离散型灰色预测模型的无偏性与自适应性及其应用

doi:10.16381/j.cnki.issn1003-207x.2022.1937

Abstract

Abstract:

Unbiasedness and self-adaptability are two important properties of grey prediction model and the basis of studying the structure and performance of the model. The existing studies have mostly verified unbiased from the perspective of examples， without strict mathematical proof， and less analysis of the relationship between unbiased and adaptive. Firstly， using matrix theory， the unbiasedness of two kinds of common discrete grey prediction models is strictly proved in theory and verified by homogeneous index/nonhomogeneous index/linear function sequence. The results show that the two parameter discrete grey prediction model DGM（1，1） is unbiased only for the homogeneous exponential sequence， while the three parameter discrete grey prediction model TDGM（1，1） is unbiased for the exponential/non-homogeneous exponential/linear function sequence. Then， the adaptability of TDGM（1，1） to different feature sequences is analyzed from the perspective of model structure， and the internal relationship between self-adaptability and unbiased of model structure is verified. Finally， the TDGM（1，1） model is applied to forecast the world sales of new energy vehicles， and the prediction results are compared and analyzed. This study has positive significance for enriching and perfecting the grey prediction theory.

Key words: discrete grey prediction model, unbiasedness, self-adaptability, prediction of world sales of new energy vehicles

CLC Number:

N941.5

Shuliang Li,Shuangyi Yang,Bo Zeng,Wei Meng,Yun Bai. Unbiased and Adaptive of Discrete Grey Prediction Model and Its Application[J]. Chinese Journal of Management Science, 2024, 32(8): 149-158.

Figures/Tables 13

$k$	$x 0 k$	TDGM(1,1)		DGM(1,1)		GM(1,1)		NGM(1,1,k)
$k$	$x 0 k$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$
$k = 1$	1.08	—	—	—	—	—	—	—	—
$k = 2$	1.94	1.94	0.00	1.94	0.00	1.87	3.65	1.13	41.71
$k = 3$	3.50	3.50	0.00	3.50	0.00	3.32	5.21	2.21	36.79
$k = 4$	6.30	6.30	0.00	6.30	0.00	5.87	6.75	4.12	34.64
$k = 5$	11.34	11.34	0.00	11.34	0.00	10.40	8.26	7.48	34.00
$k = 6$	20.41	20.41	0.00	20.41	0.00	18.42	9.75	13.43	34.21
$k = 7$	36.73	36.73	0.00	36.73	0.00	32.61	11.22	23.93	34.86
$k = 8$	66.12	66.12	0.00	66.12	0.00	57.75	12.66	42.48	35.76
$k = 9$	119.02	119.02	0.00	119.02	0.00	102.26	14.08	75.24	36.78
$k = 10$	214.23	214.23	0.00	214.23	0.00	181.09	15.47	133.11	37.87
$Δ ˉ 1$			0.00		0.00		9.67		36.29
$k = 11$	385.61	385.61	0.00	385.61	0.00	320.67	16.84	235.33	38.97
$k = 12$	694.10	694.10	0.00	694.10	0.00	567.84	18.19	415.88	40.08
$k = 13$	1249.38	1249.38	0.00	1249.38	0.00	1005.50	19.52	734.82	41.19
$k = 14$	2248.88	2248.88	0.00	2248.88	0.00	1780.60	20.82	1298.20	42.28
$k = 15$	4047.98	4047.98	0.00	4047.98	0.00	3153.10	22.11	2293.30	43.35
$Δ ˉ 2$			0.00		0.00		19.50		41.17
$Δ ˉ$			0.00		0.00		13.18		38.03

