Time-delay nonlinear model based on interval grey number and its application

doi:10.23919/JSEE.2022.000039

Journal of Systems Engineering and Electronics ›› 2022, Vol. 33 ›› Issue (2): 370-380.doi: 10.23919/JSEE.2022.000039

• SYSTEMS ENGINEERING • Previous Articles Next Articles

Time-delay nonlinear model based on interval grey number and its application

Pingping XIONG^1,*(), Shiting CHEN²(), Shuli YAN¹()

¹ School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
² College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received:2020-10-12 Online:2022-05-06 Published:2022-05-06
Contact: Pingping XIONG E-mail:xpp8125@163.com;1780458169@qq.com;yshuli@126.com
About author:|XIONG Pingping was born in 1981. She received her M.S. degree from Faculty of Mathematics, Huazhong University of Science and Technology and Ph.D. degree in School of Economics and Management from Nanjing University of Aeronautics and Astronautics. She is currently a professor in Nanjing University of Information Science and Technology. Her current research interests include grey system, system evaluation and prediction analysis, and financial statistical analysis. E-mail: xpp8125@163.com||CHEN Shiting was born in 1995. She received her B.S. degree from Nanjing University of Information Science and Technology in 2018. She received her M.S. degree from the School of Mathematics and Statistics, Nanjing University of Information Science and Technology in 2021. Her main research is grey system. E-mail: 1780458169@qq.com||YAN Shuli was born in 1982. She received her M.S. degree in School of Science from Wuhan University of Technology and Ph.D. degree in School of Economics and Management from Nanjing University of Aeronautics and Astronautics. She is currently an associate professor in Nanjing University of Information Science and Technology. Her current research interests include grey system, decision making, system evaluation, and prediction analysis. E-mail: yshuli@126.com
Supported by:
This study was supported by the National Natural Science Foundation of China (71701105; 71901191), the Major Program of the National Social Science Fund of China (17ZDA092), the Key Research Project of Philosophy and Social Sciences in Universities of Jiangsu Province (2018SJZDI111), and Jiangsu Provincial Government Scholarship for studying abroad.

Abstract

Abstract:

In this paper, an optimization model is proposed to simulate and predict the current situation of smog. The model takes the interval grey number sequence with the known possibility function as the original data, and constructs a time-delay nonlinear multivariable grey model MGM $(1,m|\tau ,\gamma )$ based on the new kernel and degree of greyness sequences considering its time-delay and nonlinearity. The time-delay parameter is determined by the maximum value of the grey time-delay absolute correlation degree, and the nonlinear parameter is determined by the minimum value of average relative error. In order to verify the feasibility of the model, this paper uses the smog related data of Nanjing city for simulation and prediction. Compared with the other four models, the new model has higher simulation and prediction accuracy.

Key words: kernel, degree of greyness, time-delay, nonlinear, smog

Pingping XIONG, Shiting CHEN, Shuli YAN. Time-delay nonlinear model based on interval grey number and its application[J]. Journal of Systems Engineering and Electronics, 2022, 33(2): 370-380.

Figures/Tables 9

Table 1

Fig 1

Table 2

Possibility function"

