Research on a novel fractional GM(α, n) model and its applications

Wenqing Wu (School of Science, Southwest University of Science and Technology, Mianyang, China) (V.C. and V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, China)
Xin Ma (School of Science, Southwest University of Science and Technology, Mianyang, China) (State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China)
Yong Wang (School of Science, Southwest Petroleum University, Chengdu, China)
Yuanyuan Zhang (School of Science, Southwest University of Science and Technology, Mianyang, China)
Bo Zeng (College of Business Planning, Chongqing Technology and Business University, Chongqing, China)

Grey Systems: Theory and Application

ISSN: 2043-9377

Publication date: 1 July 2019

Abstract

Purpose

The purpose of this paper is to develop a novel multivariate fractional grey model termed GM(a, n) based on the classical GM(1, n) model. The new model can provide accurate prediction with more freedom, and enrich the content of grey theory.

Design/methodology/approach

The GM(α, n) model is systematically studied by using the grey modelling technique and the forward difference method. The optimal fractional order a is computed by the genetic algorithm. Meanwhile, a stochastic testing scheme is presented to verify the accuracy of the new GM(a, n) model.

Findings

The recursive expressions of the time response function and the restored values of the presented model are deduced. The GM(1, n), GM(a, 1) and GM(1, 1) models are special cases of the model. Computational results illustrate that the GM(a, n) model provides accurate prediction.

Research limitations/implications

The GM(a, n) model is used to predict China’s total energy consumption with the raw data from 2006 to 2016. The superiority of the GM(a, n) model is more freedom and better modelling by fractional derivative, which implies its high potential to be used in energy field.

Originality/value

It is the first time to investigate the multivariate fractional grey GM(α, n) model, apply it to study the effects of China’s economic growth and urbanization on energy consumption.

Keywords

Citation

Wu, W., Ma, X., Wang, Y., Zhang, Y. and Zeng, B. (2019), "Research on a novel fractional GM(α, n) model and its applications", Grey Systems: Theory and Application, Vol. 9 No. 3, pp. 356-373. https://doi.org/10.1108/GS-11-2018-0052

Publisher

:

Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited


1. Introduction

Grey system theory was developed by Professor Deng (1982) in 1982 aiming to cope with uncertain problems with poor and incomplete information. The first-order univariate GM(1, 1) model is the foundation of this theory having simple modelling process and easy to understand. Over the past three decades, the classical GM(1, 1) model has significant generalized with the following aspects: the univariate linear grey models (Xie and Liu, 2009; Cui et al., 2013; Wang, Liu, Wang, Wang and Liu, 2017; Zeng, Tan and Xu, 2018; Wei et al., 2018; Zeng, Duan, Bai and Meng, 2018; Ding et al., 2018; Wang and Li, 2019); the univariate nonlinear grey models (Chen et al., 2008; Wang et al., 2011; Wang, 2013; Wu et al., 2017; Wang, Pei and Wang, 2017; Ma, 2019); and the multivariate grey models (Tien, 2005; Zeng et al., 2016; Ma and Liu, 2017; Zeng and Chuan, 2018; Ma et al., 2019). These generalized grey models are all first-order derivative which results in less freedom modelling original data sequences. Thus, the fractional order grey model is taken into account in this paper.

The idea of the fractional order derivative can be traced to a communication between Leibniz and L’Hôpital in the 1600s. The first monograph on fractional calculus was published in 1974 (Oldham and Spanier, 1974). Considering the memory principle in fractional differential equations (Granger and Joyeux, 1980; Teyssoere and Kirman, 2007; Deng, 2007), it has been developed rapidly and been applied in chemical processing systems (Flores-Tlacuahuac and Biegler, 2014), economics (Hosking, 1981) and control system (Monje et al., 2010). As grey models with fractional order outperform than classical ones, they have received considerable attention in recent years. Wu et al. (2013) studied the GM(1, 1) model with fractional accumulation, where the model properties and perturbation problems are derived. Based on the fractional order accumulation, Wu et al. then considered non-homogenous discrete grey model (Wu et al., 2014) and the GM(p/q) (2, 1) model (Wu et al., 2015). Interested readers can refer to Xiao et al. (2014), Gao et al. (2015), Zeng and Liu (2017), Wu, Gao, Yang and Chen (2018), Zeng (2018), Wu, Li and Yang (2018), Wu, Ma, Zeng, Wang and Cai (2018) and Wu and Zhao (2019), for more details on the fractional-order accumulation grey model. Among these studies, one checks easily that the fractional operator is mainly applied in the pretreatment of data rather than in the grey model structure. This means the whitening equation of grey models remains the same as the integer-order ones.

