Fractional derivative multivariable grey model for nonstationary sequence and its application

doi:10.23919/JSEE.2020.000075

Abstract

Abstract:

Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC$ \mathit{\boldsymbol{(q, N)}} $ is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization (PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC$ \mathit{\boldsymbol{(q, N)}} $ model is used to predict the municipal solid waste (MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.

Key words: fractional derivative of Caputo type, fractional accumulation generating operation (FAGO), Laplace transform, multivariable grey prediction model, particle swarm optimization (PSO)

Yuxiao KANG, Shuhua MAO, Yonghong ZHANG, Huimin ZHU. Fractional derivative multivariable grey model for nonstationary sequence and its application[J]. Journal of Systems Engineering and Electronics, 2020, 31(5): 1009-1018.

Figures/Tables 10

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Author	Name	Abbreviation	Application
Deng (2002) [16]	Multivariate grey model	GM(1, N)	Dynamic analysis of athletes' training status
Xie et al. (2009) [17]	Discrete multivariate grey model	DGM(1, N)	Mobile telecommunication customer
Wang (2014) [18]	Nonlinear grey multivariable model	NGM(1, N)	High-technology industry total output
Mao et al. (2015) [19]	Fractional accumulation time-lag model	GM(1, N, τ)	Economic development of Wuhan
Ding et al. (2017) [20]	Multivariable time-delayed discrete grey model	TDDGM(1, N)	Output value of high-tech enterprises
Wu et al. (2018) [21]	Grey multivariable convolution model	GMCN(1, N)	Industrial power consumption
Wu et al. (2018) [22]	Grey multivariable model with fractional accumulation	FGMC(1, N)	Shandong's electricity consumption
Pei et al. (2018) [23]	Transformed model of nonlinear grey models	TNGM(1, N)	Pollutant emission
Ma et al. (2019) [24]	Fractional discrete multivariate model	FDGMC(1, N)	Industrial pollutant emission
Ma et al. (2019) [25]	Nonlinear multivariate grey Bernoulli model	NGBMC(1, N)	Tourist income of China
Zeng et al. (2019) [26]	Grey model of ternary interval numbers	TIGM(1, N)	Power generation, consumer price index
This paper	Caputo fractional derivative multivariable model	CFGMC(q, N)	Municipal solid waste (MSW) yields of Wuhan

Year	y1	y2	y3	y4	y5	y6
Year	MSW yields/10 000 tons	Resident population/10 000 people	Road sweeping area/100 000 m²	Passenger capacity/10 000 people	Per capita net income/yuan	Resident consumption index/yuan
2006	211.00	875.00	4 009.00	16 193.70	4 748.00	9 182.10
2007	215.40	891.00	5 414.00	17 338.40	5 371.00	10 600.00
2008	219.10	897.00	5 994.00	18 882.10	6 349.00	11 433.00
2009	217.50	910.00	6 479.00	21 735.60	7 161.00	12 710.00
2010	219.20	978.50	6 640.00	22 896.70	8 295.00	14 490.10
2011	224.40	1 002.00	7 130.00	25 743.20	9 814.00	17 141.00
2012	225.00	1 012.00	8 857.00	27 492.20	11 190.00	18 813.10
2013	264.00	1 022.00	10 040.00	29 621.69	12 713.00	20 157.30
2014	257.36	1 033.80	15 837.00	27 899.48	16 160.00	22 002.20
2015	330.66	1 060.77	17 730.00	27 628.71	17 722.00	23 943.05
2016	356.29	1 076.62	12 486.00	29 177.15	19 152.00	26 535.00
2017	396.38	1 089.29	13 853.00	29 950.30	20 887.00	28 546.00

Year or Index	y1 MSW	CFGMC(q, N)		GM(1, N)		FGMC(1, N)		MLR
		r=0.5	q=0.6	r=1	q=1	r=0.8	q=1	MLR
		Fitting	APE	Fitting	APE	Fitting	APE	Fitting	APE
2006	211.00	211.00	0.00%	211.00	0.00%	211.00	0.00%	212.24	0.59%
2007	215.40	220.32	2.28%	194.01	9.93%	206.37	4.19%	215.97	0.27%
2008	219.10	220.50	0.64%	250.54	14.35%	211.73	3.36%	216.69	1.10%
2009	217.50	216.21	0.59%	231.33	6.36%	211.63	2.70%	217.95	0.21%
2010	219.20	223.69	2.05%	223.93	2.16%	214.12	2.32%	219.42	0.10%
2011	224.40	238.06	6.09%	226.38	0.88%	219.62	2.13%	222.87	0.68%
2012	225.00	234.22	4.10%	225.42	0.19%	220.57	1.97%	226.45	0.65%
MAPE			2.25%		4.84%		2.38%		0.51%
MAE			4.99		10.54		5.22		1.12
RMSE			6.76		15.42		5.84		1.33
2013	264.00	237.37	10.09%	188.04	28.77%	218.87	17.10%	228.16	13.58%
2014	257.36	213.46	17.06%	131.75	48.81%	214.52	16.65%	236.10	8.26%
2015	330.66	217.06	34.35%	141.09	57.33%	208.77	36.86%	240.86	27.16%
2016	356.29	366.26	2.80%	39.13	89.02%	217.71	38.90%	237.29	33.40%
2017	396.38	393.80	0.65%	22.86	94.23%	234.59	40.82%	241.07	39.18%
MAPE			12.99%		63.63%		30.06%		24.32%
MAE			39.34		216.36		102.05		84.24
RMSE			55.94		243.96		113.23		98.07

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