Data	$ \otimes _1^{(0)}$	$ \otimes _2^{(0)}$
Dec.24	$f_1^{ - 11}[74.00,90.67,107.33,124.00]$	$f_2^{ - 11}[4.20,6.60,9.00,11.40]$
Dec.25	$f_1^{{\text{ ? 10}}}[74.00,95.67,117.33,139.00]$	$f_2^{{\text{ ? 10}}}[3.70,6.27,8.83,11.40]$
Dec.26	$f_1^{{\text{ ? 9}}}[82.00,101.00,120.00,139.00]$	$f_2^{{\text{ ? 9}}}[3.70,6.27,8.83,11.40]$
Dec.27	$f_1^{ {\text{ ? 8}}}[87.00,104.33,121.67,139.00]$	$f_2^{{\text{ ? 8}}}[3.70,5.10,6.50,7.90]$
Dec.28	$f_1^{ {\text{ ? 7}}}[87.00,104.00,121.67,139.00]$	$f_2^{{\text{ ? 7}}}[3.20,4.43,5.67,6.90]$
Dec.29	$f_1^{ {\text{ ? 6}}}[87.00,104.33,121.67,139.00]$	$f_2^{{\text{ ? 6}}}[2.50,3.83,5.17,6.50]$
Dec.30	$f_1^{ {\text{ ? 5}}}[69.00,92.33,115.67,139.00]$	$f_2^{{\text{ ? 5}}}[2.50,3.83,5.17,6.50]$
Dec.31	$f_1^{{\text{ ? 4}}}[51.00,75.00,99.00,123.00]$	$f_2^{{\text{ ? 4}}}[2.50,9.00,15.50,22.00]$
Jan.1	$f_1^{{\text{ ? 3}}}[51.00,75.00,99.00,123.00]$	$f_2^{{\text{ ? 3}}}[2.50,9.60,16.70,23.80]$
Jan.2	$f_1^{{\text{ ? 2}}}[51.00,75.00,99.00,123.00]$	$f_2^{{\text{ ? 2}}}[2.50,9.60,16.70,23.80]$
Jan.3	$f_1^{{\text{ ? 1}}}[51.00,75.00,99.00,123.00]$	$f_2^{{\text{ ? 1}}}[2.50,9.60,16.70,23.80]$
Jan.4	$f_1^0[51.00,59.67,68.33,77.00]$	$f_2^0[4.60,11.00,17.40,23.80]$
Jan.5	$f_1^1[51.00,59.67,68.33,77.00]$	$f_2^1[8.80,13.80,18.80,23.80]$
Jan.6	$f_1^2[53.00,61.00,69.00,77.00]$	$f_2^2[8.80,13.80,18.80,23.80]$
Jan.7	$f_1^3[58.00,64.33,70.67,77.00]$	$f_2^3[8.80,11.30,13.80,16.30]$
Jan.8	$f_1^4[58.00,64.33,70.67,77.00]$	$f_2^4[8.80,12.47,16.13,19.80]$
Jan.9	$f_1^5[57.00,63.67,70.33,77.00]$	$f_2^3[8.80,13.93,19.07,24.20]$
Jan.10	$f_1^6[57.00,63.67,70.33,77.00]$	$f_2^6[8.80,13.93,19.07,24.20]$
Jan.11	$f_1^7[57.00,63.67,70.33,77.00]$	$f_2^7[9.50,14.40,19.30,24.20]$
Jan.12	$f_1^8[57.00,63.67,70.33,77.00]$	$f_2^8[10.60,15.13,19.67,24.20]$
Jan.13	$f_1^9[57.00,69.00,81.00,93.00]$	$f_2^9[10.60,15.13,19.67,24.20]$
Jan.14	$f_1^{10}[57.00,69.00,81.00,93.00]$	$ f_2^{10}[10.60,15.13,19.67,24.20] $
Jan.15	$f_1^{11}[57.00,71.00,85.00,99.00]$	$ f_2^{11}[10.60,13.33,16.07,18.80] $
Jan.16	$f_1^{12}[71.00,82.33,93.67,105.00]$	$ f_2^{12}[6.60,9.30,12.00,14.70] $
Jan.17	$f_1^{13}[71.00,82.33,93.67,105.00]$	$ f_2^{13}[6.60,9.30,12.00,14.70] $

Table 2

Table 3

Table 4

Table 5

Comparison of simulated and predicted values"

