# Forecasting Chinese Carbon Emissions from Fossil Energy Based on the Fractional Order Cumulative Multivariate Grey Model

Authors: Yunxin Zhang; Huan Guo; Xin Xiong
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the process of Chinese economic development shifting from a stage of high-speed growth to a stage of high-quality development, carbon emissions caused by fossil energy consumption are also increasing. To reflect the interaction between Chinese carbon emissions from fossil energy consumption and economic growth, this paper constructed the fractional order cumulative multivariate gray model with equal-dimensional recursive optimization (EFMGM(1,2)) to predict the trend of carbon emissions from fossil energy consumption and constant price GDP in China and calculated the carbon emissions from fossil energy consumption according to the IPCC accounting method. EFMGM(1,2) is compared with other gray models to verify the performance of the model. The results show that this model has better prediction performance than other models. It can be used as an effective method for forecasting carbon emissions from fossil energy.


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## Body

## 1. Introduction

With the continuous development of the economy, energy consumption is increasing and global warming is becoming more and more serious, which has become a common concern for all countries and people in the world. Since the reform and opening, China’s economy has shown rapid growth. However, more energy consumption and large amounts of carbon dioxide emissions and other pollutants have caused a series of environmental and ecological problems, especially the problem of high carbon emissions from fossil energy consumption has become increasingly prominent. High carbon emissions are a direct contributor to global warming, which causes different impacts and damages, leading to climate warming, greenhouse effect, and extreme weather, including hurricanes, flood disasters, and extreme heat. It leads to accelerated global warming and faster melting of glaciers, resulting in an annual rise in the sea level. In addition, high carbon emissions lead to massive emissions of greenhouse gases, which also endanger the human quality of life and health.According to the 2021 BP Statistical Review of World Energy, until 2019, carbon emissions had continued their unrelenting rise. Although global carbon emissions are growing at a slower rate than the global economy, they have yet to see signs of a peak. The global energy system is still dominated by fossil fuels and their consumption continues to grow. For China, the energy and petrochemical industry is a key area to achieve the carbon peaking and carbon neutrality goals. According to the statistics of the National Energy Administration, fossil energy combustion currently accounts for about 84% of China’s carbon emission sources. The use of energy, especially the consumption of fossil energy, is the main reason for the rapid increase of CO2 emissions.GDP and carbon emissions have different interactions at different time periods. There is a long-term equilibrium cointegration relationship and a short-term dynamic adjustment mechanism between total carbon emissions from fossil energy consumption and GDP, and carbon emissions from fossil energy are a one-way Granger causality of GDP [1]. Economic development has contributed exponentially to pulling China’s carbon emissions per capita [2]. However, economic development has a significant negative effect on carbon emissions from energy consumption, with an inverted “U” curve between economic growth and carbon emissions from energy consumption [3]. The relationship between carbon emissions and economic growth is a special form of environmental quality and economic development. Ma et al. [4] used the Kaya-LMDI method to construct a decomposition model of carbon emissions of major energy consumption in three northeastern provinces from 2005 to 2016, and found that the economic output effect and population size effect played a pulling role in the growth of carbon emissions in three northeastern provinces. Zhao et al. [5] found that energy consumption and economic growth are mutually important influences. Yin et al. [6] revealed a significant inverted U-shaped EKC relationship between electrification and carbon intensity in China. Xiao et al. [7]found that the relationship between renewable energy consumption and regional CO2 emissions in all six regions of China shows an inverted N-shaped EKC relationship. Wang et al. [8] found that the relationship between CO2 emissions and economic development satisfies the inverted “U” curve in the EKC hypothesis. Qiu et al. [9]analyzed typical Chinese cities through the Tapio decoupling model and found that the overall carbon intensity decoupling status of typical Chinese cities showed a strong decoupling, and predicted that each city would achieve its peak target around 2023. The overall conclusion of these studies is that there is a long-run or short-run interaction between carbon emissions and economic growth, an interactive relationship, a non-linear relationship, and an autocorrelation characteristic between carbon emissions and economic growth.Other scholars have used support vector machines and neural networks to predict carbon emissions. Sun and Liu [10] used least squares support vector machines to predict CO2 emissions from three major industries as well as residential consumption, Huang et al. [11] used gray correlation analysis as well as principal component analysis to select the influencing factors of carbon emissions. Long- and short-term memory methods were used to predict the carbon emissions in China. Wang et al. [12] constructed an extreme learning machine model based on the improved whale optimization algorithm to predict carbon emissions and carbon emission intensity. Soytas et al. [13] investigated the Granger causality between income, energy consumption, and carbon emissions and found that energy consumption contributed to the U.S. carbon emissions. Zou et al. [14] used a time series vector