目  錄

摘  要 II

Abstract IV

第1章  前言 1

1.1  本課題的目的、意義 1

1.2  論文的主要內容 2

第2章  灰建模及緩沖算子的基礎理論 3

2.1  灰建模的基本原理 3

2.2  緩沖算子的基本理論 4

第3章  灰色GM(1,1)模型及緩沖算子的研究 6

3.1  GM(1,1)模型的研究現(xiàn)狀 6

3.2  緩沖算子的研究現(xiàn)狀 8

第4章  GM（1，1）模型建模方法的改進 9

4.1  優(yōu)化灰導數的等間距GM(1,1) 9

4.2  優(yōu)化灰導數的非等間距GM(1,1) 13

第5章  一類新的緩沖算子的構造及緩沖算子新定理 19

5.1  一類新的實用強化緩沖算子的構造 19

5.2  緩沖算子新定理 22

第6章  結論與展望 25

6.1  全文總結 25

6.2  研究展望 26

參考文獻 27

致  謝 ⅰ

關于學位論文使用授權的聲明 ⅱ

關于學位論文原創(chuàng)性的聲明 ⅲ

在學期間的科研情況 ⅳ

摘  要

GM（1，1）模型是灰色系統(tǒng)預測理論的基礎與核心[1]，它已被廣泛應用于農業(yè)、工業(yè)、氣象、電力、經濟、社會等領域。它將系統(tǒng)看成一個隨時間變化而變化的指數函數，不需要大量的時間序列數據就能夠建立預測模型，其計算簡單已被普遍認同。但是一方面灰色系統(tǒng)理論還存在一些缺陷，其模型精度有待進一步提高，很多學者已在提高精度方面做了很多研究[3-7]。另一方面，由于現(xiàn)實生活中的數據往往因受到外界很多沖擊因素的干擾而失真，為了排除擾動因素的作用，劉思峰教授開創(chuàng)了對波動數據預測的新領域，他針對級比漸趨穩(wěn)定的數據序列，提出了用滿足緩沖三公理的緩沖算子作用后進行建模預測的新思路，眾多學者從不同的背景出發(fā)，提出了各種緩沖算子，大大提高了灰色預測建模精度，從而大大拓廣了灰色系統(tǒng)理論的應用范圍。文獻[41]將緩沖算子的構造與函數結合起來，為緩沖算子的構造開辟了新方向，文獻[49]對緩沖算子公理進行了補充，并構造了變權緩沖算子。

本選題在他們的工作的基礎上，主要研究成果如下：

（1）通過對不用一次累加而直接建模的等間距GM(1,1)模型的灰色微分方程中的灰導數進行優(yōu)化，提出了用（其中），代替原始灰色微分方程中的灰導數，同時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，從而獲得新模型，經過嚴格理論驗證該模型具有指數，系數，平移常數重合性。大量的數據模擬和模型比較結果表明，優(yōu)化后的模型提高了背景值的準確性以及灰預測模型的擬合精度和預測精度，且該模型既適合于低增長指數序列建模，也適合于高增長指數序列建模，同時也適合于非齊指數序列建模，可見新的建模方法大大提高了模型的模擬精度與預測精度，同時擴大了模型的適用范圍。

（2）基于完全沿用等間距一次累加的原始非等間距模型精度不盡人意，但各種改進非等間距模型一次累加表達式復雜、計算繁瑣這一基本事實，依據各種非等間距預測表達式都具有數據預測序列是時序指標的齊次指數函數的共同特征，提出不涉及非等間距的一次累加表達式，更無需其計算值，直接建立非等間距灰色微分方程，同時優(yōu)化其灰導數，用序列擬合誤差平方和最小來尋求最佳初始條件，獲得了模擬預測精度較高的非等間距灰色預測模型。

（3）文獻[41]將緩沖算子的構造與函數結合起來，為緩沖算子的構造開辟了新方向，文獻[49]對緩沖算子公理進行了補充，并構造了變權緩沖算子。本選題在他們的工作的基礎上，構造了一類緩沖算子，整合了這些常用的緩沖算子，使得常用緩沖算子更一般化了，也更加靈活了。

（4）在現(xiàn)有灰色系統(tǒng)緩沖算子公理體系下，本文得到了以下結果：設為一強化（或弱化）緩沖算子，為系統(tǒng)原始行為數據序列，其緩沖序列為，均為單調函數，并具有相同的單調性，且滿足，，，其中,則無論為單調增長序列，單調衰減序列還是振蕩序列， 均為強化（或弱化）緩沖算子。 

關鍵詞：灰色理論；GM（1，1）模型；模型的改進；緩沖算子

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