Mao et al. (2016) investigated a novel grey model FGM(q, 1) with fractional order derivative and fractional accumulation. Recently, Yang and Xue (2016, 2017) considered generalized continuous fractional-order grey models by using fractional calculus definition and Fourier series. The optimal value α is determined by genetic algorithm (GA), and the GM(α, 1) model is used to predict per capital output of electricity of China. Compared with studies being published, Mao’s and Yang’s work considered the fractional form of continuous GM(1, 1) by fractional calculus. However, to our best knowledge, no work considered the general continuous fractional order GM(α, n) model. Following the idea of Mao and Yang, this paper considers a fractional-order multivariate grey model GM(α, n) which can be viewed as generalized models of classical ones. In order to improve the accuracy of system for simulation and prediction, the parameter α is determined by the GA.

As we know, energy is the foundation strategic resource for a city or a country, and energy consumption can reflect the development of economic and the accelerating process of urbanization. Along with the energy consumption in China increases yearly, a good demand forecast of energy consumption becomes a hot issue. Recently, considerable attention has been paid to energy consumption. Various methods and techniques have been used, such as cointegrated panel analysis (Lee, 2005), artificial neural network (Neto and Fiorelli, 2008), time series analysis (Wang, 2014; Cai et al., 2018; Ma and Cai, 2019), coupling mathematical model (Wang and Yi, 2018; Wang et al., 2018), etc. Hence, this paper discusses the China’s total energy consumption by grey models. The energy consumption under the GM(α, n) model, the recursive discrete multivariate grey model RDGM(1, n), the convolutional GM(1, n) with trapezoid formula GMCT(1, n) model, and the convolutional GM(1, n) with Gaussian formula GMCG(1, n) model are systematically studied. It is noted that the results show that the GM(α, n) model outperforms the others.

The rest of this paper is organized as follows. Section 2 gives some preliminaries. Detailed discussions of the GM(α, n) are given in Section 3. Section 4 validates the GM(α, n) model. Applications are provided in Section 5. Conclusions are drawn in the last section.

2. Preliminaries

2.1 The fractional accumulated generating matrix and inverse matrix

Accumulated generating operation (AGO) is a technique to reduce the volatility of original data and improve grey exponential rate. The inverse operating of accumulated generating named inverse accumulated generating operation (IAGO). And the definition of the rth AGO and the rth IAGO can be found in the papers (Wu et al., 2013; Wu, Ma, Zeng, Wang and Cai, 2018; Mao et al., 2016):

Definition 1.

Let X(0)={x(0)(1), x(0)(2), …, x(0)(m)} be an original sequence, X(r)(r > 0) be the rth accumulated generating operation (r-AGO) sequence of X(0) where x ( r ) ( k ) = i=1 k x ( r 1 ) ( i ) , k=1,2, , m . Denote by Ar the r-AGO matrix which satisfies X(r)=X(0)Ar, and:

A r = ( [ r0 ] [ r1 ] [ r2 ] [ r m 1 ] 0 [ r0 ] [ r1 ] [ r m 2 ] 00 [ r0 ] [ r m 3 ] 000 [ r0 ] ) m × m ,
with:
[ r i ] = r ( r + 1 ) , , ( r + i 1 ) i ! = ( r + i 1 i ) = ( r + i 1 ) ! i ! ( r 1 ) ! , [ 0 i ] =0, [ 00 ] = ( 00 ) =1.
Definition 2.

The inverse accumulated generation is defined as x(r−1)(k)=x(r)(k)−x(r)(k−1), k=1, 2, …, m. Denote by Dr the rth inverse accumulated generating operation (r-IAGO) matrix that satisfies X(0)=X(r)Dr, and:

D r = ( [ r 0 ] [ r 1 ] [ r 2 ] [ r m 1 ] 0 [ r 0 ] [ r 1 ] [ r m 2 ] 00 [ r 0 ] [ r m 3 ] 000 [ r 0 ] ) m × m ,
with:
[ r i ] = r ( r + 1 ) , , ( r + i 1 ) i ! = ( 1 ) i r ( r 1 ) , , ( r i + 1 ) i ! = ( 1 ) i ( r i ) , [ r i ] =0, i > r.
Theorem 1.

The r-AGO Ar and the r-IAGO Dr satisfies ArDr=E.

Proof 1. From the definition of Ar, it is easy to know determinant det (Ar)=1 which means Ar is reversible. Based on X(r)=X(0)Ar, it is straightforward to get X(r) (Ar)−1=X(0) Ar (Ar)−1=X(0). From Definition 2, we know that X(r) Dr=X(0). Thus, (Ar)−1=Dr, namely, ArDr=E.

