基于偏最小二乘回归非线性理论在经济系统中的应用
|
摘要
偏最小二乘回归(PLS)是构造回归方程的一种较新的多元统计方法,最早由化学界的S.Wold在1983年提出,后来日益受到统计界的关注。许多统计学家都对这种方法进行了研究。这种方法不但适用于单变量的回归分析,而且可以用于多元变量回归分析。PLS具备主成分分析、典型相关分析和OLS回归三者的某些特点,是对样本数据进行建模的一种稳健统计分析方法。
本文在总结多元线性回归模型基础上,进一步认识到其存在的多重共线性问题，提出用偏最小二乘回归模型定义；从多元线性回归、主成分分析以及典型相关分析相结合三个层面上提出了偏最小二乘回归模型。对于加法模型,在各维自变量独立的情况下,使用基于基函数变换的偏最小二乘非线性回归方法得到整体模型后,可以从中提取各维自变量独立的回归关系式，分别绘出各维自变量对因变量的函数曲线,进一步分析它们之间的关系。从理论和实际应用方面证明了偏最小二乘回归模型的科学合理性；在论文最后一章中运用偏最小二乘回归方法分析了我国R&D分类投入与经济增长之间的关系,研究表明了中国R&D分类投入每一项都对经济增长产生一定贡献力：应用研究R&D支出对经济增长贡献率较高,表明其对我国经济增长的促进作用较为显著；基础研究R&D支出增长率对经济贡献率最高,表明其增长越快我国经济增长越猛。另外,通过实例分析验证了非线性偏最小二乘回归在经济系统中的应用的可行性。
Partial Least-Squares is a new multivariate analysis method to build the equation about regression, which was fist proposed by economist S.Wold in 1983.Partial Least-Squares Regression was increasingly concerned later. Many statisticians study this method. The method is not only used for single variable regression analysis, but also multiple analyses. PLS with principal component analysis, canonical correlation analysis and OLS regression of certain characteristics of the three is a sample data model of a robust statistical analysis.
This paper is based on summarizing the multivariate linear regression model, further to recognize the existence of multi-collinearity problems, giving the partial least-squares regression model to definite; from the combinations of multiple linear regression, principal component analysis and canonical correlation analysis to propose the partial least squares regression model. For additive models, in that the independent variables, the use of transform basis function based on partial least squares approach to the overall non-linear regression model can be extracted from the variable that the return of an independent relationship, respectively, all drawn from the variables on the dependent variable of the function curve, then further analyze the relationship between them. From the application of theoretical and practical field we prove scientific rationality of partial least-squares regression model; in the last chapter, by using the partial least-squares regression we analysis the relationship between our R & D classification of the input and economic growth, our research shows that each the classified input of China R & D bring about a specific contribution to economic growth; the expenditures of applying the research on R & D produces a supreme contribution rates to economic growth, and our results indicate that it produces promote more significant promoted effects in Chinese economic growth; the expenditure growth rate of basic research R & D produces a supreme contribution rates to economic, which shows that the more it increase quickly the more our national economic has grown quickly. In addition, by researching the case we verified the non-linear partial least-squares regression in the economic system feasibility.
本文在总结多元线性回归模型基础上,进一步认识到其存在的多重共线性问题，提出用偏最小二乘回归模型定义；从多元线性回归、主成分分析以及典型相关分析相结合三个层面上提出了偏最小二乘回归模型。对于加法模型,在各维自变量独立的情况下,使用基于基函数变换的偏最小二乘非线性回归方法得到整体模型后,可以从中提取各维自变量独立的回归关系式，分别绘出各维自变量对因变量的函数曲线,进一步分析它们之间的关系。从理论和实际应用方面证明了偏最小二乘回归模型的科学合理性；在论文最后一章中运用偏最小二乘回归方法分析了我国R&D分类投入与经济增长之间的关系,研究表明了中国R&D分类投入每一项都对经济增长产生一定贡献力：应用研究R&D支出对经济增长贡献率较高,表明其对我国经济增长的促进作用较为显著；基础研究R&D支出增长率对经济贡献率最高,表明其增长越快我国经济增长越猛。另外,通过实例分析验证了非线性偏最小二乘回归在经济系统中的应用的可行性。
Partial Least-Squares is a new multivariate analysis method to build the equation about regression, which was fist proposed by economist S.Wold in 1983.Partial Least-Squares Regression was increasingly concerned later. Many statisticians study this method. The method is not only used for single variable regression analysis, but also multiple analyses. PLS with principal component analysis, canonical correlation analysis and OLS regression of certain characteristics of the three is a sample data model of a robust statistical analysis.
