應(yīng)用多元分析第八章因子分析

優(yōu)選應(yīng)用多元分析第八章因子分析當(dāng)前1頁，總共78頁。例

為了評價(jià)即將進(jìn)大學(xué)的高中生的學(xué)習(xí)能力，抽了200名高中生進(jìn)行問卷調(diào)查，共50個(gè)問題。素有這些問題可以歸結(jié)為閱讀理解、數(shù)學(xué)水平和藝術(shù)素養(yǎng)三個(gè)方面。例

公司老板對48名應(yīng)聘者進(jìn)行面試，并給出他們在15個(gè)方面的得分，這15個(gè)方面是：申請書的形式(x1)、外貌(x2)、專業(yè)能力(x3)、討人喜歡(x4)、自信心(x5)、精明(x6)、誠實(shí)(x7)、推銷能力(x8)、經(jīng)驗(yàn)(x9)、積極性(x10)、抱負(fù)(x11)、理解能力(x12)、潛力(x13)、交際能力(x14)、適應(yīng)性(x15)。通過因子分析，這15個(gè)方面可歸結(jié)為應(yīng)聘者的外露能力、討人喜歡的程度、經(jīng)驗(yàn)、專業(yè)能力和外貌。當(dāng)前2頁，總共78頁。§8.2因子模型一、數(shù)學(xué)模型當(dāng)前3頁，總共78頁。

假設(shè)條件當(dāng)前4頁，總共78頁。二、因子模型的性質(zhì)當(dāng)前5頁，總共78頁。例8.2.1設(shè)隨機(jī)向量(x1,x2,x3,x4)’的協(xié)方差矩陣為當(dāng)前6頁，總共78頁。2、模型不受單位影響當(dāng)前7頁，總共78頁。3、因子載荷不唯一（相差一個(gè)正交變換）注：在實(shí)際中，利用因子載荷陣的不唯一性對因子進(jìn)行旋轉(zhuǎn)，使得新的因子有更好的實(shí)際意義。當(dāng)前8頁，總共78頁。三、因子載荷矩陣的統(tǒng)計(jì)意義2、A的行元素平方和——公因子對原始變量的方差貢獻(xiàn)1、A的元素aij——原始變量xi與公因子fj之間的協(xié)方差函數(shù)當(dāng)前9頁，總共78頁。3、A的列平方和——公共因子fj對x的貢獻(xiàn)率當(dāng)前10頁，總共78頁。Principalcomponents:主成分法Unweightedleastsquare:不加權(quán)最小平方法Generalizedleastsquares:普通最小平方法Maximumlikelihood:最大似然法Principalaxisfactoring:主因子法Alphafactoring:α因子提取法Imagefactoring:映象因子提取法常用確定q的方法是按特征根由大至小的次序抽取，直到與接近為止?！?.3參數(shù)估計(jì)當(dāng)前11頁，總共78頁。一、主成分法：

當(dāng)前12頁，總共78頁。例

在例中，分別取m=1,m=2,用主成分法估計(jì)的因子載荷和共性方差如下表：

變量m=1m=2ai1(f1)

ai1f1ai2f2

0.8170.8670.9150.9490.9590.9380.9440.880

0.6680.7520.8380.9000.9200.8790.8910.774

0.8170.5310.8670.4320.9150.2330.9490.0120.959-0.1310.938-0.2920.944-0.2870.880-0.411

0.9500.9390.8920.9000.9380.9650.9730.943所解釋的總方差的累計(jì)比例0.828

0.8280.938當(dāng)前13頁，總共78頁。相應(yīng)于m=2的解的殘差矩陣為：

當(dāng)前14頁，總共78頁。dataexamp733(type=corr);inputx1-x8;cards;1.000.......0.9231.000......0.8410.8511.000.....0.7560.8070.8701.000....0.7000.7750.8350.9181.000...0.6190.6950.7790.8640.9281.000..0.6330.6970.7870.8690.9350.9751.000.0.5200.5960.7050.8060.8660.9320.9431.000;proc

