilCfeiHI-化結(jié)果如下：nJvl=cwstd(vector)resul

8、t=cwfac(vl);cwscore(v1,result);4.程序測試4.1原始數(shù)據(jù)中國人陸35個人城市某年的10項(xiàng)社會經(jīng)濟(jì)統(tǒng)計(jì)指標(biāo)數(shù)據(jù)見下表。城名市稱年底總?cè)丝冢ㄈf人）非農(nóng)業(yè)人II比（%）農(nóng)業(yè)總產(chǎn)值（力元）工業(yè)總產(chǎn)值（力元）總量（萬人）貨運(yùn)總量（力噸）地方財(cái)政預(yù)越內(nèi)收入6元）城鄉(xiāng)居民年底儲蓄余額（萬元）在崗職工人數(shù)（力人）在崗職工工資總額（萬元）北1249.900.59781843427199997062032345562279086326806646410.805773301犬910.170.58091501136226455023259263171128073II301931202.

9、682254343右召E莊875.400.23322918680688576829291911352348709587595.60758877太299.920.656323603827377501937II895203277394310088.65654023呼和浩特207.780.441236534381645223512623105783139658842.11309337沈陽677.080.6299129541858267337782154125679199016998135.451152811人連545.310.4946187973984263851078019187709227755

10、679694.15965922長春691.230.406818532105966343481095323570964803744102.63884447哈5濱927.090.462726638554186123672075204814436450020172.791309151上海1313.120.7384206901954529()98640644485431850025971200336.845605445南京537.44().53419891991307273714269111936642995680472113.811357861杭州616.050.355614147371200079

11、617883II684449593742596796.901180947寧波538.410.25471428235106228662221510298501723524635062.15824034合肥429.950.3184628764251412548931517233628162293147.27369577福州583.130.2733215228S6555351S8517190467524503022069.59680607門128.990.4865333374575112437282570418758210833146.93657484南昌424.200.39886882892305

12、88136743189167714264046062.08479,555濟(jì)557.630.40851486302628588259151177546069041269708331756696I1!島702.970.36932382320II49203613408170386584354978045103.52961704鄭州615.360.34246774255287601104336768387252513533884.66696848武漢740.200.5869121129175060859793154426046585748055149.201314766長沙582.470.310711

13、46367309817987065718323660346124469.57596986廣州685.000.62141600738233481392200723854176149920401811182.813047594深圳119.850.793129966220368295875442741847908951990091.261890338南寧285.870,4064720486114969151303293149700219091845.09371809溝口54.380.83544481571746153-152356115174162680019.01198138重慶3072.340.

14、206741&7M)8585525524412512489&9129090969223.731606804成都1(X)3.560.3351935590589428940140196325611897479684132.891200671貴陽321.500.45573620612247934157034143197908178774855.28419681昆明473.390.38657933563605729560412042524216412790088.11842321西安674.500.409473990536659421031197664088965863980114.01885169蘭

15、州287.590.5445259444294088418324749169540264156865.83550890133.950.522765848711310174611694913485505127.21219251銀川95.380.570917160366122621()&11937475881410323.72178621烏魯木齊158.920.824478513184724126689041254870236550855.275176222運(yùn)行結(jié)果cwprinl(cwbook.lxi350)fid=6數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下:vl=0.05810.03560.04350.0680().0

16、557()1120.1194().11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.0252(101830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.0

17、2130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950-04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.U6280.01420.01840.01840.02060.02850.04550.03160.06100.04400.0488()8530.01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03

18、910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.0337().03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.0

19、1020.00630.01790.00930.01240.01590.01970.02370-01620.00780.0101().00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.022()0.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.02230.01680.03440.03490.02860.02550

20、.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.00250.04970.00II0.00240

21、.()146O.(X)570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.020()0.110()0.04790.02400.03310.035()0.02900.01490.02710.00850,00760.0430().01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.0203().03130.02440.0174

22、0.01250.02830.0238().0175().02590.0300().02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.0062().03110.0016O.(X)24O.(X)48O.(X)36().(X)210.0038().(X)720.00530.00440.03400.0040().0022().0058().00290.00320.00360.00630.00430.00740.04910.00190.00630.0073().0221().01090.01050.01460.()125相

23、關(guān)系數(shù)矩陣:std=1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.0(X)00.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390035390.58910.15071.00000.42940.19710.3

24、1820.38930.25950.62150.19710.5056056640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.9201I.()0000.93960.46890.30900.37420,86140.25950.90270.95270.9

25、7310.93961.0000特征向量(vec)：vec=-0.13670.2282-0.2628().1939().6371-0.2163().317601312-0.41910.2758-0.0329-0.0217O.(X)O90.0446-0.1447-0.44370.4058-0.55620.5487().0593-0.0522-0.02800.20400.0492-0.5472-04225034400.3188-0.44380.24010.0067-0.4176-0.28560.23890.1926-0.4915-0.41890.27260.20650.34030.04040.140

26、80.08960.0380-0.1969-0.04370.4888-0.67890.44050J861-0.03430.23600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465502-0.263102450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.09870.355003280001390.00710.3832-0.78940

27、.16280.19250.2510-0.04220.26940.03960.04560.16680.3799特征值(val)val=0.003900000000000.0240()0000000000.030700000000()00.0991000()000000()22200000000()00.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.32()6990.2566390.1232410.09909150.03070880.02403550.00393387各主成分貢獻(xiàn)率:newraie=0.62600.23510.053()().0321().02570.01230.00990.0024().0004第一、二主成分的載荷:0.690I-0.64270.1483().60070.85150.46560-94630.91170.95370-95890.95060.8414-0.680503167-0.67540.04260.32990.18620.01090.2