This paper is based on summarizing the multivariate linear regression model, further to recognize the existence of multi-collinearity problems, giving the partial least-squares regression model to definite; from the combinations of multiple linear regression, principal component analysis and canonical correlation analysis to propose the partial least squares regression model. For additive models, in that the independent variables, the use of transform basis function based on partial least squares approach to the overall non-linear regression model can be extracted from the variable that the return of an independent relationship, respectively, all drawn from the variables on the dependent variable of the function curve, then further analyze the relationship between them. From the application of theoretical and practical field we prove scientific rationality of partial least-squares regression model; in the last chapter, by using the partial least-squares regression we analysis the relationship between our R & D classification of the input and economic growth, our research shows that each the classified input of China R & D bring about a specific contribution to economic growth; the expenditures of applying the research on R & D produces a supreme contribution rates to economic growth, and our results indicate that it produces promote more significant promoted effects in Chinese economic growth; the expenditure growth rate of basic research R & D produces a supreme contribution rates to economic, which shows that the more it increase quickly the more our national economic has grown quickly. In addition, by researching the case we verified the non-linear partial least-squares regression in the economic system feasibility.
引文
[1]何晓群.回归分析与经济数据建模[J]北京：中国人民大学出版社,1997.5：142-153页
[2]于秀林,任雪松.多元统计分析[M]中国统计出版社,1999：340-351页
[3]袁志发,周静芋,多元统计分析[M].科学出版社,2002：256-300页
[4]J Iidiko E F,Jerome H F.A statistical view of some chemometrics regression toolsl J J.Technometrics,1993,35:109-135P
[5]肯道尔.多元分析.中国科学院计算中统计概率组译.北京：科学出版社.1983：1-42页
[6]汪洋.PLS回归应用实例.安徽大学学报.2001(3)：18-22页
[7]张尧庭.成分数据统计分析引论.科学出版社,2000：154-201页
[8]宁自军.成分数据的预测方法及其应用[J].统计与决策,2001(5)：6-7页
[9]张崇甫,陈述云.成分数据主成分分析及其应用[J].数理统计与管理,1996,15(3)：14-20页
[10]章栋恩.成分数据的几个估计[J].应用数学学报,2002,25(1)：58-68页
[11]张尧庭.成分数据统计分析引论[C].北京：科学出版社,2000：234-241页
[12]王惠文,黄薇.成分数据的线性回归模型[J]系统工程,2003,21(2)：103-106页
[13]Kenneth N.Berk.TU 8rBnce and Conditioin Regres sion Computations.J.of America StatisticalAssociation.Vel.72.1977.863-86P
[14]施锡铨等数据分析方法.上海上海财经大学出版社.1997：1-10页
[15]李大鹏、王惠文.偏最小二乘Logistic回归在都阳湖洪涝灾害预测中的应用.数理统计与管理.1998(3)：25-40页
[16]罗批,郭继昌,李锵,等.基于偏最小二乘回归建模的探讨[J].天津大学学报.2002,35(6)：783-786页
