﻿ 灰色GM(1,1)模型与线性回归模型的内在联系 The Internal Connection between Grey GM(1,1) Model and Simple Linear Regression Model

Computer Science and Application
Vol.08 No.05(2018), Article ID:24921,8 pages
10.12677/CSA.2018.85073

The Internal Connection between Grey GM(1,1) Model and Simple Linear Regression Model

Wen Liu, Yong Wang

College of Information Science and Engineering, Ocean University of China, Qingdao Shandong

Received: Apr. 25th, 2018; accepted: May 10th, 2018; published: May 17th, 2018

ABSTRACT

Purpose: The analog sequence of GM(1,1) model for the original sequence after translation transformation, is very similar to simple linear regression line. So, we will explore the relationship between the two models. Method: Get some original sequences from real world, construct GM(1,1) model after translation transformation and simple linear regression model, and use T test and difference measure to compare the two models. Result: The GM(1,1) model constructed by the original sequence after translation transformation, and the simple linear regression model constructed by the sequence except the first data, are the same. Conclusion: After translation transformation, the grey model becomes a special linear regression model.

Keywords:GM(1,1) Model, Simple Linear Regression Model, Translation Transformation

1. 引言

2. 研究问题与方法

2.1. 研究问题

2.2. 实验步骤

1) 准备实验数据集，对数据进行分组，整理成所需的原始序列集。

2) 对原始序列进行平移变换。

3) 对原始序列构建一元线性回归模型，对每次平移后得到的新序列分别构建GM(1,1)模型。

4) 对每一组GM(1,1)模型的模拟值与对应的一元线性回归模型的模拟值进行比较。

5) 分析随平移量的变化，两种模型的相似性的变化趋势，并得出结论。

2.3. 检验方法

1) 对平移变换后的GM(1,1)模型所得数据与一元线性回归模型所得数据进行配对样本T检验，每平移一次进行一次检验，如果验证每次平移后，配对样本T检验所得的 值均大于0.05，说明两种模型的结果不存在显著差异，来自一个总体，具有相似性。

2) 对两组实验数据进行差异度量。两组数据之间的差异可以用偏差和来表示。将GM(1,1)模型每次平移后所得的结果分别与一元线性回归模型所得的结果作差并取绝对值，将每组结果的差值绝对值相加，若差值绝对值和越小，则说明两组数据越接近，反之，两组数据的差距越大。

3. 实验与分析

3.1. 平移变换对GM(1,1)模型的作用

$\rho \left(3\right)=1.0000,\text{\hspace{0.17em}}\rho \left(4\right)=0.8000,\text{\hspace{0.17em}}\rho \left(5\right)=0.9889,\text{\hspace{0.17em}}\rho \left(6\right)=0.9888$

${X}_{A1}=\left\{1,1.5,2.5,4,8.9,17.7\right\}$

${X}_{A2}=\left\{3,3.5,4.5,6,10.9,19.7\right\}$

$\cdots$

${X}_{A10}=\left\{513,513.5,514.5,516,520.9,529.7\right\}$

${X}_{A1}$${X}_{A10}$ 10组序列进行GM(1,1)模型的构建，得到的10次建模的曲线如图1所示。

Figure 1. Curve: 10 translations of original sequence of GM(1,1) model

Figure 2. Curve: The last translational GM (1,1) model and one linear regression model

3.2. 原煤产量实验

${x}_{0}=$
{ 13.32 , 13.64 , 13.84 , 14.72 , 15.5 , 18.35 , 21.23 , 23.65 , 25.7 , 27.6 , 29.03 , 31.15 , 34.28 , 37.64 , 39.45 , 39.74 , 38.74 }

Figure 3. Curve: GM(1,1) model and simple linear regression model for no first point

Table 1. P value of paired sample T test

Table 2. The difference measure of 4 subgroups of the output of raw coal

4. 总结

(a) 5个数据一组 (b) 6个数据一组 (c) 8个数据一组 (d) 10数据一组

Figure 4. The difference measure of 4 subgroups of the output of raw coal

The Internal Connection between Grey GM(1,1) Model and Simple Linear Regression Model[J]. 计算机科学与应用, 2018, 08(05): 649-656. https://doi.org/10.12677/CSA.2018.85073

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