灰色预测技术及其应用研究
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摘要
在系统科学的研究中,由于内外扰动的存在和认识水平的局限,人们得到的信息往往带有某种不确定性。随着科学技术水平的发展和人类社会的进步,人们对系统不确定性的认识逐步深化,不确定性系统的研究日益深入。灰色系统理论着重研究概率统计、模糊数学等所难以解决的“小样本”、“贫信息”不确定性问题,并依据信息覆盖,通过从已知数据中生成、开发和提取有价值的信息,实现对事物运动规律的探索。另外,灰色系统理论对数据没有什么特殊的要求和限制,应用领域十分宽广。灰色预测是灰色系统理论中的一个重要组成部分,也是一个非常活跃的研究领域。现有的相关文献主要从初始值、背景值、灰导数、离散化、模型参数和病态性等角度对灰色预测模型进行研究,取得了丰硕的成果,但是,灰色预测技术在理论上依然存在一些殛待解决的问题。本文对灰色预测技术展开研究:根据灰色系统理论中的缓冲算子理论,通过灰色序列生成,在满足缓冲算子公理体系的基础上构建了物理意义明确的若干弱化和强化缓冲算子;总结了已有的数据变换技术条件,并提出相应的数据变换技术;针对非单调系统产生的振荡序列的灰色建模技术进行研究;把连分式理论和灰色模型有效地结合起来,提出了基于连分式理论的GM(1,1)模型和基于向量连分式理论的MGM(1,n)模型。
本文的工作主要分为以下几个部分:
(1)缓冲算子理论研究。缓冲算子理论是灰色系统理论的主要特色理论之一,也是近年来灰色系统理论研究的热点之一。灰色系统理论通过对社会、经济、生态等系统的原始数据挖掘和整理来寻求其变化规律的,这是一种从数据来寻找规律的理论体系。目前关于缓冲算子的研究基本分为,缓冲算子构建的理论研究和利用缓冲算子解决实际问题的应用研究两个方面。本文在缓冲算子公理体系下,根据灰色系统理论的“新信息优先”原理和时间序列理论中的一些思想,构造了若干具有明确物理意义的弱化和强化缓冲算子,对所获得的原始数据序列经过缓冲处理,能够弱化其随机性,显示其规律性,成功地排除了外在冲击干扰,得到了能够反映系统变化规律的数据序列,从而提高了灰色预测模型的稳定性和精度。
(2)数据变换技术研究。数据变换技术作为提高灰色预测模型精度的方法之一是行之有效的。本文全面分析了数据变换技术提高灰色预测模型精度的影响因素,指出选择数据变换来提高模型精度应从整体上综合考虑,主要与以下几个方面的因素有关:一是提高光滑比,二是级比压缩,三是保持凹凸性,四是还原误差。以这些因素为数据变换技术的准则,分析比较现有数据变换形式的优劣,同时在满足数据变换技术准则的条件下,提出了两种数据变换形式,分别是对数函数变换f(x~(0)(k))=clnx~(0)(k)+d和三角函数变换f(y~(0)(k))=cscy~(0)(k),提高了灰色预测模型的预测精度和适用性。
(3)研究了非单调系统的振荡序列灰色预测模型的建模技术。当原始数据序列带有一定振荡特征时,构建GM(1,1)模型难以获得较高的模拟、预测精度。当原始数据振荡且摆动幅度不是很大时,本文对原始数据序列进行适当的处理,把原始波动数据序列转换为单调增长序列,然后建立GM(1,1)模型,并研究了模型的一些性质。另外,针对先快速增长,后低速增长的具有“S”形振荡序列,提出了灰色离散Verhulst模型。
(4)连分式理论与灰色模型结合的新型组合模型。由于系统存在不确定性因素,利用传统的单个预测模型进行预测的缺陷表现为对模型设定形式的敏感性等,因此,仅采用传统的灰色系统预测模型往往难以取得理想的预测效果。众所周知,不同的理论和方法,来源于不同的物理背景,都是为解决实际生活中遇到的某一小类问题而诞生的。但是这些理论和方法之间并不是相互排斥,而是相互联系、相互补充的。因为这些理论和方法都有自己的特点和特色,能够挖掘到一个系统中不同的有用信息,这些有用信息对于正确预测都很重要。鉴于此,本文把连分式理论与灰色预测模型结合起来,提出了基于连分式理论的GM(1,1)模型和基于向量连分式理论的MGM(1,n)模型,从而有效地提高了灰色组合预测模型的模拟和预测精度。
In research of system, because of the existence of internal and external disturbances and the limited level of awareness, people get information with some uncertain. With the scientific and technological of development and progress of human society, people gradually deepen to understand uncertainty of all kinds of systems, and strengthen to research system’s uncertainty. Grey systems theory is developed to study problems of“small samples and poor information”. These problems studied by grey systems theory cannot be handled successfully by using either probablitity statistics or fuzzy mathematics.Grey systems theory looks for realistic patterns based on modeling a few available data. Different from fuzzy mathematics, grey systems theory focuses on such research objects that have clear extension and unclear intension.Grey system theory explores reality regulity through the generation of information, development, and extraction of valuable information. Grey systems theory has no special requirements and restrictions on data sequence. So its application is very broad. Grey prediction is an important component of grey system theory, but also a very active research field. Mainly from the initial value, the background value, grey derivative, discretization, model parameters and the pathological point of view, etc. the existing literatures have achieved fruitful results. But there are still many theoretical problems needed to be resolved as soon as possible. This paper aims to study the basic theories of grey prediction thechnology. According to the grey system theory in the buffer operator theory, some new buffer operators have been constructed through the grey sequence generated to satisfy the three buffer operator axioms. Summarized conditions of existing data transformation technology, the corresponding data transformation technologies are proposed. The paper studies how to model with the oscillation sequence of non-monotonic system. Grey models are proposed based on the theory of continued fractions GM (1,1) model and the theory of vector continued fractions MGM (1, n) model. The main innovations of the paper are follows.