factordata=examp733(type=corr);varx1-x8;proc

factordata=examp733(type=corr)n=2;varx1-x8;run;當(dāng)前15頁，總共78頁。proc

iml;x={1.000

0.923

0.841

0.756

0.700

0.619

0.633

0.520,

0.923

1.000

0.851

0.807

0.775

0.695

0.697

0.596,

0.841

0.851

1.000

0.870

0.835

0.779

0.787

0.705,

0.756

0.807

0.870

1.000

0.918

0.864

0.869

0.806,

0.700

0.775

0.835

0.918

1.000

0.928

0.935

0.866,

0.619

0.695

0.779

0.864

0.928

1.000

0.975

0.932,

0.633

0.697

0.787

0.869

0.935

0.975

1.000

0.943,

0.520

0.596

0.705

0.806

0.866

0.932

0.943

1.000};C={0.81717

0.53110,0.86729

0.43271,0.91517

0.23251,

0.94874

0.01185,0.95938-0.13148,0.93766-0.29268,

0.94397-0.28708,0.87981-0.41117};a={0.877

0.888

0.845

0.884

0.927

0.995

0.967

0.905};y=x-i(8)+diag(a);b=eigval(y);e={0.050

0.061

0.108

0.100

0.062

0.035

0.027

0.057};D=x-c*t(c)-diag(e);printybd;當(dāng)前16頁，總共78頁。1、給出共同度hi2的初步估計(jì)值hi*2

以第i個(gè)變量xi*與其它所有變量x1*,x2*,…,xi1*,xi+1*,…,xp*回歸的復(fù)相關(guān)系數(shù)的平方作為初始估計(jì)值2、求出約化相關(guān)陣計(jì)算Di*=1-hi*2,再計(jì)算出R*=R-D*3、求出特征根和特征向量由方程︱R*-λI︱=0求出，并利用特征根、特征向量求出因子載荷陣A14、求出D的估計(jì)，用估計(jì)值代替第二步的D*D的估計(jì)：D*（1）=R-A1A1′5、繼續(xù)第三步，直到A，

D的估計(jì)達(dá)到穩(wěn)定為止二、主因子法當(dāng)前17頁，總共78頁。例8.3.2在例中取m=2,選用xi與其他7個(gè)變量的復(fù)相關(guān)系數(shù)平方作為的初始估計(jì)值。計(jì)算得:當(dāng)前18頁，總共78頁。約相關(guān)矩陣為當(dāng)前19頁，總共78頁。ai1f1ai2f2

0.8070.4960.8580.4120.8900.2160.9390.0240.956-0.1140.938-0.2820.946-0.2810.874-0.378

0.8970.9060.8560.8810.9260.9600.9740.907所解釋的總方差的累計(jì)比例0.8160.914當(dāng)前20頁，總共78頁。相應(yīng)于m=2的解的殘差矩陣為：

當(dāng)前21頁，總共78頁。dataexamp733(type=corr);inputx1-x8;cards;1.000.......0.9231.000......0.8410.8511.000.....0.7560.8070.8701.000....0.7000.7750.8350.9181.000...0.6190.6950.7790.8640.9281.000..0.6330.6970.7870.8690.9350.9751.000.0.5200.5960.7050.8060.8660.9320.9431.000;proc