$k$	$x 0 k$	TDGM(1,1)		DGM(1,1)		GM(1,1)		NGM(1,1,k)
$k$	$x 0 k$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$
$k = 1$	4.38	—	—	—	—	—	—	—	—
$k = 2$	5.05	5.05	0.00	3.78	25.17	3.74	25.89	2.06	59.18
$k = 3$	5.99	5.99	0.00	4.98	16.88	4.93	17.80	3.20	46.63
$k = 4$	7.31	7.31	0.00	6.56	10.21	6.48	11.34	4.66	36.24
$k = 5$	9.15	9.15	0.00	8.65	5.52	8.53	6.85	6.54	28.53
$k = 6$	11.74	11.74	0.00	11.40	2.89	11.22	4.41	8.96	23.62
$k = 7$	15.35	15.35	0.00	15.02	2.17	14.76	3.85	12.08	21.31
$k = 8$	20.41	20.41	0.00	19.79	3.05	19.42	4.86	16.09	21.18
$k = 9$	27.49	27.49	0.00	26.07	5.16	25.55	7.08	21.25	22.72
$k = 10$	37.41	37.41	0.00	34.36	8.16	33.61	10.16	27.89	25.46
$Δ ˉ 1$			0.00		8.80		10.25		31.65
$k = 11$	51.29	51.29	0.00	45.27	11.74	13.80	10.25	36.43	28.98
$k = 12$	70.73	70.73	0.00	59.66	15.66	17.76	13.80	47.42	32.96
$k = 13$	97.95	97.95	0.00	78.61	19.74	21.86	17.76	61.57	37.14
$k = 14$	136.04	136.04	0.00	103.58	23.86	25.99	21.86	79.77	41.37
$k = 15$	189.38	189.38	0.00	136.49	27.93	30.05	25.99	103.19	45.51
$Δ ˉ 2$			0.00		19.79		21.89		37.19
$Δ ˉ$			0.00		12.72		14.41		33.63

$k$	$x 0 k$	TDGM(1,1)		DGM(1,1)		GM(1,1)		NGM(1,1,k)
$k$	$x 0 k$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$	$x ? k$	$Δ k k / %$
$k = 1$	4.80	—	—	—	—	—	—	—	—
$k = 2$	8.40	8.40	0.00	11.70	39.27	11.64	38.60	5.93	8.40
$k = 3$	12.00	12.00	0.00	13.63	13.59	13.57	13.06	9.83	12.00
$k = 4$	15.60	15.60	0.00	15.88	1.82	15.81	1.35	13.67	15.60
$k = 5$	19.20	19.20	0.00	18.51	3.61	18.42	4.04	17.43	19.20
$k = 6$	22.80	22.80	0.00	21.56	5.42	21.47	5.84	21.13	22.80
$k = 7$	26.40	26.40	0.00	25.13	4.82	25.02	5.23	24.76	26.40
$k = 8$	30.00	30.00	0.00	29.28	2.41	29.16	2.82	28.32	30.00
$k = 9$	33.60	33.60	0.00	34.11	1.53	33.98	1.12	31.83	33.60
$k = 10$	37.20	37.20	0.00	39.75	6.85	39.59	6.43	35.27	37.20
$Δ ˉ 1$			0.00		8.81		8.72		10.96
$k = 11$	40.80	0.00	46.32	13.52	46.14	13.08	38.65	40.80	0.00
$k = 12$	44.40	0.00	53.97	21.55	53.77	21.09	41.96	44.40	0.00
$k = 13$	48.00	0.00	62.88	31.01	62.65	30.53	45.22	48.00	0.00
$k = 14$	51.60	0.00	73.27	42.00	73.01	41.50	48.43	51.60	0.00
$k = 15$	55.20	0.00	85.38	54.67	85.08	54.14	51.57	55.20	0.00
$Δ ˉ 2$			0.00		32.55		32.07		5.86
$Δ ˉ$			0.00		17.29		17.06		9.14

典型数据序列	齐次指数序列			非齐次指数序列			线性函数序列
典型数据序列	$Δ ˉ 1$ (%)	$Δ ˉ 2$ (%)	$Δ ˉ$ (%)	$Δ ˉ 1$ (%)	$Δ ˉ 2$ (%)	$Δ ˉ$ (%)	$Δ ˉ 1$ (%)	$Δ ˉ 2$ (%)	$Δ ˉ$ (%)
TDGM(1,1)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
DGM(1,1)	0.00	0.00	0.00	8.80	19.79	12.72	8.81	32.55	17.29
GM(1,1)	9.67	19.50	13.18	10.25	21.89	14.41	8.72	32.07	17.06
NGM(1,1,k)	36.29	41.17	38.03	31.65	37.19	33.63	10.96	5.86	9.14

