# Forecasting China’s per Capita Living Energy Consumption by Employing a Novel DGM (1, 1,tα) Model with Fractional Order Accumulation

Authors: Shuanghua Liu; Chong Liu; Haodan Pang; Ting Feng; Zijie Dong
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

The living energy consumption of residents has become an important technical index to promote the economic and social development strategy. The country’s medium- and short-term living energy consumption is featured with both a certainty of annual increment and an uncertainty of random variation. Thus, it can be seen as a typical grey system and shall be suitable for the grey prediction model. In order to explore the future development trend of China’s per capita living energy consumption, this paper establishes a novel grey model based on the discrete grey model with time power term and the fractional accumulation (FDGM (1, 1,tα) for short) for forecasting China’s per capita living energy consumption, which makes the existing model to adapt to different time series by adjusting fractional order accumulation parameter and power term. In order to verify the feasibility and effectiveness of the novel model, the proposed and eight other existing grey prediction models are applied to the case of China’s per capita living energy consumption. The results show that the proposed model is more suitable for predicting China’s per capita energy consumption than the other eight grey prediction models. Finally, the proposed model based on metabolism mechanism is used to predict China’s per capita living energy consumption from 2018 to 2029, which can provide a reference for energy companies or government decision makers.


---

## Body

## 1. Introduction

With the advancement of urbanization, economic growth, and improvement of the living standards of residents, the demand for energy in China has been greatly increased, in which the energy consumption of residents shows the characteristics of rapid growth, accounting for a large proportion of China’s total energy consumption. Whether the future energy supply can support the sustainable growth of China’s economy has become a topic of concern at home and abroad. Therefore, it is of great practical significance to accurately predict the per capita living energy consumption in the future for maintaining the healthy, sustainable, and stable development of China’s social economy. At present, there are few studies on per capita living energy consumption [1, 2]. With the development of science and technology, various prediction methods emerge in endlessly. Among them, grey prediction models and regression prediction models are two commonly used prediction methods. The main difference between them is the different samples required for modeling. The former requires only a small amount of sample data, while the latter is based on large sample modeling. As we all know, some data in China has been lost due to various reasons. In this small sample case, the regression prediction model is obviously no longer applicable.The grey prediction models play a vital role in the grey system theory, which were originally proposed by Deng [3]. The grey prediction models preprocess the original time series by the operation of accumulation generation to make it become a series with obvious exponential law and then construct differential equation and difference equation to establish the model, so as to realize the simulation and prediction of the original time series. Because of the widespread existence of poor information and uncertain system, grey system theory has a very broad application and development prospect. At present, grey prediction models have been applied in various fields of society due to its excellent forecasting performance [4–10]. Because this type of forecasting model is developed based on the grey system theory, it is usually called the grey forecasting model (GM). The basic GM (1, 1) model is the most popular and important grey model. In order to further improve the prediction accuracy of the GM (1, 1) model, a number of scholars have improved the model from the cumulative order [11, 12], background values [13], discrete grey prediction model [14, 15], and time response function [16]. With the widespread application of the GM (1, 1) model, some scholars have found that the GM (1, 1) model cannot be applied to all time series. In fact, since the GM (1, 1) model is a model based on homogeneous exponential function, it is difficult to fit time series with approximately nonhomogeneous exponential law. In order to solve this problem, Cui et al. proposed the NGM (1, 1, k) model, which opens a new door for improving the GM (1, 1) model [17]. Qian et al. proposed the GM (1, 1, tα) model based on the research foundation of Cui et al. [18]. The GM (1, 1, tα) model can adapt to different time series by replacing the time power term α, so as to achieve the purpose of accurate prediction. It is a grey prediction model with strong adaptability. These improved methods have greatly expanded the scope of application of grey prediction models and enriched grey system theory.The fractional accumulation operation was originally proposed by Professor Wu, and it is a measure that can effectively improve the accuracy of the grey prediction model [11]. It can improve the adaptability of the prediction model by expanding the search range of the cumulative order of the grey prediction model to achieve the purpose of accurate prediction. At present, the grey prediction model with fractional accumulation has been widely used in various fields of society [19–21]. In order to further expand the application of fractional accumulation operations in grey prediction models, many scholars are committed to combining the new fractal theory with grey prediction models. For example, Zhu et al. proposed a new fractional grey prediction model-adaptive grey score weighting model and used this model to predict the electricity consumption of Jiangsu Province in China [22], Chen et al. proposed the Fractional Hausdorff grey model [23], and Ma et al. established a conformable fractional grey system model [12]. These studies have greatly expanded the grey system theory and further promoted the combination of analysis theory. Using an operator is an effective way to improve the prediction accuracy of grey prediction models.Admittedly, both the discrete grey model with time power term and fractional accumulation can improve the prediction performance of the grey models to some extent; however, few studies combine these two methods simultaneously. Based on the previous knowledge, the novel model considering the discrete grey model with time power term and fractional accumulation can not only improve the adaptability, but also fill knowledge gap of grey forecasting theory. After that, this mixed model is applied to predict China’s per capita living energy consumption, which provides a solid basis for policy makers to formulate the reasonable plans.In general, the primary contributions of this paper can be summarized as follows:(1)
A novel discrete GM (1, 1, tα) model with fractional order accumulation is proposed(2)
The final optimal nonlinear parameters are determined by the whale algorithm(3)
The novel grey prediction model based on the metabolic thought is used to predict China’s per capita living energy consumption from 2018 to 2029The rest of paper is organized as follows.Section 2 introduces the prerequisite knowledge of this article, including the fractional accumulation operation, the basic GM (1, 1, tα) model, and the discrete GM (1, 1, tα) model with fractional accumulation proposed in this article. Section 3 introduces the method of solving the FDGM (1, 1, tα) model by using the whale algorithm. The application of the FDGM (1, 1, tα) to predict China’s per capita living energy consumption is presented in Section 4, including the comparison with the other eight grey prediction models, and the conclusions are drawn in Section 5.


