# Fractional Order Accumulation NGM (1, 1,k) Model with Optimized Background Value and Its Application

Authors: Jun Zhang; Yanping Qin; Xinyu Zhang; Bing Wang; Dongxue Su; Huaqiong Duo
Journal: Journal of Mathematics
 (2021)

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## Abstract

Aiming at the problem of unstable prediction accuracy of the classic NGM (1, 1,k) model, the modeling principle and parameter estimation method of this model are deeply analyzed in this study. Taking the minimum mean absolute percentage error as the objective function, the model is improved from the two perspectives of the construction method of the background value and the fractional order accumulation generation. The fractional order accumulation NGM (1, 1, k) model based on the optimal background value (short for the FBNGM (1, 1, k) model) is proposed in the study. The particle swarm optimization algorithm is used to estimate the parameters of the proposed model. Taking two actual cases with economic significance as examples, empirical analysis of the proposed model is conducted. The simulation and prediction results show the practicality and efficiency of the FBNGM (1, 1, k) model proposed in this study, which further broadens the application scope of the grey prediction model.


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## Body

## 1. Introduction

The grey forecasting model has the advantages of no special requirements on the distribution of modeling data and easily understood modeling mechanism. So it is used in many fields such as environment [1, 2], agriculture [3], industry [4], energy [5, 6], economy [7, 8], and so on. The GM (1, 1) model plays an important role in the grey forecasting model. In order to improve the prediction accuracy of the GM (1, 1) model, some scholars have made a series of improvements from the perspectives of the background value [8], cumulative order [9], and discretization [10]. But the prediction accuracy of the improved model for nonhomogeneous exponential series is still not ideal, so the NGM (1, 1, k) model came into being [11].In order to expand the application scope of the NGM (1, 1,k) model, the researchers further optimized the grey differential equation of the NGM (1, 1, k) model in practical applications and proposed the NHGM (1, 1, k) model [12] and the grey prediction model with time power term [13]. In order to eliminate the inherent bias caused by the mismatch problem, Wang and Gong [14] proposed the intensional NGM (1, 1, k) model based on the classic NGM (1, 1, k) model. In order to compensate the lack of accuracy of GM (1, 1) in simulating nonhomogeneous exponential growth series, Cui et al. [11] constructed the NGM (1, 1, k) model to approximate the characteristics of nonhomogeneous exponential growth series by using the grey differential equation of GM (1, 1) as a deductive reasoning tool. Tong et al. [15] proposed the BNGM (1, 1, k) model with optimized grey differential equation by analyzing the cause of parameter estimation error in the classic NGM (1, 1, k) model. However, the single optimized NGM (1, 1, k) model has limited effect in improving the accuracy of model prediction. Based on fractional order accumulation, Wu et al. [16] proposed the discrete grey model with the fractional order. Zeng et al. [17] proposed the idea of multiple optimizations and verified the prediction accuracy of the dual optimized GM (1, 1) model based on function transformation, and background value reconstruction is much higher than the single optimized GM (1, 1) model through actual application examples. This shows that multiple optimizations could complement each other through different optimization measures. It makes up for the limitations of single optimization and further improves the accuracy of the grey prediction model.The main contributions of this study are as follows:(1)
Based on the previous scholars’ research, referring to the fractional order accumulation generation technology and the idea of optimizing the background value, the fractional order accumulation NGM (1, 1,k) model based on the optimal background value (short for FBNGM (1, 1, k) model) is constructed in this study.(2)
Based on the actual application cases of China’s electricity consumption and agricultural output value, the proposed FBNGM (1, 1,k) model is compared with the NGM (1, 1, k) model and two single optimization model in simulation and prediction accuracy. The results show that the FBNGM (1, 1, k) model has better performance in improving model accuracy than the other models, which verifies the practicality and efficiency of the proposed model.The remainder of this study is organized as follows: the second part introduces the basic model and accuracy measure indicator; the third part introduces the FBNGM (1, 1,k) model and its parameter solution in detail; the fourth part is the empirical study, based on two actual application cases of predicting China’s electricity consumption and agricultural output value; the fifth part is the conclusion.