error correction model and found that oil price fluctuation is the reason why carbon emissions change, while GDP fluctuation is not the reason for the growth of carbon emissions.Meanwhile, other scholars applied gray system theory to study the relationship between carbon emissions and economic growth, the field of carbon emissions prediction, including univariate gray models and multivariate gray models, and gray correlation analysis. Wang et al. [15] introduced a non-linear correction term for the non-linear relationship between carbon emissions from fossil energy and GDP, a power exponential term and a gray action quantity to improve the GM(1, n) model. Ding et al. [16] constructed a gray multivariate model based on the trend of driving variables by considering the influence of the trend of driving variables on the behavior of the system and investigated China’s CO2 emissions. Ding et al. [17] used gray correlation analysis to screen the influencing factors that have strong non-linear effects on carbon emissions and introduced the power exponent term in the model structure to improve the DGM(1, n) model and optimize the parameter estimation and solution methods. Wang et al. [18] used the GM(1,1) model to forecast the per capita carbon emissions in Jilin province and analyzed the relevant influencing factors of carbon emissions using gray correlation analysis. Xiong et al. [19] used gray correlation analysis to conduct a study on the influencing factors of carbon emissions in East China. Zeng et al. [20] considered the new information priority principle and used the new information priority-based principle. A new gray rolling mechanism model based on the new information priority principle was used to forecast and analyze CO2 emission trends. Wang et al. [21] used Simpson's formula and Boolean values to optimize the background values of MGM(1,n) and used rolling forecasts to predict economic growth, energy consumption, and urbanization levels. Zhang et al. [22] introduced the power exponential term, linear correction term, and random disturbance term on the basis of GM(1, n) model to establish a new flexible gray multivariate model for the complexity and uncertainty of energy consumption system. Xiong et al. [23] considered the strong seasonal characteristics of residential electricity consumption and constructed a whale optimization algorithm based on an improved hybrid optimization gray seasonal change index model to fit residential electricity consumption in four Chinese cities as well as the United States. Liu et al. [24] created a fractional-order cumulative linear time-varying parameter discrete gray prediction model to forecast China’s energy consumption.Currently, China is entering a “new normal” of economic growth. More stringent planning and policy support are needed to improve ecological sustainability and for sustainable economic development. The problem of carbon dioxide emissions and other forms of environmental damage is becoming increasingly serious in China. To achieve sustainable economic development, the Chinese government has developed and implemented a number of policies, such as establishing a low-carbon energy system, controlling emissions from the building and transportation sectors, and promoting low-carbon lifestyles. These policies and plans have been effective in reducing the environmental pollution. Therefore, it is necessary to analyze carbon emissions and economic growth, as well as carbon emission projections of fossil energy consumption in China.Protecting the ecological environment and developing the economy are fundamentally organic and complementary. GDP and carbon emissions from fossil energy have different interactions at different time periods, while the variables also have autocorrelation characteristics. MGM(1,n) is a state model that is used to reflect the interconnections between GDP and carbon emissions from fossil energy in the system as well as the autocorrelation between GDP and fossil energy emissions in the system. The MGM(1, n) model was proposed by Zhai et al. [25], and Luo [26] extended the fractional order accumulation to the construction of the MGM model. Wu et al. [27] proposed the GM(1,1) with fractional Hausdorff accumulation model which will not be affected by the initial value. Zeng [28] used a fractional order multivariate gray model to forecast and analyze the output value of high-tech products in Guangdong Province, China.In view of this, this paper uses data from the China Statistical Yearbook and China Energy Statistical Yearbook to predict the development trend of carbon emissions from fossil energy consumption in China based on a fractional order cumulative gray prediction model, to provide prediction support for the carbon peaking and carbon neutrality goals.To better describe the relationship between China’s economic development and carbon emissions, this paper uses the fractional cumulative multivariate gray prediction model optimized by equal-dimensional recursive optimization model, abbreviated as EFMGM(1,n)), to predict the development trend of GDP and carbon emissions from fossil energy. The main contributions of this paper are summarized as follows:(1)
Considering that GDP and carbon emissions have different interactions in different periods, this paper constructs an equal-dimensional recursive multivariate gray model to predict economic development and carbon emissions from fossil energy.(2)
To improve the prediction accuracy, a particle swarm algorithm is used to optimize the fractional order and background value of the model.The rest of this paper is organized as follows: Section2 is the model design, including the construction of the fractional order cumulative multivariate gray model with equal-dimensional recurrence optimization and parameter identification. Section 3 is an empirical study, the model proposed in this paper simulates and predicts China’s constant price GDP and carbon emissions from fossil energy consumptionand compares with other models to verify the performance of the prediction model. Section 4 is the conclusion and prospects.