## 2. Methods

### 2.1. Discrete GM (1, 1,tα) Model

As an extensive version of the traditional discrete grey model (DGM (1, 1) for short), the discrete grey model with time power term (referred as GM (1, 1,tα)) further expands the applicable scope by virtue of adjustable time-power coefficient. The detailed steps can be seen as follows.Assume a nonnegative sequence to beX0=x01,x02,…,x0n, and the first-order accumulated generating operator sequence of X0 is given as X1=x11,x12,…,x1n, where x1k=∑i=1kx0i,k=1,2,…,n.On this basis, the differential equation of the GM (1, 1,tα) model is constructed as(1)dx1tdt+ax1t=btα+c,where a,b,cT are the parameters of the GM (1,1, tα) model. It can be inferred that the discrete formula of equation (1) is calculated as(2)x0k+az1k=bkα+1−k−1α+1α+1+c,(3)z1k=0.5⋅x1k+x1k−1.By the least square method, the model parameters can be obtained as follows:(4)a,b,cT=BTB−1BTY.where(5)B=−z1221+α−11+α1−z1331+α−21+α1+α1⋮⋮⋮−z1nn1+α−n−11+α1+α1,Y=x02x03⋮x0n.According to equation (2), it can be seen that the GM (1, 1, tα) model has no specific solution formula (in general, the prediction formula of the grey prediction model is the analytical formula of the differential equation of the model). The introduction of the power term not only makes the model more adaptable but also makes the model difficult to solve. In fact, when the literature proposed the GM (1, 1, tα) model, in order to facilitate the solution, the power term α of the GM (1, 1, tα) model was set to 2, which not only restricted the prediction accuracy of the GM (1, 1, tα) model but also violated the original intention of improving the adaptability of the model. Discretization of the grey prediction model can avoid the steps of solving differential equations. In order to give full play to the performance of the GM (1, 1, tα) model, this article will establish a discrete GM (1, 1, tα) model with fractional accumulation.


### 2.2. Discrete GM (1, 1,tα) Model with Fractional Order Accumulation

As previously suggested, the fractional accumulated generating operator is crucial for processing original sequences that is affected by nonlinearity and uncertainty, thus generating satisfactory results in many applications. Therefore, we introduce the fractional order accumulation into the GM (1, 1,tα) model to improve the performance of the existing grey models.Suppose that a nonnegative sequence to beX0=x01,x02,…,x0n, and the r-order accumulated generating operator sequence of X0 is given as(6)Xr=xr1,xr2,…,xrn,where xrk=∑i=1kk−i+r−1k−ix0i,k=1,2,…,n.Then, the whitening differential equation of the GM (1, 1,tα) model with fractional accumulation (FDGM (1, 1, tα)) can be expressed as(7)dxrtdt|t=k≈limΔt⟶1xrk−xrk−ΔtΔk=xrk−xrk−1.Then, the left side of the whitening differential equation of the FDGM (1, 1,tα) model with t=k can be written as(8)xrk−xrk−1+axrk=1+axrk−xrk−1.Furthermore, we obtain(9)1+axrk−xrk−1=bkα+c,which can also be written as(10)xrk=xrk−11+a+bkα1+a+c1+a.Denotingη1=1/1+a,η2=b/1+a,andη3=c/1+a, we have(11)xrk=η1xrk−1+η2kα+η3.Equation (11) is the expression of the FDGM (1, 1, tα) model [24] proposed in this paper. The parameters in equation (10) can also be solved using the least square method, namely,(12)γ^=a^,b^,c^T=BTB−1BTY,where(13)B=xr12α1xr23α1⋮⋮⋮xrn−1nα1,Y=xr2xr3⋮xrn.The relationship between the FDGM (1, 1,tα) and the other commonly used grey prediction models is obvious. When r=1andα=0, the FDGM (1, 1, tα) yields the basic DGM (1, 1) model [14]. When r=1andα=1, the FDGM (1, 1, tα) yields the NDGM (1, 1) model [25]. When r∈0,1andα=0, the FDGM (1, 1, tα) yields the FDGM (1, 1) model [26]. When r∈0,1andα=1, the FDGM (1,1, tα) yields the FNDGM (1, 1) model [27]. Thus, it can be seen that the FDGM (1, 1, tα) model is a highly adaptive grey prediction model.The recursive formula of the FDGM (1, 1,tα) model is(14)xrk=η1k−1xr1+∑i=0k−2η1iη2k−1−iα+η3.According toSection 2.1, the prediction formula of the FDGM (1, 1, tα) model can be obtained as(15)X⌢0=x⌢01,x⌢02,…,x⌢0n=X⌢rA−r=x⌢r1,x⌢r2,…,x⌢rnA−r.


