Continuous grey model with conformable fractional derivative
aa r X i v : . [ m a t h . G M ] A ug Continuous grey model with conformable fractional derivative
Wanli Xie a † , Caixia Liu a † ,Weidong Li a , Wenze Wu b ∗ , Chong Liu c a Institute of EduInfo Science and Engineering, Nanjing Normal University, Nanjing Jiangsu 210097, China b School of Economics and Business Administration, Central China Normal University, Wuhan 430079, China c School of Science,Inner Mongolia Agricultural University,Hohhot 010018,China
Abstract
The existing fractional grey prediction models mainly use discrete fractional-orderdifference and accumulation, but in the actual modeling, continuous fractional-ordercalculus has been proved to have many excellent properties, such as hereditary. Nowthere are grey models established with continuous fractional-order calculus method,and they have achieved good results. However, the models are very complicated in thecalculation and are not conducive to the actual application. In order to further simplifyand improve the grey prediction models with continuous fractional-order derivative, wepropose a simple and effective grey model based on conformable fractional derivativesin this paper, and two practical cases are used to demonstrate the validity of theproposed model.
Key words : Fractional grey prediction models; Fractional order; Grey system model;Optimization. ∗ corresponding author, E-mail: [email protected]; † : co-first authors. Introduction
Fractional calculus has been around for hundreds of years and came around the same timeas classical calculus. After years of development, fractional calculus has been widely appliedin control theory, image processing, elastic mechanics, fractal theory, energy, medicine, andother fields [1–8]. Fractional calculus is an extension of the integer-order calculus and thecommon fractional derivatives include Grunwald-Letnikov (GL) [9], Riemann-Liouville (RL)[10], Caputo [11], and so on. Although continuous fractional-order grey models have beenapplied in various fields, it is seldom used in the grey systems, while discrete fractional-orderdifference is mostly used at present.The grey model was first proposed by Professor Deng. It solves the problem of small sam-ple modeling, and the grey model does not need to know the distribution rules of data [12].The potential rules of data can be fully mined through sequence accumulation, which hasa broad application [12]. With the development of grey theory during several decades, greyprediction models have been developed very quickly and have been applied to all walks of life.For example, Li et al. [13] used a grey prediction model to predict building settlements. Caoet al. [14] proposed a dynamic Verhulst model for commodity price and demand prediction.Zhang et al. [15] applied a grey prediction model and neural network model for stock pre-diction. Ma et al. [16] presented a multi-variable grey prediction model for China’s tourismrevenue forecast. Wu et al. [17] proposed a fractional grey Bernoulli model to forecast therenewable energy of China. Zeng et al. [18] used a new grey model to forecast naturalgas demand. Wu et al. [19] put forward a fractional grey model for air quality prediction.Ding et al. [20] presented a multivariable grey model for the prediction of carbon dioxidein China. Modeling background in the real world becomes more and more complex, whichputs forward higher requirements for modeling. Many scholars have improved various greyprediction models. For example, Xie et al. [21] proposed a grey model and the predictionformula was derived directly from the difference equation, which improved the predictionaccuracy. Cui et al. [22] presented a grey prediction model and it can fit an inhomogeneoussequence, which improved the range of application of the model. Chen et al. [23] put forwarda nonlinear Bernoulli model, which can capture nonlinear characteristics of data. Wu et al.[24] proposed a fractional grey prediction model and it successfully extended the integer-order to the fractional-order, at the same time, they proved that the fractional-order greyodel had smaller perturbation bounds integer order derivative. Ma et al. [25] put forwarda fractional-order grey prediction model that was simple in the calculation and was easyto be popularized and applied in engineering. Zeng et al. [26] proposed an adaptive greyprediction model based on fractional-order accumulation. Wei et al. [27] presented a methodfor optimizing the polynomial model and obtained expected results. Liu et al. [28] proposeda grey Bernoulli model based on the Weibull Cumulative Distribution, which improved themodeling accuracy. In [29], a mathematical programming model was established to optimizethe parameters of grey Bernoulli.Although the above models have achieved good results, they all use continuous integer-order derivatives. In fact, the continuous derivative has many excellent characteristics, suchas heritability [30]. At present, there is little work on the grey prediction model based oncontinuous fractional derivative, and the corresponding research is still in early stage. In re-cent years, a new limit-based fractional order derivative is introduced by Khalil et al. in 2014[31], which is called the conformable fractional derivative. It is simpler than the previousfractional order derivatives, such as the Caputo derivative and Riemann-Liouville derivative,so it can easily solve many problems, compared with other derivatives with complex defini-tions. In 2015, Abdeljawad [32] developed this new fractional order derivative and proposedmany very useful and valuable results, such as Taylor power series expansions, Laplace trans-forms based on this novel fractional order derivative. Atangana et al. [33] introduced thenew properties of conformable derivative and proved some valuable theorems. In 2017, Al-Rifae and Abdeljawad proposed [34] a regular fractional generalization of the Sturm-Liouvilleeigenvalue problems and got some important results. The Yavuz and Ya¸skıran [35] suggesteda new method for the approximate-analytical solution of the fractional one-dimensional ca-ble differential equation (FCE) by employing the conformable fractional derivative. In thispaper, we propose a new grey model based on conformable fractional derivative, which hasthe advantage of simplicity and efficiency. The organization of this paper is as follows:In the second section, we introduce a few kinds of fractional-order derivatives. In thethird section, we show a grey model with Caputo fractional derivative and in the fourthsection, we present a new grey prediction model containing conformable derivative. In thefifth section, we give the optimization methods of the order and background-value coefficient.In the sixth section, two practical cases are used to verify the validity of the model and the2eventh section is a summary of the whole paper.
Fractional derivatives have rich forms, three common forms are Grunwald-Letnikov (GL),Riemann-Liouville (RL), and Caputo [36].
