巧用Matlab進(jìn)行主成分降維

1、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)界尤其是自動限制領(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)開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計算相關(guān)系數(shù)矩陣rp1rp2rpp(1)在3.5.3式中, 公式為rj (i, j=1 ,2,p為原變量的xi與x

2、j之間的相關(guān)系數(shù),其計算r1 ijn(Xkik 1n(Xkik 1Xi )(XkjXj)nXi)2(XkjXj)2k 1(2)由于R是實對稱矩陣即rj=rji,所以只需計算上三角元素或下三角元素即可. 計算特征值與特征向量首先解特征方程| I R 0,通常用雅可比法Jacobi求出特征值ii 12 ,p,并使其按大小順序排列,即12, p 0;然后分別求出對應(yīng)于特征值i的特征向量peii1,2, , p.這里要求|同二1,即ej21 ,其中ej表示向量e的第j個分量j 1計算主成分奉獻(xiàn)率及累計奉獻(xiàn)率主成分Zi的奉獻(xiàn)率為累計奉獻(xiàn)率為一般取累計奉獻(xiàn)率達(dá)85-95%的特征值1, 2, , m所對應(yīng)的

3、第一、第二,第m m 0.85 break;endend%記下累積奉獻(xiàn)率大85%的特征值的序號放入newi中fprintf(主成分?jǐn)?shù)：gnn,length(newi);fprintf(主成分載荷：n)for p=1:length(newi)for q=1:length(y)result(q,p)=sqrt(newval(newi(p)*vec(q,newi(p);endend%計算載荷disp(result)3.3 cwscore.m%cwscore.m,計算得分function score=cwscore(vector1,vector2);sco=vector1*vector2;csum=s

4、um(sco,2);newcsum,i=sort(-1*csum);newi,j=sort(i);fprintf(計算得分：n)score=sco,csum,j%得分矩陣：sco為各主成分得分；csum為綜合得分；j為排序結(jié)果3.4 cwprint.m%cwprint.mfunction print=cwprint(filename,a,b);%filename為文本文件文件名,a為矩陣行數(shù)(樣本數(shù)),b為矩陣列數(shù)(變量指標(biāo)數(shù))fid=fopen(filename,r)vector=fscanf(fid,%g,a b);fprintf(標(biāo)準(zhǔn)化結(jié)果如下：n)v1=cwstd(vector)res

5、ult=cwfac(v1);cwscore(v1,result);4 .程序測試中國大陸35個大城市某年的10項社會經(jīng)濟(jì)統(tǒng)計指標(biāo)數(shù)據(jù)見下表.年底非農(nóng)工業(yè)城鄉(xiāng)居民城市農(nóng)業(yè)客運貨運地方財政在崗職在崗職工業(yè)總量總量年底儲蓄工資總額總?cè)丝诳偖a(chǎn)值總產(chǎn)值預(yù)算內(nèi)收余額工人數(shù)名稱萬人人口比%萬元萬元萬人萬噸入萬元萬人萬元萬元北 京1 249.900.597 81 843 42719 999 70620 32345 5622 790 86326 806 646410.805 773 301天 津910.170.580 91 501 13622 645 5023 25926 3171 128 07311 301

6、931202.682 254 343石家莊875.400.233 22 918 6806 885 7682 9291 911352 3487 095 87595.60758 877太 原299.920.656 3236 0382 737 7501 93711 895203 2773943 10088.65654 023呼和浩特207.780.441 2365 343816 4522 3512 623105 7831 396 58842.11309 337沈 陽677.080.629 91 295 4185 826 7337 78215 412567 9199 016 998135.451 15

7、2 811大 連545.310.494 61 879 7398 426 38510 78019 187709 2277 556 79694.15965 922長 春691.230.406 81 853 2105 966 3434 8109 532357 0964 803 744102.63884 447哈爾濱927.090.462 72 663 8554 186 1236 7207 520481 4436 450 020172.791 309 151上 海1 313.120.738 42 069 01954 529 0986 40644 4854 318 50025 971 200336.84