## 2. Preliminary Knowledge

This section presents the fractional order accumulation and inverse accumulation generation technology of modeling sequence, the related concepts of the NGM (1, 1,k) model, and the accuracy measure indicator of the model.


### 2.1. Fractional Order Accumulation and Inverse Accumulation Generation Technology [17]

(1)
AssumingX0=x01,x02,…,x0nis a nonnegative original sequence, so Xr=xr1,xr2,…,xrn, r∈R+ is called the rth order accumulation generation sequence of X0, where(1)xrk=∑i=1kΓr+k−iΓk−i+1Γrx0i,k=1,2,…,n.(2)
AssumingX0=x01,x02,…,x0n is a nonnegative original sequence, so X−r=x−r1,x−r2,…,x−rn,r∈R+ is called the rth order inverse accumulation generation sequence of X0, where(2)x−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x0k−i,k=1,2,…,n.


### 2.2. NGM (1, 1,k) Model [15]

X1=x11,x12,…,x1n is the first-order accumulation generation sequence of X0 and Z1=z12,z13,…,z1n, z1k=0.5x1k+0.5x1k−1,k=2,3,…,n, is the mean generation sequence of X1, so(3)x0k+az1k=b2k−12+cis called the grey differential equation form of the NGM (1, 1, k) model, where z1k is called the background value, a is called the development coefficient, b2k−1/2+c is named the grey action or driving term, and the least squares estimates of the parameter a,b,c is(4)P∧=a∧,b∧,c∧T=BTB−1BTY,where(5)B=−z12321−z13521⋮⋮⋮−z1n2n−121,Y=x02x03⋮x0n.Finally,(6)dx1tdt+ax1t=bt+cis called whitened differential equation form of the NGM (1, 1, k) model.Under the initial conditionx∧11=x11, the time response function of the NGM (1, 1, k) model is(7)x∧1k=x11−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n.The simulated and predicted values of the NGM (1, 1,k) model for the original sequence X0 are(8)x∧0k=x∧1k−x∧1k,k=2,3,…,x∧01=x01.


### 2.3. Accuracy Measure Indicator

In grey system theory, the absolute percentage error (APE) of each point, the mean absolute percentage error (MAPE) of all points, and the root mean square error (RMSE) of all points are usually used to measure the simulation and prediction accuracy of the grey prediction model [6].The absolute percentage error of each point is(9)APEk=x∧0k−x0kx0k×100%,k=1,2,….The mean absolute percentage error of all points is(10)MAPE=1n−1∑k=2nx∧0k−x0kx0k×100%.The root mean square error of all points is(11)RMSE=1n−1∑k=2nx∧0k−x0k2.


## 2.1. Fractional Order Accumulation and Inverse Accumulation Generation Technology [17]

(1)
AssumingX0=x01,x02,…,x0nis a nonnegative original sequence, so Xr=xr1,xr2,…,xrn, r∈R+ is called the rth order accumulation generation sequence of X0, where(1)xrk=∑i=1kΓr+k−iΓk−i+1Γrx0i,k=1,2,…,n.(2)
AssumingX0=x01,x02,…,x0n is a nonnegative original sequence, so X−r=x−r1,x−r2,…,x−rn,r∈R+ is called the rth order inverse accumulation generation sequence of X0, where(2)x−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x0k−i,k=1,2,…,n.


## 2.2. NGM (1, 1,k) Model [15]

X1=x11,x12,…,x1n is the first-order accumulation generation sequence of X0 and Z1=z12,z13,…,z1n, z1k=0.5x1k+0.5x1k−1,k=2,3,…,n, is the mean generation sequence of X1, so(3)x0k+az1k=b2k−12+cis called the grey differential equation form of the NGM (1, 1, k) model, where z1k is called the background value, a is called the development coefficient, b2k−1/2+c is named the grey action or driving term, and the least squares estimates of the parameter a,b,c is(4)P∧=a∧,b∧,c∧T=BTB−1BTY,where(5)B=−z12321−z13521⋮⋮⋮−z1n2n−121,Y=x02x03⋮x0n.Finally,(6)dx1tdt+ax1t=bt+cis called whitened differential equation form of the NGM (1, 1, k) model.Under the initial conditionx∧11=x11, the time response function of the NGM (1, 1, k) model is(7)x∧1k=x11−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n.The simulated and predicted values of the NGM (1, 1,k) model for the original sequence X0 are(8)x∧0k=x∧1k−x∧1k,k=2,3,…,x∧01=x01.


