# Prediction of China’s Sulfur Dioxide Emissions by Discrete Grey Model with Fractional Order Generation Operators

Authors: Wei Meng; Daoli Yang; Hui Huang
Journal: Complexity
 (2018)

---

## Abstract

Sulfur dioxide is an important source of atmospheric pollution. Many countries are developing policies to reduce sulfur dioxide emissions. In this paper, a novel prediction model is proposed, which could be used to forecast sulfur dioxide emissions. To improve the modeling procedure, fractional order accumulating generation operator and fractional order reducing generation operator are introduced. Based on fractional order operators, a discrete grey model with fractional operators is developed, which also makes use of genetic algorithms to optimize the modeling parameterr. The improved performance of the model is demonstrated via comparison studies with other grey models. The model is then used to predict China’s sulfur dioxide emissions. The forecast result shows that the amount of sulfur dioxide emissions is steadily decreasing and the policies of sulfur dioxide reduction in China are effective. According to the current trend, by 2020, the value of China’s sulfur dioxide emissions will be only 86.843% of emissions in 2015. Fractional order generation operators can be used to develop other fractional order system models.


---

## Body

## 1. Introduction

Sulfur dioxide (chemical formula SO2) is the most common form of sulfur oxides. The colorless gas has a strong irritating odor and is a major air pollutant. Inhaling it will lead to respiratory inflammation, bronchitis, emphysema, conjunctivitis, and other health problems. Sulfur dioxide will also weaken the immunity of young people, making them less resistant to infections. When sulfur dioxide reacts with water, sulfurous acid is formed [1]. If sulfurous acid is further oxidized by particulate matter in the atmosphere, sulfuric acid (the main component of acid rain) is rapidly and efficiently produced. With the presence of oxidants, light can catalyze the production of sulfate aerosols, which have an adverse effect on human health and could increase the mortality of patients [2]. Sulfur dioxide is the main cause of global “acid rain.” It is estimated that the direct economic losses caused by metal corrosion in industrialized countries account for between 2 and 4 per cent of the gross domestic product (GDP). Due to the serious negative impact of sulfur dioxide on flora, fauna, and buildings, sulfur dioxide emissions and the emission concentrations are under strict regulation. The concentration of sulfur dioxide is an important indicator of whether the air is polluted. Using coal as the main energy source, China’s sulfur dioxide emissions are ranked the first in the world [3]. Hence, it is of great practical significance to forecast the emissions of sulfur dioxide in order to formulate emission reduction policies.The world has exerted great efforts to reduce sulfur dioxide emissions. There has also been a lot of research focusing on sulfur dioxide emissions. Lu and colleagues have studied sulfur dioxide emissions in China and sulfur trends in East Asia since 2000 [4] and also analyzed sulfur dioxide and primary carbonaceous aerosol emissions in China and India from 1996 to 2010 [5]. Smith et al. investigated global and regional anthropogenic sulfur dioxide emissions [6]. Ma et