含可变参数的缓冲算子与GM（1，1）幂模型研究
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摘要
随着应用领域的日益广泛,现有的灰色预测理论与方法往往难以解决实际应用中不断出现的新问题。本文按照“提出问题、分析问题、解决问题”的逻辑思路,主要研究含可变参数的缓冲算子和GM(1,1)幂模型的理论与应用问题,以期进一步完善灰色预测理论,扩大灰色预测理论与方法的应用范围。本文的主要工作体现在以下几个方面:
1.提出了缓冲算子光滑性的概念,给出了缓冲算子能否提高序列光滑性的判别条件,在此基础上,分别证明了弱化缓冲算子和强化缓冲算子的光滑性特点,并通过经典算例加以验证。从提高建模序列光滑性和消除冲击扰动因素两个方面,揭示了缓冲算子提高灰色模型对冲击扰动系统预测精度的原因。
2.针对传统缓冲算子不能实现作用强度的微调,从而导致缓冲算子的作用效果过强或过弱的问题,将可变参数引入到缓冲算子的构造中,分别提出了两类含可变参数的缓冲算子:幂缓冲算子和变权缓冲算子。构造了若干实用的幂弱化缓冲算子和幂强化缓冲算子,研究了各类幂缓冲算子之间的关系,并分析了幂缓冲算子的参数与其作用强度的定性关系;研究了变权缓冲算子对原始序列作用强度的定量测算与有效控制的问题,从根本上解决了冲击扰动系统的预测过程中常常出现的定量预测结果与定性分析结论不符的问题。在此基础上,提出了“单调性不变”公理,进一步完善了缓冲算子的公理体系。
3.提出了GM(1,1)幂模型幂指数的白化方法,给出了其白化微分方法的求解过程,并分析了GM(1,1)幂模型解的性质;基于矩阵求逆条件数的定义,分三种情形研究了GM(1,1)幂模型参数估计过程可能出现的矩阵病态性问题,总结了影响GM(1,1)幂模型病态性的主要影响因素。基于非线性优化方法,依次对GM(1,1)幂模型的幂指数、背景值插值系数,以及初始条件进行优化,并通过实例验证了本文的优化效果,实现了灰色预测方法对小样本振荡序列的高精度预测。此外,基于GM(1,1)幂模型的基本定义,本文提出了GM(1,1)幂模型的五类派生模型,进一步完善了灰色系统幂模型体系。
4.基于GM(1,1)幂模型的误差分析,提出了无偏GM(1,1)幂模型,并从理论上和实例应用两个方面证明了该模型的无偏性;在此基础上,考虑参数β1和β2已知和未知两种情形下,分别研究了无偏GM(1,1)幂模型初始条件的优化问题,并通过实例验证模型的有效性。
5.以本文提出的变权缓冲算子和优化的GM(1,1)幂模型为基本工具,研究了我国31个省、市、自治区的工业废水排放达标率的预测和预警问题,并提出了相应的政策建议。
With the wider application of Grey forecast theory, there are increasingly many new issues that need to be tackled with this theory. Based on ideas of“questions raising, question analysis, and question solution”, this paper focuses on the theories and application of parameter-contained buffer operators and GM(1,1) exponential model. This research aims at improving Grey Systems Theory as well as expanding its application. The study can be concluded as the following parts.
1. The smoothness of buffer operator is defined by initiating criterion of improving the smoothness of series as well as analyzing the smoothness of weakening buffer operator and reinforcement buffer operator. The reason of buffer operator to improve the forecast accuracy of shock disturbed system has been revealed in aspects of improving smoothness of series and eliminating shock disturbed factors.
2. It is known that function strength cannot be slightly adjusted in traditional buffer operator so that its function effect is hard to co-ordinate. To solve this problem, the variable parameter is introduced in the construction of buffer operators. Such operators can be divided into two categories, exponential buffer operators and weight variable buffer operators. Several practical exponential weakening and reinforcement buffer operators are constructed. In addition, the relation of various kinds of buffer operators has been studied as well as the quantificational relation between parameters and their function strength in exponential buffer operators. Besides, the function strength is measured quantitatively and effectively controlled in weight variable buffer operator to original series. This can perfectly solve the discrepancy of results in quantitative forecast and qualitative analysis in forecast process of shock disturbed system. Based on this, the axiom of“monotony invariance”is put forward to further improve the axiom system of buffer operators.
3. The whitenization methods of power exponent in GM(1,1) Power Model is put forward with the solution in whitenized differential method and properties of solution. Based on definition of condition matrix in inverse problem, the issues of possible ill-condition of matrix are studied in the parameter estimation. The main factors affecting the ill-condition of GM(1,1) Power Model are concluded. Based on non-linear optimization methods, the power exponents, interpolation coefficient of background value, and initial condition are optimized. The optimal results are verified in case study to achieve high accuracy in fluctuant series with small sample. In addition, based on the basic definition of GM(1,1) Power Model, five derived models are put forward to perfect the system of Grey Exponential Models.
4. Based on the error analysis of GM(1,1) Power Model, unbiased GM(1,1) Power Model is put forward. Furthermore, the unbiasedness of model is proved in theory and in case study. Based on this, the optimization of initial condition in unbiased GM(1,1) Power Model is analyzed separately taking account of given parametersβ1 andβ2. The effectiveness of model is studied in case study.
