基于背景值和结构相容性改进的多维灰色预测模型

缪燕子; 王志铭; 李守军; 代伟

doi:10.16383/j.aas.c200780

基于背景值和结构相容性改进的多维灰色预测模型

doi: 10.16383/j.aas.c200780

1.
中国矿业大学信息与控制工程学院徐州 221116
2.
宿迁学院机电工程学院宿迁 223800

基金项目: 国家重点研发计划重点专项(2018YFC0808100), 国家自然科学基金(61976218, 61973306), 江苏省高等学校自然科学研究项目(19KJB440002), 江苏省自然科学基金(BK20200086), 中央高校基本科研业务费专项资金资助(2020ZDPY0303)资助

详细信息

作者简介:
缪燕子：中国矿业大学信息与控制工程学院教授. 主要研究方向为多传感器信息融合, 机器人智能感知与控制. 本文通信作者. E-mail: myz@cumt.edu.cn

王志铭：中国矿业大学信息与控制工程学院硕士研究生, 2019年获中国矿业大学电气工程及其自动化学士学位. 主要研究方向为预测控制, 煤矿安全. E-mail: 04151249@cumt.edu.cn

李守军：宿迁学院机电工程学院副教授. 主要研究方向为工业自动化, 人工智能与灰色系统理论. E-mail: lishoujunbox@126.com

代伟：中国矿业大学信息与控制工程学院教授. 主要研究方向为复杂工业过程建模, 运行优化与控制. E-mail: weidai@cumt.edu.cn

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出版历程
- 收稿日期: 2020-09-21
- 录用日期: 2020-12-31
- 网络出版日期: 2021-02-01
- 刊出日期: 2022-04-13

Improved Multi-dimensional Grey Prediction Model Based on Background Value and Structural Compatibility

1.
School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116
2.
School of Mechanical and Electrical Engineering of Suqian College, Suqian 223800

Funds: Supported by Key Project of National Key Research and Development Project (2018YFC0808100), National Natural Science Foundation of China (61976218, 61973306), Natural Science Research Project of Higher Education Institutions in Jiangsu Province (19KJB440002), Natural Science Foundation of Jiangsu Provinces (BK20200086), Fundamental Research Fund for the Central Universities (2020ZDPY0303)

More Information

Author Bio:
MIAO Yan-Zi　Professor at the School of Information and Control Engineering, China University of Mining and Technology. Her research interest covers multi-sensor information fusion, intelligent perception and control of robot. Corresponding author of this paper

WANG Zhi-Ming　Master student at the School of Information and Control Engineering, China University of Mining and Technology. She received her bachelor degree from China University of Mining and Technology in 2019. Her research interest covers predictive control, and coal mine safety

LI Shou-Jun　Associate Professor at the School of Mechanical and Electrical Engineering of Suqian College. His research interest covers industrial automation, artificial intelligence and grey system theory

DAI Wei　Professor at the School of Information and Control Engineering, China University of Mining and Technology. His research interest covers modeling, operational optimization and control for complex industrial process