Value	$n$	Actual value		Model 1		Model 2		Model 3		Model 4
Value	$n$	AQI	Visibility/km	AQI	Visibility/km	AQI	Visibility/km	AQI	Visibility/km	AQI	Visibility/km
Simulated value	1	[51.00,77.00]	[8.80,23.80]	[51.85,69.77]	[9.27,23.85]	[51.00,77.00]	[8.80,23.80]	[51.00,77.00]	[8.80,23.80]	[51.00,77.00]	[8.80,23.80]
	2	[53.00,77.00]	[8.80,23.80]	[52.96,72.38]	[9.24,23.43]	[56.32,72.71]	[8.57,21.29]	[58.28,69.78]	[11.33,20.77]	[54.90,76.28]	[9.58,17.31]
	3	[58.00,77.00]	[8.80,16.30]	[54.08,75.00]	[9.21,23.00]	[55.00,75.19]	[9.25,21.42]	[59.97,74.53]	[8.87,16.19]	[58.88,76.86]	[9.08,16.14]
	4	[58.00,77.00]	[8.80,19.80]	[55.19,77.61]	[9.17,22.58]	[55.04,77.87]	[9.47,20.99]	[59.46,76.53]	[9.20,19.58]	[58.24,76.08]	[9.55,19.74]
	5	[57.00,77.00]	[8.80,24.20]	[56.31,80.23]	[9.14,22.16]	[55.70,79.42]	[9.16,21.77]	[57.94,76.28]	[11.18,19.88]	[57.93,76.69]	[9.62,23.52]
	6	[57.00,77.00]	[8.80,24.20]	[57.42,82.84]	[9.10,21.74]	[56.45,80.64]	[8.94,22.82]	[57.87,76.30]	[10.61,20.33]	[57.90,76.39]	[9.72,23.60]
	7	[57.00,77.00]	[9.50,24.20]	[58.54,85.46]	[9.07,21.31]	[57.25,82.11]	[8.95,23.56]	[57.76,76.41]	[10.73,21.36]	[57.92,76.23]	[10.33,23.84]
	8	[57.00,77.00]	[10.60,24.20]	[59.65,88.07]	[9.04,20.89]	[57.99,83.71]	[9.22,24.13]	[58.25,76.09]	[11.13,23.41]	[58.19,76.82]	[11.18,23.83]
	9	[57.00,93.00]	[10.60,24.20]	[60.77,90.69]	[9.00,20.47]	[58.63,86.42]	[9.84,23.80]	[64.56,85.55]	[11.31,23.25]	[62.17,83.24]	[11.29,23.68]
	10	[57.00,93.00]	[10.60,24.20]	[61.88,93.30]	[8.97,20.04]	[59.57,90.10]	[10.42,22.73]	[62.32,87.02]	[11.84,21.84]	[65.27,81.23]	[11.54,23.07]
	11	[57.00,99.00]	[10.60,18.80]	[63.00,95.92]	[8.93,19.62]	[61.28,94.74]	[10.54,20.84]	[65.05,89.78]	[10.83,18.71]	[66.26,79.90]	[11.28,18.80]
	12	[71.00,105.00]	[6.60,14.70]	[64.11,98.53]	[8.90,19.20]	[65.35,101.50]	[8.94,17.24]	[73.56,104.35]	[6.92,14.56]	[77.89,86.97]	[7.03,14.60]
	13	[71.00,105.00]	[6.60,14.70]	[65.23,101.15]	[8.87,18.77]	[71.43,109.59]	[5.80,12.70]	[72.37,103.77]	[7.03,12.27]	[79.09,84.06]	[7.08,14.59]
Predicted value	14	[71.00,105.00]	[6.60,14.70]	[66.34,103.76]	[8.83,18.35]	[76.83,116.60]	[3.19,9.08]	[72.34,105.82]	[7.30,10.67]	[72.86,75.41]	[6.52,15.57]
Predicted value	15	[71.00,105.00]	[6.60,14.70]	[67.46,106.38]	[8.80,17.93]	[81.58,122.74]	[1.13,6.22]	[63.96,104.16]	[7.17,9.55]	[84.75,84.78]	[9.78,15.68]

Table 5

Table 6

Comparison of relative errors"