8、5 605 445南 京537.440.534 1989 19913 072 73714 26911 193664 2995 680 472113.811 357 861杭 州616.050.355 61 414 73712 000 79617 88311 684449 5937 425 96796.901 180 947寧 波538.410.254 71 428 23510 622 86622 21510 298501 7235 246 35062.15824 034合 肥429.950.318 4628 7642 514 1254 8931 517233 6281 622 93147.27

9、369 577福 州583.130.273 32 152 2886 555 3518 8517 190467 5245 030 22069.59680 607廈門128.990.486 5333 3745 751 1243 7282 570418 7582 108 33146.93657 484南昌424.200.398 8688 2892 305 8813 6743 189167 7142 640 46062.08479 ,555濟(jì) 南557.630.408 51 486 3026 285 8825 91511 775460 6904 126 97083.31756 696青 島702.97

10、0.369 32 382 32011 492 03613 40817 038658 4354 978 045103.52961 704鄭 州615.360.342 4677 4255 287 60110 4336 768387 2525 135 33884.66696 848武 漢740.200.586 91 211 2917 506 0859 79315 442604 6585 748 055149.201 314 766長 沙582.470.310 71 146 3673 098 1798 7065 718323 6603 461 24469.57596 986廣 州685.000.621

11、 41 600 73823 348 13922 00723 8541 761 49920 401 811182.813 047 594深 圳119.850.793 1299 66220 368 2958 7544 2741 847 9089 519 90091.261 890 338南 寧285.870.406 4720 4861 149 6915 1303 293149 7002 190 91845.09371 809?？?4.380.835 444 815717 4615 3452 356115 1741 626 80019.01198 138重 慶3 072.340.206 74 168

12、 7808 585 52552 44125 124898,9129 090 969223.731 606 804成 都1 003.560.3351 935 5905 894 28940 14019 632561 1897 479 684132.891 200 671貴 陽321.500.455 7362 0612 247 93415 7034 143197 9081 787 74855.28419 681昆明473.390.386 5793 3563 605 7295 60412 042524 2164 127 90088.11842 321西 安674.500.409 4739 9053 6

13、65 94210 3119 766408 8965 863 980114.01885 169蘭 州287.590.544 5259 4442 940 8841 8324 749169 5402 641 56865.83550 890西寧133.950.522 765 848711 3101 7461 46949 134855 05127.21219 251銀川95.380.570 9171 603661 2262 1061 19374 758814 10323.72178 621烏魯木齊158.920.824 478 5131 847 2412 6689 041254 8702 365 508

14、55.27517 6224.2運行結(jié)果 cwprint(cwbook.txt,35,10)數(shù)據(jù)標(biāo)準(zhǔn)化結(jié)果如下:fid =v1 =0.05810.03560.04350.06800.0557 0.1112 0.1194 0.1184 0.1083 0.13920.04230.03460.03540.07700.0089 0.0642 0.0483 0.0499 0.0534 0.05440.04070.01390.06880.02340.0080 0.0047 0.01510.0314 0.0252 0.01830.01390.03910.00560.00930.0053 0.0290 0.00

15、87 0.0174 0.0234 0.01580.00970.02630.00860.00280.0064 0.0064 0.0045 0.0062 0.01110.00750.03150.03750.03050.01980.0213 0.0376 0.0243 0.0398 0.0357 0.02780.02530.02950.04430.02860.0295 0.0468 0.0304 0.0334 0.0248 0.02330.03210.02420.04370.02030.0132 0.0233 0.0153 0.0212 0.0270 0.02130.04310.02760.0628

16、0.01420.0184 0.0184 0.0206 0.0285 0.0455 0.03160.06100.04400.04880.18530.0176 0.1086 0.1848 0.1148 0.0888 0.13520.02500.03180.02330.04440.0391 0.0273 0.0284 0.0251 0.0300 0.03270.02860.02120.03340.04080.0490 0.0285 0.0192 0.0328 0.0255 0.02850.02500.01520.03370.03610.0609 0.0251 0.0215 0.0232 0.0164

17、 0.01990.02000.01900.01480.00850.0134 0.0037 0.0100 0.0072 0.0125 0.00890.02710.01630.05080.02230.0243 0.0175 0.0200 0.0222 0.0183 0.01640.00600.02900.00790.01950.0102 0.0063 0.0179 0.0093 0.0124 0.01590.01970.02370.01620.00780.0101 0.0078 0.0072 0.0117 0.0164 0.01160.02590.02430.03500.02140.0162 0.

