gm(1,1)模型的优化与一类强化缓冲算子的构造.doc

目  录

摘  要 II

Abstract IV

第1章  前言 1

1.1  本课题的目的、意义 1

1.2  论文的主要内容 2

第2章  灰建模及缓冲算子的基础理论 3

2.1  灰建模的基本原理 3

2.2  缓冲算子的基本理论 4

第3章  灰色GM(1,1)模型及缓冲算子的研究 6

3.1  GM(1,1)模型的研究现状 6

3.2  缓冲算子的研究现状 8

第4章  GM（1，1）模型建模方法的改进 9

4.1  优化灰导数的等间距GM(1,1) 9

4.2  优化灰导数的非等间距GM(1,1) 13

第5章  一类新的缓冲算子的构造及缓冲算子新定理 19

5.1  一类新的实用强化缓冲算子的构造 19

5.2  缓冲算子新定理 22

第6章  结论与展望 25

6.1  全文总结 25

6.2  研究展望 26

参考文献 27

致  谢 ⅰ

关于学位论文使用授权的声明 ⅱ

关于学位论文原创性的声明 ⅲ

在学期间的科研情况 ⅳ

摘  要

GM（1，1）模型是灰色系统预测理论的基础与核心[1]，它已被广泛应用于农业、工业、气象、电力、经济、社会等领域。它将系统看成一个随时间变化而变化的指数函数，不需要大量的时间序列数据就能够建立预测模型，其计算简单已被普遍认同。但是一方面灰色系统理论还存在一些缺陷，其模型精度有待进一步提高，很多学者已在提高精度方面做了很多研究[3-7]。另一方面，由于现实生活中的数据往往因受到外界很多冲击因素的干扰而失真，为了排除扰动因素的作用，刘思峰教授开创了对波动数据预测的新领域，他针对级比渐趋稳定的数据序列，提出了用满足缓冲三公理的缓冲算子作用后进行建模预测的新思路，众多学者从不同的背景出发，提出了各种缓冲算子，大大提高了灰色预测建模精度，从而大大拓广了灰色系统理论的应用范围。文献[41]将缓冲算子的构造与函数结合起来，为缓冲算子的构造开辟了新方向，文献[49]对缓冲算子公理进行了补充，并构造了变权缓冲算子。

本选题在他们的工作的基础上，主要研究成果如下：

（1）通过对不用一次累加而直接建模的等间距GM(1,1)模型的灰色微分方程中的灰导数进行优化，提出了用（其中），代替原始灰色微分方程中的灰导数，同时用代替原始灰色微分方程中的背景值，得到新的灰色微分方程，从而获得新模型，经过严格理论验证该模型具有指数，系数，平移常数重合性。大量的数据模拟和模型比较结果表明，优化后的模型提高了背景值的准确性以及灰预测模型的拟合精度和预测精度，且该模型既适合于低增长指数序列建模，也适合于高增长指数序列建模，同时也适合于非齐指数序列建模，可见新的建模方法大大提高了模型的模拟精度与预测精度，同时扩大了模型的适用范围。

（2）基于完全沿用等间距一次累加的原始非等间距模型精度不尽人意，但各种改进非等间距模型一次累加表达式复杂、计算繁琐这一基本事实，依据各种非等间距预测表达式都具有数据预测序列是时序指标的齐次指数函数的共同特征，提出不涉及非等间距的一次累加表达式，更无需其计算值，直接建立非等间距灰色微分方程，同时优化其灰导数，用序列拟合误差平方和最小来寻求最佳初始条件，获得了模拟预测精度较高的非等间距灰色预测模型。

（3）文献[41]将缓冲算子的构造与函数结合起来，为缓冲算子的构造开辟了新方向，文献[49]对缓冲算子公理进行了补充，并构造了变权缓冲算子。本选题在他们的工作的基础上，构造了一类缓冲算子，整合了这些常用的缓冲算子，使得常用缓冲算子更一般化了，也更加灵活了。

