应用多元分析第八章因子分析

十项全能运动员得分相关矩阵X1X2X3X4X5X6

X7X8X9X10X1X2X3X4X5X6

X7X8X9X101.000.591.000.350.421.000.340.510.381.000.630.490.190.291.000.400.520.360.460.341.000.280.310.730.270.170.321.000.200.360.240.390.230.330.241.000.110.210.440.170.130.180.340.241.00-0.770.09-0.080.180.390.00-0.020.17-0.001.00当前第1页\共有79页\编于星期三\22点例

为了评价即将进大学的高中生的学习能力，抽了200名高中生进行问卷调查，共50个问题。素有这些问题可以归结为阅读理解、数学水平和艺术素养三个方面。例

公司老板对48名应聘者进行面试，并给出他们在15个方面的得分，这15个方面是：申请书的形式(x1)、外貌(x2)、专业能力(x3)、讨人喜欢(x4)、自信心(x5)、精明(x6)、诚实(x7)、推销能力(x8)、经验(x9)、积极性(x10)、抱负(x11)、理解能力(x12)、潜力(x13)、交际能力(x14)、适应性(x15)。通过因子分析，这15个方面可归结为应聘者的外露能力、讨人喜欢的程度、经验、专业能力和外貌。当前第2页\共有79页\编于星期三\22点§8.2因子模型一、数学模型当前第3页\共有79页\编于星期三\22点

假设条件当前第4页\共有79页\编于星期三\22点二、因子模型的性质当前第5页\共有79页\编于星期三\22点例8.2.1设随机向量(x1,x2,x3,x4)’的协方差矩阵为当前第6页\共有79页\编于星期三\22点2、模型不受单位影响当前第7页\共有79页\编于星期三\22点3、因子载荷不唯一（相差一个正交变换）注：在实际中，利用因子载荷阵的不唯一性对因子进行旋转，使得新的因子有更好的实际意义。当前第8页\共有79页\编于星期三\22点三、因子载荷矩阵的统计意义2、A的行元素平方和——公因子对原始变量的方差贡献1、A的元素aij——原始变量xi与公因子fj之间的协方差函数当前第9页\共有79页\编于星期三\22点3、A的列平方和——公共因子fj对x的贡献率当前第10页\共有79页\编于星期三\22点Principalcomponents:主成分法Unweightedleastsquare:不加权最小平方法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确定q的方法是按特征根由大至小的次序抽取，直到与接近为止。§8.3参数估计当前第11页\共有79页\编于星期三\22点一、主成分法：

当前第12页\共有79页\编于星期三\22点例

在例中，分别取m=1,m=2,用主成分法估计的因子载荷和共性方差如下表：

变量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所解释的总方差的累计比例0.828

0.8280.938当前第13页\共有79页\编于星期三\22点相应于m=2的解的残差矩阵为：

当前第14页\共有79页\编于星期三\22点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;当前第15页\共有79页\编于星期三\22点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;当前第16页\共有79页\编于星期三\22点1、给出共同度hi2的初步估计值hi*2

以第i个变量xi*与其它所有变量x1*,x2*,…,xi1*,xi+1*,…,xp*回归的复相关系数的平方作为初始估计值2、求出约化相关阵计算Di*=1-hi*2,再计算出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、特征向量求出因子载荷阵A14、求出D的估计，用估计值代替第二步的D*D的估计：D*（1）=R-A1A1′5、继续第三步，直到A，

D的估计达到稳定为止二、主因子法当前第17页\共有79页\编于星期三\22点例8.3.2在例中取m=2,选用xi与其他7个变量的复相关系数平方作为的初始估计值。计算得:当前第18页\共有79页\编于星期三\22点约相关矩阵为当前第19页\共有79页\编于星期三\22点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所解释的总方差的累计比例0.8160.914当前第20页\共有79页\编于星期三\22点相应于m=2的解的残差矩阵为：

当前第21页\共有79页\编于星期三\22点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;当前第22页\共有79页\编于星期三\22点

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;当前第23页\共有79页\编于星期三\22点三、极大似然法用迭代法求上述方程组的解。当前第24页\共有79页\编于星期三\22点当前第25页\共有79页\编于星期三\22点当前第26页\共有79页\编于星期三\22点当前第27页\共有79页\编于星期三\22点当前第28页\共有79页\编于星期三\22点当前第29页\共有79页\编于星期三\22点当前第30页\共有79页\编于星期三\22点

载荷矩阵A是不唯一的，有人建议添加一个计算上方便的的唯一性条件：当前第31页\共有79页\编于星期三\22点当前第32页\共有79页\编于星期三\22点当前第33页\共有79页\编于星期三\22点当前第34页\共有79页\编于星期三\22点当前第35页\共有79页\编于星期三\22点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所解释的总方差的累计比例0.8010.917例8.3.3在例中，取m=2，同,极大似然法的计算结果如下表：的初始估计值与例当前第36页\共有79页\编于星期三\22点极大似然解的残差矩阵为：