## 2. The Construction of the Model

### 2.1. Fractional Order Multivariate Gray Model Optimized by Equal-Dimensional Recursive

Assume that the sequencex0=x01,…,x0j,…,x0n is an original non-negative data sequence, the sequence xr=xr1,…,xrj,…,xrn is the r-order accumulated generation sequence of the original data sequence, which is written as [22](1)xirk=∑j=1kΓr+k−jΓk−j+1Γrxi0j,k=1,2,…,m,a system of differential equations is established as follows:(2)dx1r1dt=a11x1r1+a12x2r2+⋯+a1nxnrn+b1,dx2r2dt=a21x1r1+a22x2r2+⋯+a2nxnrn+b2,⋮dxnrndt=an1x1r1+an2x2r2+⋯+annxnrn+bn,let X0k=x10k,x20k,…,xn0kT,Xrk=x1r1k,x2r2k,…,xnr3kT,k=1,2,…,m(3)A=a11a12…a1na21a22…a2n⋮⋮⋱⋮an1an2⋯amnB=b1b2⋮bn,so equation (2) is written as follows:(4)dXrdt=AXr+B.Its continuous time response equation is(5)Xrt=eAtXt1+A−1eAt−IB,where(6)eAt=I+At+A22!t2+⋯=I+∑k=1∞Akk!tk.Discretization(7)xirik−xirik−1=∑j=1naijuxirik+1−uxirik−1+bi,i=1,2,…,n;k=2,3,…,m,the value of the model parametersai=ai1,ai2,…,ain,biT,i=1,2,…,n can be recognized by the least squares method as follows:(8)a^i=a^i1,a^i2,…,a^in,b^iT=LTL−1LTYi,i=1,2,…,n,where(9)L=z1r12z2r22⋯znrn21z1r13z2r23⋯znrn31⋮⋮⋱⋮⋮z1r1mz2r2m⋯znrnm1,(10)Yi=xiri2−xiri1,xiri3−xiri2,…,xirim−xirim−1T,i=1,2,…,n,ziri=uxirik+1−uxirik−1,0≤u≤1,k=2,3,…,n.The time response function of the EFMGM(1,n) is(11)X^irik=eA^j−1Xiri1+A^−1eA^k−1−IB^,k=1,2,…,m;thus, the restored values ofx^i0k is(12)x^i0k=∑j=0kΓk−ri−jΓk−j+1Γ−rix^irij,k=1,2,…,m,whenri=1, the model is the multivariate gray model MGM(1,n). When ri=1 and n=1, the model is the gray model GM (1,1).The absolute percentage error (APE) and the mean absolute percentage error (MAPE) were defined to evaluate the accuracy of the model and to compare the results with those of the multivariate gray model.0.(13)APEk=x^i0k−xi0kxi0k×100%,MAPE=1nm−1∑i=1n∑k=2mx^i0k−xi0kxi0k×100%,


