主成分分析法实例

1、利用Matlab编程实现主成分分析1.概述Matlab语齐是为今国际上科学界(尤其是自动控制领域)最具影响力、也是最有活力的软件。它起源于矩阵运算，并已经发展成种高度集成的计算机语言。它提供了强人的科学运算、灵活的程序设计流程、高质显的图形可视化与界血设计、与其他程序和语言的便捷接口的功能。Matlab语汗在各国高校与研究单位起着車人的作用。主成分分析是把原來多个变量划为少数儿个综合指标的-种统计分析方法，从数学和度来看，这是-种降维处理技术。1.1主成分分析计算步骤计算和关系数矩阵在(3.5.3)式中，q(ifj=l2,p)为原变量的与寸之间的相关系数，其计算公式为(2)因为R是实对称矩阵(

2、即0所以只需计算上二和元素或卜二介兀素即呗计算特征值与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)，并使其按人小顺序排列，即2,22.-,0；然后分别求出对应于特征值人的特征向量勺(山1,2,刃。这里要求|ej|=l,即其7=1中e)表示向量勺的第/个分量。计算主成分贡献率及累计贡献率工成分玄的贞献率为(心12,卩)ZaA=l累计贡献率为VI(7=1,2,丿)般取累计贡献率达8595%的特征值入A，九所对应的第-、第二，第加(mWp)个上成分n计算主成分载荷其计算公式为(3)Ijj=P(可宀)=7烏=12，p)得到各主成分的载荷以后,还可

3、以按照(3.5.2)式进步计算,得到各主成分的得分(4)2.程序结构及函数作用在软件Matlab中实现主成分分析可以采取两种方式实现：-足通过编程来实现：是门接调用Matlab种自带程丿T；实现。卜面丄要主要介绍利用Matlab的矩阵计算功能编程实现主成分分析。1程序结构2.2函数作用Cwstd.m用总和标准化法标准化矩阵Cwfac.m计算相关系数矩阵；计算特征值和特征向量；对主成分进行排序；计算各特征值贡献率;挑选主成分(累计贡献率大于85%),输出主成分个数;计算主成分载荷Cwscore.m计算各主成分得分、综合得分并排序Cwprint.m读入数拥文件：调用以匕三个函数并输出结杲3.源程序

4、3.1cwstd.m总和标准化法标准化矩阵%cwstd.mj|j总和标准化法标准化矩阵functionsid=cwstd(vector)cwsum=sum(vector,I);%对列求和a?bl=size(vector);%矩阵人小,a为行数,b为列数fori=l:aforj=l:bst(J(i.j)=vcc(or(ij)/cvvsuni(j);endend3.2cwfac.m计算相关系数矩阵%cwfac.mfunctionresu1t=cwfac(vector):fprintf(相关系数矩阵:n)std=CORRCOEF(vector)%i|算相关系数矩阵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):%对特征根述行排序，y为排序结果.i为索引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贡献率: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%记下累枳贡献率大85%的特征侑的序号放入newifprimf(主成分数：%gnnlcngth(ncwi);fprintf(1成分载荷:n)forp=l:length(newi)forq=klength(y)resull(q,p)=sqrt(newval(newi(p)*vec(qjiewi(p):endend%计算戦荷clisp(result)3cwscorem%cwscore.m,计算得分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%得分知阵：sc。为各主成分得分：csum综合得分：j为排序结果3.4cwprint.m%cwprinl.mfunctionprini=cwprini(filenames,b);%filename文本文H文fl名a为矩阵行数(样本数),b为矩阵列数(变鈕指标数)nd=fopcn(filenameJr)vector=fscanf(fid/%gab);fpriiiilCfeiHI-化结果如下：nJvl=cwstd(vector)resul

8、t=cwfac(vl);cwscore(v1,result);4.程序测试4.1原始数据中国人陆35个人城市某年的10项社会经济统计指标数据见下表。城名市称年底总人口（万人）非农业人II比（%）农业总产值（力元）工业总产值（力元）总量（万人）货运总量（力吨）地方财政预越内收入6元）城乡居民年底储蓄余额（万元）在岗职工人数（力人）在岗职工工资总额（万元）北1249.900.59781843427199997062032345562279086326806646410.805773301犬910.170.58091501136226455023259263171128073II301931202.

9、682254343右召E庄875.400.23322918680688576829291911352348709587595.60758877太299.920.656323603827377501937II895203277394310088.65654023呼和浩特207.780.441236534381645223512623105783139658842.11309337沈阳677.080.6299129541858267337782154125679199016998135.451152811人连545.310.4946187973984263851078019187709227755

10、679694.15965922长春691.230.406818532105966343481095323570964803744102.63884447哈5滨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宁波538.410.25471428235106228662221510298501723524635062.15824034合肥429.950.3184628764251412548931517233628162293147.27369577福州583.130.2733215228S6555351S8517190467524503022069.59680607门128.990.4865333374575112437282570418758210833146.93657484南昌424.200.39886882892305

12、88136743189167714264046062.08479,555济557.630.40851486302628588259151177546069041269708331756696I1!岛702.970.36932382320II49203613408170386584354978045103.52961704郑州615.360.34246774255287601104336768387252513533884.66696848武汉740.200.5869121129175060859793154426046585748055149.201314766长沙582.470.310711

13、46367309817987065718323660346124469.57596986广州685.000.62141600738233481392200723854176149920401811182.813047594深圳119.850.793129966220368295875442741847908951990091.261890338南宁285.870,4064720486114969151303293149700219091845.09371809沟口54.380.83544481571746153-152356115174162680019.01198138重庆3072.340.

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

15、州287.590.5445259444294088418324749169540264156865.83550890133.950.522765848711310174611694913485505127.21219251银川95.380.570917160366122621()&11937475881410323.72178621乌鲁木齐158.920.824478513184724126689041254870236550855.275176222运行结果cwprinl(cwbook.lxi350)fid=6数据标准化结果如下: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、关系数矩阵: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各主成分贡献率:newraie=0.62600.23510.053()().0321().02570.01230.00990.0024().0004第一、二主成分的载荷:0.690I-0.64270.1483().60070.85150.46560-94630.91170.95370-95890.95060.8414-0.680503167-0.67540.04260.32990.18620.01090.2