## 2.1. Discrete GM (1, 1,tα) Model

As an extensive version of the traditional discrete grey model (DGM (1, 1) for short), the discrete grey model with time power term (referred as GM (1, 1,tα)) further expands the applicable scope by virtue of adjustable time-power coefficient. The detailed steps can be seen as follows.Assume a nonnegative sequence to beX0=x01,x02,…,x0n, and the first-order accumulated generating operator sequence of X0 is given as X1=x11,x12,…,x1n, where x1k=∑i=1kx0i,k=1,2,…,n.On this basis, the differential equation of the GM (1, 1,tα) model is constructed as(1)dx1tdt+ax1t=btα+c,where a,b,cT are the parameters of the GM (1,1, tα) model. It can be inferred that the discrete formula of equation (1) is calculated as(2)x0k+az1k=bkα+1−k−1α+1α+1+c,(3)z1k=0.5⋅x1k+x1k−1.By the least square method, the model parameters can be obtained as follows:(4)a,b,cT=BTB−1BTY.where(5)B=−z1221+α−11+α1−z1331+α−21+α1+α1⋮⋮⋮−z1nn1+α−n−11+α1+α1,Y=x02x03⋮x0n.According to equation (2), it can be seen that the GM (1, 1, tα) model has no specific solution formula (in general, the prediction formula of the grey prediction model is the analytical formula of the differential equation of the model). The introduction of the power term not only makes the model more adaptable but also makes the model difficult to solve. In fact, when the literature proposed the GM (1, 1, tα) model, in order to facilitate the solution, the power term α of the GM (1, 1, tα) model was set to 2, which not only restricted the prediction accuracy of the GM (1, 1, tα) model but also violated the original intention of improving the adaptability of the model. Discretization of the grey prediction model can avoid the steps of solving differential equations. In order to give full play to the performance of the GM (1, 1, tα) model, this article will establish a discrete GM (1, 1, tα) model with fractional accumulation.


## 2.2. Discrete GM (1, 1,tα) Model with Fractional Order Accumulation

As previously suggested, the fractional accumulated generating operator is crucial for processing original sequences that is affected by nonlinearity and uncertainty, thus generating satisfactory results in many applications. Therefore, we introduce the fractional order accumulation into the GM (1, 1,tα) model to improve the performance of the existing grey models.Suppose that a nonnegative sequence to beX0=x01,x02,…,x0n, and the r-order accumulated generating operator sequence of X0 is given as(6)Xr=xr1,xr2,…,xrn,where xrk=∑i=1kk−i+r−1k−ix0i,k=1,2,…,n.Then, the whitening differential equation of the GM (1, 1,tα) model with fractional accumulation (FDGM (1, 1, tα)) can be expressed as(7)dxrtdt|t=k≈limΔt⟶1xrk−xrk−ΔtΔk=xrk−xrk−1.Then, the left side of the whitening differential equation of the FDGM (1, 1,tα) model with t=k can be written as(8)xrk−xrk−1+axrk=1+axrk−xrk−1.Furthermore, we obtain(9)1+axrk−xrk−1=bkα+c,which can also be written as(10)xrk=xrk−11+a+bkα1+a+c1+a.Denotingη1=1/1+a,η2=b/1+a,andη3=c/1+a, we have(11)xrk=η1xrk−1+η2kα+η3.Equation (11) is the expression of the FDGM (1, 1, tα) model [24] proposed in this paper. The parameters in equation (10) can also be solved using the least square method, namely,(12)γ^=a^,b^,c^T=BTB−1BTY,where(13)B=xr12α1xr23α1⋮⋮⋮xrn−1nα1,Y=xr2xr3⋮xrn.The relationship between the FDGM (1, 1,tα) and the other commonly used grey prediction models is obvious. When r=1andα=0, the FDGM (1, 1, tα) yields the basic DGM (1, 1) model [14]. When r=1andα=1, the FDGM (1, 1, tα) yields the NDGM (1, 1) model [25]. When r∈0,1andα=0, the FDGM (1, 1, tα) yields the FDGM (1, 1) model [26]. When r∈0,1andα=1, the FDGM (1,1, tα) yields the FNDGM (1, 1) model [27]. Thus, it can be seen that the FDGM (1, 1, tα) model is a highly adaptive grey prediction model.The recursive formula of the FDGM (1, 1,tα) model is(14)xrk=η1k−1xr1+∑i=0k−2η1iη2k−1−iα+η3.According toSection 2.1, the prediction formula of the FDGM (1, 1, tα) model can be obtained as(15)X⌢0=x⌢01,x⌢02,…,x⌢0n=X⌢rA−r=x⌢r1,x⌢r2,…,x⌢rnA−r.


## 3. Determination of Parameters for FDGM (1, 1,tα)

Since the solving method of the FDGM (1, 1,tα) model is too complicated, this section is divided into three parts to explain the solution of the FDGM (1, 1, tα) model.


### 3.1. Method for Determining Parametersα and r of the FDGM (1, 1, tα) Model

It can be seen that the FDGM (1, 1,tα) model is established based on the given conditions of parameters α and r. Therefore, how to determine the optimal values of parameters α and r is also a problem. The best parameter values should enable the model proposed in this paper to have the highest accuracy under the given sample. Therefore, we should establish a programming problem that aims to minimize the errors of the proposed model by replacing the values of parameters α and r and follows the modeling steps of the proposed model. This article chooses the mean absolute percentage error (MAPE) as the standard for evaluating model errors, and then, this planning problem can be written as(16)minfα,r=1n−1∑t=1nx0t−x⌢0tx⌢0t×100%,s.t.r∈0,1,γ^=a^,b^,c^T=BTB−1BTY,B=xr12α1xr23α1⋮⋮⋮xrn−1nα1,Y=xr2xr3⋮xrn,x⌢rk=η1k−1xr1+∑i=0k−2η1iη2k−1−iα+η3,X⌢0=x⌢01,x⌢02,…,x⌢0n=X⌢rA−r=x⌢r1,x⌢r2,…,x⌢rnA−r.It can be seen that the abovementioned planning problem is very complicated, and conventional methods cannot be used to solve this planning problem. Therefore, this article will introduce how to use the whale algorithm to solve this planning problem; see Section3.2, for details.