factorm=prinitpriors=smc;varx1-x8;proc

factorm=prinitpriors=smcn=2;varx1-x8;run;當(dāng)前22頁，總共78頁。

proc

factor

data=sasuser.exec65n=2method=prinitheywood;varx1-x8;run;proc

iml;x={1.00000

0.92264

0.84115

0.75603

0.70024

0.61946

0.63254

0.51995,0.92264

1.00000

0.85073

0.80663

0.77495

0.69538

0.69654

0.59618,0.84115

0.85073

1.00000

0.87017

0.83527

0.77861

0.78720

0.70499,

0.75603

0.80663

0.87017

1.00000

0.91804

0.86359

0.86905

0.80648,0.70024

0.77495

0.83527

0.91804

1.00000

0.92811

0.93470

0.86555,0.61946

0.69538

0.77861

0.86359

0.92811

1.00000

0.97464

0.93219,

0.63254

0.69654

0.78720

0.86905

0.93470

0.97464

1.00000

0.94318,0.51995

0.59618

0.70499

0.80648

0.86555

0.93219

0.94318

1.00000};y={0.81202

0.51251,0.86094

0.41594,0.90061

0.21099,0.93708

0.01791,0.95452-0.11798,

0.93843-0.28554,0.94696-0.28620,0.87304

-0.37739};z={1,1,1,1,1,1,1,1}-{0.92204479,0.91422910,0.85561058,0.87844540,0.92502009,0.96217654,0.97865293,0.90462155};a=diag(z);b=x-y*t(y)-a;printab;run;當(dāng)前23頁，總共78頁。三、極大似然法用迭代法求上述方程組的解。當(dāng)前24頁，總共78頁。當(dāng)前25頁，總共78頁。當(dāng)前26頁，總共78頁。當(dāng)前27頁，總共78頁。當(dāng)前28頁，總共78頁。當(dāng)前29頁，總共78頁。當(dāng)前30頁，總共78頁。

載荷矩陣A是不唯一的，有人建議添加一個(gè)計(jì)算上方便的的唯一性條件：當(dāng)前31頁，總共78頁。當(dāng)前32頁，總共78頁。當(dāng)前33頁，總共78頁。當(dāng)前34頁，總共78頁。當(dāng)前35頁，總共78頁。ai1f1ai2f2

0.731-0.6200.792-0.5450.855-0.3430.916-0.1610.958-0.0260.9720.1440.981-0.1430.923-0.249

0.9190.9240.8490.8650.9180.9660.9820.914所解釋的總方差的累計(jì)比例0.8010.917例8.3.3在例中，取m=2，同,極大似然法的計(jì)算結(jié)果如下表：的初始估計(jì)值與例當(dāng)前36頁，總共78頁。極大似然解的殘差矩陣為：

當(dāng)前37頁，總共78頁。dataexamp733(type=corr);inputx1-x8;cards;1.000.......0.9231.000......0.8410.8511.000.....0.7560.8070.8701.000....0.7000.7750.8350.9181.000...0.6190.6950.7790.8640.9281.000..0.6330.6970.7870.8690.9350.9751.000.0.5200.5960.7050.8060.8660.9320.9431.000;proc

factorm=ml;varx1-x8;proc

factorm=mln=2;varx1-x8;run;當(dāng)前38頁，總共78頁。TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodPriorCommunalityEstimates:SMCx1x2x3x4

0.877839350.888722470.844686300.88357282x5x6x7x80.927120960.955454730.967228140.90408487當(dāng)前39頁，總共78頁。PreliminaryEigenvalues:Total=101.310432Average=12.6638039EigenvalueDifferenceProportionCumulative

193.788831185.53188880.92580.9258

28.25694237.70562430.08151.0073

30.55131810.49621880.00541.0127

40.05509930.21273650.00051.0132

5-0.15763720.1387816-0.00161.0117

6-0.29641880.0631990-0.00291.0088

7-0.35961780.1684675-0.00351.0052

8-0.5280854-0.00521.00002factorswillberetainedbythePROPORTIONcriterion.當(dāng)前40頁，總共78頁。

IterationCriterionRidgeChangeCommunalities

10.33712300.00000.03300.910880.919410.854320.872500.921540.96588

0.981680.91119

20.33200700.00000.00650.917410.923640.849470.866060.919070.96667

0.982090.91339

30.33178100.00000.00130.918530.924910.848480.864860.918460.96664

0.982260.91341

40.33177200.00000.00030.918850.925070.848230.864600.918360.96665

0.982270.91345Convergencecriterionsatisfied.當(dāng)前41頁，總共78頁。SignificanceTestsBasedon10000ObservationsPr>TestDFChi-SquareChiSqH0:Nocommonfactors28142472.086<.0001HA:AtleastonecommonfactorH0:2Factorsaresufficient133315.7842<.0001HA:Morefactorsareneeded當(dāng)前42頁，總共78頁。TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodChi-SquarewithoutBartlett'sCorrection3317.3878Akaike'sInformationCriterion3291.3878Schwarz'sBayesianCriterion3197.6534TuckerandLewis'sReliabilityCoefficient0.9501SquaredCanonicalCorrelationsFactor1Factor20.992346680.92414870當(dāng)前43頁，總共78頁。EigenvaluesoftheWeightedReducedCorrelationMatrix:Total=141.845985Average=17.7307481EigenvalueDifferenceProportionCumulative