al. developed scenario analysis to study the reduction potential of sulfur dioxide emissions in China’s iron and steel industry [7]. Yang and Hu conducted research on the inventory system of sulfur dioxide emissions in China [8]. Smith et al. discussed the methods and results of historical sulfur dioxide emissions from 1850 to 2000 [9]. Li et al. observed recent large reduction in sulfur dioxide emissions from China’s power plants by using ozone monitoring instruments [10]. Hao et al. designated sulfur dioxide and acid rain pollution control zones and studied the impact on energy industry in China [11]. Mohajan studied China’s sulfur dioxide emissions and local environmental pollution [12]. Zhou and Zhang established an improved metabolism grey model for predicting small samples with a singular datum and applied it to study sulfur dioxide emissions in China [13]. An adaptive neurofuzzy logic method has been developed by Yildirim et al. to estimate the impact of meteorological factors on sulfur dioxide pollution levels [14]. Hassanzadeh and coworkers developed statistical models and time series to forecast sulfur dioxide [15]. Li et al. looked into the contribution of China’s emissions to global climate forcing [16]. Liu et al. believed that air pollutant emissions from Chinese households are a major and underappreciated source of ambient pollution [17].To improve China’s policies on sulfur dioxide emission reduction, it is necessary to predict China’s future sulfur dioxide emissions. Grey prediction model is an important part of grey system theory [18, 19]. It has been studied by many researchers and improved over the years [20–22]. Recently, Liu et al. [23], Wu et al. [24, 25], and Xiao et al. [26, 27] published results on fractional order accumulation and grey system model with the fractional order accumulation. However, all the existing research [28, 29] does not deduce the analytical expression of fractional order accumulating generation operator and fractional order reducing generation operator, nor does it prove that these fractional order generation operators satisfy mutual invertibility. Accumulating generation operator is similar to summation operator and reducing generation operator is similar to difference operator. Cheng and Wu [28, 29] proposed the analytic expression of fractional summation operator but did not propose fractional difference operator and believed that the fractional difference operator generally does not satisfy the exponential rule. In this paper, both the fractional order accumulating generation operator and fractional order reducing generation operator are studied. Furthermore, a discrete grey model with fractional operators is used to predict China’s future sulfur dioxide emissions.The rest of the paper is organized as follows: in Section2, the analytical expressions of fractional order accumulating generation operator and fractional order reducing generation operator are derived. In Section 3, the discrete grey model with fractional operators and order optimization algorithm is studied. This is followed by comparisons of the proposed model with the other discrete grey models, demonstrated via three case studies, in Section 4. In Section 5, the proposed discrete grey model with fractional operators is used to predict China’s sulfur dioxide emissions. Then, conclusions are drawn in Section 6.