It follows from X(r)=X(0)Ar that:

(1) x ( r ) ( k ) = i=1 k [ r k i ] x ( 0 ) ( i ) = i=0 k 1 [ r i ] x ( 0 ) ( k i ) ,
which means x(r)(k) is the weight of x(0)(i), i=1, 2, …, k.◼

2.2 The fractional calculus and difference

The fractional calculus is a generalization of integration and differentiation to non-integer-order fundamental operator. The differential-integral operator is:

D t γ a = d γ d t γ , γ>0, D t γ a=1, γ=0, D t γ a = a t ( d t ) γ , γ<0,
where a is a real number related to initial value, γ is a number with positive or negative value corresponding to differentiation or integration, respectively.

For fractional derivatives, the Riemann-Liouville, the Grunwald-Letnikov (GL) and the Caputo derivative are three mainly definitions (Oldham and Spanier, 1974; Monje et al., 2010; Podlubny, 1999). This paper chooses the GL definition:

(2) D t γ a f ( t ) = d γ f ( t ) d t γ = lim h 0 1 h γ j=0 ( t a ) / h ( 1 ) j ( γ j ) f ( t j h ) ,
where h is sampling interval.

From Definition 2, ( 1 ) j ( γ j ) = [ γ j ] then we have:

(3) D t γ a f ( t ) = d γ f ( t ) d t γ = lim h 0 1 h γ j=0 ( t a ) / h [ γ j ] f ( t j h ) .

In order to numerical calculate the fractional calculus, the approximation is obtained by Dγf(t)≈Δγf(t). Based on the forward difference, a=0, and zero initial conditions, we have:

(4) Δ h γ f ( t ) | t = k h = h γ j=0 k + 1 [ γ j ] f ( ( k + 1 ) h j h ) ,
or based on the backward difference and the same conditions, we have:
(5) Δ h γ f ( t ) | t = k h = h γ j=0 k [ γ j ] f ( k h j h ) ,
where k=t/h is the samples number.

With the zero condition f(0)=0, Equation (5) can be rewritten as:

(6) Δ h γ f ( t ) | t = k h = h γ j=0 k [ γ j ] f ( k h j h ) = h γ i=1 k [ γ k i ] f ( k h ( k i ) h ) = h γ i=1 k f ( i h ) [ γ k i ] ,
which is according with x ( 0 ) ( k ) = i=1 k x ( r ) ( i ) [ r k i ] obtained from Definition 2, and this is the reason that we choose the GL definition in this paper.

3. The fractional GM(α, n) model

3.1 The basis of GM(1, n) model

Suppose original non-negative data sequences of a system with m entries are X i ( 0 ) = ( x i ( 0 ) ( 1 ) , x i ( 0 ) ( 2 ) , , x i ( 0 ) ( m ) ) , i=1, 2, …, n where x i ( 0 ) ( k ) stands for the behaviour of data at the time k for k=1, 2, …, m. Generally, the observations X i ( 0 ) , i=2,3, , n are viewed as the input sequences, and X 1 ( 0 ) is viewed as the output sequence.

Let X i ( 1 ) = ( x i ( 1 ) ( 1 ) , x i ( 1 ) ( 2 ) , , x i ( 1 ) ( m ) ) , i=1,2, , n be the 1-AGO sequences of X i ( 0 ) , i=1,2, , n . From Definition 1, we have X i ( 1 ) = X i ( 0 ) A that the x i ( 1 ) ( k ) = j=1 k x i ( 0 ) ( j ) , k=1,2, , m.

Denoted by Z i ( 1 ) = ( z i ( 1 ) ( 2 ) , z i ( 1 ) ( 3 ) , , z i ( 1 ) ( m ) ) , i=1,2, , n the mean sequence generated by consecutive neighbours of X i ( 1 ) , i=1,2, , n where the z i ( 1 ) ( k ) =0.5 x i ( 1 ) ( k 1 ) + 0.5 x i ( 1 ) ( k ) , k=2,3, , m.

From Liu and Lin (2011), the mathematical form of the GM (1, n) model is:

(7) d x 1 ( 2 ) ( t ) d t + q 1 x 1 ( 1 ) ( t ) = i=2 n q i x i ( 1 ) ( t ) + ι ,
where the system parameters ω ˆ = ( q 1 , q 2 , , q n , ι ) T are given by:
(8) ω ˆ = ( χ T χ ) 1 χ T η ,
with:
χ = ( z 1 ( 1 ) ( 2 ) x 2 ( 1 ) ( 2 ) x n ( 1 ) ( 2 ) 1 z 1 ( 1 ) ( 3 ) x 2 ( 1 ) ( 3 ) x n ( 1 ) ( 3 ) 1 z 1 ( 1 ) ( υ 1 ) x 2 ( 1 ) ( υ 1 ) x n ( 1 ) ( υ 1 ) 1 z 1 ( 1 ) ( υ ) x 2 ( 1 ) ( υ ) x n ( 1 ) ( υ ) 1 ) , η = ( x 1 ( 0 ) ( 2 ) x 1 ( 0 ) ( 3 ) x 1 ( 0 ) ( υ 1 ) x 1 ( 0 ) ( υ ) ) .

