應用多元分析第八章因子分析

優(yōu)選應用多元分析第八章因子分析現(xiàn)在是1頁\一共有78頁\編輯于星期六例

為了評價即將進大學的高中生的學習能力，抽了200名高中生進行問卷調查，共50個問題。素有這些問題可以歸結為閱讀理解、數學水平和藝術素養(yǎng)三個方面。例

公司老板對48名應聘者進行面試，并給出他們在15個方面的得分，這15個方面是：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(x7)、推銷能力(x8)、經驗(x9)、積極性(x10)、抱負(x11)、理解能力(x12)、潛力(x13)、交際能力(x14)、適應性(x15)。通過因子分析，這15個方面可歸結為應聘者的外露能力、討人喜歡的程度、經驗、專業(yè)能力和外貌?，F(xiàn)在是2頁\一共有78頁\編輯于星期六§8.2因子模型一、數學模型現(xiàn)在是3頁\一共有78頁\編輯于星期六

假設條件現(xiàn)在是4頁\一共有78頁\編輯于星期六二、因子模型的性質現(xiàn)在是5頁\一共有78頁\編輯于星期六例8.2.1設隨機向量(x1,x2,x3,x4)’的協(xié)方差矩陣為現(xiàn)在是6頁\一共有78頁\編輯于星期六2、模型不受單位影響現(xiàn)在是7頁\一共有78頁\編輯于星期六3、因子載荷不唯一（相差一個正交變換）注：在實際中，利用因子載荷陣的不唯一性對因子進行旋轉，使得新的因子有更好的實際意義。現(xiàn)在是8頁\一共有78頁\編輯于星期六三、因子載荷矩陣的統(tǒng)計意義2、A的行元素平方和——公因子對原始變量的方差貢獻1、A的元素aij——原始變量xi與公因子fj之間的協(xié)方差函數現(xiàn)在是9頁\一共有78頁\編輯于星期六3、A的列平方和——公共因子fj對x的貢獻率現(xiàn)在是10頁\一共有78頁\編輯于星期六Principalcomponents:主成分法Unweightedleastsquare:不加權最小平方法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?！?.3參數估計現(xiàn)在是11頁\一共有78頁\編輯于星期六一、主成分法：

現(xiàn)在是12頁\一共有78頁\編輯于星期六例

在例中，分別取m=1,m=2,用主成分法估計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比例0.828

0.8280.938現(xiàn)在是13頁\一共有78頁\編輯于星期六相應于m=2的解的殘差矩陣為：

現(xiàn)在是14頁\一共有78頁\編輯于星期六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;現(xiàn)在是15頁\一共有78頁\編輯于星期六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;現(xiàn)在是16頁\一共有78頁\編輯于星期六1、給出共同度hi2的初步估計值hi*2

以第i個變量xi*與其它所有變量x1*,x2*,…,xi1*,xi+1*,…,xp*回歸的復相關系數的平方作為初始估計值2、求出約化相關陣計算Di*=1-hi*2,再計算出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，用估計值代替第二步的D*D的估計：D*（1）=R-A1A1′5、繼續(xù)第三步，直到A，

D的估計達到穩(wěn)定為止二、主因子法現(xiàn)在是17頁\一共有78頁\編輯于星期六例8.3.2在例中取m=2,選用xi與其他7個變量的復相關系數平方作為的初始估計值。計算得:現(xiàn)在是18頁\一共有78頁\編輯于星期六約相關矩陣為現(xiàn)在是19頁\一共有78頁\編輯于星期六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比例0.8160.914現(xiàn)在是20頁\一共有78頁\編輯于星期六相應于m=2的解的殘差矩陣為：

現(xiàn)在是21頁\一共有78頁\編輯于星期六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;現(xiàn)在是22頁\一共有78頁\編輯于星期六

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;現(xiàn)在是23頁\一共有78頁\編輯于星期六三、極大似然法用迭代法求上述方程組的解?，F(xiàn)在是24頁\一共有78頁\編輯于星期六現(xiàn)在是25頁\一共有78頁\編輯于星期六現(xiàn)在是26頁\一共有78頁\編輯于星期六現(xiàn)在是27頁\一共有78頁\編輯于星期六現(xiàn)在是28頁\一共有78頁\編輯于星期六現(xiàn)在是29頁\一共有78頁\編輯于星期六現(xiàn)在是30頁\一共有78頁\編輯于星期六

載荷矩陣A是不唯一的，有人建議添加一個計算上方便的的唯一性條件：現(xiàn)在是31頁\一共有78頁\編輯于星期六現(xiàn)在是32頁\一共有78頁\編輯于星期六現(xiàn)在是33頁\一共有78頁\編輯于星期六現(xiàn)在是34頁\一共有78頁\編輯于星期六現(xiàn)在是35頁\一共有78頁\編輯于星期六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比例0.8010.917例8.3.3在例中，取m=2，同,極大似然法的計算結果如下表：的初始估計值與例現(xiàn)在是36頁\一共有78頁\編輯于星期六極大似然解的殘差矩陣為：