摘要

摘要: 现有的多变量灰色预测模型的背景值估计误差及模型结构单一是导致该模型预测性能不稳定的重要因素, 致使该模型在实际预测领域中应用并不广泛. 本文通过分析背景值函数的几何意义, 结合积分几何面积公式, 提出一种改进的背景值优化方法, 使预测模型在背景值系数的选取上更加灵活.在此基础上, 模型中加入灰色作用量, 提出一种改进背景值及结构相容性的多维灰色预测模型(Improved background value and structure compatibility of grey prediction model, IBSGM(1, N)). 通过对模型参数的改变分析, 新模型理论上可达到与传统单变量和多变量灰色预测模型的兼容性. 为检验新模型的性能, 本文进行了三个案例对比分析, 实验结果表明, 与现有的灰色预测模型(Grey model, GM) GM(1, 1)和GM(1, N)相比较, 所提出的IBSGM(1, N)模型在背景值参数估计上误差明显减小, 结构相容性更强, 泛化性能更好, 具有更高的预测精度.
- 背景值优化 /
- 结构相容性 /
- 多维灰色预测模型 /
- IBSGM(1, N)
Abstract: The background value estimation error of the existing multivariate gray prediction model and the single model structure are important factors that lead to the unstable prediction performance of the model. Therefore, the model is not widely used in the actual prediction field. In this paper, by analyzing the geometric meaning of the background value function, combined with the integral geometric area formula, an improved background value optimization method is proposed to make the prediction model more flexible in the selection of background value coefficients. On this basis, the gray effect is added to the model, and a new multi-dimensional gray prediction model called improved background value and structure compatibility of grey prediction model (IBSGM(1, N)) is proposed. Through the analysis of the change of model parameters, the new model can theoretically achieve compatibility with traditional univariate and multivariate gray prediction models. In order to test the performance of the new model, this paper conducts a comparative analysis of three cases. The experimental results show that compared with the existing grey prediction model (GM) GM(1, 1) and GM(1, N), the proposed IBSGM(1, N) model has significantly reduced error in background parameter estimation, stronger structural compatibility, better generalization performance, and higher prediction accuracy.
- Background value optimization /
- structural compatibility /
- multi-dimensional grey prediction model /
- IBSGM(1, N)
注释:

1) 收稿日期 2020-09-21 录用日期 2020-12-31 Manuscript received September 21, 2020; accepted December31, 2020 国家重点研发计划重点专项 (2018YFC0808100), 国家自然科学基金 (61976218, 61973306), 江苏省高等学校自然科学研究项目(19KJB440002), 江苏省自然科学基金(BK20200086), 中央高校基本科研业务费专项资金资助 (2020ZDPY0303) 资助 Supported by Project of National Key Research and Development Project (2018YFC0808100), National Natural Science Foundation of China (61976218, 61973306), Natural Science Research Project of Higher Education Institutions in Jiangsu Province (19KJB440002), Natural Science Foundation of Jiangsu Provinces (BK20200086), Fundamental Research Fund for the Central Universities (2020ZDPY0303) 本文责任编委吕宜生 Recommended by Associate Editor LV Yi-Sheng

2) 1. 中国矿业大学信息与控制工程学院徐州 221116 2. 宿迁学院机电工程学院宿迁 223800 1. School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116 2. School of Mechanical and Electrical Engineering of Suqian College, Suqian 223800

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图 1 背景值几何示意图1

Fig. 1 Schematic diagram 1 of the background value

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图 2 背景值几何示意图2

Fig. 2 Schematic diagram 2 of the background value

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图 3 例1中四种模型的模拟预测结果曲线图

Fig. 3 Curves of simulated prediction results of the four models in Example 1

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图 4 例2中四种模型的模拟预测结果曲线图

Fig. 4 Curves of simulation prediction results of the four models in Example 2

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图 5 例3中四种模型的模拟预测结果曲线图

Fig. 5 Curves of simulation prediction results of the four models in Example 3

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表 1 寸草塔煤矿日均瓦斯浓度及影响因素

Table 1 Daily average gas concentration and influencing factors in Cuncaota Coal Mine

序号	X₁⁽⁰⁾	X₂⁽⁰⁾	X₃⁽⁰⁾	X₄⁽⁰⁾
1	0.34	0.34	21.7	0.34
2	0.34	0.29	18.1	0.36
3	0.26	0.29	25.3	0.31
4	0.26	0.41	21.4	0.33
5	0.23	0.51	25.3	0.28
6	0.22	0.37	22.3	0.29
7	0.21	0.38	23.2	0.23
8	0.17	0.41	22.5	0.35
9	0.17	0.36	24.1	0.19
10	0.16	0.48	22.9	0.25

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表 2 IBSGM(1, N)与GM(1, N)模型预测模拟值误差对比