The first innovation is to construct some new buffer operators with clear phisical meanings. Buffer operator theory is an inpormant aspect of grey system theory and one of the main features of the theory. Grey system theory seeks the laws of a system, such as the social, economic, ecological systems, which is a kind of data from the data to find the rules. At present, the study on the buffer operators is basically divided into two aspect: rebuilding the buffer operators and application of the existing buffer operators to solve practical problems. In this paper, some new buffer operators are constructed with economic sigificance based on grey system theory "the new information priority" principle and the theory of time series. They can weaken some randomness to show regularity successfully by excluding the impact of external interference. So stability and prediction accuracy of grey prediction model are improved.
The second innovation is to data transformation technology. Data transformation technologies as a method to improve prediction accuracy of grey model one of the methods are effective. The paper comprehensively analysizes factors of data transformation technology to improve prediction accuracy of grey model, and shows that the choice of data transformation technology should be considered as a whole. Smooth ratio, stepwise ratio, convex-concave and reductive error should be considered. The paper takes these factors as the principles of data transformation technology, and proposes two forms of data transformation technology. These two forms improve the prediction accuracy and applicability of grey model. Also the existing data transformation technolgies are analysized.
The third innovation is to how to model with the oscillation sequence of non-monotonic system. When the original data sequence features with some fluctuations, to build GM (1, 1) model for the simulation of access to higher forecast accuracy. When the raw data rate volatility is not a big swing and the raw data in this article series on the proper handling of the fluctuations in the original data sequence into a monotonous sequence of growth, and then establishment of GM (1,1) model, and to study some properties of the model. In addition, grey discrete Verhulst model is proposed with the rapid growth in the fist part and slow growth in the second part of data sequence. The Verhulst model is mainly used to study processes with saturated states (or say sigmoid processes).
The last innovation is to combine grey prediction model with continue fraction theory. Because of uncertainty of system, there is sensitivity to the setting form of traditional single model. Therefore, only using the traditional grey forecasting model is often difficult to achieve the desired prediction. As we know, Different theories and methods from different physical backgrounds, are to solve a subset of real life problem encountered. And these theories and methods are not mutually exclusive, but interrelated and mutually complementary. These theories and methods have their own characteristics and features to excavate from a different system, useful information which is very important for accurate prediction. So grey models are proposed based on the theory of continued fractions GM (1,1) model and the theory of vector continued fractions MGM (1, n) model.
本文的工作主要分为以下几个部分:
(1)缓冲算子理论研究。缓冲算子理论是灰色系统理论的主要特色理论之一,也是近年来灰色系统理论研究的热点之一。灰色系统理论通过对社会、经济、生态等系统的原始数据挖掘和整理来寻求其变化规律的,这是一种从数据来寻找规律的理论体系。目前关于缓冲算子的研究基本分为,缓冲算子构建的理论研究和利用缓冲算子解决实际问题的应用研究两个方面。本文在缓冲算子公理体系下,根据灰色系统理论的“新信息优先”原理和时间序列理论中的一些思想,构造了若干具有明确物理意义的弱化和强化缓冲算子,对所获得的原始数据序列经过缓冲处理,能够弱化其随机性,显示其规律性,成功地排除了外在冲击干扰,得到了能够反映系统变化规律的数据序列,从而提高了灰色预测模型的稳定性和精度。
(2)数据变换技术研究。数据变换技术作为提高灰色预测模型精度的方法之一是行之有效的。本文全面分析了数据变换技术提高灰色预测模型精度的影响因素,指出选择数据变换来提高模型精度应从整体上综合考虑,主要与以下几个方面的因素有关:一是提高光滑比,二是级比压缩,三是保持凹凸性,四是还原误差。以这些因素为数据变换技术的准则,分析比较现有数据变换形式的优劣,同时在满足数据变换技术准则的条件下,提出了两种数据变换形式,分别是对数函数变换f(x~(0)(k))=clnx~(0)(k)+d和三角函数变换f(y~(0)(k))=cscy~(0)(k),提高了灰色预测模型的预测精度和适用性。
(3)研究了非单调系统的振荡序列灰色预测模型的建模技术。当原始数据序列带有一定振荡特征时,构建GM(1,1)模型难以获得较高的模拟、预测精度。当原始数据振荡且摆动幅度不是很大时,本文对原始数据序列进行适当的处理,把原始波动数据序列转换为单调增长序列,然后建立GM(1,1)模型,并研究了模型的一些性质。另外,针对先快速增长,后低速增长的具有“S”形振荡序列,提出了灰色离散Verhulst模型。
(4)连分式理论与灰色模型结合的新型组合模型。由于系统存在不确定性因素,利用传统的单个预测模型进行预测的缺陷表现为对模型设定形式的敏感性等,因此,仅采用传统的灰色系统预测模型往往难以取得理想的预测效果。众所周知,不同的理论和方法,来源于不同的物理背景,都是为解决实际生活中遇到的某一小类问题而诞生的。但是这些理论和方法之间并不是相互排斥,而是相互联系、相互补充的。因为这些理论和方法都有自己的特点和特色,能够挖掘到一个系统中不同的有用信息,这些有用信息对于正确预测都很重要。鉴于此,本文把连分式理论与灰色预测模型结合起来,提出了基于连分式理论的GM(1,1)模型和基于向量连分式理论的MGM(1,n)模型,从而有效地提高了灰色组合预测模型的模拟和预测精度。
