高級生物統計第十章因子分析因子分析介紹：1、因子分析的概念2、因子分析的過程因子分析的概念由于實測的變量間存在一定的相關關系，因此有可能用較少數的綜合指標分別綜合存在于各變量中的各類信息，而綜合指標之間彼此不相關，即各指標代表的信息不重疊。綜合指標稱為因子研究相關陣或協方差陣的內部依賴關系，將多個變量綜合為少數幾個因子把這個數據的N個變量用一兩個綜合變量特征值>1累計貢獻率>0.8R型分析和Q型分析R型分析：研究指標之間的相關關系，通過對變量的相關陣或協差陣內部結構的研究，找出控制所有變量的公共因子Q型分析：研究樣品之間的相關關系，通過對樣品的相似矩陣內部結構的研究，找出控制所有樣品的幾個主要因素。因子分析的作用一.尋求幾個控制所有變量的公共因子，因子數量少于變量數量二.所得到的公共因子進一步分析（聚類分析）因子分析與主成分分析的區(qū)別1.主成分分析是通常的變量變換，因子分析需要構造因子模型，把一個變量看成由公共因子和特殊因子構成，解釋變量的內部關系2.主成分的個數與變量個數相等，是將一組相關的變量轉化為不相關的分量，因子的個數少于變量個數因子分析模型一般地，設X=(x1,x2,…,xp)’為可觀測的隨機變量，且有f=(f1,f2,…,fm)’為公共（共性）因子(commonfactor)，簡稱因子(factor)e=(e1,e2,…,ep)’為特殊因子（specificfactor）μ=(μ1,μ2,…,μp)’為隨機變量x的總體均值A=(aij)p*m為因子負荷（載荷）（factorloading）矩陣通常先對x作標準化處理，使標準化得到的新變量均值為零，方差為１．這樣就有假定（１）fi的均數為０，方差為１；（２）ei的均數為０，方差為δi；（３）fi與ei相互獨立．則稱x為具有m個公共因子的因子模型因子載荷（負荷）aij是隨機變量xi與公共因子fj的相關系數。設稱gj2為公共因子fj對x的“貢獻”，是衡量公共因子fj重要性的一個指標。因子分析的步驟輸入原始數據xn*p，計算樣本均值和方差，進行標準化計算（處理）；求樣本相關系數矩陣R=(rij)p*p；求相關系數矩陣的特征根λi

(λ1,λ2,…,λp>0)和相應的標準正交的特征向量li；12個玉米雜交種10項指標觀察值：品種代號平均畝產（x1）穗長（x2）穗行數（x3）行粒數（x4）穗粒重（x5）出粒率（x6）千粒重（x7）蛋白質（x8）全籽粒狀賴氨酸（x9）百克賴氨酸（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項指標的相關系數矩陣變量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相關系數矩陣的特征向量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相關系數矩陣的特征值及累積百分數特征值特征值占總體百分數特征值的累積百分數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

因子載荷矩陣

A=(U1*,u2*,…….).

Hi2=A112+A122+A132+A142+A152Ai2為第i個公共因子對X1變量的方差。Hi2為各公共因子對變量X1的方差總和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累積貢獻0.40930.19340.13000.10970.0781X1=0.786Y1-0.1279Y2+0.1721Y3+0.1846Y4+0.4549Y5X2=X3=X4=X5=X6=X7=X8=X9=X10=應用數據集SOCECON為洛杉基12個地區(qū)統計的五個社會經濟指標：人口總數（POP），教育程度（SCHOOL），就業(yè)數（EMPLOY），服務業(yè)人數（SERVICES），中等的房價（HOUSE）。用FACTOR過程可以進行主分量分析。DATASOCECON;TITLE'五個經濟指標的分析';INPUTPOPSCHOOLEMPLOYSERVICESHOUSE;CARDS;570012.8250027025000100010.9600101000034008.81000109000380013.6170014025000400012.816001402500082008.326006012000120011.44001016000910011.533006014000990012.5340018018000960013.736003902500096009.633008012000940011.4400010013000;PROCFACTORDATA=SOCECONSIMPLECORR;TITLE2'主分量分析';RUN;為了得到好的因子解釋，我們在上面的PROCFACTOR語句中再加上一個ROTATE=PROMAX旋轉選項，這樣將在得到主因子分析后先產生方差最大正交預旋轉（VARIMAX）然后進行斜交旋轉，并加了一個REORDER選項使輸出時把原始變量受相同因子影響的放在一起：PROCFACTORDATA=SOCECONPRIORS=SMCROTATE=PROMAXREORDER;TITLE2'主因子分析及PROMAX斜交旋轉';RUN;

五個經濟指標的分析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按照缺省的選擇因子個數的準則MINEIGEN，取大于1的特征值，所以取兩個因子。它們是用公因子預報原始變量的回歸系數。第一主分量（因子）在所有五