[17]张旸,周成武等.偏最小二乘回归方法在土地利用结构研究中的应用.自然 资源学报2004：19-24页
[18]张伏生,汪鸿等.基于偏最小二乘回归分析的短期负荷预测.电网技术.2003，27(3)：36-40页
[19]S.Wold.S.Hellberg,T.Lundstedt. M.Sjostrom.H.Wold.ProceedingsPLS model building.Theoryand Applications. Symposium FrankfurtaMain.l987:234-245P
[20]Eriksson.J.Damborsky.M.Earll,E.Johansson.J.Trygg.S.Wold.SARQSAREnvir on.Res.56(2004)481-499P
[21]王惠文.偏最小二乘回归方法及其应用[M].北京：国防工业出版社,1999：201-2063页
[22]许和连,赖明勇,许青松.基于GRPLS回归的中国经济增长影响因素分析管理科学学报.2003.12：5-11页
[23]王惠文,吴载斌等.偏最小二乘回归的线性与非线性方法[M].北京：国防工业出版社.2006：123-134页
[24]范金城,梅长林.数据分析.北京：科学出版社.2002：94-124页
[25]周复恭,黄运成.应用线性回归分析.北京：众人大学出版社.1989：1-60页
[26]刘强,尹力.一种简化递推偏最小二乘建模算法及应用.北京航空学报.2003(22)：5-7页
[27]JFrankel J A.David R.Does trade cause growthlJJ.American Economic Review.1999(7):379-399P
[28]Skuldsson H A.PLS regression methods[J].Journal of Chemo-metrics.1988 (2):211-228P
[29]JLucas R E.On the mechanics of economic development J J.Journal of Monetary Economics,1998(22):3-42P
[30]Temple J.The new growth evidence[J].Journal of Economic Literature.1999 (March):112-156P
[31]Netel Jetal.Applied Linear Statistical Models. Chicago:IRWIN Inc.1990: 65-70P
[32]Draper N r and Smith H. Applied Regression Analysis. New York:John Wiley & Son.1981:1-67P
[33]Wold S.Kettaneh-Wold N.SkagerB.NonlinearPLSmodeling[J].Chemometrics and Intelligent LaboratorySystems.l989(7):53-65P
[34]Philippe Bastien.Vincenzo EspositoVinzi,MichelTenenhaus:PLSGerneralised Linear Regression.Computational Statistics and Data Analysis.2005 Vol.48: 17-46P
[35]邓念武.单因变量的偏最小二乘回归模型及其应用[J].武汉大学学报(工学版).2001,33(4)：14-16页
[36]付强.基于偏最小二乘回归的水稻腾发量建模[J].农业工程学报,2002,18(6)：9-12页
[37]秦蓓蕾,王文圣,丁晶.偏最小二乘回归模型在水文相关分析中的应用[J].四川大学学报(工程科学版).2003,35(4)：115-118页
[38]高惠璇.两个多重相关变量组的统计分析[J].数理统计与管理,2002,21(2)：58-64页
[39]周松青.解决多重共线性问题的线性回归方法[J].江苏统计,2000,(11)：14-15页
[40]肖琳,何大卫.PLS回归在消除多元共线性中的作用.山西医科大学学报.2002,33(2)：4-10页
[41]杨杰,吴中如.观测数据拟合分析中的多重共线性问题.自然学报.2005(5)：12-15页
[42]刘燕,计东亚,蒋达火.FDI与我国经济发展的关系-对增长、出口和技术进步作用的检验[J].绍兴文理学院学报.2005(9)：89-93页
[43]高铁梅,康书隆.外商直接投资对中国经济影响的动态分析[J].世界经 济.2006(4)：22-30页
[44]李听.FDI对中国经济增长的影响及其理论预期：基于新古典经济增长模型[J].南方金融.2007(12)：33-36页
[45]Ferrer,N M. An elementary treatise on trilinear coordinates[M]. London Macmillan,1866:49-55P
[46]Quenouille M H. The design and analysis of ex periments[M]. London Griffin 1953.on Paper.2003:944-951P
[47]Cornell J A. Experiments with mixtures. New York:Wiley,1981.[J]Aitchison, The statistical analysis of compositional data[M]. London:Chapman and Hall 1986:123-128P
[48](USA)w.H.Green.USA.Econometric.Analysis[M].NewYork-Macmillan.1993 :288-313P
[49](USA)Roberts,Pindych and Daniel L,Rubinfeld.Econometric Models and ELY)nomie Forecasts[M].(4.Edition)51-106P