1129.662297117.4786080.91410.9141

212.18368911.5974400.08591.0000

30.5862490.4011620.00411.0041

40.1850870.1759800.00131.0054

50.0091070.0612700.00011.0055

6-0.0521630.239994-0.00041.0051

7-0.2921570.143967-0.00211.0031

8-0.436124-0.00311.0000T129.64951012.18376200.587805900.183956100.00909540-0.0535390-0.2925640-0.4354380當(dāng)前44頁，總共78頁。FactorPatternFactor1Factor2x10.73040-0.62079x20.79140-0.54661x30.85454-0.34348x40.91565-0.16171x50.95795-0.02600x60.972630.14371x70.980890.14189x80.922860.24854VarianceExplainedbyEachFactorFactorWeightedUnweightedFactor1129.6622976.40592383Factor212.1836890.93152570當(dāng)前45頁，總共78頁。TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodFinalCommunalityEstimatesandVariableWeightsTotalCommunality:Weighted=141.84599Unweighted=7.337450VariableCommunalityWeightx10.9188651412.3222163x20.9250959313.3456046x30.848210716.5890175x40.864559767.3855720x50.9183397412.2487688x60.9666533029.9870945x70.9822744556.4139209x80.9134505211.5537899當(dāng)前46頁，總共78頁。proc

iml;x={0.87783935

0.88872247

0.84468630

0.88357282

0.92712096

0.95545473

0.96722814

0.90408487};y={1.000

0.923

0.841

0.756

0.700

0.619

0.633

0.520,0.923

1.000

0.851

0.807

0.775

0.695

0.697

0.596,0.841

0.851

1.000

0.870

0.836

0.779

0.787

0.705,0.756

0.807

0.870

1.000

0.918

0.864

0.869

0.806,0.700

0.775

0.835

0.918

1.000

0.928

0.935

0.866,0.619

0.695

0.779

0.864

0.928

1.000

0.975

0.932,0.633

0.697

0.787

0.869

0.935

0.975

1.000

0.943,0.520

0.596

0.705

0.806

0.866

0.932

0.943

1.000};z=sqrt(inv(i(8)-diag(x)));s=z*y*z;t=eigval(s-i(8));printt;當(dāng)前47頁，總共78頁。prociml;h={0.918850.925070.848230.86460

0.91836

0.96665

0.98227

0.91345};r={1.000

0.923

0.841

0.756

0.700

0.619

0.633

0.520,0.923

1.000

0.851

0.807

0.775

0.695

0.697

0.596,0.841

0.851

1.000

0.870

0.836

0.779

0.787

0.705,0.756

0.807

0.870

1.000

0.918

0.864

0.869

0.806,0.700

0.775

0.835

0.918

1.000

0.928

0.935

0.866,0.619

0.695

0.779

0.864

0.928

1.000

0.975

0.932,0.633

0.697

0.787

0.869

0.935

0.975

1.000

0.943,0.520

0.596

0.705

0.806

0.866

0.932

0.943

1.000

};d1=sqrt(inv(i(8)-diag(h)));S=d1*r*d1;t=eigval(s-i(8));t1=eigvec(s-i(8))[,1:5];d=diag(t[1:5,1]);a=inv(d1)*t1*sqrt(d);b1=a[,1:2];b=b1[,##];c=a[##,];d1=1/(j(8,1)-b);printtt1dabcd1;特殊方差的倒數(shù)weight共性方差h2當(dāng)前48頁，總共78頁。T

