因子分析的方法

三关于中国民航客运量模型建立与研究 为研究我国民航客运量的变化趋势及其成因，我们以民航客运量作为因变量Y，以国民收入，消费额，铁路客运量，民航航线里程，来华旅游入境人数为影响民航客运量的主要因素。Y表示民航客运量（万人），X1表示国民收入（亿元），X2表示消费额（亿元），X3表示铁路客运量（万人），X4表示民航航线里程（万公里），X5表示来华旅游入境人数（万人）。统计数据如下：年份yx1x2x3x4x51978231301018888149114.89180.921979298335021958638916.00420.391980343368825319220419.53570.251981401394127999530021.82776.711982445425830549992223.27792.4319833914736335810604422.91947.7019845545652390511035326.021285.2219857447020487911211027.721783.3019869977859555210857932.432281.95198713109313638611242938.912690.231988144211738803812264537.383169.481989128313176900511380747.192450.14199016601438496639571250.682746.201991217816557109699508155.913335.651992288620223129859969383.663311.5019933383248821594910545896.084152.70 现对该数据进行分析如下：回归分析Correlationsyx1x2x3x4x5yPearson Correlation1.989*.985*.227.987*.924*Sig. (2-tailed).000.000.398.000.000N161616161616x1Pearson Correlation.989*1.999*.258.984*.930*Sig. (2-tailed).000.000.335.000.000N161616161616x2Pearson Correlation.985*.999*1.289.978*.942*Sig. (2-tailed).000.000.278.000.000N161616161616x3Pearson Correlation.227.258.2891.213.504*Sig. (2-tailed).398.335.278.428.046N161616161616x4Pearson Correlation.987*.984*.978*.2131.882*Sig. (2-tailed).000.000.000.428.000N161616161616x5Pearson Correlation.924*.930*.942*.504*.882*1Sig. (2-tailed).000.000.000.046.000N161616161616*. Correlation is significant at the 0.01 level (2-tailed).*. Correlation is significant at the 0.05 level (2-tailed).(1)从相关阵看出，y与x1，x2，x4，x5的相关系数都在0.9以上，说明所选自变量与y高度线性相关，用y与自变量作多元线性回归是适当的。 (2)y与x3的相关系数ry3=0.227偏小,P值=0.398,x3是铁路客运量，这说明铁路客运量对民航客运量无显著影响。 Model SummaryModelRR SquareAdjusted R SquareStd. Error of the Estimate1.999a.998.99749.49240a. Predictors: (Constant), x5, x3, x4, x2, x1ANOVAbModelSum of SquaresdfMean SquareFSig.1Regression1.382E752763775.3541.128E3.000aResidual24494.981102449.498Total1.384E715a. Predictors: (Constant), x5, x3, x4, x2, x1b. Dependent Variable: yCoefficientsaModelUnstandardized CoefficientsStandardized CoefficientstSig.BStd. ErrorBeta1(Constant)450.909178.0782.532.030x1.354.0852.4474.152.002x2-.561.125-2.485-4.478.001x3-.007.002-.083-3.510.006x421.5784.030.5315.354.000x5.435.052.5648.440.000a. Dependent Variable: y(3)由上图知：回归方程为y=450.909+0.354x1-0.561x2-0.007x3+21.578x4+0.435x5(4)复相关系数R=0.999，决定系数R*R=0.998，由决定系数看回归方程高度显著。(5)有方差分析表可知：F=1128.303，P值=0.000，表明回归方程高度显著，说明x1,x2,x3,x4,x5整体上对y有高度显著的线性影响。(6)回归系数的显著性检验。自变x1,x2,x3,x4,x5对y均有显著影响，其中x3铁路客运量的P值=0.006最大,但仍在1%的显著性水平上对y高度显著，这充分说明在多元线性回归中不能仅凭简单相关系数的大小而决定变量的取舍。(7)疑问:x2的回归系数为-0.561是负的,x2是消费额，负的回归系数说明与y呈负相关，显然是不合理的！故进一步分析；由关于相关系数分析表知：其原因是自变量之间的共线性（x1是国民收入，x2是消费额，二者的简单相关系数r12=0.9989，高度相关！），故在多重共线性部分作进一步改进。多重共线性分析CoefficientsaModelUnstandardized CoefficientsStandardized CoefficientstSig.Collinearity StatisticsBStd. ErrorBetaToleranceVIF1(Constant)450.909178.0782.532.030x1.354.0852.4474.152.002.0011.963E3x2-.561.125-2.485-4.478.001.0011.741E3x3-.007.002-.083-3.510.006.3153.171x421.5784.030.5315.354.000.01855.488x5.435.052.5648.440.000.04025.193a. Dependent Variable: yCollinearity DiagnosticsaModelDimensionEigenvalueCondition IndexVariance Proportions(Constant)x1x2x3x4x5115.5781.000.00.00.00.00.00.002.3783.842.00.00.00.00.00.003.03712.205.01.00.00.00.03.194.00436.431.17.00.01.09.50.045.00253.643.72.00.01.66.15.7168.080E-5262.762.10.99.99.25.31.06a. Dependent Variable: y(8)有上述图表知：VIF1=1963最大，且远大于十,故剔除x1,建立y对四个自变量x2,x3,x4,x5的回归方程后，再求回归方程。