当前第37页\共有79页\编于星期三\22点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;当前第38页\共有79页\编于星期三\22点TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodPriorCommunalityEstimates:SMCx1x2x3x4

0.877839350.888722470.844686300.88357282x5x6x7x80.927120960.955454730.967228140.90408487当前第39页\共有79页\编于星期三\22点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.当前第40页\共有79页\编于星期三\22点

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.当前第41页\共有79页\编于星期三\22点SignificanceTestsBasedon10000ObservationsPr>TestDFChi-SquareChiSqH0:Nocommonfactors28142472.086<.0001HA:AtleastonecommonfactorH0:2Factorsaresufficient133315.7842<.0001HA:Morefactorsareneeded当前第42页\共有79页\编于星期三\22点TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodChi-SquarewithoutBartlett'sCorrection3317.3878Akaike'sInformationCriterion3291.3878Schwarz'sBayesianCriterion3197.6534TuckerandLewis'sReliabilityCoefficient0.9501SquaredCanonicalCorrelationsFactor1Factor20.992346680.92414870当前第43页\共有79页\编于星期三\22点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当前第44页\共有79页\编于星期三\22点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当前第45页\共有79页\编于星期三\22点TheFACTORProcedureInitialFactorMethod:MaximumLikelihoodFinalCommunalityEstimatesandVariableWeightsTotalCommunality:Weighted=141.84599Unweighted=7.337450VariableCommunalityWeightx10.9188651412.3222163x20.9250959313.3456046x30.848210716.5890175x40.864559767.3855720x50.9183397412.2487688x60.9666533029.9870945x70.9822744556.4139209x80.9134505211.5537899当前第46页\共有79页\编于星期三\22点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;当前第47页\共有79页\编于星期三\22点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;特殊方差的倒数weight共性方差h2当前第48页\共有79页\编于星期三\22点T

129.649510

12.1837620

0.58780590

0.18395610

0.00909540

-0.0535390

-0.2925640

-0.4354380当前第49页\共有79页\编于星期三\22点

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;当前第50页\共有79页\编于星期三\22点§8.4因子旋转因子旋转的意义：使公因子易于解释。方法：正交旋转，斜交旋转。实施：对载荷矩阵作正交（斜交）变换。正交旋转：当前第51页\共有79页\编于星期三\22点最大方差旋转法：选择正交矩阵T使得A*的所有m个列元素平方和的相对方差之和达到最大。当前第52页\共有79页\编于星期三\22点m=2时，设元因子载荷矩阵为当前第53页\共有79页\编于星期三\22点当前第54页\共有79页\编于星期三\22点当前第55页\共有79页\编于星期三\22点当前第56页\共有79页\编于星期三\22点当前第57页\共有79页\编于星期三\22点变量

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

所解释的总方差的累计比例0.5230.9830.5100.9140.5120.917

例8.4.1在例中分别用最大方差旋转法，旋转后的因子载荷矩阵如下表

主成分

主因子

极大似然当前第58页\共有79页\编于星期三\22点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;当前第59页\共有79页\编于星期三\22点例8.4.2沪市604家上市公司2001年财务报表中有如下十个主要财务指标：x1:主营业务收入（元）x6：每股净资产（元）x2：主营业务利润（元）x7：净资产收益率（%）x3：利润总额（元）x8：总资产收益率（%）x4：净利润（元）x9：资产总计（元）x5：每股收益（元）x10：股本当前第60页\共有79页\编于星期三\22点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样本相关矩阵如下表：当前第61页\共有79页\编于星期三\22点

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所解释的总方差的累计比例0.4880.7450.838

m=3时的主成分解当前第62页\共有79页\编于星期三\22点

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所解释的总方差的累计比例0.4040.7120.838

m=2,3时用最大方差旋转法旋转后的主成分解当前第63页\共有79页\编于星期三\22点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;当前第64页\共有79页\编于星期三\22点§8.5因子得分

因子得分是对不可观测的因子变量的估计一、加权最小二乘法

改写因子模型为：当前第65页\共有79页\编于星期三\22点当前第66页\共有79页\编于星期三\22点Bartlett(1937)得分（加权最小二乘估计）：当前第67页\共有79页\编于星期三\22点当前第68页\共有79页\编于星期三\22点二、回归法(Thompson因子得分)

当前第69页\共有79页\编于星期三\22点当前第70页\共有79页\编于星期三\22点当前第71页\共有79页\编于星期三\22点当前第72页\共有79页\编于星期三\22点例8.5.1在例中，用回归法得到的因子得分为：当前第73页\共有79页\编于星期三\22点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;当前第74页\共有79页\编于星期三\22点

TheSASSystem16:32Thursday,November11,20061TheFACTORProcedureInitialFactorMethod:PrincipalComponentsPriorCommunalityEstimates:ONEEigenvaluesoftheCorrelationMatrix:Total=10Average=1EigenvalueDifferenceProportionCumulative14.870307642.280618630.48700.487022.589689011.656127420.25900.746030.933561590.229068640.09340.839440.704492950.307384070.07040.909850.397108880.212314950.03970.9495