129.649510

12.1837620

0.58780590

0.18395610

0.00909540

-0.0535390

-0.2925640

-0.4354380當(dāng)前49頁，總共78頁。

proc

factor

data=sasuser.exec65n=2method=mlheywood;varx1-x8;run;proc

iml;x={1.00000

0.92264

0.84115

0.75603

0.70024

0.61946

0.63254

0.51995,0.92264

1.00000

0.85073

0.80663

0.77495

0.69538

0.69654

0.59618,

0.84115

0.85073

1.00000

0.87017

0.83527

0.77861

0.78720

0.70499,

0.75603

0.80663

0.87017

1.00000

0.91804

0.86359

0.86905

0.80648,0.70024

0.77495

0.83527

0.91804

1.00000

0.92811

0.93470

0.86555,0.61946

0.69538

0.77861

0.86359

0.92811

1.00000

0.97464

0.93219,

0.63254

0.69654

0.78720

0.86905

0.93470

0.97464

1.00000

0.94318,0.51995

0.59618

0.70499

0.80648

0.86555

0.93219

0.94318

1.00000};y={0.731-0.620,0.792-0.545,0.855-0.343,

0.916-0.161,0.958-0.026,0.972

0.144,

0.981

0.143,0.923

0.249};z={1,1,1,1,1,1,1,1}-{0.919,0.924,0.849,0.865,0.918,0.966,

0.982,0.914};a=diag(z);b=x-y*t(y)-a;printab;run;當(dāng)前50頁，總共78頁。§8.4因子旋轉(zhuǎn)因子旋轉(zhuǎn)的意義：使公因子易于解釋。方法：正交旋轉(zhuǎn)，斜交旋轉(zhuǎn)。實(shí)施：對載荷矩陣作正交（斜交）變換。正交旋轉(zhuǎn)：當(dāng)前51頁，總共78頁。最大方差旋轉(zhuǎn)法：選擇正交矩陣T使得A*的所有m個(gè)列元素平方和的相對方差之和達(dá)到最大。當(dāng)前52頁，總共78頁。m=2時(shí)，設(shè)元因子載荷矩陣為當(dāng)前53頁，總共78頁。當(dāng)前54頁，總共78頁。當(dāng)前55頁，總共78頁。當(dāng)前56頁，總共78頁。當(dāng)前57頁，總共78頁。變量

f1f2f1f2f1f2

0.2740.9350.3760.8930.5430.7730.7120.6270.8130.5250.9020.3890.9030.3970.9360.2610.2870.9030.3810.8720.5410.7510.6950.6310.7990.5370.8950.3990.9000.4050.9090.2840.2880.9140.3790.8830.5410.7460.6890.6240.7970.5320.8990.3970.9060.4020.9140.281

所解釋的總方差的累計(jì)比例0.5230.9830.5100.9140.5120.917

例8.4.1在例中分別用最大方差旋轉(zhuǎn)法，旋轉(zhuǎn)后的因子載荷矩陣如下表

主成分

主因子

極大似然當(dāng)前58頁，總共78頁。dataexamp733(type=corr);inputx1-x8;cards;1.000.......0.9231.000......0.8410.8511.000.....0.7560.8070.8701.000....0.7000.7750.8350.9181.000...0.6190.6950.7790.8640.9281.000..0.6330.6970.7870.8690.9350.9751.000.0.5200.5960.7050.8060.8660.9320.9431.000;proc

factordata=examp733(type=corr)rotate=varimax;varx1-x8;proc

factordata=examp733(type=corr)n=2rotate=varimax;varx1-x8;run;當(dāng)前59頁，總共78頁。例8.4.2滬市604家上市公司2001年財(cái)務(wù)報(bào)表中有如下十個(gè)主要財(cái)務(wù)指標(biāo)：x1:主營業(yè)務(wù)收入（元）x6：每股凈資產(chǎn)（元）x2：主營業(yè)務(wù)利潤（元）x7：凈資產(chǎn)收益率（%）x3：利潤總額（元）x8：總資產(chǎn)收益率（%）x4：凈利潤（元）x9：資產(chǎn)總計(jì)（元）x5：每股收益（元）x10：股本當(dāng)前60頁，總共78頁。X1X2X3X4X5X6X7X8X9X10X1X2X3X4X5X6