## 2.3. Accuracy Measure Indicator

In grey system theory, the absolute percentage error (APE) of each point, the mean absolute percentage error (MAPE) of all points, and the root mean square error (RMSE) of all points are usually used to measure the simulation and prediction accuracy of the grey prediction model [6].The absolute percentage error of each point is(9)APEk=x∧0k−x0kx0k×100%,k=1,2,….The mean absolute percentage error of all points is(10)MAPE=1n−1∑k=2nx∧0k−x0kx0k×100%.The root mean square error of all points is(11)RMSE=1n−1∑k=2nx∧0k−x0k2.


## 3. Fractional Order Accumulation NGM (1, 1,k) Model with Optimal Background Value

### 3.1. FBNGM (1, 1,k) Model

AssumingXr=xr1,xr2,…,xrn, r∈R+ is the rth order accumulation generation sequence of the nonnegative original sequence X0=x01,x02,…,x0n, and Zr=zr2,zr3,…,zrn is the optimized background value sequence of Xr, among them, zrk=wx1k+1−wx1k−1, 0<r<1, 0<w<1, k=2,3,…,n, so(12)xr−1k+azrk=b2k−12+cis called the grey differential equation form of the NGM (1, 1, k) model with the fractional order and optimized background value, denoted as the FBNGM (1, 1, k) model. Where a is called development coefficient, zrk is named the optimized background value, w is named the optimized background value weight, and the least squares estimate of the three parameters a,b,c is(13)P∧=a∧,b∧,c∧T=BTB−1BTY,where(14)B=−zr2321−zr3521⋮⋮⋮−zrn2n−121,Y=xr−12xr−13⋮xr−1n.Thus,(15)dxrtdt+axrt=bt+cis called the whitened differential equation form of the FBNGM (1, 1, k) model.Under initial conditionx∧r1=xr1, the time response function of the FBNGM (1, 1, k) model is(16)x∧rk=xr1−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n.The simulated and predicted values of the FBNGM (1, 1,k) model for the original sequence X0 are(17)x∧0k=x∧r−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x∧rk−i,k=2,3,…,x∧01=x01.


### 3.2. Parameter Solution of the FBNGM (1, 1,k) Model

The proposed FBNGM (1, 1,k) model described above is established based on unknown parameters r and w. In order to find the best fitness for the FBNGM (1, 1, k) model, MAPE is taken as the objective function, the relationship between model parameters is taken as constraint condition, and the nonlinear programming model is conducted as follows:(18)minfr,w=1n−1∑k=2nx∧0k−x0kx0k×100%,s.t.0<r<1,0<w<1,P∧=a∧,b∧,c∧T=BTB−1BTY,B=−zr2321−zr3521⋮⋮⋮−zrn2n−121,Y=xr−12xr−13⋮xr−1n,zrk=wxrk+1−wxrk−1,k=2,3,…,n,x∧rk=xr1−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n,x∧0k=x∧r−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x∧rk−i,k=2,3,…,x∧01=x01.For the abovementioned nonlinear programming model, the particle swarm optimization algorithm [18] (PSO) is used to solve the best parameters of the proposed FBNGM (1, 1, k) model in this study.


### 3.3. Modeling Procedure of the FBNGM (1, 1,k) Model

The modeling procedure of the FBNGM (1, 1,k) model is presented clearly in Figure 1.Figure 1
Modeling procedure of the FBNGM (1, 1,k) model.