[50](USA)Mason.R.L.andGunst.R.F.Outlier-InducedCollinearlties Technometrics. Journal of statistical Associa-tion.1985(27):401-407P
[51]王济川,郭志刚：Logistic回归模型-方法与应用.教育出版社2001：45-50页
[52]Aitchison J.The Statistical Analysis of Compositional Data. CHAPMAN AND HALL. London.1986:25-20P
[53]PhilippeBastien.VincenzoEspositoVinzi.MichelTenenhaus:PLSGerneralisedL inearRegression.ComputationalStatisticsandDataAnalysis.2005:Vol.48,,17-46 P
[54]葛新权,线性化非线性回归模型预测模型质疑[J].1999(1)：77-79页
[55]Bates.DM,Wates DG.非践性回归分析及其应用[M].韦博成译北京：中国筑计出版社.1997：1-35页
[56]Wold S.Nonlinear partial least squares modelling(II):spline inner relation[J].Chemometrics and Intelligent Labora orSystems.1992(14):71-84P
[57]韦博成.近代非线性回归分析[M].南京：东南大学出版社1989：24-30页
[58]王建军.城市化、第三产业发展与消费需求扩张[J].经济与管理.2006,20(8)：13-16页
[59]罗汉,柏超.线性模型中参数的平衡LS估计及其性质[J].湖南大学学报(自然科版)2006,33(2)：122-124页
[60]侯荣华,凤翔.中国固定资产投资效益研究—理论,实证,案例[M].北京：中国计划出版社.2002：215-230页
[2]于秀林,任雪松.多元统计分析[M]中国统计出版社,1999：340-351页
[3]袁志发,周静芋,多元统计分析[M].科学出版社,2002：256-300页
[4]J Iidiko E F,Jerome H F.A statistical view of some chemometrics regression toolsl J J.Technometrics,1993,35:109-135P
[5]肯道尔.多元分析.中国科学院计算中统计概率组译.北京：科学出版社.1983：1-42页
[6]汪洋.PLS回归应用实例.安徽大学学报.2001(3)：18-22页
[7]张尧庭.成分数据统计分析引论.科学出版社,2000：154-201页
[8]宁自军.成分数据的预测方法及其应用[J].统计与决策,2001(5)：6-7页
[9]张崇甫,陈述云.成分数据主成分分析及其应用[J].数理统计与管理,1996,15(3)：14-20页
[10]章栋恩.成分数据的几个估计[J].应用数学学报,2002,25(1)：58-68页
[11]张尧庭.成分数据统计分析引论[C].北京：科学出版社,2000：234-241页
[12]王惠文,黄薇.成分数据的线性回归模型[J]系统工程,2003,21(2)：103-106页
[13]Kenneth N.Berk.TU 8rBnce and Conditioin Regres sion Computations.J.of America StatisticalAssociation.Vel.72.1977.863-86P
[14]施锡铨等数据分析方法.上海上海财经大学出版社.1997：1-10页
[15]李大鹏、王惠文.偏最小二乘Logistic回归在都阳湖洪涝灾害预测中的应用.数理统计与管理.1998(3)：25-40页
[16]罗批,郭继昌,李锵,等.基于偏最小二乘回归建模的探讨[J].天津大学学报.2002,35(6)：783-786页
[17]张旸,周成武等.偏最小二乘回归方法在土地利用结构研究中的应用.自然 资源学报2004：19-24页
[18]张伏生,汪鸿等.基于偏最小二乘回归分析的短期负荷预测.电网技术.2003，27(3)：36-40页
[19]S.Wold.S.Hellberg,T.Lundstedt. M.Sjostrom.H.Wold.ProceedingsPLS model building.Theoryand Applications. Symposium FrankfurtaMain.l987:234-245P
[20]Eriksson.J.Damborsky.M.Earll,E.Johansson.J.Trygg.S.Wold.SARQSAREnvir on.Res.56(2004)481-499P
[21]王惠文.偏最小二乘回归方法及其应用[M].北京：国防工业出版社,1999：201-2063页
[22]许和连,赖明勇,许青松.基于GRPLS回归的中国经济增长影响因素分析管理科学学报.2003.12：5-11页
[23]王惠文,吴载斌等.偏最小二乘回归的线性与非线性方法[M].北京：国防工业出版社.2006：123-134页
[24]范金城,梅长林.数据分析.北京：科学出版社.2002：94-124页
[25]周复恭,黄运成.应用线性回归分析.北京：众人大学出版社.1989：1-60页
[26]刘强,尹力.一种简化递推偏最小二乘建模算法及应用.北京航空学报.2003(22)：5-7页
[27]JFrankel J A.David R.Does trade cause growthlJJ.American Economic Review.1999(7):379-399P
[28]Skuldsson H A.PLS regression methods[J].Journal of Chemo-metrics.1988 (2):211-228P
[29]JLucas R E.On the mechanics of economic development J J.Journal of Monetary Economics,1998(22):3-42P