## 2. Fractional Order Generation Operators

Definition 1.
Assume that(1)X0=x01,x02,…,x0nis a sequence of raw data and D a sequence operator satisfying that(2)X0D=x01d,x02d,…,x0nd,where(3)x0kd=∑i=1kx0i,k=1,2,…,nand then the sequence operator D is called a (first-order) accumulating generator of X(0), denoted as 1-AGO.Definition 2.
Forr∈Z+, we define (4)Xr=xr1,xr2,…,xrnas the rth-order accumulating generation sequence of X(0), where(5)xrk=∑i=1kxr-1i=∑i=1k∑j=1ixr-2j,k=1,2,…,n.Definition 3.
Assume that the sequence of raw data is(6)X0=x01,x02,…,x0n,X1=x11,x12,…,x1nand is the first-order accumulating generation sequence of X(0), where(7)x1k=∑i=1kx0i,k=1,2,…,n,and it follows that (8)X1=x01,x01+x02,…,∑i=1kx0i,…,∑i=1nx0i.From Definition 2, it follows that(9)xr+1k=∑i=1kxri,k=1,2,…,n.So(10)X2=x11,x11+x12,…,∑i=1kx1i,…,∑i=1nx1i=x01,2x01+x02,…,nx01+n-1x02+⋯+n-k+1x0k+x0n.Theorem 4.
X
(
2
)
=
(
x
2
(
1
)
,
x
2
(
2
)
,
…
,
x
(
2
)
(
n
)
) is the second-order accumulating generation sequence of X(0), where(11)x2k=∑i=1kk+1-ix0i.Proof.
By induction onk, consider the following.
First, fork=1, x(2)(1)=∑i=1k(k+1-i)x(0)(i)=x(0)(1), so it is true.
Next, assume that, for somek=s and s≥1, the statement is true. That is, (12)x2s=∑i=1ss+1-ix0i.Then, for k=s+1, it follows that(13)x2s+1=x2s+x1s+1=∑i=1ss+1-ix0i+∑i=1s+1x0i=∑i=1s+1s+1-ix0i+∑i=1s+1x0i=∑i=1s+1s+2-ix0i.It  is  also  true.So, Theorem 4 is true.Theorem 5.
Assume that the sequence of raw data is(14)X0=x01,x02,…,x0n,where r∈Z+, and (15)Xr=xr1,xr2,…,xrnis the rth integer order accumulating generation sequence of X(0), where(16)xrk=∑i=1kk-i+1k-i+2⋯k-i+r-1r-1!x0i,k=1,2,…,n.Proof.
By induction onr, consider the following.
Forr=1, x(1)(k)=∑i=1k(k-i+1)(k-i+2)⋯(k-i+r-1)/(1-1)!x(0)(i)=∑i=1kx(0)(i). It is true.
For eachs≥1, if x(s)(k)=∑i=1kk-i+1k-i+2⋯(k-i+s-1)/(s-1)!x(0)(i) is true, it follows that(17)xs+1k=xs+1k-1+xsk=∑i=1k-1k-1-i+1k-1-i+2⋯k-1-i+s+1-1s+1-1!x0i+∑i=1kk-i+1k-i+2⋯k-i+s-1s-1!x0i=∑i=1k-1k-ik-i+1⋯k-i+s-1s!x0i+∑i=1kk-i+1k-i+2⋯k-i+s-1s-1!x0i=∑i=1kk-ik-i+1⋯k-i+s-1s!+k-i+1k-i+2⋯k-i+s-1s-1!x0i=∑i=1kk-i+1⋯k-i+s-1k-i+ss!x0i.It  is  also  true.
So Theorem5 is true for all r∈Z+.Definition 6.
Forn∈R and n∉0,-1,-2,-3,…, (18)Γn=∫0∞e-ttn-1dtis Gamma function for n.
Using integration by parts, the Gamma function satisfies the function(19)Γn+1=nΓn.In particular, if n is a positive integer,(20)Γn+1=∫0∞e-ttndt=n!For example, Γ(1/2)=π, Γ(1)=Γ(2)=1, Γ(3)=2!, Γ4=3!,…,Γ(n)=(n-1)!
The Gamma function is an extension of the factorial function.
By Gamma function, it follows that(21)xrk=∑i=1kk-i+1k-i+2⋯k-i+r-1r-1!x0i=∑i=1kr+k-i-1!k-i!r-1!x0i=∑i=1kΓr+k-iΓk-i+1Γrx0i.Definition 7.