## 3.1. FBNGM (1, 1,k) Model

AssumingXr=xr1,xr2,…,xrn, r∈R+ is the rth order accumulation generation sequence of the nonnegative original sequence X0=x01,x02,…,x0n, and Zr=zr2,zr3,…,zrn is the optimized background value sequence of Xr, among them, zrk=wx1k+1−wx1k−1, 0<r<1, 0<w<1, k=2,3,…,n, so(12)xr−1k+azrk=b2k−12+cis called the grey differential equation form of the NGM (1, 1, k) model with the fractional order and optimized background value, denoted as the FBNGM (1, 1, k) model. Where a is called development coefficient, zrk is named the optimized background value, w is named the optimized background value weight, and the least squares estimate of the three parameters a,b,c is(13)P∧=a∧,b∧,c∧T=BTB−1BTY,where(14)B=−zr2321−zr3521⋮⋮⋮−zrn2n−121,Y=xr−12xr−13⋮xr−1n.Thus,(15)dxrtdt+axrt=bt+cis called the whitened differential equation form of the FBNGM (1, 1, k) model.Under initial conditionx∧r1=xr1, the time response function of the FBNGM (1, 1, k) model is(16)x∧rk=xr1−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n.The simulated and predicted values of the FBNGM (1, 1,k) model for the original sequence X0 are(17)x∧0k=x∧r−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x∧rk−i,k=2,3,…,x∧01=x01.


## 3.2. Parameter Solution of the FBNGM (1, 1,k) Model

The proposed FBNGM (1, 1,k) model described above is established based on unknown parameters r and w. In order to find the best fitness for the FBNGM (1, 1, k) model, MAPE is taken as the objective function, the relationship between model parameters is taken as constraint condition, and the nonlinear programming model is conducted as follows:(18)minfr,w=1n−1∑k=2nx∧0k−x0kx0k×100%,s.t.0<r<1,0<w<1,P∧=a∧,b∧,c∧T=BTB−1BTY,B=−zr2321−zr3521⋮⋮⋮−zrn2n−121,Y=xr−12xr−13⋮xr−1n,zrk=wxrk+1−wxrk−1,k=2,3,…,n,x∧rk=xr1−b∧a∧−c∧a∧+b∧a∧2e−a∧k−1+b∧a∧k+c∧a∧−b∧a∧2,k=2,3,…,n,x∧0k=x∧r−rk=∑i=0k−1−1iΓr+1Γi+1Γr−i+1x∧rk−i,k=2,3,…,x∧01=x01.For the abovementioned nonlinear programming model, the particle swarm optimization algorithm [18] (PSO) is used to solve the best parameters of the proposed FBNGM (1, 1, k) model in this study.


## 3.3. Modeling Procedure of the FBNGM (1, 1,k) Model

The modeling procedure of the FBNGM (1, 1,k) model is presented clearly in Figure 1.Figure 1
Modeling procedure of the FBNGM (1, 1,k) model.


## 4. Empirical Applications

In this section, two cases will be used to verify the feasibility and effectiveness of the proposed FBNGM (1, 1,k) model.


