【作者】 张军亮；

【导师】 刘新平；

【作者基本信息】 陕西师范大学 ， 应用数学， 2003， 硕士

【摘要】 非线性回归分析(Nonlinear Regression Analysis)是近二十年来首先在国外迅速发展起来的重要统计方法，是在传统的线性回归分析(Linear Regression Analysis)的基础上借助计算机来处理复杂非线性问题的常用手段。其实质是通过构建变量间的非线性回归模型来达到对变量变化关系的拟合和控制预测目的。在许多实际问题当中，观测数据所反映出的变量间的函数关系往往是较为复杂的非线性关系，如果仍采用线性回归方法或经过变量和参数变换线性化后再采用线性回归方法，则常会使问题变得无法处理或因为随机项假设条件的改变而难以取得满意的拟合效果。在非线性回归模型的构建中，模型期望函数的选取和参数迭代初值的设定是两个不可或缺的重要环节，针对具体的观测数据选取合适的模型期望函数和接近真实值的参数初估值是保证参数迭代估计收敛、提高收敛速度和避免模型失拟的前提。这一点也是目前非线性回归分析中灵活性较高、人为因素复杂且无一般程式可供遵循的难点问题之一。 为此，本文首先根据以往许多著名的S形曲线如Logistic曲线、Gompertz 曲线、Richards曲线和Bertalanffy曲线等，对一类具有S形特征的观测数据推导出建立非线性回归模型所需的一般S形期望函数——广义S形曲线表达式，并详细地给出各类常见S形曲线和广义S形曲线非线性回归模型参数迭代初估值的基本计算方法。最后结合实际观察数据采用SAS统计软件编程处理方法，建立了广义S形曲线非线性回归模型(Nonlinear Regression Models of Generalized S-Curve)，并将这一结果成功地应用于文献计量学中，对科技文献增长过程进行了非线性回归拟合和预测控制。 第一章，简要介绍了非线性回归分析的一般方法，给出了建立非线性回归模型的主要步骤和基本流程图，总结出在非线性状态进行模型期望函数选择的意义和一般的选取原则和方法，说明了参数迭代初估值的设定在各种迭代法中的重要性及其与迭代收敛和收敛速度之间的密切关系。除此之外，还指出了非线性回归与线性回归在分析方法以及参数和模型检验方面的差异和存在的问题。 第二章，由常见的几种S形曲线归纳出一般S形函数的基本特征关系式，在此基础上进一步分析了各类S形函数增长率与响应变量增长的变化关系，以Richards曲线的增长率为原形，Gompertz曲线的初值为渐近值，通过求解方程初值问题推导出具有一般S形特征的广义S形函数表达式。这样，在对S形特征观测数据的非线性回归分析中，可首先考虑直接选取广义S形函数作为模型期望函数。 第三章，给出了S形函数参数初估值的六种设定方法和广义S形函数及其它几种常见S形函数参数初估值的设定结果。采用这些方法并通过SAS／TATA统计软件编程处理方式，成功地建立了国内出版机构增长的广义S形曲线非线性回归模型，对三十年来国内出版机构增长的变化过程进行了有效的拟合和预测。其中包括有关参数估计和模型检验的 SAS／TATA语言程序输出结果和几种 S形曲线模型拟合效果的对比分析。 第四章，分析了文献计量学中不同时期关于科技文献增长模型的产生机理和模型的结构缺点。借用广义S形曲线非线性回归模型，对至今一直使用的文献增氏综合模型进行了结构调整，给出了科技文献增长的广义综合模型。作为广义S形曲线非线性回归模型的一个具体应用，最后通过对几类文献观测数据分别建立的不同模型的拟合效果分析，肯定了广义S形曲线非线性回归模型的有效性和实用性。 本文给出的广义S形函数表达式和S形函数参数初估值的设定方法，能够在局部范围内解决S形曲线非线性回归分析中两个较为困难的问题：期望函数和参数初估值的设定，从而使得对具有S形特征的一类观测数据的非线性回归分析变得相对容易，实例分析结果也说明广义S形曲线非线性回归模型拟合误差和稳定性通常要优于其它同类S形曲线模型，而且还可在参数估计结果的基础上，根据调节因子参数d的估计值进一步给出所对应的三参数S形曲线模型，以起到简化模型的作用。更多还原

【Abstract】 Nonlinear Regression Models (NRM) play a central part in modern statistical methods. These models have been one kind of most heavily-used methods first in abroad since recent twenty years. By comparison, Nonlinear Regression Analysis (NRA) based on the classical Linear Regression Analysis (LRA) can be put to use in dealing with the nonlinear relations between variables that can not be solved by linear regression method even under the simple linear transformation on variables and parameters for its unsatisfying goodness of fit. It has been proved that Nonlinear Regression Analysis is a powerful method in proceeding nonlinear problems with the aid of computer and some statistical software (SAS, SPSS); however, the good fitting result can not be gained every time for the complexities itself and the assumption of random disturbances errors of model. The selections of expectation function and the parameter initial value are two main factors contributing to this. These are two kinds of difficult problems in Nonlinear Regression Analysis for their complexities and flexibilities in proceeding by hand without normal method to abide by. In fact, it’s a important factor for getting a rapid convergence criterion and a satisfying goodness of fit to select a fitting expectation function and an idea parameter initial value.This paper discusses and researches the Nonlinear Regression Model of Generalized S-Curve on the set of observations in the shape of S. It gives out a more widely used expectation function S-Curve on account of some famous S-Curve functions exited. Some general estimating methods of parameter initial value are obtained in detail. As an important application and test of the model, a Nonlinear Regression Model in bibliometrics are made out successfully with the help of computer and the statistical software SAS.In chapter 1, This paper gives out a brief introduction to Nonlinear Regression Analysis, staring with the basic concepts, general methods and the graph of assembly-line, then describes the general principle of selection of fitting expectation function and idea parameter initial value for Nonlinear Regression Model, up to somedifficult problems to be solved in the future.Chapter 2 presents the expression of generalized S-Curve function based on some kinds ordinary S-Curve functions widely used in the past in many branches of sciences such as Logistic Curve, Gompertz Curve, Bertalanffy Curve and Richards Curve and so on. It draws out the fundamental property representative formula of generalized S-Curve function. On account of the flexible relative increment rate of Richards Curve and the adaptive initial value of Gompertz Curve, the generalized S-Curve function can be concluded by solving the initial value problem and should be chosen as the expectation function in Nonlinear Regression Analysis first for its generalized properties of S-Curve.Chapter 3 contains the standard procedure for Nonlinear Regression Analysis on the set of observations of the national publishing organizations number from year 1971 to 2000 including the six general estimating methods of parameter initial value and the concrete number for this kind of S-Curve. By the aid of the SAS program, it shows that the Nonlinear Regression Model is a more idea model than the other S-Curve models about this set of observations.Chapter 4 is devoted to the applications of Nonlinear Regression Model in bibliometrics. This chapter analyses the advantages and the disadvantages of the increment of scientific document models existed already for many years, It points out that comprehensive model given in 1988 can not be put to use in the case of asymmetric increment. So a new Nonlinear Regression Model of Generalized S-Curve is made to fit and predict the increment of document instead of it by changing the Generalized S-Curve for the Logistic Curve in the last chapter.The Generalized S-Curve and general estimating methods of parameter initial value on S-Curve presented in this paper can solves two relative difficult problems in S-C更多还原