参数	$a$	$b$	$c$	$α$	$β$	$γ$	$?$
参数值	-0.16	22.95	-6.53	1.17	24.94	-19.57	34.06

年份	$x 0 k$	TDGM(1,1)		DGM(1,1)		GM(1,1)		NGM(1,1,k)
年份	$x 0 k$	$x ? k$	$Δ k k$	$x ? k$	$Δ k k$	$x ? k$	$Δ k$	$x ? k$	$Δ k k$
2013	21.50	—	—	—	—	—	—	—	—
2014	33.50	34.06	1.67%	58.41	74.37%	57.14	70.58%	32.04	4.35%
2015	68.90	64.93	5.76%	78.88	14.48%	77.14	11.96%	57.43	16.65%
2016	93.40	101.17	8.32%	106.51	14.03%	104.13	11.49%	88.31	5.45%
2017	139.90	143.72	2.73%	143.82	2.80%	140.57	0.48%	125.86	10.04%
2018	218.50	193.68	11.36%	194.20	11.12%	189.75	13.16%	171.52	21.50%
2019	228.40	252.32	10.47%	262.24	14.81%	256.15	12.15%	227.06	0.59%
2020	328.40	321.17	2.20%	354.10	7.83%	345.78	5.30%	294.59	10.29%
$Δ ˉ$			6.07%		19.92%		17.87%		9.84%

模型	DGM(1,1)	GM(1,1)	NGM(1,1,k)
参数值	$a = 1.35$	$a = - 0.30$	$a = - 0.20$
参数值	$b = 50.88$	$b = 42.33$	$b = 16.72$