Assume that the sequence of raw data is(22)X0=x01,x02,…,x0n,where r=R+, and (23)Xr=xr1,xr2,…,xrnis the rth integer order accumulating generation sequence of X(0), where(24)xrk=∑i=1kΓr+k-iΓk-i+1Γrx0i,k=1,2,…,n.Definition 8.
Assume that(25)X0=x01,x02,…,x0nis a sequence of raw data and D a sequence operator satisfying that(26)X0D=x01d,x02d,…,x0nd,where(27)x0kd=x0k+1-x0k,k=1,2,…,n,and the sequence operator D is called the first-order forward reducing generation operator.
If(28)x0kd=x0k-x0k-1,k=1,2,…,n,the sequence operator D is called the first-order backward reducing generation operator.
By default, the reducing generation operator is a backward reducing generation operator. The first-order reducing generation operator is the first-order inverse accumulating generation operator (IAGO), written as 1-RGO.Definition 9.
Forr∈Z+, we write (29)X-r=x-r1,x-r2,…,x-rnas the rth-order reducing generation sequence of X(0), where(30)x-rk=x-r+1k-x-r+1k-1,k=1,2,…,n.Definition 10.
Assume that the sequence of raw data is(31)X0=x01,x02,…,x0n,X-1=x-11,x-12,…,x-1nand is the first-order reducing generation sequence of X(0), where(32)x-1k=x0k-x0k-1,k=1,2,…,n,and then it follows that (33)X-1=x01,x02-x01,…,x0k-x0k-1,…,x0n-x0n-1.From Definition 9, it obviously follows that(34)x-r-1k=x-rk-x-rk-1,k=1,2,…,n.Then, it follows that(35)X-2=x-11,x-12-x-11,…,x-1k-x-1k-1,…,x-1n-x-1n-1=x01,x02-x01-x01,…,x0k-x0k-1-x0k-1-x0k-2,…,x0n-x0n-1-x0n-1-x0n-2=x01,x02-2x01x0,…,x0k-2x0k-1+x0k-2,…,x0n-2x0n-1+x0n-2So,x-2k=x0k-2x0k-1+x0k-2,k=3,4,…,n.Theorem 11.
X
(
-
2
)
=
(
x
-
2
(
1
)
,
x
-
2
(
2
)
,
…
,
x
(
-
2
)
(
n
)
) is the second-order reducing generation sequence of X(0), where(36)x-2k=∑i=0k-1-1i22-i!x0k-ik=1,2∑i=02-1i22-i!x0k-ik=3,4,…,n.Proof.
By induction onk, consider the following.
Fork=1, (37)x-21=∑i=0k-1-1i22-i!x0k-i=∑i=00-1i22-i!x01-i=x01.It  is  true.For k=2, (38)x-22=∑i=0k-1-1i22-i!x0k-i=∑i=01-1i22-i!x02-i=x02-2x01.It  is  true.For k=3,4,…,n, (39)x-22=∑i=0r-1ir!i!r-i!x0k-i=∑i=02-1i2i!2-i!x0k-i=x0k-2x0k-1+x0k-2.It  is  also  true.So, Theorem 11 is true.Theorem 12.
Assume that(40)X0=x01,x02,…,x0n,where r∈Z+, and (41)X-r=x-r1,x-r2,…,x-rnis the rth integer order reducing generation sequence of X(0), where(42)x-rk=∑i=0k-1-1ir!i!r-i!x0k-ik=1,2,…,r∑i=0r-1ir!i!r-i!x0k-ik=r+1,r+2,…,n.Proof.
By induction onr, consider the following.
Forr=1, ifk=1,(43)x-11=∑i=0k-1-1ir!i!r-i!x0k-i=x01.It  is  true;ifk=2,3,…,n, (44)x-1k=∑i=0r-1ir!i!r-i!x0k-i=∑i=01-1i1!i!1-i!x0k-i=x0k-x0k-1.It  is  true.For r=2, ifk=1, (45)x-21=∑i=0k-1-1ir!i!r-i!x0k-i=x01.It  is  true;ifk=2, (46)x-22=∑i=0k-1-1ir!i!r-i!x0k-i=∑i=01-1i2!i!2-i!x02-i=x02-2x01.It  is  true;ifk=3,4,…,n, (47)x-2k=∑i=0r-1ir!i!r-i!x0k-i=∑i=02-1i2!i!2-i!x0k-i=x0k-2x0k-1+x0k-2.It  is  true.For each s≥1, if (48)x-sk=∑i=0k-1-1is!i!s-i!x0k-ik=1,2,…,r∑i=0s-1is!i!s-i!x0k-ik=s+1,s+2,…,nis true, then it follows that(49)x-s-1k=x-sk-x-sk-1=∑i=0s-1is!i!s-i!x0k-i-∑i=0s-1is!i!s-i!x0k-i-1=s!s!x0k-s!1!s-1!x0k-1+s!2!s-2!x0k-2+-1is!i!s-i!x0k-i+-1ss!s!s-s!x0k-s-s!s!x0k-1+s!1!s-1!x0k-2-s!2!s-2!x0k-3--1i+1s!i+1!s-i-1!x0k-i--1ss!s!s-s!x0k-s-1=s!s!x0k-s!1!s-1!