### 2.2. Determination of Optimal Fractional Order Value and Background Value

The fractional order valueri and the background value u as the design variables and MAPE as the objective function are used to solve the EFMGM(1,n) using the particle swarm optimization algorithm.The optimization of the background value can reduce the fitting and prediction errors of the model, and the optimization of the fractional order can make the fractional order cumulative data better meet the EFMGM(1,n) model design and comply with the new information priority principle.The objective function is defined as(14)MinMAPE=1nm−1∑i=1n∑k=2mx^i0k−xi0kxi0k×100%s.t.a^i=a^i1,a^i2,…,a^in,b^iT=LTL−1LTYi,i=1,2,…,nX^irik=eA^k−1Xiri1+A^−1eA^k−1−IB^,k=1,2,…,mx^i0k=∑j=0kΓk−ri−jΓk−j+1Γ−rix^irij,k=1,2,…,m.The above non-linear optimization model can be solved by particle swarm optimization algorithm. Once the fractional order valueri and the background value u are determined, the other parameters A and B can be determined. Then, the optimal values of these parameters are introduced into the EFMGM(1, n) model to obtain simulation and prediction results. The pseudo code of the particle swarm optimization algorithm is shown in Table 1.


## 2.1. Fractional Order Multivariate Gray Model Optimized by Equal-Dimensional Recursive

Assume that the sequencex0=x01,…,x0j,…,x0n is an original non-negative data sequence, the sequence xr=xr1,…,xrj,…,xrn is the r-order accumulated generation sequence of the original data sequence, which is written as [22](1)xirk=∑j=1kΓr+k−jΓk−j+1Γrxi0j,k=1,2,…,m,a system of differential equations is established as follows:(2)dx1r1dt=a11x1r1+a12x2r2+⋯+a1nxnrn+b1,dx2r2dt=a21x1r1+a22x2r2+⋯+a2nxnrn+b2,⋮dxnrndt=an1x1r1+an2x2r2+⋯+annxnrn+bn,let X0k=x10k,x20k,…,xn0kT,Xrk=x1r1k,x2r2k,…,xnr3kT,k=1,2,…,m(3)A=a11a12…a1na21a22…a2n⋮⋮⋱⋮an1an2⋯amnB=b1b2⋮bn,so equation (2) is written as follows:(4)dXrdt=AXr+B.Its continuous time response equation is(5)Xrt=eAtXt1+A−1eAt−IB,where(6)eAt=I+At+A22!t2+⋯=I+∑k=1∞Akk!tk.Discretization(7)xirik−xirik−1=∑j=1naijuxirik+1−uxirik−1+bi,i=1,2,…,n;k=2,3,…,m,the value of the model parametersai=ai1,ai2,…,ain,biT,i=1,2,…,n can be recognized by the least squares method as follows:(8)a^i=a^i1,a^i2,…,a^in,b^iT=LTL−1LTYi,i=1,2,…,n,where(9)L=z1r12z2r22⋯znrn21z1r13z2r23⋯znrn31⋮⋮⋱⋮⋮z1r1mz2r2m⋯znrnm1,(10)Yi=xiri2−xiri1,xiri3−xiri2,…,xirim−xirim−1T,i=1,2,…,n,ziri=uxirik+1−uxirik−1,0≤u≤1,k=2,3,…,n.The time response function of the EFMGM(1,n) is(11)X^irik=eA^j−1Xiri1+A^−1eA^k−1−IB^,k=1,2,…,m;thus, the restored values ofx^i0k is(12)x^i0k=∑j=0kΓk−ri−jΓk−j+1Γ−rix^irij,k=1,2,…,m,whenri=1, the model is the multivariate gray model MGM(1,n). When ri=1 and n=1, the model is the gray model GM (1,1).The absolute percentage error (APE) and the mean absolute percentage error (MAPE) were defined to evaluate the accuracy of the model and to compare the results with those of the multivariate gray model.0.(13)APEk=x^i0k−xi0kxi0k×100%,MAPE=1nm−1∑i=1n∑k=2mx^i0k−xi0kxi0k×100%,


## 2.2. Determination of Optimal Fractional Order Value and Background Value

The fractional order valueri and the background value u as the design variables and MAPE as the objective function are used to solve the EFMGM(1,n) using the particle swarm optimization algorithm.The optimization of the background value can reduce the fitting and prediction errors of the model, and the optimization of the fractional order can make the fractional order cumulative data better meet the EFMGM(1,n) model design and comply with the new information priority principle.The objective function is defined as(14)MinMAPE=1nm−1∑i=1n∑k=2mx^i0k−xi0kxi0k×100%s.t.a^i=a^i1,a^i2,…,a^in,b^iT=LTL−1LTYi,i=1,2,…,nX^irik=eA^k−1Xiri1+A^−1eA^k−1−IB^,k=1,2,…,mx^i0k=∑j=0kΓk−ri−jΓk−j+1Γ−rix^irij,k=1,2,…,m.The above non-linear optimization model can be solved by particle swarm optimization algorithm. Once the fractional order valueri and the background value u are determined, the other parameters A and B can be determined. Then, the optimal values of these parameters are introduced into the EFMGM(1, n) model to obtain simulation and prediction results. The pseudo code of the particle swarm optimization algorithm is shown in Table 1.