[30]Temple J.The new growth evidence[J].Journal of Economic Literature.1999 (March):112-156P
[31]Netel Jetal.Applied Linear Statistical Models. Chicago:IRWIN Inc.1990: 65-70P
[32]Draper N r and Smith H. Applied Regression Analysis. New York:John Wiley & Son.1981:1-67P
[33]Wold S.Kettaneh-Wold N.SkagerB.NonlinearPLSmodeling[J].Chemometrics and Intelligent LaboratorySystems.l989(7):53-65P
[34]Philippe Bastien.Vincenzo EspositoVinzi,MichelTenenhaus:PLSGerneralised Linear Regression.Computational Statistics and Data Analysis.2005 Vol.48: 17-46P
[35]邓念武.单因变量的偏最小二乘回归模型及其应用[J].武汉大学学报(工学版).2001,33(4)：14-16页
[36]付强.基于偏最小二乘回归的水稻腾发量建模[J].农业工程学报,2002,18(6)：9-12页
[37]秦蓓蕾,王文圣,丁晶.偏最小二乘回归模型在水文相关分析中的应用[J].四川大学学报(工程科学版).2003,35(4)：115-118页
[38]高惠璇.两个多重相关变量组的统计分析[J].数理统计与管理,2002,21(2)：58-64页
[39]周松青.解决多重共线性问题的线性回归方法[J].江苏统计,2000,(11)：14-15页
[40]肖琳,何大卫.PLS回归在消除多元共线性中的作用.山西医科大学学报.2002,33(2)：4-10页
[41]杨杰,吴中如.观测数据拟合分析中的多重共线性问题.自然学报.2005(5)：12-15页
[42]刘燕,计东亚,蒋达火.FDI与我国经济发展的关系-对增长、出口和技术进步作用的检验[J].绍兴文理学院学报.2005(9)：89-93页
[43]高铁梅,康书隆.外商直接投资对中国经济影响的动态分析[J].世界经 济.2006(4)：22-30页
[44]李听.FDI对中国经济增长的影响及其理论预期：基于新古典经济增长模型[J].南方金融.2007(12)：33-36页
[45]Ferrer,N M. An elementary treatise on trilinear coordinates[M]. London Macmillan,1866:49-55P
[46]Quenouille M H. The design and analysis of ex periments[M]. London Griffin 1953.on Paper.2003:944-951P
[47]Cornell J A. Experiments with mixtures. New York:Wiley,1981.[J]Aitchison, The statistical analysis of compositional data[M]. London:Chapman and Hall 1986:123-128P
[48](USA)w.H.Green.USA.Econometric.Analysis[M].NewYork-Macmillan.1993 :288-313P
[49](USA)Roberts,Pindych and Daniel L,Rubinfeld.Econometric Models and ELY)nomie Forecasts[M].(4.Edition)51-106P
[50](USA)Mason.R.L.andGunst.R.F.Outlier-InducedCollinearlties Technometrics. Journal of statistical Associa-tion.1985(27):401-407P
[51]王济川,郭志刚：Logistic回归模型-方法与应用.教育出版社2001：45-50页
[52]Aitchison J.The Statistical Analysis of Compositional Data. CHAPMAN AND HALL. London.1986:25-20P
[53]PhilippeBastien.VincenzoEspositoVinzi.MichelTenenhaus:PLSGerneralisedL inearRegression.ComputationalStatisticsandDataAnalysis.2005:Vol.48,,17-46 P
[54]葛新权,线性化非线性回归模型预测模型质疑[J].1999(1)：77-79页
[55]Bates.DM,Wates DG.非践性回归分析及其应用[M].韦博成译北京：中国筑计出版社.1997：1-35页
[56]Wold S.Nonlinear partial least squares modelling(II):spline inner relation[J].Chemometrics and Intelligent Labora orSystems.1992(14):71-84P
[57]韦博成.近代非线性回归分析[M].南京：东南大学出版社1989：24-30页
[58]王建军.城市化、第三产业发展与消费需求扩张[J].经济与管理.2006,20(8)：13-16页
[59]罗汉,柏超.线性模型中参数的平衡LS估计及其性质[J].湖南大学学报(自然科版)2006,33(2)：122-124页
[60]侯荣华,凤翔.中国固定资产投资效益研究—理论,实证,案例[M].北京：中国计划出版社.2002：215-230页