+s!s!x0k-1+s!2!s-2!+s!1!s-1!x0k-2+-1is!i!s-i!+-1i+1s!i-1!s-i+1!x0k-i--1ss!s!s-s!x0k-s-1.Then, it follows that(50)x-s-1k=x-sk-x-sk-1=∑i=0s-1is!i!s-i!x0k-i-∑i=0s-1is!i!s-i!x0k-i-1.The coefficient of x(0)(k-i) for the expansion of x(-s-1)(k) is (51)-1is!i!s-i!--1i-1s!i-1!s-i+1!=-1is!i!s-i!+-1is!i-1!s-i+1!=-1is!i!s-i!+s!i-1!s-i+1!=-1is!s-i+1i!s-i!s-i+1+i·s!i·i-1!s-i+1!=-1is!s-i+1+ii!s-i+1!=-1is!s+1i!s-i+1!=-1is+1!i!s-i+1!.So, x(-s-1)(k)=∑i=0s-1i(s+1)!/i!(s-i+1)!x(0)(k-i).
So, Theorem12 is true.Proposition 13.
Forr∈Z+, the rth integer order reducing generation operator(52)x-rk=∑i=0k-1-1ir!i!r-i!x0k-ik=1,2,…,r∑i=0r-1ir!i!r-i!x0k-ik=r+1,r+2,…,ncan be simplified as(53)x-rk=∑i=0k-1-1iΓr+1Γi+1Γr-i+1x0k-i,k=1,2,…,n.Proof.
According to the property of Gamma function,Γ(n+1)=n!
When1≤k≤r, (54)x-rk=∑i=0k-1-1ir!i!r-i!x0k-i=∑i=0k-1-1iΓr+1Γi+1Γr-i+1x0k-i.It  is  true.When r+1≤k≤n,fork-i≥1, then i≤k-1;forr+1≤i≤k-1, then r!/i!(r-i)!=0.
Thus, it follows that(55)x-rk=∑i=0r-1ir!i!r-i!x0k-i=∑i=0r-1ir!i!r-i!x0k-i+∑i=r+1k-1-1ir!i!r-i!x0k-i=∑i=0k-1-1iΓr+1Γi+1Γr-i+1x0k-i.It  is  also  true.So, Proposition 13 is true.Definition 14.
Assume that(56)X0=x01,x02,…,x0n,where r∈R+, and (57)X-r=x-r1,x-r2,…,x-rnis the rth integer order reducing generation sequence of X(0), where(58)x-rk=∑i=0k-1-1iΓr+1Γi+1Γr-i+1x0k-i,k=1,2,…,n.Theorem 15.
Assume that(59)X0=x01,x02,…,x0nis a sequence of raw data, where p∈R+ and q∈R+. X(p) is the sequence of the pth-order accumulating generation of X(0). X(-q) is the sequence of the qth-order reducing generation of X(0). (X(p))(-q) is the sequence of the qth-order reducing generation of X(p). (X(-q))(p) is the sequence of the pth-order accumulating generation of X(-q). The following holds true [19]:(1)
Ifp-q>0, X(p-q) is the p-q order accumulating generation of X(0).(2)
Ifp-q<0, X(p-q) is the q-p order accumulating generation of X(0).(3)
The fractional accumulating generation operator and the fractional reducing generation operator satisfy the exchange law and the exponential rate.(60)Xp-q=Xp-q=X-qp.Theorem 16.
Assume that(61)X0=x01,x02,…,x0nis a sequence of raw data and r∈R+. X(r) is the sequence of the rth-order accumulating generation of X(0). X(-r) is the sequence of the rth-order reducing generation of X(0). The rth order accumulating generation operator and the rth-order reducing generation operator are inverse operations [19].(62)X0=Xr-r=X-rr.