Definition 1 (See [36])
GL derivative with α order of function f ( t ) is defined as GLa D αt f ( t ) = n X k =0 f ( k ) ( a )( t − a ) − α + k Γ( − α + k + 1) + 1Γ( n − α + 1) Z ta ( t − τ ) n − α f ( n +1) ( τ ) dτ (1) where GLa D αt is the form of fractional derivative of GL, α > , n − < α < n, n ∈ N , [ a, t ] is the integral interval of f ( t ) , Γ( · ) is Gamma function, which has the following properties: Γ( α ) = R ∞ t α − e − t dt . Definition 2 (See [36])
RL derivative with order α of function f ( t ) is defined as RLa D αt f ( t ) = d n dt n a D − ( n − α ) t f ( t ) = 1Γ( n − α ) d n dt n Z ta ( t − τ ) n − α − f ( τ ) dτ (2) where RLa D αt f ( t ) is the fractional derivative of RL, a is an initial value, α is the order, Γ( · ) is Gamma function. Definition 3 (See [36])
Caputo derivative with α -order of function f ( t ) is defined as Ca D αt f ( t ) = 1Γ( n − α ) Z ta ( t − τ ) n − α − f ( n ) ( τ ) dτ, Amongthem , Ca D αt f ( t ) (3) where a is an initial value, α is the order, Γ( · ) is Gamma function. In particular, if thederivative order is ranged from 0 to 1, the Caputo derivative can be written as follows Ca D αt f ( t ) = 1Γ(1 − α ) Z ta ( t − τ ) − α f ′ ( τ ) dτ (4)Although the above derivatives have been successfully applied in various fields, it is difficultto be applied in engineering practice due to the complicated definition in the calculation.In recent years, some scholars have proposed a simpler fractal derivative called conformablederivative [37] defined as follows. 3 efinition 4 (See [37]) Assume T α ( f )( t ) is the derivative operator of f : [0 , ∞ ) → R , t > , α ∈ (0 , , and T α ( f )( t ) is defined as T α ( f )( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε (5) when α ∈ ( n, n + 1] , f is differentiable at t ( t > , the α -order derivative of the function f is T α ( f )( t ) = lim ε → f ( ⌈ α ⌉− (cid:0) t + εt ( ⌈ α ⌉− α ) (cid:1) − f ( ⌈ α ⌉− ( t ) ε (6) where ⌈ α ⌉ is the smallest integer greater than or equal to α . The conformable derivative has the following properties,
Definition 5 (See [37])
Let α ∈ (0 , and f , g be α -differentiable at a point t > , then(1) T α ( af + bg ) = aT α ( f ) + bT α ( g ) for all a, b ∈ R .(2) T α ( t p ) = pt p − α for all p ∈ R .(3) T α ( λ ) = 0 , for all constant functions f ( t ) = λ .(4) T α ( f g ) = f T α ( g ) + gT α ( f ) .(5) T α (cid:16) fg (cid:17) = gT α ( f ) − fI α ( g ) g . where T α is a -order conformable derivative. Theorem 1 (See [37])
Let α ∈ (0 , and f , g be α -differentiable at a point t > . Then T α ( f )( t ) = t − α dfdt ( t ) (7) Proof.
Let h = εt − α , then T α ( f )( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε = t − α lim h → f ( t + h ) − f ( t ) h = t − α df ( t ) dt ,where dfdt is first-order Riemann derivative, T α ( f )( t ) is a -order conformable derivative. Definition 6 (See [37]) I aα ( f )( t ) = I a ( t α − f ) = R ta f ( x ) x − α dx , where the integral is the usualRiemann improper integral, and α ∈ (0 , .Based on the above definitions, we give the definitions of conformable fractional-orderdifference and derivative. Definition 7 (See [38])
The conformable fractional accumulation (CFA) of f with α -order s expressed as ∇ α f ( k ) = ∇ ( k α − f ( k )) = k P i =1 f ( i ) i − α , α ∈ (0 , , k ∈ N + ∇ α f ( k ) = ∇ ( n +1) (cid:0) k α − [ α ] f ( k ) (cid:1) , α ∈ ( n, n + 1] , k ∈ N + . (8) The conformable fractional difference (CFD) of f with α -order is given by ∆ α f ( k ) = k − α ∆ f ( k ) = k − α [ f ( k ) − f ( k − , α ∈ (0 , , k ∈ N + ∆ α f ( k ) = k [ α ] − α ∆ n +1 f ( k ) , α ∈ ( n, n + 1] , k ∈ N + . (9)In the next section, We give a brief introduce for the fractional grey model with Caputoderivative. This model uses continuous fractional derivative for modeling at the first timeand achieves good results. Most of the previous grey models were based on integer-order derivatives. Wu first proposeda grey prediction model based on the Caputo fractional derivative, and the time responsesequence of the model was directly derived from the fractional derivative of Caputo, whichachieved good results [39]. In this section, we will introduce the modeling mechanism of thismodel.
Definition 8 (See [39])
Assume X (0) = (cid:8) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:9) is a non-negativesequence, the grey model with univariate of p (0 < p < order equation is α (1) x (1 − p ) ( k ) + az (0) ( k ) = b (10)where z (0) ( k ) = x (1 − p ) ( k )+ x (1 − p ) ( k − , α (1) x (1 − p ) ( k ) is a p -order difference of x (0) ( k ), the leastsquares estimation of GM ( p,
1) satisfies ab = (cid:0) B T B (cid:1) − B T Y , where5 = − z (0) (2) 1 − z (0) (3) 1... ... − z (0) ( n ) 1 , Y = α (1) x (1 − p ) (2) α (1) x (1 − p ) (3)... α (1) x (1 − p ) ( n ) (11)The winterization equation of GM ( p,
1) isd p x (0) ( t )d t p + ax (0) ( t ) = b. (12)Assume ˆ x (0) (1) = x (0) (1), the solution of the fractional equation calculated by the Laplacetransform is x (0) ( t ) = (cid:18) x (0) (1) − ba (cid:19) ∞ X k =0 ( − at p ) k Γ( pk + 1) + ba (13)Then, the restored values of can be obtained x (0) ( k ) = (cid:18) x (0) (1) − ba (cid:19) ∞ X i =0 ( − ak p ) i Γ( pi + 1) + ba (14)Although many fractional grey models have achieved good results, most of the fractional grayprediction models use fractional difference and fractional accumulation, while those modelstill use integer derivative. Although there are some studies on grey models with fractionalderivatives, they are more complicated to calculate than previous grey models. In order tosimplify calculation, we will propose a novel fractional prediction model with conformablederivative. In this section, based on the conformable derivative, we propose a simpler grey model, namedcontinuous conformable fractional grey model, abbreviated as CCFGM(1,1). Wu et al. [40]first gives the unified form of conformable fractional accumulation Eq. (8). On this basis,we use the matrix method to give the equivalent form of unified conformable fractional order6ccumulation.