In research of system, because of the existence of internal and external disturbances and the limited level of awareness, people get information with some uncertain. With the scientific and technological of development and progress of human society, people gradually deepen to understand uncertainty of all kinds of systems, and strengthen to research system’s uncertainty. Grey systems theory is developed to study problems of“small samples and poor information”. These problems studied by grey systems theory cannot be handled successfully by using either probablitity statistics or fuzzy mathematics.Grey systems theory looks for realistic patterns based on modeling a few available data. Different from fuzzy mathematics, grey systems theory focuses on such research objects that have clear extension and unclear intension.Grey system theory explores reality regulity through the generation of information, development, and extraction of valuable information. Grey systems theory has no special requirements and restrictions on data sequence. So its application is very broad. Grey prediction is an important component of grey system theory, but also a very active research field. Mainly from the initial value, the background value, grey derivative, discretization, model parameters and the pathological point of view, etc. the existing literatures have achieved fruitful results. But there are still many theoretical problems needed to be resolved as soon as possible. This paper aims to study the basic theories of grey prediction thechnology. According to the grey system theory in the buffer operator theory, some new buffer operators have been constructed through the grey sequence generated to satisfy the three buffer operator axioms. Summarized conditions of existing data transformation technology, the corresponding data transformation technologies are proposed. The paper studies how to model with the oscillation sequence of non-monotonic system. Grey models are proposed based on the theory of continued fractions GM (1,1) model and the theory of vector continued fractions MGM (1, n) model. The main innovations of the paper are follows.
The first innovation is to construct some new buffer operators with clear phisical meanings. Buffer operator theory is an inpormant aspect of grey system theory and one of the main features of the theory. Grey system theory seeks the laws of a system, such as the social, economic, ecological systems, which is a kind of data from the data to find the rules. At present, the study on the buffer operators is basically divided into two aspect: rebuilding the buffer operators and application of the existing buffer operators to solve practical problems. In this paper, some new buffer operators are constructed with economic sigificance based on grey system theory "the new information priority" principle and the theory of time series. They can weaken some randomness to show regularity successfully by excluding the impact of external interference. So stability and prediction accuracy of grey prediction model are improved.
The second innovation is to data transformation technology. Data transformation technologies as a method to improve prediction accuracy of grey model one of the methods are effective. The paper comprehensively analysizes factors of data transformation technology to improve prediction accuracy of grey model, and shows that the choice of data transformation technology should be considered as a whole. Smooth ratio, stepwise ratio, convex-concave and reductive error should be considered. The paper takes these factors as the principles of data transformation technology, and proposes two forms of data transformation technology. These two forms improve the prediction accuracy and applicability of grey model. Also the existing data transformation technolgies are analysized.
The third innovation is to how to model with the oscillation sequence of non-monotonic system. When the original data sequence features with some fluctuations, to build GM (1, 1) model for the simulation of access to higher forecast accuracy. When the raw data rate volatility is not a big swing and the raw data in this article series on the proper handling of the fluctuations in the original data sequence into a monotonous sequence of growth, and then establishment of GM (1,1) model, and to study some properties of the model. In addition, grey discrete Verhulst model is proposed with the rapid growth in the fist part and slow growth in the second part of data sequence. The Verhulst model is mainly used to study processes with saturated states (or say sigmoid processes).
The last innovation is to combine grey prediction model with continue fraction theory. Because of uncertainty of system, there is sensitivity to the setting form of traditional single model. Therefore, only using the traditional grey forecasting model is often difficult to achieve the desired prediction. As we know, Different theories and methods from different physical backgrounds, are to solve a subset of real life problem encountered. And these theories and methods are not mutually exclusive, but interrelated and mutually complementary. These theories and methods have their own characteristics and features to excavate from a different system, useful information which is very important for accurate prediction. So grey models are proposed based on the theory of continued fractions GM (1,1) model and the theory of vector continued fractions MGM (1, n) model.
引文
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