X7X8X9X101.0000.7231.0000.4270.7431.0000.4070.6970.9821.0000.1710.3250.5390.5591.0000.1490.2280.2840.2740.5851.0000.0960.1770.3620.4020.7760.2181.0000.0060.2040.4550.5000.8490.2900.8331.0000.7480.7680.5740.5670.1250.1380.0670.0581.0000.6220.6190.4850.5000.002-0.0660.0330.0510.8611.000樣本相關(guān)矩陣如下表：當(dāng)前61頁，總共78頁。

f1f2f3

0.695-0.4720.1210.835-0.3460.0970.8860.003-0.0370.8880.037-0.0820.6660.6920.0190.3910.3670.8410.5270.670-0.3250.5810.703-0.2600.747-0.5640.0190.636-0.596-0.219

0.6720.8260.7860.7960.9340.9510.8320.8990.8770.808所解釋的總方差的累計(jì)比例0.4880.7450.838

m=3時(shí)的主成分解當(dāng)前62頁，總共78頁。

f1f2f3

0.809-0.0290.1290.8740.1710.1820.7060.5090.1670.6880.5520.1350.1150.8490.4470.0820.1990.9510.0220.9120.0040.0450.9430.0870.936-0.0120.0280.869-0.013-0.228

0.6720.8260.7860.7950.9340.9510.8320.8990.8770.808所解釋的總方差的累計(jì)比例0.4040.7120.838

m=2,3時(shí)用最大方差旋轉(zhuǎn)法旋轉(zhuǎn)后的主成分解當(dāng)前63頁，總共78頁。dataexamp842(type=corr);inputx1-x10;cards;1.000.........0.7231.000........0.4270.7431.000.......0.4070.6970.9821.000......0.1710.3250.5390.5591.000.....0.1490.2280.2840.2740.5851.000....0.0960.1770.3620.4020.7760.2181.000...0.0660.2040.4550.5000.8490.2900.8331.000..0.7480.7680.5740.5670.1250.1380.0670.0581.000.0.6220.6190.4850.5000.002-0.0660.0330.0510.8611.000;procfactordata=examp842method=prinn=3rotate=varimax;varx1-x10;run;當(dāng)前64頁，總共78頁。§8.5因子得分

因子得分是對不可觀測的因子變量的估計(jì)一、加權(quán)最小二乘法

改寫因子模型為：當(dāng)前65頁，總共78頁。當(dāng)前66頁，總共78頁。Bartlett(1937)得分（加權(quán)最小二乘估計(jì)）：當(dāng)前67頁，總共78頁。當(dāng)前68頁，總共78頁。二、回歸法(Thompson因子得分)

當(dāng)前69頁，總共78頁。當(dāng)前70頁，總共78頁。當(dāng)前71頁，總共78頁。當(dāng)前72頁，總共78頁。例8.5.1在例中，用回歸法得到的因子得分為：當(dāng)前73頁，總共78頁。dataexamp842(type=corr);inputx1-x10;cards;1.000.........0.7231.000........0.4270.7431.000.......0.4070.6970.9821.000......0.1710.3250.5390.5591.000.....0.1490.2280.2840.2740.5851.000....0.0960.1770.3620.4020.7760.2181.000...0.0060.2040.4550.5000.8490.2900.8331.000..0.7480.7680.5740.5670.1250.1380.0670.0581.000.0.6220.6190.4850.5000.002-0.0660.0330.0510.8611.000;procfactordata=examp842n=3rotate=varimaxscore;varx1-x10;run;當(dāng)前74頁，總共78頁。

TheSASSystem16:32Thursday,November11,20061TheFACTORProcedureInitialFactorMethod:PrincipalComponentsPriorCommunalityEstimates:ONEEigenvaluesoftheCorrelationMatrix:Total=10Average=1Eigenvalue