Theorem 2
The conformable fractional accumulation is x ( α ) ( k ) = n X i = k ⌈ α ⌉ k − i x (0) ( i ) i ⌈ α ⌉ − α , α ∈ R + (15) where ⌈ α ⌉ is the smallest integer greater than or equal to α , ⌈ α ⌉ k − i = ⌈ α ⌉ ( ⌈ α ⌉ +1) ··· ( ⌈ α ⌉ + i − k − i )! = k − i + ⌈ α ⌉ − k − i = ( k − i + ⌈ α ⌉− k − i )!( ⌈ α ⌉− . α is the order of the model. Theoretically, the orderof grey model can be any positive number. In order to simplify the calculation, we will makethe order of the model between 0 and 1 in the later modeling. Proof. if α ∈ (0 , , ⌈ α ⌉ = 1, x ( α ) ( k ) = k P i =1 x (0) ( i ) i − α x (0) ( i ) = (cid:2) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:3) · · · − α · · · − α − α ... ... ... ... ...0 0 · · · n − − α n − − α · · · n − α = (cid:2) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:3) · · · − α · · · · · · n − − α
00 0 · · · n − α · · · · · · · · · · · · = k P i =1 k − ik − i x ( i ) i − α , k = 1 , , · · · , n. if α ∈ (1 , , ⌈ α ⌉ = 2. 7 ( α ) ( j ) = n P j = k n P i = j x (0) ( i ) i − r = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α
00 0 · · · n − α · · · · · · · · · · · · · · · · · · · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α
00 0 · · · n − α · · · n − n · · · n − n − · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α
00 0 · · · n − α · · · n − n − nn − · · · n − n − n − n − ... ... ... ... ...0 0 · · · · · · = k P i =1 k − i + 1 k − i x ( i ) i − α , k = 1 , , · · · , n. Assuming that the equation holds when α ∈ ( m − , m ], then ⌈ α ⌉ = m , x ( α ) ( k ) = n P i = k mk − i x (0) ( i ) i ⌈ α ⌉− α , α ∈ R + , when α ∈ ( m, m + 1], ⌈ α ⌉ = m + 1,let · · · m +1 − α · · · · · · n − m +1 − α
00 0 · · · n m +1 − α = A, we have8 α ( k ) = x (0) (1) x (0) (2) · · · x (0) ( n ) T A · · · · · · · · · · · · m +1 = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m · · · m + n − n − · · · m + n − n − ... ... ... ...0 0 · · · m · · · · · · · · · · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m + m · · · n − P i =0 m + ii + 1 n − P i =0 m + ii + 1 · · · n − P i =0 m + ii + 1 n − P i =0 m + ii + 1 ... ... ... ... ...0 0 · · · m + m · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m + 11 . . . m + n − n − m + n − n − · · · m + n − n − m + n − n − ... ... ... ... ...0 0 · · · m + 11 · · · = k P i =1 k − i + mk − i x (0) ( i ) .9o the result is proved. Remark 1
Similarly, the Refs. [25, 40] give the other two methods to get the same result.It can be proved that the definitions of these accumulation are essentially the same. Usingthe matrix method can help us better understand the fractional accumulation. Secondly, itcan better help us write computer programs.
Next, we will derive the grey differential equation with continuous conformable fractionalderivatives.
Definition 9
Assume X (0) = (cid:8) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:9) is a non-negative sequence, r (0 If r=1 and q=1, the equation (16) is equivalent to GM(1,1) (see [12]), if r ∈ [0 , and q =0, the equation equation (16) is equivalent to the equation (12) in form, if r=0 and q ∈ [0 , , the equation equation (16) is equivalent to the FGM(1,1) (see [24]) in form. Theorem 3 The exact solution of the conformable fractional-order differential equation is ˆ x ( r ) ( k ) = ˆ b + (cid:16)b ax (0) (1) − ˆ b (cid:17) e b a ( − kr ) r b a , k = 1 , , , ..., n ( n > 4) (17) Proof. Using equation (7) to convert the fractional order derivative into integer orderderivative, we can find the exact solution of equation (16). d r x ( q ) ( t )d t r + ax ( q ) ( t ) = b , t − r d x ( q ) ( t )d t + ax ( q ) ( t ) = b , by integrating the two sides, we have R d x ( q ) ( t )( b − ax ( q ) ( t )) = R dtt − r , soln (cid:12)(cid:12) b − ax ( q ) ( t ) (cid:12)(cid:12) = (cid:0) − ar (cid:1) t r + C , b − ax ( q ) ( t = ± e C e ( − ar ) t r , it can be sorted out, x ( q ) ( t ) = b + Ce ( − ar ) tr a , assume ˆ a , ˆ b is estimated parameters, ˆ x (0) ( k ) is an estimated valueof x (0) ( k ), k is a discrete variable with respect to t , with ˆ x ( q ) (0) = x (0) (1), then C = (cid:16)b ax (0) (1) − ˆ b (cid:17) e b ar , so the time response function of the CCFGM model is Eq. (17).10 emark 3 If r=1 and q=1, the equation (17) is equivalent to response function of GM(1,1)(see [12]), if r ∈ [0 , and q =0, the equation equation (17) is equivalent to the equation(14) in form (Mittag Leffler is a direct generalization of exponential function.), if r=0 and q ∈ [0 , , the equation equation (17) is equivalent to the response function of FGM(1,1) (see[24]) in form. Next, we will derive the discrete form of CCFGM(1,1) model. Through the discrete differenceequation, we can use least squares algorithm to get the parameters of the model. Thepredicted value can be obtained by q-order difference of the obtained predicted value, asfollows, ˆ x (0) ( k ) = ∆ ∇ − q ˆ x ( q ) ( k ). x (1 − q ) ( t ) stands for 1 − q -order accumulation, and it isequal to ∆ ∇ q x (0) ( t ). ∇ q x (0) ( t ) is the q-order accumulation of x (0) ( t ), ∆ r x ( q ) ( t ) is the r-order difference of x ( q ) ( t ), q ∈ [0 , Theorem 4 The difference equation of the continuous conformable grey model is x ( q − r ) ( t ) + a (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) = b, q ∈ [0 , , r ∈ [0 , . (18) Proof. Integrate CCFGM with r -order on both sides of Eq. (16): Z Z · · · Z kk − d r x ( q ) dt r dt r + a Z Z · · · Z kk − x ( q ) ( t ) dt r = b Z Z · · · Z kk − dt r (19)where Z Z · · · Z kk − d r x ( q ) ( t ) dt r dt r ≈ ∆ r x ( q ) = x ( q − r ) ( t ) (20) x ( q − r ) ( t ) stands for q − r -order accumulation, and it is equal to ∆ r ∇ q x (0) ( t ). ∇ q x (0) ( t ) isthe q-order accumulation of x (0) ( t ), ∆ r x (0) ( t ) is the r-order difference of x (0) ( t ), r ∈ [0 , r ∈ [0 , Z Z · · · Z kk − x ( q ) ( t ) dt r ≈ (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) (21)According to equation (10) and equation (12),we have Z Z · · · Z kk − bdt r = b Z Z · · · Z kk − dt r ≈ Z kk − bdt ≈ b. (22)11y equation (20), equation (21), and equation (22), the basic form of CCFGM(1,1) can bewritten as equation (18).Through the least square method, we can get the parameter of the CCFGM(1,1) isˆ a = ab = (cid:0) B T B (cid:1) − B T Y (23)where B = − (cid:2) x ( q ) (1) + x ( q ) (2) (cid:3) − (cid:2) x ( q ) (2) + x ( q ) (3) (cid:3) − (cid:2) x ( q ) ( n − 1) + x ( q ) ( n ) (cid:3) , Y = x ( q − r ) (2) x ( q − r ) (3)... x ( q − r ) ( n ) (24)Let ε = Y − B ˆ a be the error sequence and s = ε · ε T = Y − B ˆ a T ( Y − B ˆ a ) = n P k =2 (cid:8) x ( q − r ) ( t ) + a (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) − b (cid:9) dx , when s is minimized, values of a and bsatisfy ∂s∂a = n P k =2 (cid:8) x ( q − r ) ( t ) + a (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) − b (cid:9) (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) dx = 0 ∂s∂b = − n P k =2 (cid:8) x ( q − r ) ( t ) + a (cid:2) x ( q ) ( k − 1) + x ( q ) ( k ) (cid:3) − b (cid:9) = 0 , (25)where ˆ a is defined in the Eq. (23), B and Y defined in the Eq. (24). r andaccumulation order q The accumulative order is usually given by default, but in fact, the difference order r andaccumulation order q as part of the model greatly affect the model accuracy. Their valuescan be dynamically adjusted according to different modeling content. So the correct orderof the model are particularly important. In the following, we first established the followingmathematical programming model to optimize the two super parameters and used a whaleoptimization algorithm for optimization [42].12in r,q n n P i =1 (cid:12)(cid:12)(cid:12) ˆ x (0) ( k i ) − x (0) ( k i ) x (0) ( k i ) (cid:12)(cid:12)(cid:12) × . t r ∈ [0 , , q ∈ [0 , x ( q ) ( k ) = n P i = k ⌈ q ⌉ k − i x (0) ( i ) x ( i ) i ⌈ q ⌉− q , q > B = − (cid:2) x ( q ) (1) + x ( q ) (2) (cid:3) − (cid:2) x ( q ) (2) + x ( q ) (3) (cid:3) − (cid:2) x ( q ) ( n − 1) + x ( q ) ( n ) (cid:3) , Y = x ( q − r ) (2) x ( q − r ) (3)... x ( q − r ) ( n ) ˆ x ( q ) ( k ) = ˆ b + ( b ax (0) (1) − ˆ b ) e b a ( − kr ) r b a , k = 2 , , , ..., n ( n > x (0) ( k ) = ∆ ∇ − q ˆ x ( q ) ( k ) (26) In order to verify the validity of the model, we test the model with two actual cases, andcompare it with other forecasting models. Case 1. Prediction of domestic energy consumption in China (Ten thousand ton stan-dard coal)In this case, we select the data of domestic energy consumption in China from 2005 to2015 for fitting and the data from 2016 to 2017 for testing. The corresponding results areshown in Table 1 and Figure 1. Year D o m e s t i c ene r g y c on s u m p t i on i n C h i na Raw dataCCFGMCFGMPR(2)ANNSVR Data for building model Figure 1: Test results of five models. Test Simulation Figure 2: Error comparison of five grey mod-els.13able 1: Comparison of test results of five grey models.Year Raw data FGM PR(2) ANN SVR CCFGM2005 27573 27573.00 27461.01 27576.24 27572.90 27573.002006 27765 28776.69 28414.85 28801.47 29574.73 27207.682007 30814 30529.07 29920.42 30403.47 31576.57 28992.482008 31898 32510.82 31879.33 32409.59 33578.40 31373.762009 33843 34650.97 34193.17 34790.27 35580.23 33965.072010 36470 36925.23 36763.53 37441.64 37582.07 36671.072011 39584 39324.40 39492.00 40193.56 39583.90 39459.342012 42306 41845.55 42280.18 42848.65 41585.73 42317.232013 45531 44488.93 45029.65 45235.79 43587.57 45239.622014 47212 47256.57 47642.01 47249.65 45589.40 48224.632015 50099 50151.64 50018.85 48859.33 47591.23 51271.90MAPE 1.4358 0.9604 1.8158 3.6857 1.59422016 54209 52721.73 53852.78 50091.35 49593.07 54381.792017 57620 55350.93 57254.38 51003.38 51594.90 57555.00MAPE 3.3408 0.6458 9.5395 9.4858 0.2158The test errors of five grey models are shown in Figure 2. The experimental resultsshow that the fitting error and test error of the proposed model are 1.5942% and 0.2158%respectively, and the fitting error and test error of the FGM model are 1.4358% and 3.3408%respectively. The fitting error and test error of PR(2) are 0.9604% and 0.6458%, respectively,ANN are 1.8158% and 9.5395%, respectively, SVR are 3.6857% and 9.4858%, respectively.The fitting errors of PR(2) are slight lower than ours. However, the test error of our modelare smaller than other models. Case 2. Prediction of domestic coal consumption in China (ten thousand tons). Coalconsumption is related to the sustainable development of society. Accurate and effectiveprediction of coal consumption can contribute to effective decision-making and early warning.14 004 2006 2008 2010 2012 2014 2016 20188800900092009400960098001000010200 Year D o m e s t i c c oa l c on s u m p t i on i n C h i na Raw dataCCFGMCFGMPR(2)ANNSVRData for building model Figure 3: Test results of five models. Test Simulation Figure 4: Error comparison of five grey mod-els.Table 2: Comparison of test results of five grey models.Year Raw data FGM PR(2) ANN SVR CCFGM2005 10039.00 10039.00 10039.00 10031.67 9917.90 10039.002006 10036.00 9633.62 9600.63 10029.69 9839.40 9687.542007 9761.00 9464.86 9545.90 9755.68 9760.90 9434.762008 9148.00 9372.94 9491.48 9232.88 9682.40 9326.972009 9122.00 9319.89 9437.36 9225.60 9603.90 9274.312010 9159.00 9290.99 9383.56 9225.57 9525.40 9248.882011 9212.00 9279.07 9330.07 9225.57 9446.90 9238.872012 9253.00 9280.18 9276.87 9225.57 9368.40 9238.352013 9290.00 9291.95 9223.99 9225.57 9289.90 9243.992014 9253.00 9312.86 9171.40 9225.57 9211.40 9253.822015 9347.00 9341.96 9119.11 9225.57 9132.90 9266.55MAPE 1.4856 2.1776 0.5641 2.3623 1.32372016 9492.00 9378.60 9620.06 9225.57 9054.40 9281.342017 9283.00 9422.38 9860.18 9225.57 8975.90 9297.62MAPE 1.3481 3.7834 1.7128 3.9592 1.1884Table 2, Figure 3 and Figure 4 show the prediction of carbon dioxide emission with ourmodel. From Table 2, we can see that the fitting error and test error of our model are 1.3237%and 1.1884%, respectively. The fitting error and test error of FGM model are 1.4856% and15.3481%, respectively, PR(2) are 2.1776% and 3.7834%, respectively, ANN are 0.5641% and1.7128%, respectively, SVR are 2.3623% and 3.9592%, respectively. It can be seen that ourmodel has smaller test errors compared with other models, which means that our model issuperior to other models. In this paper, we propose a grey forecasting model with a conformable fractional derivative.Compared with integer derivatives, continuous fractional derivatives have been proved tohave many excellent properties. However, the most existing grey models are modeled byinteger derivatives. Secondly, it has been proved that the integer derivative cannot simulatesome special development laws in nature, the model can be further optimized by extendingthe grey model with the integer derivative to the fractional derivative. The existing frac-tional order grey model with continuous fractional-order derivative, achieved good result,but its calculation is complicated. This paper proposes a new grey model with conformablefractional-order derivative, further to simplify the calculation. Two actual cases show thatour model has high precision, and it can be easily promoted in engineering. The contributionsof this paper are as follows:(1) We constructed a fractional-order differential equation with a conformable derivativeas a whitening form of our model.