### 3.2. The Whale Optimization Algorithm

In the past, people often use genetic algorithm (GA) or particle swarm optimization algorithm (PSO) to solve grey forecasting models, but this method has certain defects [28, 29]. GA and PSO are prone to the phenomenon of too many iterations, slow convergence, and falling into local extreme value, but whale algorithm (WOA) does not have such a situation. Therefore, this paper chooses to use WOA to solve the model proposed in this paper. Inspired by the social behavior of humpback whale groups, Mirijalili and Lewis proposed the whale optimization algorithm (WOA) in 2016 [30]. At present, WOA has been widely used in bioinformatics [31], image processing [32], and other fields due to its excellent performance. At the same time, WOA is also used to solve nonlinear programming problems which are more complex than problem 14 [33]. Therefore, this paper chooses WOA to solve the nonlinear programming problem 14. The main idea of WOA is as follows.When whales prey, they move in a spiral to surround the school of fish currently considered the best target. Then, these whales update their positions based on the candidate target. This behavior can be expressed by a mathematical formula, namely,(17)P⟶i+1=P∗⟶i−2fi⋅r⟶−fi⋅D⟶,ifξ<0.5a,P∗⟶i−P⟶i⋅eβl⋅cos2πl+P∗⟶i,ifξ≥0.5b,where P⟶i represents the current position of the whales, P∗⟶i represents the current best position of the whales, r⟶ is a random number in the interval 0,1, l is a stochastic number in the interval −1,1, β is an arbitrary constant which determines the shape of the spiral movement, T is the maximum number of iterations of the algorithm, and ξ is a probability to choose a movement strategy from encircling and spiral moving behaviors. When the norm of D⟶ is greater than 1, the position of all whales is updated based on the position of a whale randomly selected. This model can also be expressed by mathematical formulas, namely,(18)P⟶i+1=P⟶ri−C⟶⋅2P⟶ri⋅r⟶−P⟶i,where P⟶ri is the position of a randomly selected whale in the herd.Since WOA is designed to solve unconstrained programming problems, we cannot directly use it to solve the model in this paper. Therefore, we need to establish a fitness function to calculate the fitness of each whale agent. According to the nonlinear programming problem described inSection 4.1, the fitness function can be described as(19)fitness=1n−1∑t=1nx10t−x⌢10tx10t×100%.The revised WOA is presented in detail in Algorithm1.Algorithm 1: Algorithm of WOA to search for the nonlinear parameters α and r of the FDGM (1, 1, tα) model.
Input: the raw data X0 and lower and upper bound of α and rOutput: the optimal value of the nonlinear parameters α and r(1)
Initialize the maximum number of iterationsT and the number of humpback whales(2)
Initialize the locationsP⟶ of the humpback population(3)
Compute the fitness of each humpback by equation (19)(4)
Determine the best candidateP⟶ based on fitness of each whale agent(5)
fork=1;k<T;k=k+1do(6)
foreach humpback whaledo(7)
Update the parametersr,p,l,β;(8)
ifζ≤0.5then(9)
ifC⟶≤0.5then(10)
Update the location of each humpback by equation (17);(11)
else(12)
DeterminePr⟶ by randomly choosing a whale;(13)
Update the location of each humpback by equation (18);(14)
end(15)
else(16)
Update the location of each humpback by equation (17);(17)
end(18)
Compute the fitness of each humpback by equation (19);(19)
end(20)
UpdateP∗⟶ if a better solution exists;(21)
end(22)
return the optimum valueP∗⟶;


### 3.3. The Computational Steps

This section will introduce the specific steps to solve the FDGM (1, 1,tα) model, as shown below:Step 1: calculate the parametersα and r of the model according to the method described in Section 3.1Step 2: bring the parametersα and r obtained according to Step 1 into the FDGM (1, 1, tα) model, and then, calculate the parameters a, b, and c of the FDGM (1, 1, tα) model according to equation (12)Step 3: put the parameters obtained in Step 2 into equations (14) and (15) to get the predicted results of the model


## 3.1. Method for Determining Parametersα and r of the FDGM (1, 1, tα) Model

It can be seen that the FDGM (1, 1,tα) model is established based on the given conditions of parameters α and r. Therefore, how to determine the optimal values of parameters α and r is also a problem. The best parameter values should enable the model proposed in this paper to have the highest accuracy under the given sample. Therefore, we should establish a programming problem that aims to minimize the errors of the proposed model by replacing the values of parameters α and r and follows the modeling steps of the proposed model. This article chooses the mean absolute percentage error (MAPE) as the standard for evaluating model errors, and then, this planning problem can be written as(16)minfα,r=1n−1∑t=1nx0t−x⌢0tx⌢0t×100%,s.t.r∈0,1,γ^=a^,b^,c^T=BTB−1BTY,B=xr12α1xr23α1⋮⋮⋮xrn−1nα1,Y=xr2xr3⋮xrn,x⌢rk=η1k−1xr1+∑i=0k−2η1iη2k−1−iα+η3,X⌢0=x⌢01,x⌢02,…,x⌢0n=X⌢rA−r=x⌢r1,x⌢r2,…,x⌢rnA−r.It can be seen that the abovementioned planning problem is very complicated, and conventional methods cannot be used to solve this planning problem. Therefore, this article will introduce how to use the whale algorithm to solve this planning problem; see Section3.2, for details.