## 3. Empirical Studies

The data sources in this paper are as follows: constant price GDP and energy consumption data in China are from the China Statistical Yearbook; carbon emission factors related data are from IPCC Guidelines for National Greenhouse Gas Emission Inventories and China Energy Statistical Yearbook.The calculation formula is(15)Carbon emission factor=IPCCCO2emission factor/Avearage Low calorific value∗12/44Conversion Factor.The calculated carbon emission factors are multiplied by the corresponding energy consumption standard coal equivalent to calculate the carbon emissions from each energy source, and then the carbon emissions from each energy source are summed to obtain the carbon emissions from fossil energy consumption in China.The Carbon Emission factors of fossil energy consumption in China are shown in Table 2.In this paper, constant price GDP (billion RMB) is chosen and converted to the price level with 1978 as the base period to reduce the effect of inflation. Constant price GDP is calculated from the GDP index (1978 = 100).Overall, the total carbon emissions of fossil energy consumption in China from 1985 to 2020 basically showed a continuous growth trend. It is mainly divided into three stages, 1985–2001 as a stable development stage, 2002–2013 as a rapid growth stage, and 2014–2020 as a somewhat declining but growing stage again. Overall, the GDP shows a continuous growth trend from 1985 to 2020.To avoid the pathological nature of the matrix in the identification of the model parameters, the data need to be preprocessed to build the original data before the multivariate gray model. In this paper, we use the initial value transformation to solve this problem and preprocess the Chinese constant price GDP and carbon emissions from fossil energy consumption with the same dimensionality and magnitude of the time series data to better identify the parameters of the model.Starting from the theory of gray modeling, we know that gray forecasting only needs fewer observations to build the model, and too much sample data will reduce the adaptability of the gray forecasting model. To improve the adaptability of the gray model to long-term trends, an equal-dimensional recursive sample partitioning method is used for the sample data. For the observation data from 1985 to 2017, while considering that the gray model solution requires the first data point as the initial condition, which will lead to the loss of a sample point, we divide the last observation point of the previous stage as the first data point of the next stage into 8 stages, and finally, the 4 years of observation data contained in each stage as well as the data of the previous year to construct EFMGM(1,n).For the different characteristics of the two types of data in China: constant price GDP and carbon emissions from fossil energy, GDP data is growing year by year, while carbon emissions from fossil energy consumption are influenced by relevant policies and show certain fluctuations. Therefore, the two variables should be accumulated using different fractional orders to reach the effect of new information priority.Based on the existence of long-term coordination and short-term dynamic adjustment mechanism between carbon emissions and economic growth, which has nonlinear relationship and autocorrelation characteristics, this paper constructs a MGM(1,2) model to reflect the interaction between carbon emissions of fossil energy consumption and economic growth. Then, this paper constructs a fractional order MGM(1,2) model with background value optimization based on the new information priority principle. Finally, considering the different characteristics of carbon emissions and economic growth in different time stages, this paper constructs the EFMGM(1,2) model, which is the variable-order multivariate gray model. In addition, the GM(1,1) model with equal-dimensional recursive optimization is also used as a comparison model.This section presents detailed results for accurate prediction of different types of carbon emissions Tables1 and 2 from fossil energy consumption in China using classical gray models as benchmark models, as well as EFMGM(1,2) model. As shown in Table 3 and Figure 1, the superiority of the EFMGM(1,2) model constructed in this paper in terms of prediction accuracy is demonstrated. In the fitting and prediction of carbon emissions from fossil energy consumption, the MAPE of the EFMGM(1,2) model in the fitting part is only 0.1749%, while the MAPE of the MGM(1,2) model in the fitting part is 12.4077%, the MAPE of the FMGM(1,2) model in the fitting part is 8.0662%, and the MAPE of the EGM(1,1) model in the fitting part is 0.8156%. The EFMGM(1,2) model has better results than the conventional MGM(1,2) model and the FMGM(1,2) model in the fitting part. In the forecasting part, the EFMGM(1,2) model achieves a one-step forecasting error of 0.6686%, the MAPE in the forecasting part is 0.6002%, and the MAPE is less than 10%, which is a better forecasting result.Table 1
The pseudo code of PSO.