## 3. Discrete Grey Model with Fractional Operators

Definition 17.
Assume thatX(0) and X(r) are defined as in Definition 7, and(63)xrk+1=β1xrk+β2is the discrete grey model with fractional operators. In particular, consider the following.Ifr=1, x(r)(k+1)=β1x(r)(k)+β2 is DGM(1,1).(64)x1k+1=β1x1k+β2.Ifr=0, x(r)(k+1)=β1x(r)(k)+β2 is direct modeling DGM(1,1).(65)x0k+1=β1x0k+β2.Theorem 18.
Assume that(66)X0=x01,x02,…,x0nis a sequence of raw data and (67)xrk=∑i=1kΓr+k-iΓk-i+1Γrx0i,k=1,2,…,nis the sequence of the rth-order accumulating generation of X(0), where r∈R+.(68)Xr=xr1,xr2,…,xrnis the rth-order accumulating generation of X(0), where(69)xrk=∑i=1kΓr+k-iΓk-i+1Γrx0i,k=1,2,…,n.The parameter vector β^=β1,β2Τ of discrete grey model with fractional operators x(r)(k+1)=β1x(r)(k)+β2 can be calculated by the least squares method.(70)β^=BΤB-1BΤY.It follows that(71)Y=xr2xr3⋮xrnB=xr11xr21⋮⋮xrn-11,where(72)xrk=∑i=1kΓr+k-iΓk-i+1Γrx0i,k=1,2,…,n.So, it follows that(73)Y=xr2xr3⋮xrn=∑i=12Γr+2-iΓ2-i+1Γrx0i∑i=13Γr+3-iΓ3-i+1Γrx0i⋮∑i=1nΓr+n-iΓn-i+1Γrx0i=rx01+x02rr+12x01+rx02+x03⋮∑i=1nΓr+n-iΓn-i+1Γrx0i.B=xr11xr21⋮⋮xrn-11=∑i=11Γr+1-iΓ1-i+1Γrx0i1∑i=12Γr+2-iΓ2-i+1Γrx0i1⋮⋮∑i=1n-1Γr+n-1-iΓn-1-i+1Γrx0i1=x011rx01+x021⋮⋮∑i=1n-1Γr+n-1-iΓn-1-i+1Γrx0i1Theorem 19.
Assume thatB, Y, and β^ are the same as in Theorem 18. If(74)β^=β1,β2Τ=BΤB-1BΤY,then the following is true.
(1) The time response sequence of the discrete grey model with fractional operators(75)xrk+1=β1xrk+β2is given by(76)x^rk=x01-β21-β1β1k-1+β21-β1,k=2,3,…,nor(77)x^rk=x01β1k-1+1-β1k-11-β1β2,k=2,3,…,n.
(2) The restored values ofx(0)(k) can be given by(78)x^0k=x^r-rk=∑i=0k-1-1iΓr+1Γi+1Γr-i+1x^rk-i,k=2,3,…,nx^01=x01.Theorem 20.
Discrete grey model DGM(1,1) is the special case of the discrete grey model with fractional operators, where r=1.The conclusion is clearly true.To determine the optimal order of the discrete grey model with fractional operators, we can solve the following optimization problem:(79)min⁡fr=1n-1∑k=2nx0k-x^0kx0k,r∈R+.Genetic algorithms have obvious advantages at solving nonlinear optimization problems [30]. So, we can apply genetic algorithms to solve the optimized value of order r.Step 1 (start).
Initialize the parameters of population(t): initial value of order r, range of value for r, population size N, crossover probability pc, mutation probability pm, value accuracy represented as ee, fitness function results accuracy re, and generations of terminating evolution T.Step 2 (fitness).
Determine the fitness of population(t).
Step 2.1. Compute the r order accumulating generation sequence X(r).
Step 2.2. Compute the parameter vector β^.
Step 2.3. Compute the time response sequence x^(r)(k).
Step 2.4. Compute the forecast value x^(0)(k).
Step 2.5. Compute the fitness f(r).Step 3.
New population.
Step 3.1 (selection). Select parents from population (t).
Step 3.2 (crossover). Perform crossover on parents creating population (t+1).
Step 3.3 (mutation). Perform mutation of population (t+1).
Step 3.4 (accepting). Place new offspring in a new population.Step 4 (replace).
Use new generated population for a further run of algorithm.Step 5 (test).
If the end condition is satisfied, stop, and return the best solution in current population; otherwise go to Step2.Step 6 (result).
Display the result data of the model, the optimized orderr, the parameter vector β^, and the forecast value x^(0)(k).The overall modeling steps of the discrete grey model with optimized fractional operators are shown in Figure1.Figure 1
The modeling steps of discrete grey model with optimized fractional operators.


