巧用Matlab进行主成分降维

辽宁科技大学理学院12haidongkangPAGEPAGE58巧用Matlab实现主成分分析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，并已经发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。1.1主成分分析计算步骤PCA=1\*GB3①计算相关系数矩阵（1）在（3.5.3）式中，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计算主成分贡献率及累计贡献率主成分的贡献率为累计贡献率为一般取累计贡献率达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）result=cwfac(v1);cwscore(v1,result);4.程序测试4.1原始数据中国大陆35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)北京1249.900.59781843427199997062032345562279086326806646410.805773301天津910.170.5809150113622645502325926317112807311301931202.682254343石家庄875.400.23322918680688576829291911352348709587595.60758877太原299.920.65632360382737750193711895203277394310088.65654023呼和浩特207.780.441236534381645223512623105783139658842.11309337沈阳677.080.6299129541858267337782154125679199016998135.451152811大连545.310.4946187973984263851078019187709227755679694.15965922长春691.230.406818532105966343481095323570964803744102.63884447哈尔滨927.090.462726638554186123672075204814436450020172.791309151上海1313.120.7384206901954529098640644485431850025971200336.845605445南京537.440.53419891991307273714269111936642995680472113.811357861杭州616.050.35561414737120007961788311684449593742596796.901180947宁波538.410.25471428235106228662221510298501723524635062.15824034合肥429.950.3184628764251412548931517233628162293147.27369577福州583.130.27332152288655535188517190467524503022069.59680607厦门128.990.4865333374575112437282570418758210833146.93657484南昌424.200.3988688289230588136743189167714264046062.08479,555济南557.630.408514863026285882591511775460690412697083.31756696青岛702.970.369323823201149203613408170386584354978045103.52961704郑州615.360.34246774255287601104336768387252513533884.66696848武汉740.200.5869121129175060859793154426046585748055149.201314766长沙582.470.31071146367309817987065718323660346124469.57596986广州685.000.62141600738233481392200723854176149920401811182.813047594深圳119.850.793129966220368295875442741847908951990091.261890338南宁285.870.4064720486114969151303293149700219091845.09371809海口54.380.83544481571746153452356115174162680019.01198138重庆3072.340.2067416878085855255244125124898,9129090969223.731606804成都1003.560.3351935590589428940140196325611897479684132.891200671贵阳321.500.45573620612247934157034143197908178774855.28419681昆明473.390.38657933563605729560412042524216412790088.11842321西安674.500.409473990536659421031197664088965863980114.01885169兰州287.590.5445259444294088418324749169540264156865.83550890西宁133.950.522765848711310174614694913485505127.21219251银川95.380.5709171603661226210611937475881410323.72178621乌鲁木齐158.920.824478513184724126689041254870236550855.275176224.2运行结果>>cwprint('cwbook.txt',35,10)fid=6数据标准化结果如下：v1=0.05810.03560.04350.06800.05570.11120.11940.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.02130.03760.02430.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03910.02730.02840.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.01020.00630.01790.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.02230.01680.03440.03490.02860.02550.02680.03770.02590.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.00250.04970.00110.00240.01460.00570.00490.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.01460.0125相关系数矩阵:std=1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390-0.35390.58910.15071.00000.42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46890.30900.37420.86140.25950.90270.95270.97310.93961.0000特征向量(vec)：vec=-0.13670.2282-0.26280.19390.6371-0.21630.3176-0.1312-0.41910.2758-0.0329-0.02170.00090.0446-0.1447-0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.24010.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.20650.34030.04040.14080.08960.0380-0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.09870.35500.3280-0.01390.00710.3832-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.16680.3799特征值(val)val=0.003900000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.530000000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分贡献率：newrate=0.62600.23510.05300.03210.02570.01230.00990.00310.00240.0004第一、二主成分的载荷：0.6901-0.64270.14830.84140.6007-0.68050.85150.31670.4656-0.67540.94630.04260.91170.32990.95370.18620.95890.01090.95060.2558第一、二、三、四主成分的得分：score=0.71850.04990.76842.00000.38060.03860.41924.00000.1848-0.04330.141421.00000.11860.03110.149720.00000.05490.01150.066433.00000.22880.00700.23587.00000.2364-0.00810.228310.00000.1778-0.01670.161116.00000.2292-0.03370.195514.00000.83820.