Table 2 Comparison of prediction and simulation errors between IBSGM(1, N) and GM(1, N) model

实际值	0.34	0.34	0.26	0.26	0.23	0.22	0.21	0.17	0.17	0.16	平均误差
GM(1, N)	0.34	0.259	0.364	0.367	0.188	0.269	0.135	0.347	0.054	0.113	0.38
IBSGM(1, N)	0.34	0.33	0.27	0.25	0.247	0.219	0.204	0.173	0.16	0.165	0.0337

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表 3 一种热处理钢在400℉至1100℉的抗拉强度及布氏硬度

Table 3 The tensile strength and Brinell hardness of a heat-treated steel from 400°F to 1100°F

序号	X₁⁽⁰⁾	X₂⁽⁰⁾	X₃⁽⁰⁾
1	897	514	400
2	897	495	500
3	890	444	600
4	876	401	700
5	848	352	800
6	814	293	900
7	779	269	1000
8	738	235	1100

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表 4 IBSGM(1, N)模型的参数值

Table 4 Parameter values of IBSGM(1, N) model

$ a $	$ {b}_{1} $	$ {b}_{2} $	$ \gamma $	$ \lambda $
0.1711	0.2974	0.0247	728.1782	0

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表 5 四种模型下预测结果和误差对比

Table 5 Comparison of prediction results and errors under the four models

序号	原始数据	IBSGM(1, N)模型		OGM(1, N)模型		GM(1, N)模型		GM(1, 1)模型
序号	原始数据	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差
1	897	897	0	897	0	897	0	897	0
2	897	897.013	0.0015%	896.782	0.0243%	791.446	11.7674%	911.544	1.6214%
3	890	890.421	0.0473%	890.882	0.0991%	1013.103	13.8317%	886.265	0.4197%
4	876	874.707	0.1476%	874.589	0.1611%	919.923	5.0140%	861.687	1.6340%
5	848	849.283	0.1513%	848.921	0.1086%	854.567	0.7744%	837.790	1.2040%
6	814	813.571	0.0527%	813.797	0.0250%	797.161	2.0686%	914.556	0.0683%
7	779	779.005	0.0007%	778.952	0.0062%	798.870	2.5507%	791.967	1.6646%
平均拟合误差			0.0573%		0.0606%		5.1438%		0.9446%
预测结果		预测值	相对误差	预测值	相对误差	预测值	相对误差	预测值	相对误差
8	738	735.263	0.3709%	742.147	0.5619%	787.425	6.6972%	770.004	4.3366%

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表 6 中国无线通信用户数量和相关因素

Table 6 Number of wireless communication users and related factors in China

序号	X₁⁽⁰⁾	X₂⁽⁰⁾	X₃⁽⁰⁾	X₄⁽⁰⁾	X₅⁽⁰⁾
1	8453.3	13985.6	99241.6	563.5498	17825.6
2	14522.2	21926.3	109655.2	703.5769	25566.3
3	20600.5	27400.3	120322.7	773.01	28656.8
4	26995.3	33698.4	135822.8	869.3998	35082.5
5	33482.4	39684.3	159878.3	1262.998	42346.9
6	39340.6	48241.7	184937.4	1371.631	47196.1
7	46105.8	61032	216314.4	1442.343	50279.9
8	54730.6	85496.1	265810.3	1709.221	51034.6
9	64124.5	114531.4	314045.4	1690.719	50863.2
10	74721.4	144084.7	340902.8	1684.903	49265.6
11	85900.3	150284.9	401.202	1641.464	46537.3

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表 7 IBSGM(1, N)模型的参数值

Table 7 Parameter values of IBSGM(1, N) model

$ a $	$ {b}_{1} $	$ {b}_{2} $	$ {b}_{3} $	$ {b}_{4} $	$ \gamma $	$ \lambda $
0.5083	0.2095	−0.0067	0.7883	0.2811	−533.748	0