(2) We built the mathematical programming model to optimize the order and of CCFGM(1,1)by whale optimizer, which further improved the prediction accuracy of the model.(3) We verify the validity of the model in this paper through two actual cases. This modelwith a simpler structure can achieve similar or even better accuracy than other models.Although the model in this paper has some advantages, it can be further improved fromthe following aspects:(1) In order to improve the modeling accuracy of the model, a more efficient optimizationalgorithm can be used to optimize parameters.(2) The model proposed in this paper is linear and cannot capture the nonlinear charac-teristics of the data. Accordingly, nonlinear characteristics can be studied for establishing amore universal and robust grey prediction model.16 Conflicts of Interest No potential conflict of interest was reported by the authors. The work was supported by grants from the Postgraduate Research & Practice InnovationProgram of Jiangsu Province [Grant No.KYCX19 0733]; grants from the Postgraduate Re-search & Practice Innovation Program of Jiangsu Province [Grant No.KYCX20 1144]. References [1] Abdeljawad T, Al-Mdallal QM. 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G M ] A ug Grey model with conformable fractional derivative Wanli Xie a † , Caixia Liu a † ,Weidong Li a , Wenze Wu b ∗ , Chong Liu c a Institute of EduInfo Science and Engineering, Nanjing Normal University, Nanjing Jiangsu 210097, China b School of Economics and Business Administration, Central China Normal University, Wuhan 430079, China c School of Science,Inner Mongolia Agricultural University,Hohhot 010018,China Abstract The existing fractional grey prediction models mainly use discrete fractional-orderdifference and accumulation, but in the actual modeling, continuous fractional-ordercalculus has been proved to have many excellent properties, such as hereditary. Nowthere are grey models established with continuous fractional-order calculus method,and they have achieved good results. However, the models are very complicated in thecalculation and are not conducive to the actual application. In order to further simplifyand improve the grey prediction models with continuous fractional-order derivative, wepropose a simple and effective grey model based on conformable fractional derivativesin this paper, and two practical cases are used to demonstrate the validity of theproposed model. Key words : Fractional grey prediction models; Fractional order; Grey system model;Optimization. ∗ corresponding author, E-mail: [email protected]; † : co-first authors. Introduction Fractional calculus has been around for hundreds of years and came around the same timeas classical calculus. After years of development, fractional calculus has been widely appliedin control theory, image processing, elastic mechanics, fractal theory, energy, medicine, andother fields [1, 2, 3, 4, 5, 6, 7, 8]. Fractional calculus is an extension of the integer-ordermodel and the common fractional derivatives include Grunwald-Letnikov (GL) [9], Riemann-Liouville (RL) [10], Caputo [11], and so on. Although continuous fractional-order grey modelshave been applied in various fields, it is seldom used in the grey systems, while discretefractional-order is mostly used at present.The grey model was first proposed by Professor Deng. It solves the problem of smallsample modeling, and the grey model does not need to know the distribution rules of data[12]. The potential rules of data can be fully mined through sequence accumulation, whichhas a broad application [12]. With the development of grey theory during several decades,grey prediction models have been developed very quickly and have been applied to all walksof life. For example, Li et al. [13] used a grey prediction model to predict building settle-ments. Cao et al. [14] proposed a dynamic Verhulst model for commodity price and demandprediction. Zhang et al. [15] applied a grey prediction model and neural network modelfor stock prediction. Ma et al. [16] presented a multi-variable grey prediction model forChina’s tourism revenue forecast. Wu et al. [17] proposed a fractional grey Bernoulli modelto forecast the renewable energy of China. Zeng et al. [18] used a new grey model to forecastnatural gas demand prediction. Wu et al. [19] put forward a fractional grey model for airquality prediction. Ding et al. [20] presented a multivariable grey prediction model for theprediction of carbon dioxide in China. Modeling background in the real world becomes moreand more complex, which puts forward higher requirements for modeling. Many scholarshave improved various grey prediction models. For example, Xie et al. [21] proposed agrey model and the prediction formula was derived directly from the difference equation,which improved the prediction accuracy. Cui et al. [22] presented a grey prediction modeland it can fit an inhomogeneous sequence, which improved the range of application of themodel. Chen et al. [23] put forward a nonlinear Bernoulli model, which can capture nonlin-ear characteristics of data. Wu et al. [24] proposed a fractional grey prediction model andit successfully extended the integer-order to the fractional-order. At the same time, theyroved that the fractional-order model had smaller perturbation bounds. Wu et al. [25] putforward a fractional-order grey prediction model that was simple in the calculation and waseasy to be popularized and applied in engineering. Zeng et al. [26] proposed an adaptivegrey prediction model based on fractional-order accumulation. Wei et al. [27] presented amethod for optimizing the polynomial model and obtained expected results. Liu et al. [28]proposed a grey Bernoulli model based on the Weibull Cumulative Distribution, which im-proved the modeling accuracy. In [29], a mathematical programming model was establishedto optimize the parameters of grey Bernoulli.Although the above models have achieved good results, they all use continuous integer-order derivatives. In fact, the continuous derivative has many excellent characteristics, suchas heritability [30]. At present, there is little work on the grey prediction model based oncontinuous fractional derivative, and the corresponding research is still in an early stage.In