5. The forecast and warning of water discharge rate in 31 provinces and administrative regions are studied by method of weight variable buffer operator and optimized GM(1,1) Power Model, followed by policy analysis.
1.提出了缓冲算子光滑性的概念,给出了缓冲算子能否提高序列光滑性的判别条件,在此基础上,分别证明了弱化缓冲算子和强化缓冲算子的光滑性特点,并通过经典算例加以验证。从提高建模序列光滑性和消除冲击扰动因素两个方面,揭示了缓冲算子提高灰色模型对冲击扰动系统预测精度的原因。
2.针对传统缓冲算子不能实现作用强度的微调,从而导致缓冲算子的作用效果过强或过弱的问题,将可变参数引入到缓冲算子的构造中,分别提出了两类含可变参数的缓冲算子:幂缓冲算子和变权缓冲算子。构造了若干实用的幂弱化缓冲算子和幂强化缓冲算子,研究了各类幂缓冲算子之间的关系,并分析了幂缓冲算子的参数与其作用强度的定性关系;研究了变权缓冲算子对原始序列作用强度的定量测算与有效控制的问题,从根本上解决了冲击扰动系统的预测过程中常常出现的定量预测结果与定性分析结论不符的问题。在此基础上,提出了“单调性不变”公理,进一步完善了缓冲算子的公理体系。
3.提出了GM(1,1)幂模型幂指数的白化方法,给出了其白化微分方法的求解过程,并分析了GM(1,1)幂模型解的性质;基于矩阵求逆条件数的定义,分三种情形研究了GM(1,1)幂模型参数估计过程可能出现的矩阵病态性问题,总结了影响GM(1,1)幂模型病态性的主要影响因素。基于非线性优化方法,依次对GM(1,1)幂模型的幂指数、背景值插值系数,以及初始条件进行优化,并通过实例验证了本文的优化效果,实现了灰色预测方法对小样本振荡序列的高精度预测。此外,基于GM(1,1)幂模型的基本定义,本文提出了GM(1,1)幂模型的五类派生模型,进一步完善了灰色系统幂模型体系。
4.基于GM(1,1)幂模型的误差分析,提出了无偏GM(1,1)幂模型,并从理论上和实例应用两个方面证明了该模型的无偏性;在此基础上,考虑参数β1和β2已知和未知两种情形下,分别研究了无偏GM(1,1)幂模型初始条件的优化问题,并通过实例验证模型的有效性。
5.以本文提出的变权缓冲算子和优化的GM(1,1)幂模型为基本工具,研究了我国31个省、市、自治区的工业废水排放达标率的预测和预警问题,并提出了相应的政策建议。
With the wider application of Grey forecast theory, there are increasingly many new issues that need to be tackled with this theory. Based on ideas of“questions raising, question analysis, and question solution”, this paper focuses on the theories and application of parameter-contained buffer operators and GM(1,1) exponential model. This research aims at improving Grey Systems Theory as well as expanding its application. The study can be concluded as the following parts.
1. The smoothness of buffer operator is defined by initiating criterion of improving the smoothness of series as well as analyzing the smoothness of weakening buffer operator and reinforcement buffer operator. The reason of buffer operator to improve the forecast accuracy of shock disturbed system has been revealed in aspects of improving smoothness of series and eliminating shock disturbed factors.
2. It is known that function strength cannot be slightly adjusted in traditional buffer operator so that its function effect is hard to co-ordinate. To solve this problem, the variable parameter is introduced in the construction of buffer operators. Such operators can be divided into two categories, exponential buffer operators and weight variable buffer operators. Several practical exponential weakening and reinforcement buffer operators are constructed. In addition, the relation of various kinds of buffer operators has been studied as well as the quantificational relation between parameters and their function strength in exponential buffer operators. Besides, the function strength is measured quantitatively and effectively controlled in weight variable buffer operator to original series. This can perfectly solve the discrepancy of results in quantitative forecast and qualitative analysis in forecast process of shock disturbed system. Based on this, the axiom of“monotony invariance”is put forward to further improve the axiom system of buffer operators.
3. The whitenization methods of power exponent in GM(1,1) Power Model is put forward with the solution in whitenized differential method and properties of solution. Based on definition of condition matrix in inverse problem, the issues of possible ill-condition of matrix are studied in the parameter estimation. The main factors affecting the ill-condition of GM(1,1) Power Model are concluded. Based on non-linear optimization methods, the power exponents, interpolation coefficient of background value, and initial condition are optimized. The optimal results are verified in case study to achieve high accuracy in fluctuant series with small sample. In addition, based on the basic definition of GM(1,1) Power Model, five derived models are put forward to perfect the system of Grey Exponential Models.
4. Based on the error analysis of GM(1,1) Power Model, unbiased GM(1,1) Power Model is put forward. Furthermore, the unbiasedness of model is proved in theory and in case study. Based on this, the optimization of initial condition in unbiased GM(1,1) Power Model is analyzed separately taking account of given parametersβ1 andβ2. The effectiveness of model is studied in case study.
5. The forecast and warning of water discharge rate in 31 provinces and administrative regions are studied by method of weight variable buffer operator and optimized GM(1,1) Power Model, followed by policy analysis.
引文
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