基于序列算子的灰色预测模型研究与应用
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摘要
灰色系统理论的研究对象是“小样本”、“贫信息”不确定性问题,主要通过对部分已知信息的生成、开发,提取有价值的信息,实现对系统运行行为、演化规律的正确描述和有效监控。灰色预测模型是灰色系统理论的重要组成部分,其突出的特点是灰生成和灰微分方程,灰生成是使灰过程变白的数据变换方法,能为建立灰色预测模型提供中间信息,并弱化原始数据的随机性。本文系统分析了序列算子在灰色预测模型的预处理、灰生成及灰建模过程中的作用,基于序列算子建立了初始化的GM(1,1)模型参数最小二乘估计,并基于序列算子对灰色系统预测模型进行了相关研究。本文的主要创新点有:
(1)构造比较分析和整合研究了实用的强化缓冲算子和弱化缓冲算子,并用C++builder开发了缓冲算子计算软件。缓冲算子主要适用于系统数据失真情况下的建模,实现了定性分析结果与定量模型的有机结合。
(2)通过多变量线性序列算子对应的矩阵及其运算,提出GM建模数据的(广义)累加-均值生成,分析了(广义)累加-均值生成和线性弱(强)化缓冲生成的算子矩阵及其特性,并推导了线性弱(强)化缓冲算子多阶作用序列的初始化、统一化、程序化公式。
(3)建立了基于累加-均值生成GM(1,1)模型参数的最小二乘估计,该参数估计式提高了模型参数求解的初始化程度;另外针对冲击扰动系统预测问题,运用参数估计式归纳了其GM(1,1)建模过程,由此其它特殊序列(如异常值序列、阶段变化序列、随机振荡序列)的灰建模问题,运用该公式也可以减少灰色模型的建模步骤,提高模型计算速度。
(4)在灰色系统预测模型方面,提出了求解GM(1,1)模型群参数最小二乘估计的整体算法,该方法避免了在建模过程中的信息丢失和重新建模求解带来的计算量,同时增强了子模型间的有机联系和GM(1,1)模型群的整体分析解释能力;根据数乘变换序列的固有性质,进一步研究了数乘变换对灰色系统模型(GM ( n ,h)模型、GM(1,1)幂模型及离散灰色模型)参数和预测值的影响,分别给出了数乘变换序列GM ( n ,h)模型、GM(1,1)幂模型及离散灰色模型参数特性的简捷证明。
The grey system theory deals with the uncertain problems with little data and scare information. It extracts the valuable information by creating and developing the partial known information to describe and control the rules of systems well. Grey forecasting model is the core of grey system theory, and the characteristics of grey forecasting model are grey generating and grey differential equation. Grey generating is a method of transferring the grey process into white process. It can provide middle information for grey forecasting model and weaken the randomness of the raw data. This paper has analysed the roles of sequence operators in grey generating, grey modeling and its pretreatment. The initial parameter estimate of least square has been established and some properties of grey system forecasting models are discussed. The main innovations of this paper are as follows.
(1)Some compared weak buffer operators and strengthening buffer operators are established and analysed which have the universality and practicability. The buffer operators program has compiled by C + + builder. The buffer operator sequences are researched and their compared relations are required. These buffer operators are applied for vibration data sequences, some contradictions between qualitative analysis and quantitative forecast can be resolved effectively.
(2)The definition of linear sequence operator matrix is given, their characters and inherent relations are studied. The generating of data’s accumulating - mean is suggested. The matrix of the generating and some linear buffer operators are described. The initialed and unified formula of multi-step buffer operator was proved by the buffer operator matrix.
(3)The least square regression of GM(1,1) model parameter based on the generating of data’s accumulating - mean is proposed. The estimate improves the initial level of solving GM(1,1) model parameter. The modeling steps of vibration data sequences have summed up. Based on the method, the modeling steps of other special data sequences ( such as unusual value sequence and stage sequence, or random vibration sequence) can also reduce the modeling steps and raise the modeling speed.
