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1、第9章非线性回归9.1在非线性回归线性化时,对因变量作变换应注意什么问题?答:在对非线性回归模型线性化时,对因变量作变换时不仅要注意回归函数的形 式,还要注意误差项的形式。如:(1)乘性误差项,模型形式为y AK L e,(2)加性误差项,模型形式为y AK L o对乘法误差项模型(1)可通过两边取对数转化成线性模型,(2)不能线性化。 一般总是假定非线性模型误差项的形式就是能够使回归模型线性化的形式, 为了方便通常省去误差项,仅考虑回归函数的形式。9.2为了研究生产率与废料率之间的关系,记录了如表9.14所示的数据,请画出散点图,根据散点图的趋势拟合适当的回归模型。表 9.14生产率X (单
2、位/周)1000200030003500400045005000废品率y (%5.26.56.88.110.210.313.0解:先画出散点图如下图:12.0010.008.006.00从散点图大致可以判断出X和y之间呈抛物线或指数曲线,由此采用二次方程式和指数函数进行曲线回归(1)二次曲线SPSS输出结果如下:Mode l Sum maryRR SquareAdjusted R SquareStd. Error of the Estimate.981.962.942.651The in depe ndent variable is x.ANOVASum of SquaresdfMean Sq
3、uareFSig.Regressi on42.571221.28650.160.001Residual1.6974.424Total44.2696The in depe ndent variable is x.Coe fficientsUn sta ndardized Coefficie ntsStan dardized Coefficie ntstSig.BStd. ErrorBetax-.001.001-.449-.891.423x * 24.47E-007.0001.4172.812.048(Co nsta nt)5.8431.3244.414.012从上表可以得到回归方程为:y? 5.
4、843 0.087x 4.47 10 7x2由x的系数检验P值大于0.05,得到x的系数未通过显著性检验。 由x2的系数检验P值小于0.05,得到x2的系数通过了显著性检验。(2)指数曲线Mode l Sum maryRR SquareAdjusted R SquareStd. Error of the Estimate.970.941.929.085The in depe ndent variable is x.ANOVASum of SquaresdfMean SquareFSig.Regressi on.5731.57379.538.000Residual.0365.007Total.6
5、096The in depe ndent variable is x.Coe fficientsUn sta ndardized Coefficie ntsStan dardized Coefficie ntstSig.BStd. ErrorBetax.000.000.9708.918.000(Co nsta nt)4.003.34811.514.000The depe ndent variable is ln( y).从上表可以得到回归方程为:y? 4.003評0皿由参数检验P值00.05,得到回归方程的参数都非常显著。0 bserved勾畑韭址Es.pcierLtELl从R2值,的估计值和
6、模型检验统计量 F值、t值及拟合图综合考虑, 指数拟合效果更好一些。9.3已知变量x与y的样本数据如表9.15,画出散点图,试用a来拟合回归模型,假设:(1) 乘性误差项,模型形式为y=a e3/xeR /x(2) 加性误差项,模型形式为y= a e + 。表 9.15序号xy序号xy序号xy14.200.08663.200.150112.200.35024.060.09073.000.170122.000.44033.800.10082.800.190131.800.62043.600.12092.600.220141.600.94053.400.130102.400.240151.401.
7、620解:散点图:1.5001.0000.5000.000-1.D02.003.004.00(1)乘性误差项,模型形式为y=a eR/xe线性化:Iny=ln a + R /x + e 令 y1=lny, a=ln a ,x1=1/x .做y1与x1的线性回归,SPSS俞出结果如下:Model SummarModelRR SquareAdjusted R SquareStd. E rror of the Estimate1.999 a.997.997.04783a. Predictors: (Con sta nt), x1b. Depe ndent Variable: y1ANOVA bMod
8、elSum of SquaresdfMean SquareFSig.1Regressi on10.930110.9304778.305.000aResidual.03013.002Total10.96014a. Predictors: (Con sta nt), x1b. Depe ndent Variable: y1Coe fficients aModelUn sta ndardized Coefficie ntsStan dardized Coefficie ntstSig.BStd.ErrorBeta1(Co nsta nt)-3.856.037-103.830.000x16.080.0
9、88.99969.125.000a. Depe nde nt Variable: y1从以上结果可以得到回归方程为:y仁-3.856+6.08x1F检验和t检验的P值00,0b11初值请读者自己选择。y函数线性化得到:1tuIny盘)1.851 0.264 ln(丄 丄)ln b。tlndy 100,八(1命,竹3关于t的线性回归分析,SPSS输出结果如下:Model SummarjbModelRR SquareAdjusted R SquareStd. E rror of the Estimate1.994 a.988.987.16820a. Predictors: (Con sta nt)
10、, tb. Depe nde nt Variable: y3ANOVA bModelSum of SquaresdfMean SquareFSig.1Regressi on39.839139.8391408.165.000aResidual.48117.028Total40.32018a. Predictors: (Con sta nt), tb. Depe nde nt Variable: y3Coe fficients aModelUn sta ndardized CoefficientsStan dardized CoefficientstSig.BStd.ErrorBeta1(Co nsta nt)-1.851.080-23.039.000t-.264.007-.994-37.526.000a. Depe nde nt Variable: y3由表Mo