第十章因子分析

高级生物统计第十章因子分析因子分析介绍：1、因子分析的概念2、因子分析的过程因子分析的概念由于实测的变量间存在一定的相关关系，因此有可能用较少数的综合指标分别综合存在于各变量中的各类信息，而综合指标之间彼此不相关，即各指标代表的信息不重叠。综合指标称为因子研究相关阵或协方差阵的内部依赖关系，将多个变量综合为少数几个因子把这个数据的N个变量用一两个综合变量特征值>1累计贡献率>0.8R型分析和Q型分析R型分析：研究指标之间的相关关系，通过对变量的相关阵或协差阵内部结构的研究，找出控制所有变量的公共因子Q型分析：研究样品之间的相关关系，通过对样品的相似矩阵内部结构的研究，找出控制所有样品的几个主要因素。因子分析的作用一.寻求几个控制所有变量的公共因子，因子数量少于变量数量二.所得到的公共因子进一步分析（聚类分析）因子分析与主成分分析的区别1.主成分分析是通常的变量变换，因子分析需要构造因子模型，把一个变量看成由公共因子和特殊因子构成，解释变量的内部关系2.主成分的个数与变量个数相等，是将一组相关的变量转化为不相关的分量，因子的个数少于变量个数因子分析模型一般地，设X=(x1,x2,…,xp)’为可观测的随机变量，且有f=(f1,f2,…,fm)’为公共（共性）因子(commonfactor)，简称因子(factor)e=(e1,e2,…,ep)’为特殊因子（specificfactor）μ=(μ1,μ2,…,μp)’为随机变量x的总体均值A=(aij)p*m为因子负荷（载荷）（factorloading）矩阵通常先对x作标准化处理，使标准化得到的新变量均值为零，方差为１．这样就有假定（１）fi的均数为０，方差为１；（２）ei的均数为０，方差为δi；（３）fi与ei相互独立．则称x为具有m个公共因子的因子模型因子载荷（负荷）aij是随机变量xi与公共因子fj的相关系数。设称gj2为公共因子fj对x的“贡献”，是衡量公共因子fj重要性的一个指标。因子分析的步骤输入原始数据xn*p，计算样本均值和方差，进行标准化计算（处理）；求样本相关系数矩阵R=(rij)p*p；求相关系数矩阵的特征根λi

(λ1,λ2,…,λp>0)和相应的标准正交的特征向量li；12个玉米杂交种10项指标观察值：品种代号平均亩产（x1）穗长（x2）穗行数（x3）行粒数（x4）穗粒重（x5）出粒率（x6）千粒重（x7）蛋白质（x8）全籽粒状赖氨酸（x9）百克赖氨酸（x10）194723.414.845.30.4685.23739.540.373.88293523.216.241.70.485.33057.90.384.813918.220.914.843.30.3882.63209.510.430.524910.723.416.1440.4685.23388.60.333.84590522.91739.80.4580.43489.530.424.46890.622.315.7440.4185.42868.670.394.57853.420.915.941.60.3585.42739.790.424.298837.820.214.437.30.3382.53267.620.364.739833.322.215.238.30.3782.23107.840.45.110760.920.415.540.70.3284.22687.750.354.5211760.320.815.144.80.3579.53728.910.455.0512742.523.414.743.10.3579.53109.130.44.3610项指标的相关系数矩阵变量x1x2x3x4x5x6x7x8x9x10X11X20.421X30.36160.34871X40.16280.3523-0.04741X50.77810.76560.48470.40431X60.51740.02610.14820.16990.28501X70.61560.5711-0.00990.04090.72660.02641X80.23700.21820.13540.54240.3751-0.04370.24111X9-0.1717-0.2255-0.00340.1266-0.1962-0.5585-0.27610.53531x10-0.4096-0.4491-0.1622-0.4569-0.5906-0.5142-0.5138-0.50320.45871相关系数矩阵的特征向量U1U2U3U4U5U6U7U8U9U100.3885-0.09190.15100.17630.5149-0.3694-0.0204-0.5885-0.18990.04850.36660.06530.2755-0.175-0.5215-0.1606-0.6687-0.09220.0228-0.03130.19630.00940.38010.7053-0.26060.28350.2403-0.08510.3236-0.03860.22750.3499-0.4828-0.0156-0.3456-0.48260.3790-0.13690.2773-0.02480.46470.05370.20510.028-0.0221-0.16850.29020.6049-0.50130.08990.2274-0.3908-0.45010.36180.2247-0.1371-0.37680.39920.2883-0.09170.3556-0.03670.277-0.51590.29910.12480.18550.13070.5978-0.12450.22110.5625-0.21050.06320.22460.3774-0.22170.02250.04940.5843-0.16160.62430.14040.1730.2806-0.1394-0.19560.1720.004-0.6071-0.39410.0410.37200.09410.0775-0.5473-0.05570.21630.2960.5027相关系数矩阵的特征值及累积百分数特征值特征值占总体百分数特征值的累积百分数14.09320.40930.409321.93410.19340.602731.30030.13000.732841.09710.10970.842650.78050.0780.920560.54310.05430.974870.18480.01850.993380.06010.0060.999390.00690.00071.000100.0000.0001.000

因子载荷矩阵

A=(U1*,u2*,…….).