3.2 Formulating the fractional GM(α, n) model

This subsection studies the fractional GM(α, n) model, which is an extension of the GM(1, n) model in that the first-order differential equation is transformed into the fractional differential equation.

The mathematical form of fractional GM(α, n) is defined as follows:

(9) d α x 1 ( 1 ) ( t ) d t α + b 1 x 1 ( 1 ) ( t ) = i=2 n b i x i ( 1 ) ( t ) + u ,
where α, b1, b2, …, bn, u are system parameters to be determined, in which α can be any number more than zero, b1, b2, …, bn reflects the effect of corresponding sequences X i ( 1 ) , i=2,3, , n on the X 1 ( 1 ) .

For α=1, the fractional GM(α, n) model reduces to the classical GM(1, n) model (Liu and Lin, 2011) as:

(10) d x 1 ( 1 ) ( t ) d t + b 1 x 1 ( 1 ) ( t ) = i=2 n b i x i ( 1 ) ( t ) + u.

For n=1, the fractional GM(α, n) model reduces to the fractional GM(α, 1) model (Yang and Xue, 2016) as:

(11) d α x 1 ( 1 ) ( t ) d t α + b 1 x 1 ( 1 ) ( t ) = u.

For α=1, n=1, the fractional GM(α, n) model reduces to the classical GM(1, 1) model (Deng, 1982) as:

(12) d x 1 ( 1 ) ( t ) d t + b 1 x 1 ( 1 ) ( t ) = u.

3.3 Parameters estimation of fractional GM(α, n) model

Using the difference instead of differential form, the GL definition in the fractional calculus, and the sampling interval T=1, the approximation of the differential operator can be obtained by:

(13) d α x 1 ( 1 ) ( t ) d t α | t = k = D t α x 1 ( 1 ) ( t ) | t = k Δ α x 1 ( 1 ) ( t ) | t = k = j=0 k [ α j ] x 1 ( 1 ) ( k j ) .

Therefore, Equation (9) can be rewritten as:

(14) Δ α x 1 ( 1 ) ( k ) = b 1 x 1 ( 1 ) ( k ) + i=2 n b i x i ( 1 ) ( k ) + u , k=1,2, , m.

Employing the least squares estimation method, from Equation (14) by considering k=2, 3, …, υ(υm), the model parameters ξ ˆ = ( b 1 , b 2 , , b n , u ) T can be given as follows:

(15) ξ ˆ = ( B T B ) 1 B T Y ,
where B and Y are:
B = ( x 1 ( 1 ) ( 2 ) x 2 ( 1 ) ( 2 ) x n ( 1 ) ( 2 ) 1 x 1 ( 1 ) ( 3 ) x 2 ( 1 ) ( 3 ) x n ( 1 ) ( 3 ) 1 x 1 ( 1 ) ( υ 1 ) x 2 ( 1 ) ( υ 1 ) x n ( 1 ) ( υ 1 ) 1 x 1 ( 1 ) ( υ ) x 2 ( 1 ) ( υ ) x n ( 1 ) ( υ ) 1 ) , Y = ( Δ α x 1 ( 1 ) ( 2 ) Δ α x 1 ( 1 ) ( 3 ) Δ α x 1 ( 1 ) ( υ 1 ) Δ α x 1 ( 1 ) ( υ ) ) .

It is worth mentioning that the order of the differential equation is a fractional order, not the first order. For this extension, we must first determine the value α of the differential equation. Once the value α is determined, the system parameters b1, b2, …, bn, u can be derived directly. However, given the analytical solution of α is a tedious job, and even impossible. This paper will use the GA to search the optimal solution of α in what follows.

3.4 Solution of the fractional GM(α, n) model

Apply the forward difference technique discretizing the continuous fractional order differentiation operator by using the GL definition in order to get a model where x 1 ( 1 ) ( ( k + 1 ) T ) appears. Setting a=0, γ=α, h=T, and f(t)=x1(t), then from Equation (4), we known that:

(16) D t α x 1 ( 1 ) ( t ) | t = k T 1 T α j=0 k + 1 ( 1 ) j ( α j ) x 1 ( 1 ) ( ( k + 1 j ) T ) ,
where T is the sampling interval.

Substituting Equation (16) in Equation (9), we get:

(17) 1 T α j=0 k + 1 ( 1 ) j ( α j ) x 1 ( 1 ) ( ( k + 1 j ) T ) + b 1 x 1 ( 1 ) ( k T ) = i=2 n b i x i ( 1 ) ( k T ) + u ,
which is the basic form of the fractional GM(α, n) model.