References 28

1	Deng J L. The control problem of grey systems［J］. System Control Letter， 1982， 1（5），288-294.
2	刘思峰.灰色系统理论及其应用（第九版）［M］.北京：科学出版社，2021.
	Liu S F. Grey system theory and its application［M］. Beijing： Science Press， 2021.
3	熊萍萍，陈诗婷，周依凡，等.基于新型核与灰度序列的MGM（1，m，N）模型及其应用［J］.中国管理科学，2022，30（7）：130-139.
	Xiong P P， Chen S T， Zhou Y F， et al. MGM （1，m，N） model based on the new kernel and degree of greyness sequences and lts application［J］. Chinese Journal of Management Science， 2022，30（7）：130-139.
4	李惠，曾波，周文浩.基于灰色参数组合优化新模型的生活垃圾清运量预测研究［J］.中国管理科学，2022，30（4）：96-107.
	Li H， Zeng B， Zhou W H. Forecasting domestic waste clearing and transporting volume by employing a new grey parameter combination optimization model［J］. Chinese Journal of Management Science， 2022，30（4）：96-107.
5	曾波，刘思峰，白云，等.基于灰色系统建模技术的人体疾病早期预测预警研究［J］.中国管理科学，2020，28（1）：144-152.
	Zeng B， Liu S F， Bai Y， et al. Grey system modeling technology for early prediction and warning of human diseases［J］. Chinese Journal of Management Science， 2020，28（1）：144-152.
6	Xie N M， Liu S F. Discrete grey forecasting model and its optimization［J］. Applied Mathematical Modeling， 2009， 33： 1173-1186.
7	曾波，李树良，孟伟.灰色预测理论及其应用［M］.北京：科学出版社，2020.
	Zeng B， Li S L， Meng W. Grey prediction theory and its application［M］. Beijing： Science Press， 2020.
8	张鹏.引入新息项的离散灰色预测模型［J］.统计与决策，2020，36（5）：29-32.
	Zhang P. A discrete grey prediction model with innovation term［J］. Statistics & Decision， 2020，36（5）：29-32.
9	曾亮，骆世广.分数阶累加的离散GM（2，1）模型与应用［J］.重庆师范大学学报（自然科学版），2021，38（5）：73-80+2.
	Zeng L， Luo S G. A discrete GM（2，1）model with fractional-order accumulation and its application［J］. Journal of Chongqing Normal University（Natural Science），2021，38（5）：73-80+2.
10	曹阳，梁爽，沈琴琴，等.阻尼累加离散GM（1，1）模型及其应用［J］.控制与决策，2023，38（6）：1687-1694.
	Cao Y， Liang S， Shen Q Q， et al. Damping accumulated discrete gm（1，1） model and its application［J］. Control and Decision，2023，38（6）：1687-1694.
11	傅家俊，尹泉，傅鹤林，等.基于优化的灰色离散Verhulst新陈代谢模型的基坑沉降预测［J］.公路工程，2019，44（2）：19-22+120.
	Fu J J， Yin Q， Fu H L， et al. Settlement prediction of foundation pit based on optimized grey discrete verhulst metabolic model［J］. Highway Engineerin， 2019，44（2）：19-22+120.
12	Xie N M， Liu S F， Yang Y J， et al. On novel grey forecasting model based on non-homogeneous index sequence［J］. Applied Mathematical Modelling，2013，37（7）： 5059-5068.
13	谢乃明，刘思峰．近似非齐次指数序列的离散灰色预测模型特性研究［J］．系统工程与电子技术，2008，30（5）：863-868.
	Xie N M， Liu S F. Research on the non-homogenous discrete grey model and its parameters properties［J］. Systems Engineering and Electronics，2008，30（5）： 863-868.
14	曹邦兴.差分方程灰色DEGM（2，1）模型的优化及应用［J］.统计与决策， 2020， 36（13）：57-60.
	Cao B X. Forecast model of ecological economic warning degree based on gray theory and its applications［J］. Statistics & Decision， 2020， 36（13）：57-60.
15	孟伟，曾波.基于互逆分数阶算子的离散灰色模型及阶数优化［J］.控制与决策，2016，31（10）：1903-1907.
	Meng W， Zeng B. Discrete grey model with inverse fractional operators and optimized order［J］. Control and Decision， 2016， 31（10）：1903-1907.
16	Gou X Y， Zeng B， Gong Y. An improved multi-variable grey model for forecasting China’s finished products from comprehensive waste utilization［J］. Environmental Science and Pollution Research， 2021， 28： 42901-42915.
17	邹国焱，魏勇.广义离散灰色预测模型及其应用［J］.系统工程理论与实践，2020，40（3）：736-747.
	Zou G Y， Wei Y. Generalized discrete grey model and its application［J］. Systems Engineering-Theory & Practice， 2020， 40（3）：736-747.
18	罗党，韦保磊.一类离散灰色预测模型的统一处理方法及应用［J］.系统工程理论与实践，2019，39（2）：451-462.
	Luo D， Wei B L. A unified treatment approach for a class of discrete grey forecasting models and its application［J］. Systems Engineering-Theory & Practice， 2019，39（2）：451-462.
19	谢乃明，刘思峰.离散灰色模型的仿射特性研究［J］.控制与决策，2008（2）：200-203.
	Xie N M， Liu S F. Research on the affine properties of discrete grey model［J］. Control and Decision， 2008（2）：200-203.
20	吴正朋，刘思峰，党耀国，等.再论离散GM（1，1）模型的病态问题研究［J］.系统工程理论与实践，2011，31（1）：108-112.
	Wu Z P， Liu S F， Dang Y G， et al. study on the morbidity problem in the grey model［J］. Systems Engineering-Theory & Practice， 2011， 31（1）：108-112.
21	Liu Y T， Yang Y， Pan F， et al. A conformable fractional unbiased grey model with a flexible structure and it’s application in hydroelectricity consumption prediction［J］. Journal of Cleaner Production， 2022， 367：133029.
22	Wei B L. Parameter estimation strategies for separable grey system models with comparisons and applications［J］. Applied Mathematical Modelling， 2023， 116： 32-44.
23	Wang Y， Yang Z S， Ye L L， et al. A novel self-adaptive fractional grey Euler model with dynamic accumulation order and its application in energy production prediction of China［J］. Energy， 2023， 265： 126384.
24	吉培荣，黄巍松，胡翔勇. 无偏灰色预测模型［J］. 系统工程与电子技术， 2000（6）： 6-7+80.
	Ji P R， Huang W S， Hu X Y. An unbiased grey forecasting model［J］. Systems Engineering and Electronics， 2000（6）： 6-7+80.
25	刘晓梅，周钢.基于精细积分法的无偏非齐次灰色模型构建［J］.控制与决策，2022，37（11）：3058-3064.
	Liu X M， Zhou G. An unbiased non-homogeneous grey model based on high precise direct integration method［J］. Control and Decision， 2022，37（11）：3058-3064.
26	Zheng C L， Wu W Z， Xie W L， et al. Forecasting the hydroelectricity consumption of China by using a novel unbiased nonlinear grey Bernoulli model［J］. Journal of Cleaner Production， 2021， 278： 123903.
27	白旻，张旻昱，王晓超.碳中和背景下全球新能源汽车产业发展政策与趋势［J］.信息技术与标准化，2021（12）：13-20.
	Bai M， Zhang M Y， Wang X C. Development policies and trends of global new energy vehicle industry under the background of carbon neutrality［J］. Information Technology & Standardization， 2021（12）：13-20.
28	中国汽车技术研究中心，日产（中国）投资有限公司，东风汽车有限公司.中国新能源汽车产业发展报告（2021）［M］.北京：社会科学文献出版社，2021.
	China Automotive Technology and Research Center Co.， Ltd.， Nissan Group of China Co.， Ltd.， DONGFENG Motor Company Limited. Annual report on new energy vehicle industry in China（2020）［M］. Beijing： Social Sciences Academic Press （CHINA）， 2021.

