數(shù)學(xué)建模-主成分分析法模板

1、根據(jù)主成分分析的方法，分析的數(shù)據(jù)。步驟如下：Step1：為了消除不同變量的量綱的影響，首先需要對變量進行標準化，設(shè)檢測數(shù)據(jù)樣本共有n個，指標共有P個,分別設(shè)Xi%，令訂匸12磚12,p)為第i個樣本第j個指標的值。作變換X.-E(X.)=jjjVar(X)jX-X得到標準化數(shù)據(jù)矩陣yij-sL，其中X二-,X,s2二-,(x-X)2jnijjnijjJi=1i=1Step2：在標準化數(shù)據(jù)矩陣Y=(y)的基礎(chǔ)上計算p個原始指標相關(guān)系數(shù)矩陣iJnxpR=(r)=ijpxpr11r21r12r22r1pr2pR=(r)=ijpxpr11r21r12r22r1pr2prp1rp2rpp其中，r=ij,

2、(x-X)(x-Xkiikjk=1,(X-X)2工(X-Xkiikjk=1k=1(i,j=1,2;：p)2jStep3：求相關(guān)系數(shù)矩陣R的特征值并排序0,再求出R的特征12p值相應(yīng)的正則化特征向量e=(e,e,.,e)，則第i個主成分表示為各指標X的ii1i2ipk組合Z=Ee-X。iikki=1Step4：計算累積貢獻率確定主成分的數(shù)目。主成分Z的貢獻率為iw=(i=1,2,p)i，kk=1累計貢獻率為,k呂(i=1,2,p)，kk=1一般取累計貢獻率達85%95%的特征值,所對應(yīng)的第1、第2,，12m第m(mWp)個主成分。Step5：計算主成分載荷，確定綜合得分。當主成分之間不相關(guān)時，主

3、成分載荷是主成分和各指標的相關(guān)系數(shù)，相關(guān)系數(shù)越大，說明主成分對該指標變量的代表性就越好，計算公式為l二p(z,x)二e(i,j二1,2,p)ijijiij+lx+lx+lxm22mppZ=lx+1x+lx11111221ppZ=lx+1x+lx22112222ppm=lxm11ZZ1112ZZZ=(z)=2122ijnxm:zZn1n2z1mz2m，其中Z表示第i個樣本第j個主成分得分，則:ijZnm第i個樣本的綜合得分f二込w-z(i=1,2,n);ikikk=1附件中共有28個月的數(shù)據(jù)，這里僅隨機選擇2005年4月的數(shù)據(jù)來說明利分析進行水質(zhì)綜合評價的過程(同理可進行其他月份的數(shù)據(jù)分析)。調(diào)

4、用MATLAB統(tǒng)計工具箱princomp函數(shù)，格式為：pc,score,latent,tsquare=princomp(ingredients)其中ingredients指標準化后的樣本指標矩陣，pc是指各主成分關(guān)于指標的線性組合的系數(shù)矩陣，score為各主成分得分，latent是方差矩陣的特征值,tsquare為HotellingT2統(tǒng)計量。各種指標的相關(guān)系數(shù)矩陣：填充數(shù)據(jù)）各個主成分的貢獻率：表1主成分的貢獻率表特征值貝獻率累積貝獻率第1主成分第2主成分由表可看出，前三個主成分的累積貢獻率已達到（填充數(shù)據(jù)），取控制參數(shù)a=0.06（因為28個月中前三個成分貢獻率最低為94%）,因此取前三個

5、主成分對（填充文字）進行綜合評價。根據(jù)R的特征值的相應(yīng)的正則化單位特征向量，前m個主成分關(guān)于指標的線性組合為：（填充表達式）（分析）根據(jù)線性表達式中的系數(shù)及符號，可對各主成分的實際意義作如下解釋第1主成分為除（變量）之外的三項指標的綜合；第2主成分與（變量）成正相關(guān)，與（變量）成負相關(guān)；第3主成分為除（變量）之外的三項指標的綜合。以各個主成分的方差貢獻率為權(quán)重可得到（文字）的最終綜合評價。表2XX綜合評價表第1主成分第2主成分綜合得分排名樣本1樣本2上表給出了XX的綜合評價，綜合得分越高說明（文字），排名越高x=0.05810.03560.04350.06800.05570.11120.119

6、40.11840.10830.13920.04230.03460.03540.07700.00890.06420.04830.04990.05340.05440.04070.01390.06880.02340.00800.00470.01510.03140.02520.01830.01390.03910.00560.00930.00530.02900.00870.01740.02340.01580.00970.02630.00860.00280.00640.00640.00450.00620.01110.00750.03150.03750.03050.01980.02130.03760.024

7、30.03980.03570.02780.02530.02950.04430.02860.02950.04680.03040.03340.02480.02330.03210.02420.04370.02030.01320.02330.01530.02120.02700.02130.04310.02760.06280.01420.01840.01840.02060.02850.04550.03160.06100.04400.04880.18530.01760.10860.18480.11480.08880.13520.02500.03180.02330.04440.03910.02730.028

8、40.02510.03000.03270.02860.02120.03340.04080.04900.02850.01920.03280.02550.02850.02500.01520.03370.03610.06090.02510.02150.02320.01640.01990.02000.01900.01480.00850.01340.00370.01000.00720.01250.00890.02710.01630.05080.02230.02430.01750.02000.02220.01830.01640.00600.02900.00790.01950.01020.00630.017

9、90.00930.01240.01590.01970.02370.01620.00780.01010.00780.00720.01170.01640.01160.02590.02430.03500.02140.01620.02870.01970.01820.02200.01820.03270.02200.05620.03910.03670.04160.02820.02200.02730.02320.02860.02040.01600.01800.02860.01650.01660.02270.02230.01680.03440.03490.02860.02550.02680.03770.025

10、90.02540.03930.03170.02710.01850.02700.01050.02390.01400.01390.01530.01830.01440.03180.03700.03770.07930.06030.05820.07540.09010.04820.07350.00560.04720.00710.06920.02400.01040.07910.04210.02400.04560.01330.02420.01700.00390.01410.00800.00640.00970.01190.00900.00250.04970.00110.00240.01460.00570.004

11、90.00720.00500.00480.14280.01230.09830.02920.14370.06130.03850.04020.05900.03870.04660.01990.04560.02000.11000.04790.02400.03310.03500.02900.01490.02710.00850.00760.04300.01010.00850.00790.01460.01010.02200.02300.01870.01230.01540.02940.02240.01820.02320.02030.03130.02440.01740.01250.02830.02380.01750.02590.03000.02130.01340.03240.00610.01000.00500.01160.00730.01170.01730.01330.00620.03110.00160.00240.00480.00360.00210.00380.00720.00530.00440.03400.00400.00220.00580.00290.00320.00360.00630.00430.0074