## 3.2. The Whale Optimization Algorithm

In the past, people often use genetic algorithm (GA) or particle swarm optimization algorithm (PSO) to solve grey forecasting models, but this method has certain defects [28, 29]. GA and PSO are prone to the phenomenon of too many iterations, slow convergence, and falling into local extreme value, but whale algorithm (WOA) does not have such a situation. Therefore, this paper chooses to use WOA to solve the model proposed in this paper. Inspired by the social behavior of humpback whale groups, Mirijalili and Lewis proposed the whale optimization algorithm (WOA) in 2016 [30]. At present, WOA has been widely used in bioinformatics [31], image processing [32], and other fields due to its excellent performance. At the same time, WOA is also used to solve nonlinear programming problems which are more complex than problem 14 [33]. Therefore, this paper chooses WOA to solve the nonlinear programming problem 14. The main idea of WOA is as follows.When whales prey, they move in a spiral to surround the school of fish currently considered the best target. Then, these whales update their positions based on the candidate target. This behavior can be expressed by a mathematical formula, namely,(17)P⟶i+1=P∗⟶i−2fi⋅r⟶−fi⋅D⟶,ifξ<0.5a,P∗⟶i−P⟶i⋅eβl⋅cos2πl+P∗⟶i,ifξ≥0.5b,where P⟶i represents the current position of the whales, P∗⟶i represents the current best position of the whales, r⟶ is a random number in the interval 0,1, l is a stochastic number in the interval −1,1, β is an arbitrary constant which determines the shape of the spiral movement, T is the maximum number of iterations of the algorithm, and ξ is a probability to choose a movement strategy from encircling and spiral moving behaviors. When the norm of D⟶ is greater than 1, the position of all whales is updated based on the position of a whale randomly selected. This model can also be expressed by mathematical formulas, namely,(18)P⟶i+1=P⟶ri−C⟶⋅2P⟶ri⋅r⟶−P⟶i,where P⟶ri is the position of a randomly selected whale in the herd.Since WOA is designed to solve unconstrained programming problems, we cannot directly use it to solve the model in this paper. Therefore, we need to establish a fitness function to calculate the fitness of each whale agent. According to the nonlinear programming problem described inSection 4.1, the fitness function can be described as(19)fitness=1n−1∑t=1nx10t−x⌢10tx10t×100%.The revised WOA is presented in detail in Algorithm1.Algorithm 1: Algorithm of WOA to search for the nonlinear parameters α and r of the FDGM (1, 1, tα) model.
Input: the raw data X0 and lower and upper bound of α and rOutput: the optimal value of the nonlinear parameters α and r(1)
Initialize the maximum number of iterationsT and the number of humpback whales(2)
Initialize the locationsP⟶ of the humpback population(3)
Compute the fitness of each humpback by equation (19)(4)
Determine the best candidateP⟶ based on fitness of each whale agent(5)
fork=1;k<T;k=k+1do(6)
foreach humpback whaledo(7)
Update the parametersr,p,l,β;(8)
ifζ≤0.5then(9)
ifC⟶≤0.5then(10)
Update the location of each humpback by equation (17);(11)
else(12)
DeterminePr⟶ by randomly choosing a whale;(13)
Update the location of each humpback by equation (18);(14)
end(15)
else(16)
Update the location of each humpback by equation (17);(17)
end(18)
Compute the fitness of each humpback by equation (19);(19)
end(20)
UpdateP∗⟶ if a better solution exists;(21)
end(22)
return the optimum valueP∗⟶;


## 3.3. The Computational Steps

This section will introduce the specific steps to solve the FDGM (1, 1,tα) model, as shown below:Step 1: calculate the parametersα and r of the model according to the method described in Section 3.1Step 2: bring the parametersα and r obtained according to Step 1 into the FDGM (1, 1, tα) model, and then, calculate the parameters a, b, and c of the FDGM (1, 1, tα) model according to equation (12)Step 3: put the parameters obtained in Step 2 into equations (14) and (15) to get the predicted results of the model


## 4. Application

### 4.1. Raw Data Collection

The raw data of the per capita living energy consumption of China are collected from the official website National Bureau of Statistics of China (http://www.stats.gov.cn/english/), as shown in Table 1. The points from 2002 to 2011 are used for building the prediction models, and the last 6 points are used for testing the prediction accuracy of the models.Table 1
The raw data of the per capita living energy consumption of China (kg ce).
YearDataYearDataYearDataYearData2002146200623020102742014346.12003166200725020112942015365.42004191200825420123132016393.22005211200926420133352017415.6


### 4.2. Evaluating Indicator

In this section, we describe several metrics that are commonly used to evaluate the performance of a prediction model, as shown in Table2.Table 2
Metrics for evaluating effectiveness of the models.
IndicatorFormulaThe absolute percentage error (APE)APEk=x0k−x⌢0k/x0k×100%The mean absolute percentage error (MAPE)MAPE=1/n∑k=1nx0k−x⌢0k/x0k×100%The mean absolute error (MAE)MAE=1/n∑k=1nx0k−x⌢0kThe mean squares error (MSE)MSE=1/n∑k=1nx0k−x⌢0k2The root mean squares percentage error (RMSPE)RMSPE=1/n∑k=1nx0k−x⌢0k/x0k2×100%The index of agreement (IA)IA=1−∑k=1nx0k−x⌢0k2/∑k=1nx0k−x¯+x⌢0k−x¯2The correlation coefficient (R)R=Covx0k,x⌢0k/Varx⌢0kVarx0k