CoefficientsaModelUnstandardized CoefficientsStandardized CoefficientstSig.Collinearity StatisticsBStd. ErrorBetaToleranceVIF1(Constant)695.039264.5252.627.024x2-.053.042-.233-1.262.233.01377.546x3-.012.003-.134-4.207.001.4312.319x432.0374.951.7886.471.000.03033.812x5.399.080.5174.988.000.04124.469a. Dependent Variable: y(9)由上述输出结果知：x2的方差扩大因子VIF=77.5为最大，远大于10，并且x2的回归系数B2=-0.053仍然是负值，说明此回归模型仍然存在强多共线性，应继续剔除变量。故再剔除x2,用y与三个自变量x3,x4,x5建立回归方程。CoefficientsaModelUnstandardized CoefficientsStandardized CoefficientstSig.Collinearity StatisticsBStd. ErrorBetaToleranceVIF1(Constant)591.876257.7302.296.040x3-.010.003-.119-3.934.002.5041.984x426.4362.249.65011.754.000.1506.650x5.317.048.4116.568.000.1178.514a. Dependent Variable: y(10)从图表中可看到，3个方差扩大因子都小于10，回归系数也都有合理经济解释，说明此时的回归模型不存在强多重共线性，可以作为最终回归模型！(11)最终回归方程为y=591.876-0.01037x3+26.436x4+0.317x5为了避免量纲的影响对其标准化，则标准化回归方程为y=-0.119x3+0.650x4+0.411x5(12)由标准化回归系数看到，对民航客运量影响最大因素是民航航运里程，其次是来华旅游入境人数。民航航运里程每增加1%，民航客运量会增加0.650%。来华旅游入境人数每增加1%，名航客运量会增加0.411%。而铁路客运量对民航客运量影响较小，铁路客运量每增加1%，民航客运量会减少0.119%。Model SummaryModelRR SquareAdjusted R SquareStd. Error of the Estimate1.997a.994.99379.78835a. Predictors: (Constant), x5, x3, x4(13)由上图表可知：此回归方程的样本决定系数R*R=0.994;调整的样本决定系数R1*R1=0.993，其拟合优度仍保持很高，且回归系数有相应合理的经济解释。因子分析法 因子分析法Total Variance ExplainedComponentInitial EigenvaluesExtraction Sums of Squared LoadingsTotal% of VarianceCumulative %Total% of VarianceCumulative %13.99179.82679.8263.99179.82679.8262.93218.64198.468.93218.64198.4683.0651.30399.771.0651.30399.7714.011.22499.995.011.22499.9955.000.005100.000.000.005100.000Extraction Method: Principal Component Analysis.(14)由图表知：第一个主成分方差百分比是79.826%，即含有原始五个变量信息量接近80%；第二个主成分方差百分比是18.641%，即含有原始五个变量信息近19%。故二者主成分累计含有原始五个变量信息近98.46%，则可取上述二者主成分足够！ Component MatrixaComponent12345x1.985-.165.018.047.012x2.990-.132.000.055-.011x3.413.908.066.007.000x4.963-.214.150-.064-.001x5.972.128-.195-.043.000Extraction Method: Principal Component Analysis.a. 5 components extracted.(15)由上表知：Y1=-2.464+0.00003714x1+0.00005831x2+0.000009394x3+0.01022x4+0.0001957x5Y2=-8.426-0.00002672x1-0.00003332x2+0.00008851x3-0.009708x4+0.0001105x5y=1159.125+936.781Y1-185.876Y2再进行替换知：y=416+0.03976x1+0.06082x2-0.007652x3+11.37x4+0.1628x5.(16)和多重线性分析作比较，我们可以发现，各因素对y影响程度相近，且相应经济解释也合理！CommunalitiesInitialExtractionx11.000.997x21.000.997x31.000.996x41.000.973x51.000.960Extraction Method: Principal Component Analysis.(17)由上述图表知：特殊因子的方差分别为0.03,0.03,0.04,0.27,0.04。故可忽略特殊因子作用。Total Variance ExplainedComponentInitial EigenvaluesExtraction Sums of Squared LoadingsTotal% of VarianceCumulative %Total% of VarianceCumulative %13.99179.82679.8263.99179.82679.8262.93218.64198.468.93218.64198.4683.0651.30399.7714.011