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表 8 四种模型下预测结果和误差对比

Table 8 Comparison of prediction results and errors under the four models

序号	原始数据	IBSGM(1, N)模型		OBGM(1, N)模型		GM(1, N)模型		GM(1, 1)模型
序号	原始数据	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差
1	8453.3	8453.3	0	8453.3	0	8453.3	0	8453.3	0
2	14522.2	14487.37	0.24%	14522.13	0	13547.07	6.71%	18836.59	29.71%
3	20600.5	20703.43	0.49%	20767.87	0.81%	26762.83	29.91%	22465.68	9.05%
4	26995.3	26927.63	0.25%	27021.5	0.10%	36603.21	35.59%	26793.96	0.75%
5	33482.4	33346.36	0.41%	33260.96	0.66%	44119.87	31.77%	31956.14	4.56%
6	39340.6	39540.97	0.51%	39664.41	0.82%	50502.83	28.37%	38112.87	3.12%
7	46105.8	46149.15	0.09%	46512.32	0.88%	57002.66	23.63%	45455.77	1.41%
8	54730.6	54533.32	0.36%	54578.75	0.28%	66192.81	20.94%	54213.37	0.95%
9	64124.5	64228.94	0.16%	64095.54	0.05%	77398.69	20.70%	64658.22	0.83%
10	74721.4	74706.12	0.02%	74999.45	0.37%	88385.60	18.29%	77115.40	3.20%
平均拟合误差			0.25%		0.4%		21.59%		5.36%
预测结果		预测值	相对误差	预测值	相对误差	预测值	相对误差	预测值	相对误差
11	85900.3	86179.62	0.32%	85586.72	0.37%	95722.40	11.43%	91972.60	7.07%

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表 9 2003-2011年浙江省经济总产值与固定资产投资额

Table 9 2003-2011 Zhejiang province′s total economic output value and fixed asset investment

序号	X₁⁽⁰⁾	X₂⁽⁰⁾
1	9705.02	4180.38
2	11648.7	5384.38
3	13417.7	6138.39
4	15718.47	6964.28
5	18753.73	7704.9
6	21462.69	8550.71
7	22990.35	9906.46
8	27722.31	11451.98
9	32318.85	14077.25

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表 10 IBSGM(1, N)模型的参数值

Table 10 Parameter values of IBSGM(1, N) model

$ a $	$ {b}_{1} $	$ \gamma $	$ \lambda $
0.0048	0.3268	8.6375	1

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表 11 四种模型下预测结果和误差对比

Table 11 Comparison of prediction results and errors under the four models

序号	原始数据	IBSGM(1, N)模型		时滞GM(1, N)模型		GM(1, N)模型		GM(1, 1)模型
序号	原始数据	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差	模拟值	相对误差
1	9705.02	9705.02	0	9705.02	0	9705.02	0	9705.02	0
2	11648.7	11660.78	0.10%	9554.96	17.97%	9991.74	20.43%	11562.08	0.74%
3	13417.7	13602.45	1.38%	12461.41	7.13%	16856.45	33.23%	13911.74	3.68%
4	15718.47	15802.97	0.54%	15718.47	0.00%	18698.06	10.94%	15882.74	1.05%
5	18753.73	18230.98	2.79%	20665.31	10.19%	19884.38	2.99%	18133.00	3.31%
6	21462.69	20922.41	2.52%	20773.85	3.21%	21595.31	8.91%	20702.07	3.54%
7	22990.35	24049.57	4.61%	25883.56	12.58%	25006.36	8.77%	23892.54	3.92%
8	27722.31	27659.11	0.23%	27355.39	1.32%	28491.67	2.78%	27293.70	1.55%
平均拟合误差			1.52%		6.55%		11.01%		2.22%
预测结果		预测值	相对误差	预测值	相对误差	预测值	相对误差	预测值	相对误差
9	32318.85	32104.52	0.66%	31523.81	2.46%	34864.8	7.88%	31179.03	3.53%

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