[1]	Yunjie Mei,Xiangyan Zeng,Shuli Yan. A Matrix Autoregressive Time-delay Grey Multivariable Model for Three-parameter Interval Grey Number Sequences [J]. Chinese Journal of Management Science, 2024, 32(9): 48-58.
[2]	Lianyi Liu,Sifeng Liu,Lifeng Wu. New Energy Vehicle Sales Forecast Based on Siscrete Time Grey Power Model [J]. Chinese Journal of Management Science, 2024, 32(1): 106-114.
[3]	ZHOU Wei-jie, JIANG Hui-min, CHENG Yu-ke, DANG Yao-guo, DING Song. Forecasting Chinese Hydropower Consumption Forecasting by Using the Repeatability Fractional Grey Time Power Model [J]. Chinese Journal of Management Science, 2023, 31(5): 279-286.
[4]	ZENG Bo, GOU Xiao-yi, ZHANG Zhi-wei. Research on the Non-Uniqueness of Solutions of the Mean Difference GM(1,1) Model Based on Interval Grey Interaction [J]. Chinese Journal of Management Science, 2022, 30(10): 247-255.
[5]	XIONG Ping-ping, CHEN Shi-ting, ZHOU Yi-fan, LIU Yu-chun, DING Song. MGM (1, m, N) Model Based on the New Kernel and Degree of Greyness Sequences and Its Application [J]. Chinese Journal of Management Science, 2022, 30(7): 130-139.
[6]	LI Hui, ZENG Bo, ZHOU Wen-hao. Forecasting Domestic Waste Clearing and Transporting Volume by Employing A New Grey Parameter Combination Optimization Model [J]. Chinese Journal of Management Science, 2022, 30(4): 96-107.
[7]	LI Ye, DING Yuan-ping. Construction of Linear Time-varying Parameters DLDGM(1,N) Model Based on Driving Factors Control [J]. Chinese Journal of Management Science, 2022, 30(3): 221-229.
[8]	DUAN Hui-ming, XIAO Xin-ping. Research on Grey Inertial Model and Its Application Property [J]. Chinese Journal of Management Science, 2022, 30(3): 269-279.
[9]	XIA Jie, MA Xin, WU Wen-qing. The Improved FAGM(1, 1) Model Based on Simpson Formula and Its Applications [J]. Chinese Journal of Management Science, 2021, 29(5): 240-248.
[10]	QIAN Li-li, LIU Si-feng, DENG Gui-feng. Grey Off-target Deviation Degree Method for Group Decision-making with Regret Aversion [J]. Chinese Journal of Management Science, 2020, 28(6): 193-200.
[11]	ZENG Bo, LIU Si-feng, BAI Yun, ZHOU Meng. Grey System Modeling Technology for Early Prediction and Warning of Human Diseases [J]. Chinese Journal of Management Science, 2020, 28(1): 144-152.
[12]	DING Song, LI Ruo-jin, DANG Yao-guo. Construction and Application of GM(1,1) Power Model based on the Optimized Initial Condition [J]. Chinese Journal of Management Science, 2020, 28(1): 153-161.
[13]	WU Li-feng, YU Liang, WEN Zhao-xia. GM(0,N) Model for Its Application on Forecasting the Development Cost of Complicated Equipment [J]. Chinese Journal of Management Science, 2019, 27(7): 203-207.
[14]	ZHOU Wei-jie, ZHANG Hong-ru, DANG Yao-guo, WANG Zheng-xin. New Information Priority Accumulated Grey Discrete Model and Its Application [J]. Chinese Journal of Management Science, 2017, 25(8): 140-148.
[15]	ZENG Bo, LIU Si-feng, QU Xue-xin. Researching on A Grey Common Prediction Modeling with Strong Compatibility and Its Properties [J]. Chinese Journal of Management Science, 2017, 25(5): 150-156.

Unbiased and Adaptive of Discrete Grey Prediction Model and Its Application

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 13

References 28

Related Articles 15

Metrics

Comments

Recommended 0