Algorithm. Particle swarm optimization to solve the fractional order value and background valueInput: the sample setOutput: the optimal value of the fractional order valueri and the background value ufor each particle iInitialize velocityVi and position Xi for particle iEvaluate particlei and set pBesti = Xiend forgBest = min {pBesti}while not stopfori = 1 to NUpdate the velocity and position for particleiEvaluate particleiFitness ← objective function in Eq. (10)if Fitness (Xi) < Fitness (pBesti)pBesti = Xi;if Fitness (pBesti) < Fitness (gBest)gBest = pBesti;end forend whilereturn optimal value ofri and uTable 2
Carbon Emission factors of fossil energy consumption in China.
EnergyCarbon emission factors (104 t/104 t coal equivalent)Raw coal0.7552Crude oil0.5851Natural gas0.4481Table 3
Actual carbon emissions and MGM(1,2), FMGM(1,2), EGM(1,1), and EFMGM(1,2) predictions.
YearActual dataMGM(1,2)FMGM(1,2)EFMGM(1,2)EGM(1,1)ForecastsAPE (%)ForecastsAPE (%)ForecastsAPE (%)ForecastsAPE (%)198552323.6652323.66NA52323.66NA52323.66NA52323.660.0000198655251.2237437.3332.241657713.004.455655196.230.099555644.390.7116198759285.1841025.7030.799459005.390.471959465.840.304759118.500.2812198863640.9744890.6329.462761302.403.674663636.040.007762809.511.3065198966202.0749046.9425.913364127.883.133166482.440.423566730.960.7989199067314.7553509.4920.508567335.320.030667334.120.028867279.580.0523199170958.0458292.9917.848670867.170.128170836.210.171770878.400.1122199274518.0463411.9014.904074702.490.247574675.930.211974669.720.2035199378774.4968880.2112.560378830.830.071578760.600.017678663.840.1405199483058.0674711.2410.049483252.180.233783059.820.002184548.321.7942199588390.6080917.418.454787977.000.467988345.590.050986962.101.6161199690922.9587509.913.753893005.312.290290984.530.067789444.801.6257199790601.7694498.464.300998358.028.560890598.540.003691998.381.5415199890589.62101890.8412.4752104045.6014.853890619.880.033490477.380.1239199993889.81109692.5716.8312110088.9817.253493473.840.443093694.590.2079200096392.21117906.3822.3194116503.8620.864496762.680.384397026.200.65772001100845.64126531.6825.4706123311.1722.2771100703.860.1406100476.280.36632002110307.57135563.9922.8964130531.8418.3344110244.460.0572111130.660.74622003129692.12144994.2711.7988138192.026.5539130010.200.2453128899.150.61142004151268.84154808.112.3397146312.653.2764151268.470.0003149508.601.16372005172937.96164984.974.5988154925.1310.4158173101.550.0946173413.260.27482006189426.58175497.187.3535164050.3813.3963189418.950.0040192223.511.47652007205684.00186308.959.4198173725.0215.5379203997.620.8199201063.552.24642008209330.09197375.155.7110183985.6912.1074209324.110.0029210310.120.46822009219274.39208640.094.8498194853.3211.1372218662.760.2789219981.930.32272010231653.34220036.075.0149206374.9910.9121231641.470.0051235880.001.82462011251212.15231481.807.8541218582.1012.9890250257.860.3799245566.022.24762012256677.96242880.705.3753231521.749.8007256682.600.0018255649.770.40062013263823.35254118.983.6784245235.777.0455261660.130.8199266147.600.88102014266952.14265063.550.7075259766.052.6919267015.610.0238265887.010.39902015267180.26275559.743.1363275164.902.9885266778.170.1505267785.430.22652016267755.79285428.756.6004291479.428.8602268308.350.2064269697.400.72512017273109.52294464.957.8194308772.3913.0581272793.340.1158271623.020.5443MAPE12.40778.06620.17490.81562018278532.87302432.808.5806327096.1417.4354280395.000.6686273562.391.78452019283740.20309063.658.9249346513.4522.1235286037.280.8096275515.612.89862020287434.59314052.119.2604367087.1127.7115288361.710.3225277482.773.4623MAPE8.922022.42350.60022.7151Figure. 1
Actual carbon emissions and predicted curves: MGM(1,2), FMGM(1,2), EGM(1,1), and EFMGM(1, 2).To make the results more intuitive and visual, the carbon emission curves of China's fossil energy consumption curves calculated by the four models are shown in Figure1.The EFMGM(1,2) model is constructed according to the characteristics of the original data series, which plays an important role in enriching and developing the theory and method of gray prediction. In general, the model in this paper can predict constant price GDP and carbon emissions well. To further test the performance of the model, the model in this paper is used to forecast and analyze constant price GDP and carbon emissions from fossil energy consumption in China for the next five years. The predicted results are shown in Table4.Table 4
Forecast value of Constant price GDP and Carbon emissions in the next five years.
Year20212022202320242025Constant price GDP (RMB hundred million)159115.35165833.83172155.88178123.23183788.04Carbon emissions (104 tons)292312.60295809.67298906.32301682.62304165.27As illustrated in Table4, the development trend of carbon emissions from fossil energy consumption in China still shows an upward trend. However, the growth rate of carbon emissions from fossil energy consumption in China is decreasing year by year, which is mainly caused by two aspects: firstly, the gradual increase of non-fossil energy consumption, which leads to a lower growth rate of carbon emissions; and secondly, the increase of energy utilization rate of coal, crude oil, and other fossil energy sources, which is also an important reason for the decrease in the growth rate of carbon emissions.