### 4.3. Numerical Results

The numerical results of the FDGM (1, 1,tα) model are compared to the commonly used prediction models, including the GM (1, 1) model, DGM (1, 1) model [14], FDGM (1, 1) model [26], NGM (1, 1, k) model [17], NDGM (1, 1, k) model [25], FNDGM (1, 1, k) model [27], ARGM (1, 1) model [15], and DGM (1, 1, tα) model. The results of these models are shown in Table 3. The metrics of the models for fitting and prediction are listed in Table 4 and plotted in Figure 1. According to the data shown in Table 3, we can see that the GM (1, 1), DGM (1, 1), and FDGM (1, 1, tα) models overestimated the actual values, and the other six prediction models underestimated the actual values. According to the prediction indicators shown in Table 4, we can see that the six indicators of the FDGM (1, 1, tα) model are the best of the five prediction models. Therefore, the FDGM (1, 1, tα) model shows the best performance in this case. Calculation formulas of nine prediction models are listed as follows.(1)
GM (1, 1) model:(20)x^1t=2914.9837e0.061064t−1−2768.9837,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(2)
DGM (1, 1) model:(21)x^1t=146⋅1.062862t+174.534876⋅1−1.062862t−0.0628626,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(3)
FDGM (1, 1) model:(22)x^0.602843t=146⋅0.968297t+94.79946⋅1−0.968297t0.031703,t=1,2,…,16.(1)
NDGM (1, 1,k):(23)x^1t=1471.193656⋅0.86313337t+46.625869t0.1368666−1464.5028,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(2)
FNDGM (1, 1,k):(24)x^0.817046t=366.7525⋅0.8003855t+40.211239t0.19961447−348.98792767,t=1,2,…,16.(3)
ARGM (1, 1):(25)x^0t=−253.6⋅0.908097t+399.6,t=1,2,…,16.(4)
NGM (1, 1,k):(26)x^1t=1325.1184e−0.142698t−1+343.6577t−1522.77615,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(5)
DGM (1, 1,tα):(27)x^1t=115.3365198+53.44286⋅t−10.62504+0.98096x^1t−1,t=2,3,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(6)
FDGM (1, 1,tα):(28)x^0.35742t=87.95390053+0.025791068⋅t−13+0.905629116x^1t−1,t=2,3,…,16.x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.Table 3
Numerical results by the nine models.
YearDataGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM(2)FDGM2002146146.00146.00146.00146.00146.00146.00146.00146.00146.002003166183.55183.71166.30167.44166.87167.79169.31166.00168.022004191195.11195.26190.77190.87190.66191.46190.47191.82192.112005211207.39207.54211.21211.19211.19211.09209.69211.94211.842006230220.45220.58228.58228.81228.91227.92227.14228.83228.052007250234.33234.45243.69244.08244.20242.75242.99243.50241.882008254249.09249.19257.04257.32257.41256.07257.39256.50254.382009264264.77264.85268.98268.80268.80268.21270.46268.18266.482010274281.44281.50279.78278.76278.64279.41282.32278.78279.002011294299.16299.20289.61287.39287.13289.82293.10288.45292.712012313318.00318.00298.62294.87294.46299.57302.89297.32308.332013335338.02338.00306.92301.36300.78308.76311.78305.50326.512014346.1359.31359.24314.58306.98306.24317.46319.85313.05347.912015365.4381.93381.83321.69311.86310.95325.73327.18320.05373.122016393.2405.98405.83328.30316.09315.02333.63333.83326.55402.732017415.6431.55431.34334.47319.76318.53341.18339.88332.59437.32Table 4
Performance evaluation indicators of the nine prediction models.
FittingGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM (2)FDGMMAPE3.50603.50801.13601.22501.19701.21901.5631.13101.0720RMSPE4.59604.61001.43301.48101.46701.47501.85401.39401.4010MAE764.1110764.0000296.2220315.1110311.2220306.0000378.7780293.7780257.6670MSE931.7600931.8150378.2150392.9960393.1800378.6850463.7160396.0050347.5530IA0.98570.98570.99780.99760.99760.99780.99670.99790.9981R0.98140.98140.99700.99670.99670.99690.99590.99710.9974PredictionGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM (2)FDGM (1,1,tα)MAPE2.98802.964011.679014.077014.299010.719010.254012.11702.3850RMSPE3.25803.231012.721015.238015.465011.664011.441013.12802.7860MAE1108.1701099.0004395.3305289.6705372.0004032.8303881.5004554.000899.000MSE1224.4701213.7204948.6405914.4606000.4254536.8504481.3605098.6801094.950IA0.97280.97320.58550.51770.51260.62060.61310.57670.9808R0.99610.99610.98690.97890.97830.98940.98450.98680.9968Note. DGM (1) = DGM (1, 1); DGM (2) = DGM ((1, 1,tα).Figure 1
Six performance indicators of the nine prediction models (1 = GM; 2 = DGM; 3 = FDGM; 4 = NGM; 5 = NDGM; 6 = FNDGM; 7 = ARGM; 8 = DGM(2); 9 = FDGM (1, 1,tα)).
(a)(b)(c)(d)(e)(f)Note: DGM (1) = DGM (1, 1); DGM (2) = DGM ((1, 1,tα).


### 4.4. Metabolic Mechanism

As we all know, the prediction accuracy of the prediction model decreases as the number of predicted values increases. The FDGM (1, 1,tα) model proposed in this paper also has this problem. In order to solve this problem, this article will introduce the metabolism mechanism into the FDGM (1, 1, tα) model. The metabolism mechanism is an effective measure to solve the problem that the prediction model cannot make long-term prediction. Its definition is as follows.Step 1: establish the FDGM (1, 1,tα) model based on the original sequence X0=x01,x02,…,x0n to predict d values x⌢0n+1,x⌢0n+2,…,x⌢0n+dStep 2: use the newx0d,…,x0n,x⌢0n+1,…,x⌢0n+d data to establish the FDGM (1, 1, tα) model to predict d prediction values againStep 3: repeat Step 2 until the number of predicted values meets the requirements


### 4.5. Forecasting Results of China’s per Capita Living Energy Consumption from 2018 to 2029

As the FDGM (1, 1,tα) model proposed in this paper shows the best prediction performance in the previous section, this section will use the FDGM (1, 1, tα) model based on metabolism mechanism to predict China’s per capita living energy consumption from 2018 to 2029 (According to the prediction results shown in Section 3.3, we can see that when d = 6, the prediction accuracy of the FDGM (1, 1, tα) model is the highest. Therefore, this section sets d = 6. The prediction results are shown in Table 5, and the corresponding results are also shown in Figure 2. It can be seen in Figure 2 that China’s per capita living energy consumption will still increase rapidly in the next 12 years, and the annual consumption will exceed 1000 (kg ce) by the year of 2028. This means that China’s demand for energy is still very large in the next few years; thus, the production ability should be enhanced to meet the highly growing demand. At the same time, we can see that the growth rate of per capita energy living consumption in the next nine years will still increase. Increasing energy consumption will bring huge pressure and adjustment to society. Facing the future of high energy consumption, China needs to continue to adopt effective policies to reduce the consumption of living energy and build an energy-saving society. At the same time, China should optimize the energy consumption structure and increase the proportion of clean and high-quality energy in the consumption structure. Developing a low-carbon economy and building a low-carbon life will be an important strategic choice for sustainable development in the future.Table 5
The predicted values of China’s per capita living energy consumption from 2018 to 2029.
YearDataGrowth rate (%)YearDataGrowth rate (%)2018443.466.70362024722.519.77732019475.257.16862025796.3610.22132020511.757.68022026880.8710.61202021553.788.21302027970.7510.20292022602.268.754420281073.6410.59902023658.169.281720291188.8110.7271Figure 2
The predicted values and growth rate of China’s per capita living energy consumption from 2018 to 2029.