this paper, we propose a new grey model based on a uniform fractional derivative. Theorganization of this paper is as follows:In the second section, we introduce a few kinds of fractional-order derivatives. In thethird section, we show a grey model with Caputo fractional derivative and in the fourthsection, we present a new grey prediction model containing conformable derivative. In thefifth section, we give the optimization methods of the order and background-value coefficient.In the sixth section, two practical cases are used to verify the validity of the model and theseventh section is a summary of the whole paper. A fractional derivative has a variety of definitions and three common forms of it are Grunwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo [31]. Definition 2.1 GL derivative with the order of function f ( t ) is defined as GLa D αt f ( t ) = n X k =0 f ( k ) ( a )( t − a ) − α + k Γ( − α + k + 1) + 1Γ( n − α + 1) Z ta ( t − τ ) n − α f ( n +1) ( τ ) dτ (1)where GLa D αt is the form of fractional derivative of GL, α > , n − < α < n, n ∈ N , [ a, t ] isthe integral interval of f ( t ), Γ( · ) have the following properties: Γ( α ) = R ∞ t α − e − t dt .2 efinition 2.2 RL derivative with the order of function f ( t ) is defined as RLa D αt f ( t ) = d n dt n a D − ( n − α ) t f ( t ) = 1Γ( n − α ) d n dt n Z ta ( t − τ ) n − α − f ( τ ) dτ (2)where RLa D αt f ( t ) is the fractional derivative of RL, a is an initial value, a is the order, Γ( · )is Gamma function. Definition 2.3 Caputo derivative with α -order of function f ( t ) is defined as caputoa D αt f ( t ) = 1Γ( n − α ) Z ta ( t − τ ) n − α − f ( n ) ( τ ) dτ, Amongthem , caputoa D αt f ( t ) (3)where a is an initial value, α is the order, Γ( · ) is Gamma function. In particular, if thederivative order is ranged from 0 to 1, the Caputo derivative can be written as follows caputoa D αt f ( t ) = 1Γ(1 − α ) Z ta ( t − τ ) − α f ′ ( τ ) dτ (4)Although the above derivatives have been successfully applied in various fields, it isdifficult to be applied in engineering practice due to the complicated definition and difficultyin the calculation. In recent years, some scholars have proposed a simpler fractal derivativecalled conformable derivative [32] defined as follows. Definition 2.4 [32] Assume T α ( f )( t ) is the derivative operator of f : [0 , ∞ ) → R , t > α ∈ (0 , T α ( f )( t ) is defined as T α ( f )( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε (5)when α ∈ ( n, n + 1], f is differentiable at t ( t > α -order derivative of the function f is T α ( f )( t ) = lim ε → f ([ α ] − (cid:0) t + εt ([ α ] − α ) (cid:1) − f ([ α ] − ( t ) ε (6)where dαe is the smallest integer greater than or equal to α .The conformable derivative has the following properties. Definition 2.5 [32] Let α ∈ (0 , 1] and f , g be α -differentiable at a point t > 0, then(1) T α ( af + bg ) = aT α ( f ) + bT α ( g ) for all a, b ∈ R .32) T α ( t p ) = pt p − α for all p ∈ R .(3) T α ( λ ) = 0, for all constant functions f ( t ) = λ .(4) T α ( f g ) = f T α ( g ) + gT α ( f ).(5) T α (cid:16) fg (cid:17) = gT α ( f ) − fI α ( g ) g . where T α is a -order conformable derivative. Theorem 2.1 [32] Let α ∈ (0 , 1] and f , g be α -differentiable at a point t > 0. Then T α ( f )( t ) = t − α dfdt ( t ) (7) Proof. Let h = εt − α , then T α ( f )( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε = t − α lim h → f ( t + h ) − f ( t ) h = t − α df ( t ) dt Definition 2.6 [32] I aα ( f )( t ) = I a ( t α − f ) = R ta f ( x ) x − α dx , where the integral is the usualRiemann improper integral, and α ∈ (0 , Definition 2.7 [33] The conformable fractional accumulation (CFA) of f with α -orderis expressed as ∇ α f ( k ) = ∇ ( k α − f ( k )) = k P i =1 f ( i ) i − α , α ∈ (0 , , k ∈ N + ∇ α f ( k ) = ∇ ( n +1) (cid:0) k α − [ α ] f ( k ) (cid:1) , α ∈ ( n, n + 1] , k ∈ N + (8)The conformable fractional difference (CFD) of f with α -order is given by∆ α f ( k ) = k − α ∆ f ( k ) = k − α [ f ( k ) − f ( k − , α ∈ (0 , , k ∈ N + ∆ α f ( k ) = k [ α ] − α ∆ n +1 f ( k ) , α ∈ ( n, n + 1] , k ∈ N + (9)On the basis of the above conclusions, we propose a conformable derivative based on thedifferential mean value theorem and use it in the following model analysis. Theorem 2.2 Assume f ( x ) is continuous in the interval [a , b], and ∃ ξ ∈ [ a, b ], we have Z Z · · · Z ab f ( x ) dx r d x = f ( ξ ) (cid:18) b r − a r r (cid:19) (10)4here RR · · · f ( x ) dx r stands for the r -order integral of conformable f ( x ). Proof. Because f ( x ) is continuous in the interval [ a, b ], and m f ( x ) M , where m , M is the minimum and maximum of f ( x ) in the interval [ a, b ], we have Z Z · · · Z ab mdx r Z Z · · · f ( x ) dx r Z Z · · · Z ba M dx r = m Z Z · · · Z ba dx r Z Z · · · Z ab f ( x ) dx r M Z Z · · · Z ba dx r = m Z ba x − r dx Z ba f ( x ) dxM Z ba x − r dx = m (cid:18) b r − a r r (cid:19) Z ba f ( x ) dx M (cid:18) b r − a r r (cid:19) Let λ = (cid:0) b r − a r r (cid:1) , we have m R ba f ( x )d xλ M . According to the differential mean valuetheorem and ∃ ξ ∈ [ a, b ], f ( ξ ) = R ba f ( x )d xλ . Most of the previous grey models were based on integer-order derivatives. Wu first proposeda grey prediction model based on the Caputo fractional derivative, and the time responsesequence of the model was directly derived from the fractional derivative of Caputo, whichachieved good results [34]. In this section, we will introduce the modeling mechanism of thismodel. Definition 3.1 [34] Assume X (0) = (cid:8) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:9) is a non-negative se-quence, the gray model with univariate of p (0 < p < 1) order equation is α (1) x (1 − p ) ( k ) + az (0) ( k ) = b (11)where z (0) ( k ) = x (1 − p ) ( k )+ x (1 − p ) ( k − , α (1) x (1 − p ) ( k ) is a p -order difference of x (0) ( k ), the leastsquares estimation of GM ( p, 1) satisfies ab = (cid:0) B T B (cid:1) − B T Y , where B = − z (0) (2) 1 − z (0) (3) 1... ... − z (0) ( n ) 1 , Y = α (1) x (1 − p ) (2) α (1) x (1 − p ) (3)... α (1) x (1 − p ) ( n ) (12)The winterization equation of GM ( p, 1) is d p x (0) ( t )d t p + ax (0) ( t ) = b .5ssume ˆ x (0) (1) = x (0) (1), the solution of the fractional equation calculated by the Laplacetransform is x (0) ( t ) = (cid:18) x (0) (1) − ba (cid:19) ∞ X k =0 ( − at p ) k Γ( pk + 1) + ba (13)Then, the continuous gray model with a single variable is x (0) ( k ) = (cid:18) x (0) (1) − ba (cid:19) ∞ X i =0 ( − ak p ) i Γ( pi + 1) + ba (14) For the first time, the grey prediction model is constructed with continuous fractional deriva-tive and achieves a very good modeling effect. Nevertheless, its calculation is complicated.In this section, based on the conformable derivative, we propose a simpler grey model withcontinuous fractional differential equations, named continuous conformable fractional greymodel, abbreviated as CCFGM. We first define a discrete fractional accumulation. Theorem 4.1 The conformable fractional accumulation is x ( r ) ( k ) = n X i = k k − i + [ α ] − k − i x ( i ) i [ α ] − α , α > k − i + [ α ] − k − i = ( k − i +[ α ] − k − i )!