It follows from the basic form of GM(α, n) model that:

(18) 1 T α ( ( α0 ) x 1 ( 1 ) ( ( k + 1 ) T ) ( α1 ) x 1 ( 1 ) ( k T ) + j=2 k + 1 ( 1 ) j ( α j ) x 1 ( 1 ) ( ( k j + 1 ) T ) ) = b 1 x 1 ( 1 ) ( k T ) + i=2 n b i x i ( 1 ) ( k T ) + u , k=1,2, , m 1.

Noting ( α0 ) =1 , Equation (18) leads to:

(19) x ˆ 1 ( 1 ) ( ( k + 1 ) T ) = [ ( α1 ) T α b 1 ] x 1 ( 1 ) ( k T ) j=2 k + 1 ( 1 ) j ( α j ) x 1 ( 1 ) ( ( k j + 1 ) T ) + i=2 n b i T α x i ( 1 ) ( k T ) + u T α , k=1,2, , m 1.

The 1-AGO sequence X ˆ 1 ( 1 ) can be evaluated using Equation (19), and the simulative and predictive sequence X ˆ 1 ( 0 ) is:

(20) x ˆ 1 ( 0 ) ( k + 1 ) = x ˆ 1 ( 1 ) ( k + 1 ) x ˆ 1 ( 1 ) ( k ) , k=1,2, , m 1.

3.5 Optimization of the fractional order of the GM(α, n) model

In this work, the order of the differential equation is a fractional order instead of an integer order. To evaluate the forecasting accuracy of the GM(α, n) model, the root mean squared percentage error (RMSPE), respectively, for the priori-sample period (RMSPEPR) and the post-sample period (RMSPEPO) are used. Generally, RMSPEPR, RMSPEPO and RMSPE are defined, respectively, as:

RMSPEPR = 1 υ k=1 υ ( x ˆ 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) ) 2 × 100%,
RMSPEPO = 1 m υ k = υ + 1 m ( x ˆ 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) ) 2 × 100%,
RMSPE = 1 m k=1 m ( x ˆ 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) ) 2 × 100%,
where m is the number of total samples, and υ is the number samples that applied to build the prediction model.

We choose α minimizing the RMSPE as the optimal model parameter. The minimization problem is illustrated mathematically as:

(21) min α RMSPE = 1 m k=1 m ( x ˆ 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) x 1 ( 0 ) ( k ) ) 2 × 100%,
s.t.:
{ 0 α , x ˆ 1 ( 0 ) ( k ) = x ˆ 1 ( 1 ) ( k ) x ˆ 1 ( 1 ) ( k 1 ) , k=2,3, , m , x ˆ 1 ( 1 ) ( k ) = [ ( α1 ) b 1 ] x 1 ( 1 ) ( k 1 ) j=2 k + 1 ( 1 ) j ( α j ) x 1 ( 1 ) ( k j ) + i=2 n b i x i ( 1 ) ( k 1 ) + u , ( b 1 , b 2 , , b n , u ) T = ( B T B ) 1 B T Y.

The objective is to find the optimal parameter α which minimizes the RMSPE of our model. It would have been a difficult task to derive analytic results for the optimal value α because Equation (21) is a nonlinear programming with nonlinear objective function and nonlinear constraints. Similar the argument to Wang (2013) and Wang and Hsu (2008), here the optimum value α may be found by using the GA.

In fact, GA first published by Holland (1975) is a powerful stochastic search and optimization technique based on the mechanics of natural selection and natural genetics (Goldberg, 1989). In recent years, the GA has been applied in a variety of fields, such as forecasting high technology industrial output (Wang and Hsu, 2008), forecasting integrated circuit industry (Hsu, 2010) and sales forecasting (Yu and Chen, 2017). To research the optimum parameters, the GA should have the following contents: a chromosomal representation of solution; an initial population size; an appropriate fitness function; genetic operators include initialization, mutation, crossover and comparison; and stopping criteria. Goldberg (1989) suggested that the GA parameters of the cross rate are in [0.5, 1.0], the mutation rate is in [0.001, 0.1] and the population size is in [30, 200]. This algorithm has successfully applied in many optimization problems, and the results more robust and accurate. So, this paper utilizes the GA to determine the optimum value of fractional order.

3.6 Detailed modelling steps of the fractional GM(α, n) model

The detailed modelling steps for fractional order of the GM(α, n) model is provided below:

  • Step 1: determine the original data sequences X i ( 0 ) , i=1,2, , n , and the first order accumulated generating operating sequences X i ( 1 ) , i=1,2, , n by Definition 1.