（4）在现有灰色系统缓冲算子公理体系下，本文得到了以下结果：设为一强化（或弱化）缓冲算子，为系统原始行为数据序列，其缓冲序列为，均为单调函数，并具有相同的单调性，且满足，，，其中,则无论为单调增长序列，单调衰减序列还是振荡序列， 均为强化（或弱化）缓冲算子。 

关键词：灰色理论；GM（1，1）模型；模型的改进；缓冲算子

Abstract

GM (1, 1) is the foundation and core of grey system prediction theory [1-2]. And it has widely applied in numerous fields, such as agriculture, industry, meteorology, electric power, economy, society and so on. It regards a system as the exponential function which changes with the time variation, and does not need the massive time series data to establish the forecast model. The calculating simpleness for GM (1, 1) has been accepted by people. However, on the one hand, there are still some deficiencies in grey system theory, the accuracy of model need to be further improved. Many scholars have done a lot of research in improving the model accuracy [3~7]. On the other hand, due to real-life data tend to be under a lot of the impact of external interference factors, in order to exclude the impact of disturbance factors, Professor Liu Sifeng created a new field in prediction of fluctuated data, he aimed at the data series whose grade radio is becoming more and more stable, and presented a new idea to model for prediction after using the buffer operator based on the 3 axioms ,many scholars started from different backgrounds, and proposed a variety of buffer operators, then greatly increased the accuracy of grey prediction model, thus significantly broadened the field of application of grey system theory. Literature [41] connected the structure of buffer operator with functions, and opened a new direction for the structure of buffer operator .Literature [49] was supplemented for the buffer operator axioms, and constructed a variable weight buffer operator.

In this paper, on the basis of their work, the work in this dissertation mainly consists of following parts:

(1) This paper presents a new method to establish the direct model through optimizing the grey derivative, replacing the derivative  by  and the background value by, then we get. The new model has been proven strictly to have the property of exponent, coefficient and translation constants superposition. The results of data simulation and model comparison show that the improved model in this paper raises the accuracy of background value, the fitting precision and forecasting precision. Moreover, it is not only suitable for the low growth sequence, but also suitable for the high growth sequence. What’s more, it is suitable for the nonhomogeneous exponential sequence. The new method not only improves the simulation and prediction precision, but also extends the application scope of GM (1, 1) model.

(2)Based on the truth that the accuracy of the original non-equidistance model ,which completely adherence to 1-Ago of equidistance sequence ,is not satisfactory, but the 1-Ago expressions in the ways to improve the non-equidistance model are very complex and the calculation is very complicated, according to a variety of non-equidistance expressions have the common features that forecast sequence is the homogeneous exponential function about timing indicator, this paper proposes a method to establish gray differential equation of non-equidistant sequence directly, which does not involve the 1-Ago expressions of non-equidistance sequence , even without its calculated value, optimizing its gray derivative, with the sequence of squares and the smallest fitting error to find the best initial conditions, then we obtain a higher prediction accuracy of non-equidistant gray prediction model.

 (3) Literature [41] connected the structure of buffer operator with functions, and opened a new direction for the structure of buffer operator .Literature [49] was supplemented for the buffer operator axioms, and constructed a variable weight buffer operator. This paper, on the basis of their work, constructs a class of buffer operator to integrate these common buffer operators, and make the buffer operator is more general and commonly used, and also more flexible.

(4)Based on the present theories of buffer operators in grey system, the following results are obtained in this paper: Assume that is a Strengthening (or weakening) Buffer Operator,  is a sequence of raw data, is a buffer sequence, are all monotonously functions, and have the same monotonicity,satisfying ，，，, then whenever  is a monotonously increasing sequence, a monotonously decreasing sequence, or a vibration sequence,  is a strengthening(or weakening) operator. 

Key words: grey system theory; GM (1, 1); improvement of model; buffer operators