### 4.1. Case 1

China is the most electricity consuming country in the world, accounting for 27.5% of global electricity consumption. The shortage of power supply will have a serious negative impact on the stable development of China’s economy. Therefore, the ability to reasonably predict China’s electricity consumption plays an important role in formulating the Chinese government’s energy policy. The national electricity consumption data from 2010 to 2016 are used for modeling in this study (http://www.stats.gov.cn/tjsj/ndsj/2020/indexch.htm), while the data from 2017 to 2020 are used to measure the accuracy of the model. In order to illustrate the superiority of the FBNGM (1, 1, k) model in simulation and prediction, the NGM (1, 1, k) model (abbreviated as NGM), BNGM (1, 1, k) model (abbreviated as BNGM), FNGM (1, 1, k) model (abbreviated as FNGM), and FBNGM (1, 1, k) model (abbreviated as FBNGM) together are used to simulate and predict the electricity consumption in China. The selection or optimization results of hyperparameters as well as estimation results of system parameters for all models are given in Table 1. The seeking process of hyperparameters based on PSO is shown in Figure 2. The simulation and prediction results of all models are given in Table 2 and Figures 3 and 4.Table 1
The parameters for all models in Case 1.
ModelParameterrwabcNGM1.00000.50000.00413.05164.2635BNGM1.00000.46200.00433.06694.2623FNGM0.20910.50000.91096.30854.8645FBNGM0.89860.5017−0.2269−8.98174.1691Figure 2
Track of the seeking process of PSO algorithms for the three models in Case 1.Table 2
Simulation and prediction results of China’s electricity consumption (billion kilowatt hours).
YearActual valueNGMBNGMFNGMFBNGMForecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)2010419241920.0041920.0041920.0041920.002011469346950.0446930.0046800.2846800.282012495949800.4349790.3950000.8249950.732013532252651.0852631.1152691.0052651.082014552355480.4555460.4255300.1255270.062015580258300.4958280.4558040.0458020.002016613061110.3061090.3461120.2961070.38Modeling period (2010–2016)MAPE (%)0.46530.45990.53930.4220RMSE (10−2)0.30320.30330.33370.2981Testing period (2017–2020)2017648263911.4063881.4564730.1464560.402018715166706.7366666.7869093.3968673.972019728569474.6469424.7174472.2273601.022020751172243.8272183.9181228.1479575.94MAPE (%)4.14854.17683.12382.8328RMSE (10−2)3.30293.32192.61232.6727Figure 3
Simulation and prediction results for the four grey models in Case 1.Figure 4
Comparison of simulation and prediction accuracy for the four grey models in Case 1.In the case, the estimation results and search process of the unknown parameters of the models are given in Table1 and Figure 2. From Table 2 and Figures 3 and 4, it is easy to see that the simulation accuracy of the FBNGM (1, 1, k) model is higher than that of the other three prediction models, and the prediction accuracy of the FBNGM (1, 1, k) model is higher than that of the NGM (1, 1, k) model and the BNGM (1, 1, k) model. According to the accuracy measure indicator, the proposed FBNGM (1, 1, k) model has good performance in simulation and prediction, which shows that the FBNGM (1, 1, k) model is suitable and effective for predicting China’s electricity consumption.


### 4.2. Case 2

Agriculture is the basic industry to support the construction and development of national economy, and accurately predicting the development trend of agricultural production is the premise and foundation for formulating national economic plan and social development strategy. China is a large population country; its agriculture is related to national livelihood and social stability and has a fundamental position and role in the national economy. In the study, country agriculture gross output value data in 2007–2014 (http://data.stats.gov.cn/easyquery.htm) is used to establish the model, and the data in 2015–2018 is used to test the prediction accuracy. In order to illustrate the superiority of the FBNGM (1, 1, k) model in simulation and prediction, the NGM (1, 1, k) model (abbreviated as NGM), BNGM (1, 1, k) model (abbreviated as BNGM), FNGM (1, 1, k) model (abbreviated as FNGM), and FBNGM (1, 1, k) model (abbreviated as FBNGM) together are used to simulate and predict the agriculture gross output value in China. The selection or optimization results of hyperparameters as well as estimation results of system parameters for all models are given in Table 3. The seeking process of hyperparameters based on PSO is shown in Figure 5. The simulation and prediction results of all models are given in Table 4 and Figures 6 and 7.Table 3
The parameters for all models in Case 2.
ModelParameterrwabcNGM1.00000.50000.00594.50832.4532BNGM1.00000.62320.00324.39292.4631FNGM0.12300.5000−0.2754−1.98031.2934FBNGM0.10410.5017−0.2749−1.85957.7389Figure 5
Track of the seeking process of PSO algorithms for the three models in Case 2.Table 4
Simulation and prediction results of agriculture gross output value in China (100 million yuan).
YearActual valueNGMBNGMFNGMFBNGMForecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)200727674276740.00276740.00276740.00276740.00200832464310444.38310835.32311664.00312423.76200933584353575.28353693.15356276.09356586.18201038431396463.16396421.96401634.51401554.49201144781439091.95439021.91445860.44445590.49201249085481471.91481481.22488070.56487770.62201353028523611.26523801.75527560.51527220.58201455626565501.66565994.25563511.30562861.19Modeling period (2007–2014)MAPE (%)2.79902.79532.48682.4885RMSE (10−2)11.696311.699711.697411.8765Testing period (2015–2018)201557774607145.09608045.24594932.97593242.68201660139648547.84649968.08620533.18616342.492017620996897011.066917511.39638602.84629291.342018647347306212.877334013.30646910.07628082.97MAPE (%)9.21509.50272.26492.7201RMSE (10−2)60.715462.632215.589217.7565Figure 6
Simulation and prediction results for the four grey models in Case 2.Figure 7
Comparison of simulation and prediction accuracy for the four grey models in Case 2.In the case, the estimation results and search process of the unknown parameters of the models are given in Table3 and Figure 5. From Table 4 and Figures 6 and 7, it is easy to see that the simulation accuracy of the FBNGM (1, 1, k) model is higher than that of the other three prediction models, and the prediction accuracy of the FBNGM (1, 1, k) model is higher than that of the NGM (1, 1, k) model and the BNGM (1, 1, k) model. According to the accuracy measure indicator, the proposed FBNGM (1, 1, k) model has good performance in simulation and prediction, which shows that the FBNGM (1, 1, k) model is suitable and effective for predicting the agriculture gross output value in China.