## 4.1. Raw Data Collection

The raw data of the per capita living energy consumption of China are collected from the official website National Bureau of Statistics of China (http://www.stats.gov.cn/english/), as shown in Table 1. The points from 2002 to 2011 are used for building the prediction models, and the last 6 points are used for testing the prediction accuracy of the models.Table 1
The raw data of the per capita living energy consumption of China (kg ce).
YearDataYearDataYearDataYearData2002146200623020102742014346.12003166200725020112942015365.42004191200825420123132016393.22005211200926420133352017415.6


## 4.2. Evaluating Indicator

In this section, we describe several metrics that are commonly used to evaluate the performance of a prediction model, as shown in Table2.Table 2
Metrics for evaluating effectiveness of the models.
IndicatorFormulaThe absolute percentage error (APE)APEk=x0k−x⌢0k/x0k×100%The mean absolute percentage error (MAPE)MAPE=1/n∑k=1nx0k−x⌢0k/x0k×100%The mean absolute error (MAE)MAE=1/n∑k=1nx0k−x⌢0kThe mean squares error (MSE)MSE=1/n∑k=1nx0k−x⌢0k2The root mean squares percentage error (RMSPE)RMSPE=1/n∑k=1nx0k−x⌢0k/x0k2×100%The index of agreement (IA)IA=1−∑k=1nx0k−x⌢0k2/∑k=1nx0k−x¯+x⌢0k−x¯2The correlation coefficient (R)R=Covx0k,x⌢0k/Varx⌢0kVarx0k


## 4.3. Numerical Results

The numerical results of the FDGM (1, 1,tα) model are compared to the commonly used prediction models, including the GM (1, 1) model, DGM (1, 1) model [14], FDGM (1, 1) model [26], NGM (1, 1, k) model [17], NDGM (1, 1, k) model [25], FNDGM (1, 1, k) model [27], ARGM (1, 1) model [15], and DGM (1, 1, tα) model. The results of these models are shown in Table 3. The metrics of the models for fitting and prediction are listed in Table 4 and plotted in Figure 1. According to the data shown in Table 3, we can see that the GM (1, 1), DGM (1, 1), and FDGM (1, 1, tα) models overestimated the actual values, and the other six prediction models underestimated the actual values. According to the prediction indicators shown in Table 4, we can see that the six indicators of the FDGM (1, 1, tα) model are the best of the five prediction models. Therefore, the FDGM (1, 1, tα) model shows the best performance in this case. Calculation formulas of nine prediction models are listed as follows.(1)
GM (1, 1) model:(20)x^1t=2914.9837e0.061064t−1−2768.9837,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(2)
DGM (1, 1) model:(21)x^1t=146⋅1.062862t+174.534876⋅1−1.062862t−0.0628626,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(3)
FDGM (1, 1) model:(22)x^0.602843t=146⋅0.968297t+94.79946⋅1−0.968297t0.031703,t=1,2,…,16.(1)
NDGM (1, 1,k):(23)x^1t=1471.193656⋅0.86313337t+46.625869t0.1368666−1464.5028,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(2)
FNDGM (1, 1,k):(24)x^0.817046t=366.7525⋅0.8003855t+40.211239t0.19961447−348.98792767,t=1,2,…,16.(3)
ARGM (1, 1):(25)x^0t=−253.6⋅0.908097t+399.6,t=1,2,…,16.(4)
NGM (1, 1,k):(26)x^1t=1325.1184e−0.142698t−1+343.6577t−1522.77615,t=1,2,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(5)
DGM (1, 1,tα):(27)x^1t=115.3365198+53.44286⋅t−10.62504+0.98096x^1t−1,t=2,3,…,16,x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.(6)
FDGM (1, 1,tα):(28)x^0.35742t=87.95390053+0.025791068⋅t−13+0.905629116x^1t−1,t=2,3,…,16.x^01=x^11;x^0t=x^1t−x^1t−1,t=2,3,…,16.Table 3
Numerical results by the nine models.
YearDataGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM(2)FDGM2002146146.00146.00146.00146.00146.00146.00146.00146.00146.002003166183.55183.71166.30167.44166.87167.79169.31166.00168.022004191195.11195.26190.77190.87190.66191.46190.47191.82192.112005211207.39207.54211.21211.19211.19211.09209.69211.94211.842006230220.45220.58228.58228.81228.91227.92227.14228.83228.052007250234.33234.45243.69244.08244.20242.75242.99243.50241.882008254249.09249.19257.04257.32257.41256.07257.39256.50254.382009264264.77264.85268.98268.80268.80268.21270.46268.18266.482010274281.44281.50279.78278.76278.64279.41282.32278.78279.002011294299.16299.20289.61287.39287.13289.82293.10288.45292.712012313318.00318.00298.62294.87294.46299.57302.89297.32308.332013335338.02338.00306.92301.36300.78308.76311.78305.50326.512014346.1359.31359.24314.58306.98306.24317.46319.85313.05347.912015365.4381.93381.83321.69311.86310.95325.73327.18320.05373.122016393.2405.98405.83328.30316.09315.02333.63333.83326.55402.732017415.6431.55431.34334.47319.76318.53341.18339.88332.59437.32Table 4
Performance evaluation indicators of the nine prediction models.
FittingGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM (2)FDGMMAPE3.50603.50801.13601.22501.19701.21901.5631.13101.0720RMSPE4.59604.61001.43301.48101.46701.47501.85401.39401.4010MAE764.1110764.0000296.2220315.1110311.2220306.0000378.7780293.7780257.6670MSE931.7600931.8150378.2150392.9960393.1800378.6850463.7160396.0050347.5530IA0.98570.98570.99780.99760.99760.99780.99670.99790.9981R0.98140.98140.99700.99670.99670.99690.99590.99710.9974PredictionGMDGM (1)FDGMNGMNDGMFNDGMARGMDGM (2)FDGM (1,1,tα)MAPE2.98802.964011.679014.077014.299010.719010.254012.11702.3850RMSPE3.25803.231012.721015.238015.465011.664011.441013.12802.7860MAE1108.1701099.0004395.3305289.6705372.0004032.8303881.5004554.000899.000MSE1224.4701213.7204948.6405914.4606000.4254536.8504481.3605098.6801094.950IA0.97280.97320.58550.51770.51260.62060.61310.57670.9808R0.99610.99610.98690.97890.97830.98940.98450.98680.9968Note. DGM (1) = DGM (1, 1); DGM (2) = DGM ((1, 1,tα).Figure 1
Six performance indicators of the nine prediction models (1 = GM; 2 = DGM; 3 = FDGM; 4 = NGM; 5 = NDGM; 6 = FNDGM; 7 = ARGM; 8 = DGM(2); 9 = FDGM (1, 1,tα)).
(a)(b)(c)(d)(e)(f)Note: DGM (1) = DGM (1, 1); DGM (2) = DGM ((1, 1,tα).