(4)The whole algorithm of GM(1,1) models is established. The benefits of the algorithm have avoided the lose of modeling information and the work of remodeling. It has strengthened the connection between GM(1,1) models and has improved the whole ability of GM(1,1) models’explanation. The Characteristic properties of grey system forecasting models are researched. The study has been made on the data processing of grey system forecasting models. The parameter properties of a series of GM(n, h) model, grey power model and discrete grey model are proved simply based on the data’s inherent multiply nature.
(1)构造比较分析和整合研究了实用的强化缓冲算子和弱化缓冲算子,并用C++builder开发了缓冲算子计算软件。缓冲算子主要适用于系统数据失真情况下的建模,实现了定性分析结果与定量模型的有机结合。
(2)通过多变量线性序列算子对应的矩阵及其运算,提出GM建模数据的(广义)累加-均值生成,分析了(广义)累加-均值生成和线性弱(强)化缓冲生成的算子矩阵及其特性,并推导了线性弱(强)化缓冲算子多阶作用序列的初始化、统一化、程序化公式。
(3)建立了基于累加-均值生成GM(1,1)模型参数的最小二乘估计,该参数估计式提高了模型参数求解的初始化程度;另外针对冲击扰动系统预测问题,运用参数估计式归纳了其GM(1,1)建模过程,由此其它特殊序列(如异常值序列、阶段变化序列、随机振荡序列)的灰建模问题,运用该公式也可以减少灰色模型的建模步骤,提高模型计算速度。
(4)在灰色系统预测模型方面,提出了求解GM(1,1)模型群参数最小二乘估计的整体算法,该方法避免了在建模过程中的信息丢失和重新建模求解带来的计算量,同时增强了子模型间的有机联系和GM(1,1)模型群的整体分析解释能力;根据数乘变换序列的固有性质,进一步研究了数乘变换对灰色系统模型(GM ( n ,h)模型、GM(1,1)幂模型及离散灰色模型)参数和预测值的影响,分别给出了数乘变换序列GM ( n ,h)模型、GM(1,1)幂模型及离散灰色模型参数特性的简捷证明。
The grey system theory deals with the uncertain problems with little data and scare information. It extracts the valuable information by creating and developing the partial known information to describe and control the rules of systems well. Grey forecasting model is the core of grey system theory, and the characteristics of grey forecasting model are grey generating and grey differential equation. Grey generating is a method of transferring the grey process into white process. It can provide middle information for grey forecasting model and weaken the randomness of the raw data. This paper has analysed the roles of sequence operators in grey generating, grey modeling and its pretreatment. The initial parameter estimate of least square has been established and some properties of grey system forecasting models are discussed. The main innovations of this paper are as follows.
(1)Some compared weak buffer operators and strengthening buffer operators are established and analysed which have the universality and practicability. The buffer operators program has compiled by C + + builder. The buffer operator sequences are researched and their compared relations are required. These buffer operators are applied for vibration data sequences, some contradictions between qualitative analysis and quantitative forecast can be resolved effectively.
(2)The definition of linear sequence operator matrix is given, their characters and inherent relations are studied. The generating of data’s accumulating - mean is suggested. The matrix of the generating and some linear buffer operators are described. The initialed and unified formula of multi-step buffer operator was proved by the buffer operator matrix.
(3)The least square regression of GM(1,1) model parameter based on the generating of data’s accumulating - mean is proposed. The estimate improves the initial level of solving GM(1,1) model parameter. The modeling steps of vibration data sequences have summed up. Based on the method, the modeling steps of other special data sequences ( such as unusual value sequence and stage sequence, or random vibration sequence) can also reduce the modeling steps and raise the modeling speed.
(4)The whole algorithm of GM(1,1) models is established. The benefits of the algorithm have avoided the lose of modeling information and the work of remodeling. It has strengthened the connection between GM(1,1) models and has improved the whole ability of GM(1,1) models’explanation. The Characteristic properties of grey system forecasting models are researched. The study has been made on the data processing of grey system forecasting models. The parameter properties of a series of GM(n, h) model, grey power model and discrete grey model are proved simply based on the data’s inherent multiply nature.
引文
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