## 4.1. Case 1

China is the most electricity consuming country in the world, accounting for 27.5% of global electricity consumption. The shortage of power supply will have a serious negative impact on the stable development of China’s economy. Therefore, the ability to reasonably predict China’s electricity consumption plays an important role in formulating the Chinese government’s energy policy. The national electricity consumption data from 2010 to 2016 are used for modeling in this study (http://www.stats.gov.cn/tjsj/ndsj/2020/indexch.htm), while the data from 2017 to 2020 are used to measure the accuracy of the model. In order to illustrate the superiority of the FBNGM (1, 1, k) model in simulation and prediction, the NGM (1, 1, k) model (abbreviated as NGM), BNGM (1, 1, k) model (abbreviated as BNGM), FNGM (1, 1, k) model (abbreviated as FNGM), and FBNGM (1, 1, k) model (abbreviated as FBNGM) together are used to simulate and predict the electricity consumption in China. The selection or optimization results of hyperparameters as well as estimation results of system parameters for all models are given in Table 1. The seeking process of hyperparameters based on PSO is shown in Figure 2. The simulation and prediction results of all models are given in Table 2 and Figures 3 and 4.Table 1
The parameters for all models in Case 1.
ModelParameterrwabcNGM1.00000.50000.00413.05164.2635BNGM1.00000.46200.00433.06694.2623FNGM0.20910.50000.91096.30854.8645FBNGM0.89860.5017−0.2269−8.98174.1691Figure 2
Track of the seeking process of PSO algorithms for the three models in Case 1.Table 2
Simulation and prediction results of China’s electricity consumption (billion kilowatt hours).
YearActual valueNGMBNGMFNGMFBNGMForecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)2010419241920.0041920.0041920.0041920.002011469346950.0446930.0046800.2846800.282012495949800.4349790.3950000.8249950.732013532252651.0852631.1152691.0052651.082014552355480.4555460.4255300.1255270.062015580258300.4958280.4558040.0458020.002016613061110.3061090.3461120.2961070.38Modeling period (2010–2016)MAPE (%)0.46530.45990.53930.4220RMSE (10−2)0.30320.30330.33370.2981Testing period (2017–2020)2017648263911.4063881.4564730.1464560.402018715166706.7366666.7869093.3968673.972019728569474.6469424.7174472.2273601.022020751172243.8272183.9181228.1479575.94MAPE (%)4.14854.17683.12382.8328RMSE (10−2)3.30293.32192.61232.6727Figure 3
Simulation and prediction results for the four grey models in Case 1.Figure 4
Comparison of simulation and prediction accuracy for the four grey models in Case 1.In the case, the estimation results and search process of the unknown parameters of the models are given in Table1 and Figure 2. From Table 2 and Figures 3 and 4, it is easy to see that the simulation accuracy of the FBNGM (1, 1, k) model is higher than that of the other three prediction models, and the prediction accuracy of the FBNGM (1, 1, k) model is higher than that of the NGM (1, 1, k) model and the BNGM (1, 1, k) model. According to the accuracy measure indicator, the proposed FBNGM (1, 1, k) model has good performance in simulation and prediction, which shows that the FBNGM (1, 1, k) model is suitable and effective for predicting China’s electricity consumption.