## 4.4. Metabolic Mechanism

As we all know, the prediction accuracy of the prediction model decreases as the number of predicted values increases. The FDGM (1, 1,tα) model proposed in this paper also has this problem. In order to solve this problem, this article will introduce the metabolism mechanism into the FDGM (1, 1, tα) model. The metabolism mechanism is an effective measure to solve the problem that the prediction model cannot make long-term prediction. Its definition is as follows.Step 1: establish the FDGM (1, 1,tα) model based on the original sequence X0=x01,x02,…,x0n to predict d values x⌢0n+1,x⌢0n+2,…,x⌢0n+dStep 2: use the newx0d,…,x0n,x⌢0n+1,…,x⌢0n+d data to establish the FDGM (1, 1, tα) model to predict d prediction values againStep 3: repeat Step 2 until the number of predicted values meets the requirements


## 4.5. Forecasting Results of China’s per Capita Living Energy Consumption from 2018 to 2029

As the FDGM (1, 1,tα) model proposed in this paper shows the best prediction performance in the previous section, this section will use the FDGM (1, 1, tα) model based on metabolism mechanism to predict China’s per capita living energy consumption from 2018 to 2029 (According to the prediction results shown in Section 3.3, we can see that when d = 6, the prediction accuracy of the FDGM (1, 1, tα) model is the highest. Therefore, this section sets d = 6. The prediction results are shown in Table 5, and the corresponding results are also shown in Figure 2. It can be seen in Figure 2 that China’s per capita living energy consumption will still increase rapidly in the next 12 years, and the annual consumption will exceed 1000 (kg ce) by the year of 2028. This means that China’s demand for energy is still very large in the next few years; thus, the production ability should be enhanced to meet the highly growing demand. At the same time, we can see that the growth rate of per capita energy living consumption in the next nine years will still increase. Increasing energy consumption will bring huge pressure and adjustment to society. Facing the future of high energy consumption, China needs to continue to adopt effective policies to reduce the consumption of living energy and build an energy-saving society. At the same time, China should optimize the energy consumption structure and increase the proportion of clean and high-quality energy in the consumption structure. Developing a low-carbon economy and building a low-carbon life will be an important strategic choice for sustainable development in the future.Table 5
The predicted values of China’s per capita living energy consumption from 2018 to 2029.
YearDataGrowth rate (%)YearDataGrowth rate (%)2018443.466.70362024722.519.77732019475.257.16862025796.3610.22132020511.757.68022026880.8710.61202021553.788.21302027970.7510.20292022602.268.754420281073.6410.59902023658.169.281720291188.8110.7271Figure 2
The predicted values and growth rate of China’s per capita living energy consumption from 2018 to 2029.


## 5. Conclusion

In this paper, a novel FDGM (1, 1,tα) model based on the metabolism mechanism (abbreviated as FDGM (1, 1, tα) model) has been proposed to study China’s per capita living energy consumption. The FDGM (1, 1, tα) model is developed based on the grey system theory combined with fractional accumulation. The numerical results show that the FDGM (1, 1, tα) model is more suitable to predict China’s per capita living energy consumption than other grey prediction models.The FDGM (1, 1,tα) model is used to predict China’s per capita living energy consumption from 2018 to 2029. According to the prediction results of the FDGM (1, 1, tα) model, we can see that China’s per capita living energy consumption will exceed 5000 (kg ce) in the next few decades, and the growth rate will gradually balance, which is obviously consistent with China’s national conditions. Because the new normal of the Chinese economy weakens the driving force of macroeconomic growth on energy demand which is the general trend of future development, the growth rate of energy consumption will also change, and energy consumption will enter a period of medium and low-speed growth. Therefore, it can be seen that the FDGM (1, 1, tα) model proposed in this paper can well describe the future development trend of China’s per capita energy living consumption and has certain practical significance.



---

*Source: 6635462-2021-03-29.xml*