## 4. Verification of Discrete Grey Model with Fractional Operators

In this section, the advantage of the discrete grey model with fractional operators over the other discrete grey models is demonstrated by three case studies. Mean absolute percentage error (MAPE) is used to compare the actual values with forecasted values to evaluate the precision. MAPE is defined as(80)MAPE=1n∑k=1nxk-x^kxk×100%.Case 1.
In [31], a modified DGM(1,1) was proposed to predict China’s foreign exchange reserves. We consider the example from this paper [31] which provides the sample data. The actual values and the forecast values of the three compared models are presented in Table 1. As can be seen from Table 1, the discrete grey model with fractional operators where r=0.020 and r=0.662 achieved the lowest MAPE compared to DGM(1,1) and the modified DGM(1,1).Table 1
The fitted values and MAPE of different discrete grey models (unit: $ billion).
Year
Actual valuex(0)(k)
D
G
M
(
1,1
)
ModifiedDGM(1,1)
r
=
0.020
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
1994
51.620
1995
73.597
25.687
65.098
94.227
28.031
82.846
12.567
1996
105.049
34.109
67.530
106.679
1.552
107.697
2.521
1997
139.890
45.294
67.622
120.778
13.662
127.487
8.866
1998
144.959
60.145
58.509
136.740
5.670
143.251
1.178
1999
154.675
79.867
48.365
154.811
0.088
155.805
0.731
2000
165.574
106.055
35.947
175.271
5.856
165.796
0.134
MAPE (%)
9.143
4.333
2001
212.165
140.831
33.622
r
=
0.662
2002
286.407
187.009
34.705
288.721
0.808
303.083
5.823
2003
403.251
248.329
38.418
407.365
1.020
431.186
6.927
2004
609.932
329.756
45.936
574.762
5.766
594.877
2.468
2005
818.872
437.882
46.526
810.949
0.968
806.887
1.464
2006
1066.300
581.463
45.469
1144.191
7.305
1083.648
1.627
MAPE (%)
4.846
3.662
MAPE (%)
48.979
7.190
4.028Case 2.
In [32], the initial condition of DGM(1,1) was optimized and the results of direct modeling DGM(1,1) were tested against the original data, in respect of a geometric shape with up, down, concave, and convex sequences. Sample data from [32] were used in this case study. The actual values and the forecast values of three compared models are presented in Table 2. Again, the discrete grey model with fractional operators achieved the lowest MAPE compared to DGM(1,1) (where r=1) and direct modeling DGM(1,1).Table 2
The fitted values and MAPE of different discrete grey models.
Sequence
Sample data
Geometric shape of sequence
Models
MAPE (%)
X
1
(
1.2,2.9,4.2,5.1,5.8
)
Up, convex
Direct modelingDGM(1,1)
0.2958
r
=
1
6.1492
r
=
0
0.29549
X
2
(
8.5,16.4,32.3,64.2,128.1
)
Up, concave
Direct modelingDGM(1,1)
0.1754
r
=
1
0.5621
r
=
0.018
0.0086
X
3
(
5.8,5.1,4.2,2.9,1.2
)
Down, convex
Direct modelingDGM(1,1)
0.3321
r
=
1
19.1751
r
=
0
0.33136
X
4
(
128.1,64.2,32.3,16.4,8.5
)
Down, concave
Direct modelingDGM(1,1)
0.0756
r
=
1
0.7371
r
=
0.985
0.0258
X
5
(
5,11,29,83,245
)
Strict nonhomogeneous
Direct modelingDGM(1,1)
0.0000
r
=
1
13.8393
r
=
0
0.0000
X
6
(
1.4,2
,
2.8,3.9,5.4
)
Approximate nonhomogeneous
Direct modelingDGM(1,1)
0.1099
r
=
1
0.2766
r
=
0.943
0.0467Case 3.
The authors of [25] proposed a discrete grey model based on fractional order accumulation and studied the turnover of goods in Jiangsu Province, China. We consider the sample data from this paper as an example. The actual values and the forecast values of the different DGM models being compared are presented in Tables 3 and 4. The results show that the discrete grey model with fractional operators has a lower MAPE than the discrete grey model based on fractional order accumulate.Table 3
The actual values of the turnover of goods in Jiangsu Province, China.
Year
2003
2004
2005
2006
2007
2008
2009
Actual value (unit: billion tons-km)
181.74
239.81
306.83
364.41
409.84
470.75
515.45Table 4
The forecast values and errors of different discrete grey models (year 2009).
Model
D
G
M
1
/
2
(
1,1
)
D
G
M
2
/
3
(
1,1
)
D
G
M
(
1,1
)
r
=
0.27
Forecast value (unit: billion tons-km)
5236.45
5303.33
5585.60
5185.92
Absolute percentage error (%)
1.59
2.89
8.36
0.61