  • Step 2: with α as parameter, compute the value of the difference operator Δ α x 1 ( 1 ) ( t ) | t = k using Equation (14), and the values of b1, b2, …, bn, u by the least squares estimation method.

  • Step 3: obtain the simulated and predicted value sequence X ˆ 1 ( 1 ) using Equation (19), and the reverted value sequence X ˆ 1 ( 0 ) using Equation (20).

  • Step 4: find the optimal value that corresponds to the smallest RMSPE applying the GA.

  • Step 5: substituting optimal α into Step 2 to compute the Δ α x 1 ( 1 ) ( t ) | t = k , b1, b2, …, bn, u and obtain the sequence X ˆ 1 ( 0 ) .

4. Validation of GM(α, n) model

This section provides numerical experiments to validate the accuracy of the fractional order GM(α, n) model compared to the multivariate grey models which are recursive discrete multivariate grey model RDGM(1, n), the convolutional GM(1, n) model with trapezoid formula GMCT(1, n) and the convolution GM(1, n) model with Gaussian formula GMCG(1, n). For convenience, the three models are abbreviated to RDGM, GMCT and GMCG models.

We choose two independent sequences as the input sequences which listed in Table I. Initial point x 1 ( 0 ) ( 1 ) is randomly generated in the interval (1, 2) by the uniform distribution, and the other x 1 ( 0 ) ( k ) ( k>1 ) are generated applying Equation (19). Parameters α and b1 vary in the interval [0.15, 1.5] and [−1.3, 1.3], respectively. Other parameters bi, i=2, 3, …, n and u are, respectively, randomly generated in the interval (0, 5) and (0, 100) by the uniform distribution.

All the data for experiment are explanation in Figure 1.

The following three cases are considered to validate the accuracy of GM(α, n) model compared to the classical RDGM, GMCT and GMCG models. In all cases, the first six samples are used for building the models, and the last four samples are used for testing:

  • Case 1: the X 2 ( 0 ) is viewed as input sequence, the results of RMSPE are displayed in Figure 2.

  • Case 2: the X 3 ( 0 ) is viewed as input sequence, the results of RMSPE are displayed in Figure 3.

  • Case 3: the X 2 ( 0 ) and X 3 ( 0 ) are viewed as input sequences, the results of RMSPE are displayed in Figure 4.

We observe from Figures 2–4 that the three-dimensional graphics of three cases have the similar tendency. As we know, the three cases are independent to each other, and all parameters b2, b3, …, bn, u and initial points x 1 ( 0 ) ( 1 ) are randomly generated. This implies that the input sequences, the values of parameters b2, b3, …, bn, u and x 1 ( 0 ) ( 1 ) have no influence on the output sequence. Here, the values of b1 and α are the most important factors which affect the accuracy of grey models.

Further, the maximum RMSPEs of the GM(α, n) model of Cases 1–3 are 1.8389×10−12, 3.0009×10−11 and 2.9221×10−12 per cent, respectively. The values of the RDGM model of Cases 1–3 are 393.0707, 2,216.5403 and 995 per cent, respectively, while the maximum RMSPEs of GMCT model, and GMCG model are nearly 4×10150 and 5×10147 per cent, respectively. It is obviously seen that the error of the GM(α, n) model is a truncation error caused by computer. This indicates that these data are not suitable for simulating and predicting by classical grey models, and the fractional grey model has more freedom and better modelling.

5. Applications

In this section, the GM(α, n) model is applied to predict the energy consumption of China by taking the gross domestic product (GDP) and the urbanization as exogenous variables. Computational results of the GM(α, n) model are compared to the RDGM(1, n), the GMCT(1, n) and the GMCG(1, n) model.

5.1 Background and data collection

As the largest developing country in the world, China is still the world’s largest energy consumer. In 2016, China’s energy consumption totalled 4.36bn tons of standard coal occupying 23 per cent of global energy consumption, an increase of 1.4 per cent over the previous year. China has been the fastest growing energy market for 16 consecutive years according to the economist. The accurate prediction of China’s energy consumption may not only guide the reasonable development of energy in China, but also provide a scientific decision basis for formulating energy security strategy. From Lee (2005), Neto and Fiorelli (2008), Wang (2014) and Shen et al. (2005), the energy consumption is related to the GDP, urbanization, economic growth and others. The higher energy consumption appears with the rapid growth of GDP and the higher level of the urbanization. Therefore, this study takes the GDP and the urbanization as the input sequences, and the energy consumption as the output sequence.

The raw data of the energy consumption, the GDP and the urbanization of China were collected from the website of the National Bureau of Statistics of China (www.stats.gov.cn), which are shown in Table II. The total energy consumption, the GDP and the urbanization are measured in 10,000 tons of standard coal, billion and 10,000 people, respectively. The first five samples are used to build models, and the left six samples are used to test.