Error	$n$	Model 1				Model 2				Model 3				Model 4
		AQI		Visibility		AQI		Visibility		AQI		Visibility		AQI		Visibility
		L/%	U/%	L/%	U/%	L/%	U/%	L/%	U/%	L/%	U/%	L/%	U/%	L/%	U/%	L/%	U/%
Simulated error	1	1.66	9.39	5.39	0.21	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	2	0.07	5.99	5.00	1.57	6.26	5.58	2.61	10.55	9.95	9.37	28.73	12.71	3.59	0.93	8.88	27.28
	3	6.77	2.60	4.61	41.13	5.17	2.34	5.09	31.43	3.40	3.20	0.84	0.65	1.51	0.19	3.17	0.97
	4	4.84	0.80	4.23	14.05	5.11	1.14	7.65	6.02	2.51	0.60	4.51	1.12	0.42	1.19	8.47	0.31
	5	1.22	4.19	3.84	8.44	2.29	3.14	4.09	10.06	1.65	0.94	27.05	17.85	1.63	0.40	9.29	2.83
	6	0.74	7.59	3.45	10.19	0.96	4.73	1.62	5.70	1.52	0.91	20.53	15.97	1.58	0.80	10.46	2.46
	7	2.69	10.99	4.53	11.93	0.44	6.64	5.77	2.63	1.33	0.77	12.96	11.74	1.62	1.00	8.78	1.50
	8	4.65	14.38	14.75	13.68	1.74	8.72	12.98	0.28	2.19	1.19	5.04	3.27	2.08	0.23	5.47	1.53
	9	6.61	2.48	15.08	15.43	2.86	7.07	7.14	1.65	13.25	8.01	6.70	3.91	9.07	10.49	6.47	2.16
	10	8.56	0.33	15.40	17.18	4.51	3.12	1.72	6.06	9.33	6.43	11.71	9.76	14.52	12.66	8.86	4.69
	11	10.52	3.11	15.72	4.36	7.51	4.31	0.58	10.83	14.12	9.32	2.20	0.47	16.25	19.29	6.41	0.02
	12	9.70	6.16	34.85	30.59	7.96	3.34	35.39	17.30	3.60	0.61	4.91	0.94	9.70	17.17	6.49	0.66
	13	8.13	3.67	34.33	27.71	0.61	4.37	12.19	13.61	1.92	1.17	6.47	16.52	11.39	19.95	7.31	0.75
Average simulation error		5.09	5.51	12.40	15.11	3.49	4.19	7.45	8.93	4.98	3.27	10.13	7.30	5.64	6.48	6.93	3.47
Predicted error	14	6.56	1.18	33.82	24.84	8.22	11.05	51.69	38.24	1.89	0.78	10.59	27.43	2.62	28.18	1.24	5.90
Predicted error	15	4.99	1.31	33.30	21.96	14.90	16.90	82.95	57.65	9.92	1.10	8.69	35.00	19.37	19.26	48.20	6.69
Average prediction error		5.78	1.25	33.56	23.40	11.56	13.97	67.32	47.95	5.90	0.94	9.64	31.22	10.99	23.72	24.72	6.30

Table 6

Fig 2

Fig 3

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Average relative error/%	Prediction accuracy
<10	Higher
10?20	High
20?50	Medium
>50	Low

Value	n	AQI	Visibility /km
Simulated value	1	[51.00,77.00]	[8.80,23.80]
	2	[53.37,76.62]	[9.14,19.91]
	3	[58.28,76.75]	[8.86,16.25]
	4	[58.19,76.73]	[8.83,19.67]
	5	[57.56,76.81]	[8.90,21.89]
	6	[57.55,76.97]	[9.18,19.83]
	7	[57.03,76.63]	[10.12,20.31]
	8	[57.34,76.74]	[10.63,22.80]
	9	[59.19,83.46]	[10.63,24.06]
	10	[61.40,81.84]	[10.64,24.17]
	11	[58.00,80.94]	[10.65,18.72]
	12	[71.72,90.83]	[6.67,14.63]
	13	[73.40,91.35]	[6.66,14.65]
Predicted value	14	[72.69,90.34]	[6.51,15.39]
Predicted value	15	[74.54,90.14]	[6.66,15.75]

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Time-delay nonlinear model based on interval grey number and its application

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Erorr	$n$	AQI		Visibility
Erorr	$n$	L/%	U/%	L/%	U/%
Simulated error	1	0.00	0.00	0.00	0.00
	2	0.70	0.50	3.83	16.34
	3	0.48	0.32	0.70	0.29
	4	0.32	0.35	0.34	0.64
	5	0.97	0.25	1.18	9.55
	6	0.97	0.04	4.31	18.05
	7	0.05	0.49	6.54	16.09
	8	0.60	0.34	0.30	5.80
	9	3.83	10.26	0.31	0.59
	10	7.71	12.00	0.33	0.12
	11	1.76	18.25	0.48	0.44
	12	1.02	13.49	1.07	0.45
	13	3.38	13.00	0.84	0.34
Average simulation error		1.68	5.33	1.56	5.28
Predicted error	14	2.37	13.96	1.38	4.72
Predicted error	15	4.99	14.15	0.90	7.15
Average prediction error		3.68	14.06	1.14	5.94