matlab主成分分析法1

第页1.概述Matlab语言是当今国际上科学界(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，并已经开展成一种高度集成的计算机语言。它提供了强大的科学运算、灵活的程序设计流程、高质量的图形可视化与界面设计、与其他程序和语言的便捷接口的功能。Matlab语言在各国高校与研究单位起着重大的作用。主成分分析是把原来多个变量划为少数几个综合指标的一种统计分析方法，从数学角度来看，这是一种降维处理技术。=1\*GB3①计算相关系数矩阵〔1〕在〔，rij〔i，j=1，2，…，p〕为原变量的xi与xj之间的相关系数，其计算公式为〔2〕因为R是实对称矩阵〔即rij=rji〕，所以只需计算上三角元素或下三角元素即可。=2\*GB3②计算特征值与特征向量首先解特征方程，通常用雅可比法〔Jacobi〕求出特征值，并使其按大小顺序排列，即；然后分别求出对应于特征值的特征向量。这里要求=1，即，其中表示向量的第j个分量。=3\*GB3③计算主成分奉献率及累计奉献率主成分的奉献率为累计奉献率为一般取累计奉献率达85—95%的特征值所对应的第一、第二，…，第m〔m≤p〕个主成分。=4\*GB3④计算主成分载荷其计算公式为〔3〕得到各主成分的载荷以后，还可以按照〔，得到各主成分的得分〔4〕在软件Matlab中实现主成分分析可以采取两种方式实现：一是通过编程来实现；二是直接调用Matlab种自带程序实现。下面主要主要介绍利用Matlab的矩阵计算功能编程实现主成分分析。主函数子函数——用总和标准化法标准化矩阵——计算相关系数矩阵；计算特征值和特征向量；对主成分进展排序；计算各特征值奉献率；挑选主成分〔累计奉献率大于85%〕，输出主成分个数；计算主成分载荷——计算各主成分得分、综合得分并排序——读入数据文件；调用以上三个函数并输出结果总和标准化法标准化矩阵%cwstd.m,用总和标准化法标准化矩阵functionstd=cwstd(vector)cwsum=sum(vector,1);%对列求和[a,b]=size(vector);%矩阵大小,a为行数,b为列数fori=1:aforj=1:bstd(i,j)=vector(i,j)/cwsum(j);endend计算相关系数矩阵functionresult=cwfac(vector);fprintf('相关系数矩阵:\n')std=CORRCOEF(vector)%计算相关系数矩阵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);%对特征根进展排序，y为排序结果，i为索引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奉献率：\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%记下累积奉献率大85%的特征值的序号放入newi中fprintf('主成分数：%g\n\n',length(newi));fprintf('主成分载荷：\n')forp=1:length(newi)forq=1:length(y)result(q,p)=sqrt(newval(newi(p)))*vec(q,newi(p));endend%计算载荷disp(result)%cwscore.m,计算得分functionscore=cwscore(vector1,vector2);sco=vector1*vector2;csum=sum(sco,2);[newcsum,i]=sort(-1*csum);[newi,j]=sort(i);fprintf('计算得分：\n')score=[sco,csum,j]%得分矩阵：sco为各主成分得分；csum为综合得分；j为排序结果functionprint=cwprint();%为文本文件文件名，a为矩阵行数(样本数)，b为矩阵列数(变量指标数)fid=fopen(,'r')vector=fscanf(fid,'%g',[ab]);fprintf('标准化结果如下：\n')v1=cwstd(vector)result=cwfac(v1);cwscore(v1,result);中国大陆35个大城市某年的10项社会经济统计指标数据见下表。城市名称年底总人口(万人)非农业人口比(%)农业总产值(万元)工业总产值(万元)客运总量(万人)货运总量(万吨)地方财政预算内收入(万元)城乡居民年底储蓄余额(万元)在岗职工人数(万人)在岗职工工资总额(万元)北京1818434271999970620323455622790863268066465773301天津91501136226455023259263171128073113019312254343石家庄229186806885768292919113523487095875758877太原323603827377501937118952032773943100654023呼和浩特2365343816452235126231057831396588309337沈阳91295418582673377821541256791990169981152811大连61879739842638510780191877092277556796965922长春818532105966343481095323570964803744884447哈尔滨7266385541861236720752048144364500201309151上海142069019545290986406444854318500259712005605445南京198919913072737142691119366429956804721357861杭州6141473712000796178831168444959374259671180947宁波714282351062286622215102985017235246350824034合肥46287642514125489315172336281622931369577福州321522886555351885171904675245030220680607厦门53333745751124372825704187582108331657484南昌86882892305881367431891677142640460479,555济南5148630262858825915117754606904126970756696青岛323823201149203613408170386584354978045961704郑州467742552876011043367683872525135338696848武汉91211291750608597931544260465857480551314766长沙711463673098179870657183236603461244596986广州416007382334813922007238541761499204018113047594深圳12996622036829587544274184790895199001890338南宁47204861149691513032931497002190918371809海口444815717461534523561151741626800198138重庆37416878085855255244125124898,91290909691606804成都119355905894289401401963256118974796841200671贵阳736206122479341570341431979081787748419681昆明579335636057295604120425242164127900842321西安473990536659421031197664088965863980885169兰州52594442940884183247491695402641568550890西宁7658487113101746146949134855051219251银川91716036612262106119374758814103178621乌鲁木齐4785131847241266890412548702365508517622>>cwprint('cwbook.txt',35,10)fid=6数据标准化结果如下：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相关系数矩阵: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