18、0287 0.0197 0.0182 0.0220 0.01820.03270.02200.05620.03910.0367 0.0416 0.0282 0.0220 0.0273 0.02320.02860.02040.01600.01800.0286 0.0165 0.0166 0.0227 0.0223 0.01680.03440.03490.02860.02550.0268 0.0377 0.0259 0.0254 0.0393 0.03170.02710.01850.02700.01050.0239 0.0140 0.0139 0.0153 0.0183 0.01440.03180.

19、03700.03770.07930.0603 0.0582 0.0754 0.0901 0.0482 0.07350.00560.04720.00710.06920.0240 0.0104 0.0791 0.0421 0.0240 0.04560.01330.02420.01700.00390.0141 0.0080 0.0064 0.0097 0.0119 0.00900.00250.04970.00110.00240.0146 0.0057 0.0049 0.0072 0.0050 0.00480.14280.01230.09830.02920.1437 0.0613 0.0385 0.0

20、402 0.0590 0.03870.04660.01990.04560.02000.1100 0.0479 0.0240 0.0331 0.0350 0.02900.01490.02710.00850.00760.0430 0.0101 0.0085 0.0079 0.0146 0.01010.02200.02300.01870.01230.0154 0.0294 0.0224 0.0182 0.0232 0.02030.03130.02440.01740.01250.0283 0.0238 0.0175 0.0259 0.0300 0.02130.01340.03240.00610.010

21、00.0050 0.0116 0.0073 0.0117 0.0173 0.01330.00620.03110.00160.00240.0048 0.0036 0.00210.0038 0.0072 0.005360.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.00740.04910.00190.00630.00730.02210.01090.01050.01460.0125相關(guān)系數(shù)矩陣：std =1.0000-0.34440.84250.36030.73900.62150.40390.49670.67610.4689

22、-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

23、.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

24、.3176-0.1312-0.41910.2758-0.0329-0.02170.00090.0446-0.1447 -0.44370.4058-0.55620.54870.0593-0.0522-0.02800.2040-0.0492-0.5472-0.42250.34400.3188-0.44380.2401 0.0067-0.4176-0.2856-0.23890.1926-0.4915-0.41890.27260.20650.3403 0.04040.14080.08960.0380-0.1969-0.0437-0.4888-0.6789-0.44050.1861-0.03430.23

25、600.0640-0.82940.03770.26620.1356-0.12900.02780.37820.29810.47390.56850.23580.1465-0.1502-0.26310.12450.21520.36440.15670.3464-0.64850.2489-0.40430.2058-0.07040.04620.12140.38120.4879-0.57070.12170.17610.0987 0.3550 0.3280-0.01390.00710.3832-0.7894-0.16280.19250.2510-0.04220.26940.03960.04560.16680.

26、3799特征值(val)val =0.003900000000000.024000000000000.030700000000000.099100000000000.123200000000000.256600000000000.320700000000000.5300C0000000002.351400000000006.2602特征根排序：6.260222.351380.5300470.3206990.2566390.1232410.09909150.03070880.02403550.00393387各主成分奉獻(xiàn)率：newrate =0.00040.6260 0.23510.0530 0.03210.0257 0.0123 0.0099 0.00310.0024第一、二主成分的載荷：0.690 1-0.64270.148 30.84140.600 7-0.68050.851 50.31670.465 6-0.67540.946 30.04260.911 70.32990.953 70.18620.958 90.01090.950 6A-A-弟 、0.2558三、四主成分的得分：score =0.718 50.049