現(xiàn)在是37頁\一共有78頁\編輯于星期六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;現(xiàn)在是38頁\一共有78頁\編輯于星期六TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodPriorCommunalityEstimates:SMCx1x2x3x4

0.877839350.888722470.844686300.88357282x5x6x7x80.927120960.955454730.967228140.90408487現(xiàn)在是39頁\一共有78頁\編輯于星期六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.現(xiàn)在是40頁\一共有78頁\編輯于星期六

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.現(xiàn)在是41頁\一共有78頁\編輯于星期六SignificanceTestsBasedon10000ObservationsPr>TestDFChi-SquareChiSqH0:Nocommonfactors28142472.086<.0001HA:AtleastonecommonfactorH0:2Factorsaresufficient133315.7842<.0001HA:Morefactorsareneeded現(xiàn)在是42頁\一共有78頁\編輯于星期六TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodChi-SquarewithoutBartlett'sCorrection3317.3878Akaike'sInformationCriterion3291.3878Schwarz'sBayesianCriterion3197.6534TuckerandLewis'sReliabilityCoefficient0.9501SquaredCanonicalCorrelationsFactor1Factor20.992346680.92414870現(xiàn)在是43頁\一共有78頁\編輯于星期六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現(xiàn)在是44頁\一共有78頁\編輯于星期六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現(xiàn)在是45頁\一共有78頁\編輯于星期六TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodFinalCommunalityEstimatesandVariableWeightsTotalCommunality:Weighted=141.84599Unweighted=7.337450VariableCommunalityWeightx10.9188651412.3222163x20.9250959313.3456046x30.848210716.5890175x40.864559767.3855720x50.9183397412.2487688x60.9666533029.9870945x70.9822744556.4139209x80.9134505211.5537899現(xiàn)在是46頁\一共有78頁\編輯于星期六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;現(xiàn)在是47頁\一共有78頁\編輯于星期六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weight共性方差h2現(xiàn)在是48頁\一共有78頁\編輯于星期六T

129.649510

12.1837620

0.58780590

0.18395610

0.00909540

-0.0535390

-0.2925640

-0.4354380現(xiàn)在是49頁\一共有78頁\編輯于星期六

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;現(xiàn)在是50頁\一共有78頁\編輯于星期六§8.4因子旋轉因子旋轉的意義：使公因子易于解釋。方法：正交旋轉，斜交旋轉。實施：對載荷矩陣作正交（斜交）變換。正交旋轉：現(xiàn)在是51頁\一共有78頁\編輯于星期六最大方差旋轉法：選擇正交矩陣T使得A*的所有m個列元素平方和的相對方差之和達到最大。現(xiàn)在是52頁\一共有78頁\編輯于星期六m=2時，設元因子載荷矩陣為現(xiàn)在是53頁\一共有78頁\編輯于星期六現(xiàn)在是54頁\一共有78頁\編輯于星期六現(xiàn)在是55頁\一共有78頁\編輯于星期六現(xiàn)在是56頁\一共有78頁\編輯于星期六現(xiàn)在是57頁\一共有78頁\編輯于星期六變量

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法，旋轉后的因子載荷矩陣如下表

主成分

主因子

極大似然現(xiàn)在是58頁\一共有78頁\編輯于星期六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;現(xiàn)在是59頁\一共有78頁\編輯于星期六例8.4.2滬市604家上市公司2001年財務報表中有如下十個主要財務指標：x1:主營業(yè)務收入（元）x6：每股凈資產（元）x2：主營業(yè)務利潤（元）x7：凈資產收益率（%）x3：利潤總額（元）x8：總資產收益率（%）x4：凈利潤（元）x9：資產總計（元）x5：每股收益（元）x10：股本現(xiàn)在是60頁\一共有78頁\編輯于星期六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樣本相關矩陣如下表：現(xiàn)在是61頁\一共有78頁\編輯于星期六

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比例0.4880.7450.838

m=3時的主成分解現(xiàn)在是62頁\一共有78頁\編輯于星期六

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比例0.4040.7120.838

m=2,3時用最大方差旋轉法旋轉后的主成分解現(xiàn)在是63頁\一共有78頁\編輯于星期六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;現(xiàn)在是64頁\一共有78頁\編輯于星期六§8.5因子得分

因子得分是對不可觀測的因子變量的估計一、加權最小二乘法

改寫因子模型為：現(xiàn)在是65頁\一共有78頁\編輯于星期六現(xiàn)在是66頁\一共有78頁\編輯于星期六Bartlett(1937)得分（加權最小二乘估計）：現(xiàn)在是67頁\一共有78頁\編輯于星期六現(xiàn)在是68頁\一共有78頁\編輯于星期六二、回歸法(Thompson因子得分)

現(xiàn)在是69頁\一共有78頁\編輯于星期六現(xiàn)在是70頁\一共有78頁\編輯于星期六現(xiàn)在是71頁\一共有78頁\編輯于星期六現(xiàn)在是72頁\一共有78頁\編輯于星期六例8.5.1在例中，用回歸法得到的因子得分為：現(xiàn)在是73頁\一共有78頁\編輯于星期六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;現(xiàn)在是74頁\一共有78頁\編輯于星期六

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