Hi2=A112+A122+A132+A142+A152Ai2为第i个公共因子对X1变量的方差。Hi2为各公共因子对变量X1的方差总和Y1Y2Y3Y4Y5Hi2δ210.7860-0.12790.17210.18460.45490.90480.095220.74170.09070.3141-0.1833-0.46070.90290.097130.39720.01310.43340.7387-0.23030.94450.055540.46020.4866-0.5506-0.0164-0.30530.84520.154850.94010.07470.23880.0293-0.01950.94530.054760.4601-0.5435-0.51320.3790.19850.95350.046570.7194-0.05100.3158-0.54040.26420.98170.018380.44730.7823-0.24000.06620.19840.91340.08669-0.32700.86820.16010.18120.24790.98060.019410-0.79730.0570.42420.09860.06850.83330.1667gi24.09321.93411.30031.09710.7805累积贡献0.40930.19340.13000.10970.0781X1=0.786Y1-0.1279Y2+0.1721Y3+0.1846Y4+0.4549Y5X2=X3=X4=X5=X6=X7=X8=X9=X10=应用数据集SOCECON为洛杉基12个地区统计的五个社会经济指标：人口总数（POP），教育程度（SCHOOL），就业数（EMPLOY），服务业人数（SERVICES），中等的房价（HOUSE）。用FACTOR过程可以进行主分量分析。DATASOCECON;TITLE'五个经济指标的分析';INPUTPOPSCHOOLEMPLOYSERVICESHOUSE;CARDS;570012.8250027025000100010.9600101000034008.81000109000380013.6170014025000400012.816001402500082008.326006012000120011.44001016000910011.533006014000990012.5340018018000960013.736003902500096009.633008012000940011.4400010013000;PROCFACTORDATA=SOCECONSIMPLECORR;TITLE2'主分量分析';RUN;为了得到好的因子解释，我们在上面的PROCFACTOR语句中再加上一个ROTATE=PROMAX旋转选项，这样将在得到主因子分析后先产生方差最大正交预旋转（VARIMAX）然后进行斜交旋转，并加了一个REORDER选项使输出时把原始变量受相同因子影响的放在一起：PROCFACTORDATA=SOCECONPRIORS=SMCROTATE=PROMAXREORDER;TITLE2'主因子分析及PROMAX斜交旋转';RUN;

五个经济指标的分析11:29Monday,March20,20001

主分量分析

TheFACTORProcedureMeansandStandardDeviationsfrom12ObservationsVariableMeanStdDevPOP6241.6673439.9943SCHOOL11.4421.7865EMPLOY2333.3331241.2115SERVICES120.833114.9275HOUSE17000.0006367.5313CorrelationsPOPSCHOOLEMPLOYSERVICESHOUSEPOP1.000000.009750.972450.438870.02241SCHOOL0.009751.000000.154280.691410.86307EMPLOY0.972450.154281.000000.514720.12193SERVICES0.438870.691410.514721.000000.77765HOUSE0.022410.863070.121930.777651.00000

InitialFactorMethod:PrincipalComponentsPriorCommunalityEstimates:ONEEigenvaluesoftheCorrelationMatrix:Total=5Average=1EigenvalueDifferenceProportionCumulative12.873313591.076653500.57470.574721.796660091.581823210.35930.934030.214836890.114902830.04300.977040.099934050.084678680.02000.996950.015255370.00311.00002factorswillberetainedbytheMINEIGENcriterion.FactorPatternFactor1Factor2POP0.580960.80642SCHOOL0.76704-0.54476EMPLOY0.672430.72605SERVICES0.93239-0.10431HOUSE0.79116-0.55818VarianceExplainedbyEachFactorFactor1Factor22.87331361.7966601FinalCommunalityEstimates:Total=4.669974POPSCHOOLEMPLOYSERVICESHOUSE0.987826290.885105550.979305830.880235620.93750041按照缺省的选择因子个数的准则MINEIGEN，取大于1的特征值，所以取两个因子。它们是用公因子预报原始变量的回归系数。第一主分量（因子）在所有五