## 5. Prediction of China’s Sulfur Dioxide Emissions

According to China’s environmental status report published in 2016, from 2007 to 2015, the sulfur dioxide emission data (unit, million tons) are as follows:(81)X0=24.681,23.212,22.144,21.851,22.179,21.176,20.440,19.744,18.591.The order of discrete grey model with fractional order operators can be resolved by the algorithm of particle swarm optimization. The minimum mean relative error is thus obtained. The results of discrete grey model with fractional order operators wherer=0.07, DGM(1,1) model (discrete grey model with fractional order operators where r=1), direct modeling DGM(1,1) (discrete grey model with fractional order operators where r=0), and GM(1,1) are shown in Table 5. From Tables 5 and 6, we can see that the discrete grey model with fractional order operators where r=0.07 achieves the best fitting accuracy and test accuracy among the four models. The average relative error of discrete grey model with fractional order operators where r=0.07 is 1.118%. By using discrete grey model with fractional order operators where r=0.07, the predicted values of sulfur dioxide emissions in China in the next five years are shown in Table 7.Table 5
Fitting accuracy of China’s sulfur dioxide emissions (unit: million tons).
k
Year
x
(
0
)
(
k
)
G
M
(
1,1
)
Fractional orderDGM(1,1)
r
=
1
r
=
0
r
=
0.07
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
1
2007
24.681
2
2008
23.212
23.021
0.824
23.024
0.811
23.246
0.147
23.21
0.007
3
2009
22.144
22.532
1.752
22.534
1.762
22.299
0.701
22.387
1.097
4
2010
21.851
22.054
0.927
22.055
0.933
21.674
0.808
21.862
0.050
5
2011
22.179
21.585
2.676
21.586
2.675
21.262
4.134
21.478
3.161
6
2012
21.176
21.127
0.231
21.127
0.233
20.99
0.878
21.177
0.003
7
2013
20.440
20.679
1.167
20.677
1.161
20.811
1.813
20.929
2.392
Average relative errorΔ
1.263%
1.262%
1.413%
1.118%Table 6
Model testing result of China’s sulfur dioxide emissions (unit: million tons).
k
Year
x
(
0
)
(
k
)
G
M
(
1,1
)
Fractional OrderDGM(1,1)
r
=
1
r
=
0
r
=
0.07
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
x
^
(
0
)
(
k
)
Δ
k (%)
8
2014
19.744
20.240
2.512
20.237
2.497
20.369
3.166
19.642
0.517
9
2015
18.591
19.413
4.405
19.807
6.541
19.686
5.890
18.848
1.382Table 7
Prediction of China’s sulfur dioxide emissions (unit: million tons).
Year
2016
2017
2018
2019
2020
Forecast value
17.926
17.637
17.145
16.569
16.145According to the predicted results of China’s sulfur dioxide emissions shown in Table7, the emissions show a steady decline in the five years from 2016 to 2020. By 2020, the amount of sulfur dioxide emissions is 16.145 million tons, only 86.843% of emissions in 2015. The result indicates that China’s current sulfur dioxide emission reduction policies have been well implemented and achieved a positive outcome.


## 6. Conclusion

In this paper, the discrete grey model with fractional operators is firstly studied and then used to predict China’s sulfur dioxide emissions. This paper introduces a new prediction model by changing and optimizing the values of fractional order, giving rise to a discrete grey model with fractional operators. The steps of modeling and genetic algorithms of order optimization for the proposed model are explained. Case studies show that the discrete grey model with fractional operators can achieve the best precision with optimized order. Then, China’s future sulfur dioxide emissions are predicted by the discrete grey model with fractional operators. Genetic algorithm can be used to determine the optimal order under the condition of minimum mean error. The results show that the proposed discrete grey model with fractional operators can obtain a higher fitting accuracy than the grey modelGM(1,1), discrete grey model DGM(1,1), and direct modeling DGM(1,1). The forecast results show that China’s sulfur dioxide emissions will decrease steadily, indicating that China’s sulfur dioxide reduction policies are effective. It is noted that the fractional order accumulating generation operator and fractional order reducing generation operator may have the same expression. The unified expression of fractional order accumulating/reducing generation operators is an attractive topic in the grey system theory.



---

*Source: 8610679-2018-01-29.xml*