5.2 Modelling for China’s energy consumption

Employing the grey system theory, and based on the actual values of the energy consumption, the GDP, and the urbanization of China from 2006 to 2016, four grey models are built. We first derive the optimal value α=0.5860 with the minimization problem given in Equation (21) and the GA. Further, the grey coefficients of four grey models are displayed in Table III.

Then four kinds of grey models can be expressed as:

  • GM(α, 3) model:

    d 0.5860 x 1 ( 1 ) ( t ) d t 0.5860 1.3062 x 1 ( 1 ) ( t ) = 0.3350 x 2 ( 1 ) ( t ) 3.4881 x 3 ( 1 ) ( t ) + 331 , 231.0388.

  • RDGM(1, 3) model:

    x 1 ( 1 ) ( k + 1 ) =1.3124 x 1 ( 1 ) ( k ) + 0.2078 z 2 ( 1 ) ( k + 1 ) 2.4284 z 3 ( 1 ) ( k + 1 ) + 363 , 453.3191.

  • GMCT(1, 3) or GMCG(1, 3) model:

d x 1 ( 1 ) ( t ) d t + 0.2702 x 1 ( 1 ) ( t ) =0.1797 x 2 ( 1 ) ( t ) 2.1004 x 3 ( 1 ) ( t ) + 314 , 351.6072.

Then the predicted values X 1 ( 0 ) are listed in Table IV and Figure 5, and the error values obtained which are listed in Table V and Figure 6.

5.3 Discussions and suggestions

The raw data of China’s energy consumption from 2006 to 2010 are used to build the grey models, and the data from 2011 to 2016 are used for validation according to the established grey models. It can be seen from Table V that the RDGM(1, 3) performs best in simulation while the GMCT(1, 3) performs worst, the GM(0.5860, 3) performs best in prediction while GMCT(1, 3) performs worst. Table IV and Figure 5 also indicate that the predicted values by GM(0.5860, 3) are much close to the raw data than those by RDGM(1, 3), GMCT(1, 3) and GMCG(1, 3) models. From results, the GM(α, 3) outperforms the other three models which imply it is efficient to deal with uncertain problems with small samples.

We observe from Table V and Figure 6 that the minimum absolute percentage predictive error (APPE) of GM(0.5860, 3) is as small as 0.1475 per cent, and the maximum value is only 2.6403 per cent. The minimum APPE of RDGM(1, 3), GMCT(1, 3) and GMCG(1, 3) is as high as 3.8263, 6.2016 and 1.8505 per cent, respectively, which are even much larger than the minimum APPE of GM(0.5860, 3). The maximum APPE of RDGM(1, 3), GMCT(1, 3) and GMCG(1, 3) is 142.6454, 147.6399 and 134.5819 per cent, respectively. Further, the RMSPEPO and RMSPE of GM(0.5860, 3) are as low as 1.5997 and 1.2928 per cent, respectively. The RMSPEPO and RMSPE of RDGM(1, 3) are as high as 75.2823 and 55.5997 per cent, those of GMCT(1, 3) are 78.5613 and 58.0337 per cent, and those of GMCG(1, 3) are 70.5763 and 52.1296 per cent, respectively. The results show that the GM(0.5860, n) outperforms the RDGM(1, 3), the GMCT(1, 3) and the GMCG(1, 3), and the GMCT(1, 3) has the worst performance. The results indicate that the fractional GM(0.5860, 3) achieve better results due to its curve is nearly the same with the practical data. This is due to the reason that the fractional derivatives could have more freedom and better modelling.

The prediction results will give a guidance for China Government to formulate and adjust the energy policy. With the rapidly development of China’s economy and the level of urbanization, the need for energy will be increasing continuously. As we know, the fossil energy can produce harmful gases which pollute environment and lead to ecological problems. So, the China Government should reduce the traditional energy consumption, and greatly increase the clean energy consumption in the future.

6. Conclusions

In this paper, we studied the fractional GM(α, n) based on the grey modelling technique, the forward difference method and the GA. The simulative and predictive values are calculated by transforming fractional differential equation to fractional difference equation. The newly model is applied to predict the energy consumption of China. The results show that the GM(α, n) model has high potential in the Chins’s total energy consumption with higher accuracy than the RDGM(1, n), the GMCT(1, n) and the GMCG(1, n) models. Besides, the GM(α, n) has some limitations. The analytical results of the time response function and the restored values are not obtained. Further, we only take the GDP and the urbanization as the input sequences, not consider other factors such as China’s energy strategies, energy structure and energy policies which may affect the status of the energy consumption of China.

