巧用Matlab進行主成分降維

遼寧科技大學(xué)理學(xué)院12haidongkangPAGEPAGE58巧用Matlab實現(xiàn)主成分分析1.概述Matlab語言是當(dāng)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。它起源于矩陣運算，并已經(jīng)發(fā)展成一種高度集成的計算機語言。它提供了強大的科學(xué)運算、靈活的程序設(shè)計流程、高質(zhì)量的圖形可視化與界面設(shè)計、與其他程序和語言的便捷接口的功能。Matlab語言在各國高校與研究單位起著重大的作用。主成分分析是把原來多個變量劃為少數(shù)幾個綜合指標(biāo)的一種統(tǒng)計分析方法，從數(shù)學(xué)角度來看，這是一種降維處理技術(shù)。1.1主成分分析計算步驟PCA=1\*GB3①計算相關(guān)系數(shù)矩陣（1）在（3.5.3）式中，rij（i，j=1，2，…，p）為原變量的xi與xj之間的相關(guān)系數(shù)，其計算公式為（2）因為R是實對稱矩陣（即rij=rji），所以只需計算上三角元素或下三角元素即可。=2\*GB3②計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個分量。=3\*GB3③計算主成分貢獻率及累計貢獻率主成分的貢獻率為累計貢獻率為一般取累計貢獻率達85—95%的特征值所對應(yīng)的第一、第二，…，第m（m≤p）個主成分。=4\*GB3④計算主成分載荷其計算公式為（3）result=cwfac(v1);cwscore(v1,result);4.程序測試4.1原始數(shù)據(jù)中國大陸35個大城市某年的10項社會經(jīng)濟統(tǒng)計指標(biāo)數(shù)據(jù)見下表。城市名稱年底總?cè)丝?萬人)非農(nóng)業(yè)人口比(%)農(nóng)業(yè)總產(chǎn)值(萬元)工業(yè)總產(chǎn)值(萬元)客運總量(萬人)貨運總量(萬噸)地方財政預(yù)算內(nèi)收入(萬元)城鄉(xiāng)居民年底儲蓄余額(萬元)在崗職工人數(shù)(萬人)在崗職工工資總額(萬元)北京1249.900.59781843427199997062032345562279086326806646410.805773301天津910.170.5809150113622645502325926317112807311301931202.682254343石家莊875.400.23322918680688576829291911352348709587595.60758877太原299.920.65632360382737750193711895203277394310088.65654023呼和浩特207.780.441236534381645223512623105783139658842.11309337沈陽677.080.6299129541858267337782154125679199016998135.451152811大連545.310.4946187973984263851078019187709227755679694.15965922長春691.230.406818532105966343481095323570964803744102.63884447哈爾濱927.090.462726638554186123672075204814436450020172.791309151上海1313.120.7384206901954529098640644485431850025971200336.845605445南京537.440.53419891991307273714269111936642995680472113.811357861杭州616.050.35561414737120007961788311684449593742596796.901180947寧波538.410.25471428235106228662221510298501723524635062.15824034合肥429.950.3184628764251412548931517233628162293147.27369577福州583.130.27332152288655535188517190467524503022069.59680607廈門128.990.4865333374575112437282570418758210833146.93657484南昌424.200.3988688289230588136743189167714264046062.08479,555濟南557.630.408514863026285882591511775460690412697083.31756696青島702.970.369323823201149203613408170386584354978045103.52961704鄭州615.360.34246774255287601104336768387252513533884.66696848武漢740.200.5869121129175060859793154426046585748055149.201314766長沙582.470.31071146367309817987065718323660346124469.57596986廣州685.000.62141600738233481392200723854176149920401811182.813047594深圳119.850.793129966220368295875442741847908951990091.261890338南寧285.870.4064720486114969151303293149700219091845.09371809?？?4.380.83544481571746153452356115174162680019.01198138重慶3072.340.2067416878085855255244125124898,9129090969223.731606804成都1003.560.3351935590589428940140196325611897479684132.891200671貴陽321.500.45573620612247934157034143197908178774855.28419681昆明473.390.38657933563605729560412042524216412790088.11842321西安674.500.409473990536659421031197664088965863980114.01885169蘭州287.590.5445259444294088418324749169540264156865.83550890西寧133.950.522765848711310174614694913485505127.21219251銀川95.380.5709171603661226210611937475881410323.72178621烏魯木齊158.920.824478513184724126689041254870236550855.275176224.2運行結(jié)果>>cwprint('cwbook.txt',35,10)fid=6數(shù)據(jù)標(biāo)準化結(jié)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=1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689-0.34441.0000-0.47500.3096-0.35390.19710.35710.26000.15700.30900.8425-0.47501.00000.33580.58910.50560.32360.44560.55750.37420.36030.30960.33581.00000.15070.76640.94120.84800.73200.86140.7390-0.35390.58910.15071.00000.42940.19710.31820.38930.25950.62150.19710.50560.76640.42941.00000.83160.89660.93020.90270.40390.35710.32360.94120.19710.83161.00000.92330.83760.95270.49670.26000.44560.84800.31820.89660.92331.00000.92010.97310.67610.15700.55750.73200.38930.93020.83760.92011.00000.93960.46890.30900.37420.86140.25950.90270.95270.97310.93961.0000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各主成分貢獻率：newrate=0.62600.23510.05300.03210.02570.01230.00990.00310.00240.0004第一、二主成分的載荷：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.