## 4.2. Case 2

Agriculture is the basic industry to support the construction and development of national economy, and accurately predicting the development trend of agricultural production is the premise and foundation for formulating national economic plan and social development strategy. China is a large population country; its agriculture is related to national livelihood and social stability and has a fundamental position and role in the national economy. In the study, country agriculture gross output value data in 2007–2014 (http://data.stats.gov.cn/easyquery.htm) is used to establish the model, and the data in 2015–2018 is used to test the prediction accuracy. In order to illustrate the superiority of the FBNGM (1, 1, k) model in simulation and prediction, the NGM (1, 1, k) model (abbreviated as NGM), BNGM (1, 1, k) model (abbreviated as BNGM), FNGM (1, 1, k) model (abbreviated as FNGM), and FBNGM (1, 1, k) model (abbreviated as FBNGM) together are used to simulate and predict the agriculture gross output value in China. The selection or optimization results of hyperparameters as well as estimation results of system parameters for all models are given in Table 3. The seeking process of hyperparameters based on PSO is shown in Figure 5. The simulation and prediction results of all models are given in Table 4 and Figures 6 and 7.Table 3
The parameters for all models in Case 2.
ModelParameterrwabcNGM1.00000.50000.00594.50832.4532BNGM1.00000.62320.00324.39292.4631FNGM0.12300.5000−0.2754−1.98031.2934FBNGM0.10410.5017−0.2749−1.85957.7389Figure 5
Track of the seeking process of PSO algorithms for the three models in Case 2.Table 4
Simulation and prediction results of agriculture gross output value in China (100 million yuan).
YearActual valueNGMBNGMFNGMFBNGMForecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)Forecasting valueAPE (%)200727674276740.00276740.00276740.00276740.00200832464310444.38310835.32311664.00312423.76200933584353575.28353693.15356276.09356586.18201038431396463.16396421.96401634.51401554.49201144781439091.95439021.91445860.44445590.49201249085481471.91481481.22488070.56487770.62201353028523611.26523801.75527560.51527220.58201455626565501.66565994.25563511.30562861.19Modeling period (2007–2014)MAPE (%)2.79902.79532.48682.4885RMSE (10−2)11.696311.699711.697411.8765Testing period (2015–2018)201557774607145.09608045.24594932.97593242.68201660139648547.84649968.08620533.18616342.492017620996897011.066917511.39638602.84629291.342018647347306212.877334013.30646910.07628082.97MAPE (%)9.21509.50272.26492.7201RMSE (10−2)60.715462.632215.589217.7565Figure 6
Simulation and prediction results for the four grey models in Case 2.Figure 7
Comparison of simulation and prediction accuracy for the four grey models in Case 2.In the case, the estimation results and search process of the unknown parameters of the models are given in Table3 and Figure 5. From Table 4 and Figures 6 and 7, it is easy to see that the simulation accuracy of the FBNGM (1, 1, k) model is higher than that of the other three prediction models, and the prediction accuracy of the FBNGM (1, 1, k) model is higher than that of the NGM (1, 1, k) model and the BNGM (1, 1, k) model. According to the accuracy measure indicator, the proposed FBNGM (1, 1, k) model has good performance in simulation and prediction, which shows that the FBNGM (1, 1, k) model is suitable and effective for predicting the agriculture gross output value in China.


## 5. Conclusion

In order to further improve the prediction accuracy of the NGM (1, 1,k) model, this study optimizes the model from the weight of the background value and fractional order accumulation generation. Then, the parameter solution and modeling process of the proposed FBNGM (1, 1, k) model is described in detail. In order to verify the feasibility and effectiveness of the FBNGM (1, 1, k) model proposed in this study, the novel model and other three grey forecasting models together are applied to the two actual cases with economic significance. It is obvious from the empirical results of the application examples that the simulation and prediction accuracy of the FBNGM (1, 1, k) model are higher than NGM (1, 1, k) and BNGM (1, 1, k) models. It confirms that the proposed FBNGM (1, 1, k) model has certain effectiveness in improving prediction accuracy. Therefore, the thought of optimizing the NGM (1, 1, k) model proposed in the study based on the weight of the background value and fractional order accumulation generation has a certain application value in the prediction of electricity consumption and agriculture gross output value.



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*Source: 5406547-2021-09-03.xml*