In the future, we will discuss the multivariate grey models where the first-order differential equation is transformed into fractional differential equation, and the first AGO is transformed into fractional accumulated generating operation.

Figures

The diagram of the data for experiment

Figure 1

The diagram of the data for experiment

The values of the RMSPE for Case 1

Figure 2

The values of the RMSPE for Case 1

The values of the RMSPE for Case 2

Figure 3

The values of the RMSPE for Case 2

The values of the RMSPE for Case 3

Figure 4

The values of the RMSPE for Case 3

Computational results of energy consumption obtained from four grey models

Figure 5

Computational results of energy consumption obtained from four grey models

Error values among four grey models for China’s energy consumption

Figure 6

Error values among four grey models for China’s energy consumption

Two independent input sequences used for validation

k 1 2 3 4 5
x 2 ( 0 ) ( k ) 0.4072 2.5428 8.1428 2.4352 9.2926
x 3 ( 0 ) ( k ) 0.3793 0.2697 2.6539 3.8958 4.6701
k 6 7 8 9 10
x 2 ( 0 ) ( k ) 3.4998 1.9659 2.5108 6.1604 4.7328
x 3 ( 0 ) ( k ) 0.6495 2.8441 2.3469 0.0595 1.6856

Raw data of energy consumption, GDP and urbanization

Year Energy consumption ( X 1 ( 0 ) ) GDP ( X 2 ( 0 ) ) Urbanization ( X 3 ( 0 ) )
2006 286,467 219,438.5 58,288
2007 311,442 270,232.3 60,633
2008 320,611 319,515.5 62,403
2009 336,126 349,081.4 64,512
2010 360,648 413,030.3 66,978
2011 387,043 489,300.6 69,079
2012 402,138 540,367.4 71,182
2013 416,913 595,244.4 73,111
2014 425,806 643,947.0 74,916
2015 429,905 689,052.1 77,116
2016 436,000 744,127.2 79,298

Grey coefficients of four grey models (α = 0.5860)

Grey coefficients
Models b1 b2 b3 u
GM (α, 3) −1.3062 −0.3350 −3.4881 331,231.0388
RDGM (1, 3) 1.3124 0.2078 −2.4284 363,453.3191
GMCT(1, 3) 0.2702 0.1797 −2.1004 314,351.6072
GMCG(1, 3) 0.2702 0.1797 −2.1004 314,351.6072

Prediction results of China’s energy consumption by four grey models

Year GM(0.5860, 3) RDGM(1, 3) GMCT(1, 3) GMCG(1, 3)
2006 286,467 286,467 286,467 286,467
2007 310,002.0831 311,442 316,718.9845 308,600.2025
2008 319,326.2888 320,611 326,530.7431 317,089.2838
2009 333,520.3891 336,126 343,194.6619 331,658.6728
2010 355,479.4682 360,648 368,763.2095 354,866.3367
2011 376,823.8408 401,852.6059 411,045.8909 394,205.2913
2012 393,651.3214 464,053.9113 474,913.8470 453,791.2093
2013 409,853.0407 551,797.5884 564,618.4311 537,911.5361
2014 423,308.7778 673,182.1028 688,440.5393 654,295.1883
2015 433,492.1515 837,369.7885 855,668.0906 811,617.3068
2016 436,642.9458 1,057,933.8205 1,079,710.1197 1,022,777.2963

Relative error values of energy consumption by four grey models

Year GM(0.5860, 3) RDGM(1, 3) GMCT(1, 3) GMCT(1, 3)
2006 0 0 0 0
2007 0.4623 7.4759E−14 1.6944 0.9125
2008 0.4007 0 1.8464 1.0984
2009 0.7752 0 2.1030 1.3291
2010 1.4331 6.4559E−14 2.2502 1.6031
2011 2.6403 3.8263 6.2016 1.8505
2012 2.1104 15.3967 18.0972 12.8446
2013 1.6934 32.3532 35.4284 29.0225
2014 0.5865 58.0960 61.6794 53.6604
2015 0.8344 94.7802 99.0366 88.7899
2016 0.1475 142.6454 147.6399 134.5819
RMSPEPR 0.7783 4.4174E−14 1.7757 1.1292
RMSPEPO 1.5997 75.2823 78.5613 70.5763
RMSPE 1.2928 55.5997 58.0337 52.1296

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos 71771033, 71571157, 11601357), the Longshan academic talent research supporting programme of SWUST (No. 17LZXY20), the Open Fund (PLN 201710) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), applied basic research programme of science and technology commission foundation of Sichuan province (2017JY0159), and the funding of V.C. & V.R. Key Lab of Sichuan Province Grant Numbers (SCVCVR2018.08VS, SCVCVR2018.10VS).

Corresponding author

Xin Ma can be contacted at: cauchy7203@gmail.com