([ α ] − . Proof. α ∈ (0 , α ] = 1. 6 ( α ) ( k ) = k P i =1 x (0) ( i ) i − α x (0) ( i ) = (cid:2) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:3) · · · − α · · · − α − α ... ... ... ... ...0 0 · · · n − − α n − − α · · · n − α = (cid:2) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:3) · · · − α · · · · · · n − − α 00 0 · · · n − α · · · · · · · · · · · · = k P i =1 k − ik − i x ( i ) i − α , k = 1 , , · · · , n. where α ∈ (1 , , [ α ] = 2. 7 ( k ) ( j ) = n P j = k n P i = j x (0) ( i ) i − r = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α 00 0 · · · n − α · · · · · · · · · · · · · · · · · · · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α 00 0 · · · n − α · · · n − n · · · n − n − · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T · · · − α · · · · · · n − − α 00 0 · · · n − α · · · n − n − nn − · · · n − n − n − n − ... ... ... ... ...0 0 · · · · · · = k P i =1 k − i + 1 k − i x ( i ) i − α , k = 1 , , · · · , n. when α ∈ ( m − , m ], then,when α ∈ ( m, m + 1], [ α ] = m + 1.let · · · m +1 − α · · · · · · n − m +1 − α 00 0 · · · n m +1 − α = A8 α ( k ) = x (0) (1) x (0) (2) · · · x (0) ( n ) T A · · · · · · · · · · · · m +1 = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m · · · m + n − n − · · · m + n − n − ... ... ... ...0 0 · · · m · · · · · · · · · · · · · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m + m · · · n − P i =0 m + ii + 1 n − P i =0 m + ii + 1 · · · n − P i =0 m + ii + 1 n − P i =0 m + ii + 1 ... ... ... ... ...0 0 · · · m + m · · · = x (0) (1) x (0) (2) · · · x (0) ( n ) T A m + 11 . . . m + n − n − m + n − n − · · · m + n − n − m + n − n − ... ... ... ... ...0 0 · · · m + 11 · · · = k P i =1 k − i + mk − i x (0) ( i ) 9heorem 4.1 is proved. Similarly, the literature [33, 35] gives the other two methods to get thesame result. Next, we will derive the grey differential equation with continuous conformablefractional derivatives.Assume X (0) = (cid:8) x (0) (1) , x (0) (2) , · · · , x (0) ( n ) (cid:9) is a non-negative data sequence, its frac-tional r (0 < p < r x (0) ( t )d t r + ax (0) ( t ) = b (16)When d r x (0) ( t )d t r = T α ( x ( r ) )( t ) is a conformable derivative, we call it continuous conformablefractional-order derivative. Theorem 4.2 The general solution of the conformable fractional-order differential equa-tion is ˆ x (0) ( k ) = ˆ b − (cid:16) ˆ b − b ax (0) (1) (cid:17) e b a ( − kr ) r b a , k = 1 , , , ..., n ( n > 4) (17) Proof. d r x (0) ( t )d t r + ax (0) ( t ) = bt − r d x (0) ( t )d t + ax (0) ( t ) = b R d x (0) ( t )( b − ax (0) ( t )) = R dtt − r ln (cid:0) b − ax (0) ( t ) (cid:1) = (cid:0) − ar (cid:1) t r − Cb − ax (0) ( t ) = e ( − ar ) t r − C x (0) ( t ) = b − Ce ( − ar ) tr a Assume ˆ a , ˆ b is estimated parameters, ˆ x (0) ( k ) is an estimated value of x (0) ( k ), k is adiscrete variable with respect to t , then C = (cid:16) ˆ b − b ax (0) (1) (cid:17) e b ar (18)The time response function of the continuous comfortable grey model isˆ x (0) ( k ) = ˆ b − (cid:16) ˆ b − b ax (0) (1) (cid:17) e b a ( − kr ) r b a , k = 2 , , , ..., n ( n > 4) (19)10 heorem 4.3 The difference equation of the continuous conformable grey model is x ( − r ) ( t ) + ax (0) ( k ) (cid:18) ξ r − ( ξ − r r (cid:19) = b k r − ( k − r r (20) Proof. Integrate CCFGM with r -order on both sides of Eq(16): Z Z · · · Z kk − d r x (0) dt r dt r + a Z Z · · · Z kk − x (0) ( t ) dt r = b Z Z · · · Z kk − dt r (21)where Z Z · · · Z kk − d r x (0) ( t ) dt r dt r = ∇ r x (0) = x ( − r ) ( t ) = k X i =1 f ( i ) i − α − k − X i =1 f ( i ) i − α (22) x ( − r ) ( t ) stands for r -order difference, and it is equal to ∇ ∆ r x (0) . We need to calculate r -orderconformable fractional-order accumulation first, and then calculate 1-order difference. Z Z · · · Z kk − d r x (0) dt r dt r = x ( − r ) ( t ) (23)According to the integral mean value Theorem 2.2, Z Z · · · Z kk − x (0) ( t ) dt r = x (0) ( ξ ) (cid:18) k r − ( k − r r (cid:19) = x (0) ( k ) (cid:18) ξ r − ( ξ − r r (cid:19) (24) Z Z · · · Z kk − dt r = Z kk − t − r dt = k r − ( k − r r (25)The basic form of CCFGM(1,1) can be written as x ( − r ) ( t ) + ax (0) ( k ) (cid:18) ξ r − ( ξ − r r (cid:19) = b k r − ( k − r r (26)The parameter estimation of the model is s = ab = (cid:0) B T B (cid:1) − B T Y (27)11here B = − ξ r − ( ξ − r r x (0) (2) r − (2 − r r − ξ r − ( ξ − r r x (0) (3) r − (3 − r r ... ... − ξ r − ( ξ − r r x (0) ( n ) n r − ( n − r r , Y = x ( − r ) (2) x ( − r ) (3)... x ( − r ) ( n ) (28)So the smallest a , b in s satisfies n P k =2 (cid:16) x ( − r ) ( t ) + ax (0) ( k ) (cid:16) ξ r − ( ξ − r r (cid:17) − b k r − ( k − r r (cid:17) dx ∂s∂a = 2 n P k =2 (cid:16) x ( − r ) ( t ) + ax (0) ( k ) (cid:16) ξ r − ( ξ − r r (cid:17) − b k r − ( k − r r (cid:17) x (0) ( k ) (cid:16) ξ r − ( ξ − r r (cid:17) = 0 ∂s∂b = − n P k =2 (cid:16) x ( − r ) ( t ) + ax (0) ( k ) (cid:16) ξ r − ( ξ − r r (cid:17) − b k r − ( k − r r (cid:17) k r − ( k − r r = 0 The accumulative order is usually given by default, but in fact, the order and the backgroundvalue as part of the model greatly affect the model accuracy. Their values can be dynamicallyadjusted according to different modeling content. So the correct order and the backgroundvalue coefficient of the model are particularly important. In the following, we first establishedthe following mathematical programming model to optimize the two super parameters andused a whale optimization algorithm for optimization [36].min α,ξ n n P i =1 (cid:12)(cid:12)(cid:12) ˆ x (0) ( k i ) − x (0) ( k i ) x (0) ( k i ) (cid:12)(cid:12)(cid:12) × . t . r x ( r ) ( k ) = k P i =1 k − i + [ r ] − k − i x ( i ) i [ r ] − r B = − ξ r − ( ξ − r r x (0) (2) r − (2 − r r − ξ r − ( ξ − r r x (0) (3) r − (3 − r r ... ... − ξ r − ( ξ − r r x (0) ( n ) n r − ( n − r r , Y = x ( − r ) (2) x ( − r ) (3)... x ( − r ) ( n ) ˆ x (0) ( k ) = ˆ b − ( ˆ b − b ax (0) (1) ) e b a ( − kr ) r b a , k = 1 , , , ..., n ( n > 4) (29)12 Application In order to verify the validity of the model, we test the model with two practical cases. Case 1. Prediction of domestic energy consumption in China (Ten thousand ton stan-dard coal)Hepatitis poses a great threat to people’s health. Accurate prediction of the numberof hepatitis infections is helpful for decision-makers to carry out effective analysis on thesituation of the disease, and then to take a reasonable intervention. In this example, weselect the data from 2011 to 2016 for fitting and the data from 2017 to 2018 for testing. Thetest results are shown in Table 1 and Figure 1. Year D o m e s t i c ene r g y c on s u m p t i on i n C h i na Raw dataCCFGMCFGMPR(2)ANNSVR Data for building model Figure 1: Test results of four models. Test Simulation Figure 2: Error comparison of five grey mod-els.13able 1: Comparison of test results of five grey models.Year Raw data FGM PR(2) ANN SVR CCFGM2005 27573 27573.00 27461.01 27576.24 27572.90 27573.002006 27765 28776.69 28414.85 28801.47 29574.73 27207.682007 30814 30529.07 29920.42 30403.47 31576.57 28992.482008 31898 32510.82 31879.33 32409.59 33578.40 31373.762009 33843 34650.97 34193.17 34790.27 35580.23 33965.072010 36470 36925.23 36763.53 37441.64 37582.07 36671.072011 39584 39324.40 39492.00 40193.56 39583.90 39459.342012 42306 41845.55 42280.18 42848.65 41585.73 42317.232013 45531 44488.93 45029.65 45235.79 43587.57 45239.622014 47212 47256.57 47642.01 47249.65 45589.40 48224.632015 50099 50151.64 50018.85 48859.33 47591.23 51271.90MAPE 1.4358 0.9604 1.8158 3.6857 1.59422016 54209 52721.73 53852.78 50091.35 49593.07 54381.792017 57620 55350.93 57254.38 51003.38 51594.90 57555.00MAPE 3.3408 0.6458 9.5395 9.4858 0.2158The test errors of four grey models are shown in Figure 2. The experimental resultsshow that the fitting error and test error of the proposed model are 1.7274% and 6.601%respectively, and the fitting error and prediction error of the FGM model are 1.928% and8.454% respectively. The fitting error and test error of GM(1,1) are 2.566% and 11.178%respectively, while the fitting error and prediction error of DGM model are 2.566% and11.239% respectively. Accordingly, the errors of our model are smaller than other models,which means that our model is superior to other models. Case 2. Prediction of domestic coal consumption in China (ten thousand tons).Coal consumption is related to the sustainable development of society. Accurate andeffective prediction of coal consumption can contribute to effective decision-making and earlywarning. 14 004 2006 2008 2010 2012 2014 2016 20188800900092009400960098001000010200 Year D o m e s t i c c oa l c on s u m p t i on i n C h i na Raw dataCCFGMCFGMPR(2)ANNSVRData for building model Figure 3: Test results of four models. Test Simulation Figure 4: Error comparison of five grey mod-els.Table 2: Comparison of test results of five grey models.Year Raw data FGM PR(2) ANN SVR CCFGM2005 10039.00 10039.00 10039.00 10031.67 9917.90 10039.002006 10036.00 9633.62 9600.63 10029.69 9839.40 9687.542007 9761.00 9464.86 9545.90 9755.68 9760.90 9434.762008 9148.00 9372.94 9491.48 9232.88 9682.40 9326.972009 9122.00 9319.89 9437.36 9225.60 9603.90 9274.312010 9159.00 9290.99 9383.56 9225.57 9525.40 9248.882011 9212.00 9279.07 9330.07 9225.57 9446.90 9238.872012 9253.00 9280.18 9276.87 9225.57 9368.40 9238.352013 9290.00 9291.95 9223.99 9225.57 9289.90 9243.992014 9253.00 9312.86 9171.40 9225.57 9211.40 9253.822015 9347.00 9341.96 9119.11 9225.57 9132.90 9266.55MAPE 1.4856 2.1776 0.5641 2.3623 1.32372016 9492.00 9378.60 9620.06 9225.57 9054.40 9281.342017 9283.00 9422.38 9860.18 9225.57 8975.90 9297.62MAPE 1.3481 3.7834 1.7128 3.9592 1.1884Table 2, Figure 3 and Figure 4 show the prediction of carbon dioxide emission withour model. From Table 2, we can see that the fitting error and test error of our model are1.7145% and 0.990%, respectively. The fitting error and test error of FGM model are 0.910%15nd 3.025%, and GM(1,1) are 1.4342% and 1.185% respectively. The fitting error and testerror of DGM are 1.4354% and 1.1853% respectively. It can be seen that in the testing stage,our model has smaller test errors compared with other models. In this paper, we propose a grey predictive model with a conformable fractional derivative.Compared with integer derivatives, continuous fractional derivatives have been proved tohave many excellent properties. However, the most existing grey models are modeled byinteger derivatives. Secondly, it has been proved that the integer derivative cannot simulatesome special development laws in nature, and it is not necessarily the optimal parameter ofthe model., the model can be further optimized by extending the grey model with the integerderivative to the fractional derivative. In current, there is only one grey model containingcontinuous fractional-order derivative. Although it achieved a good result, its calculation isvery complicated and it is not conducive to the promotion. Accordingly, this paper proposesa new grey model with a conformable fractional-order derivative, further to simplify thecalculation of the current grey model. Two practical examples show that our model has highprecision, and it can be easily promoted in engineering. The contributions of this paper areas follows:(1) We constructed a fractional-order differential equation with a conformable derivativeas a whitening form of our model. The basic form of the model in this paper was establishedby deducing the first differential mean value theorem.(2) We built the mathematical programming model, optimized the order and backgroundvalues of the model by a whale algorithm, which further improved the prediction accuracyof the model.(3) We verify the validity of the model in this paper through two practical cases,. Thismodel with a simpler structure can achieve similar or even better accuracy than other models.Although the model in this paper has some advantages, it can be further improved fromthe following aspects:(1) In order to improve the modeling accuracy of the model, a more efficient optimizationalgorithm can be used to optimize parameters.162) The model proposed in this paper is linear and cannot capture the nonlinear charac-teristics of the data. Accordingly, nonlinear characteristics can be studied for establishing amore universal and robust grey prediction model. 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