## 4. Summary and Prospects

In this paper, a fractional order cumulative multivariate gray prediction model with equal-dimensional recursive optimization is constructed and its modeling process and parameter solving methods are discussed. This model was applied to the prediction of carbon emissions from fossil energy consumption in China. Compared with other gray models, the new model has the smallest fitting and prediction errors and higher prediction accuracy, and the results show that the new model can better reflect the complex relationship between China's constant price GDP and carbon emissions from fossil energy consumption.From the fitted and forecasted trends, the development trend of carbon emissions from fossil energy consumption in China is still on an upward trend. Therefore, China should promote green development, build an ecological civilization system, promote a comprehensive green transformation of economic and social development, and build a modernization in which people and nature coexist harmoniously.The government should accelerate the promotion of green and low-carbon development; strengthen the legal and policy guarantee of green development, promote the clean, low-carbon, safe, and efficient use of energy; and improve the energy utilization rate of fossil energy. At the same time, the government should accelerate the development of non-fossil energy; promote the development and utilization of environmentally friendly energy; develop zero-carbon and low-carbon energy sources such as wind, solar, and biomass; promote the diversification of energy structure; realize the transformation of petrochemical raw materials to low carbon, lightweight, and diversification; adhere to the innovation drive as the first; and increase the investment in research of key core technologies. The government will improve the industrial structure in terms of economic growth and technological progress, reduce the growth rate of carbon emissions, and achieve sustainable development of economy and environment.For personal: start from yourself, from small daily things, actively promote low-carbon life, green travel, and reduce food waste.


### 4.1. Further Perspectives of the Study

(1)
The relationship between carbon emissions and economic growth in different provinces and cities and different regions is at different stages of development, so it is necessary to conduct fitting prediction and robustness research by provinces, cities, and regions.(2)
The model introduces fewer kinds of variables, and several control variables should be introduced and screened through the method of gray correlation analysis to construct the model.


## 4.1. Further Perspectives of the Study

(1)
The relationship between carbon emissions and economic growth in different provinces and cities and different regions is at different stages of development, so it is necessary to conduct fitting prediction and robustness research by provinces, cities, and regions.(2)
The model introduces fewer kinds of variables, and several control variables should be introduced and screened through the method of gray correlation analysis to construct the model.



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*Source: 5623519-2022-08-27.xml*