Estimating China’s CO₂ emissions under the influence of COVID-19 epidemic using a novel fractional multivariate nonlinear grey model

Abstract

The COVID-19 pandemic has posed a severe challenge to the global economic recovery and China’s economic growth. Although China has achieved stage victory against the epidemic, the impact and influence of the epidemic on China’s economy will linger. On the positive side, the epidemic has led to a dramatic reduction in air pollution levels and a sharp slowdown in greenhouse gas emissions such as CO₂. Forecasting China’s CO₂ emissions under the impact of the COVID-19 epidemic is crucial for formulating policies that contribute to smooth economic development and rational energy consumption. To this end, this paper improves on the traditional grey model, GM(1, N), by developing a novel fractional multivariate nonlinear grey model, FMNGM(q, N). These improvements include two key points. First, different power exponents are assigned to each relevant factor variable to explain their nonlinear, uncertain, and complicated relationships with the system characteristic variables. Second, the integer accumulation is changed to fractional accumulation to preprocess the data, and the Caputo-type fractional derivative is used to characterize the endogenous relationships among the system characteristic data. In addition, the quantum particle swarm optimization (QPSO) algorithm is used to optimize the above parameters with the goal of minimizing MAPE. The collected data on China’s CO₂ emission from 2001 to 2018 were divided into two parts according to different stages of economic development: 2001–2009 and 2010–2018. Both sets of data were modelled and analysed separately and compared with other models. Results showed that the proposed model had better prediction accuracy than other models. Finally, the model built using the 2010–2018 data was used to forecast China’s CO₂ emissions. Results show that China’s CO₂ emissions in 2020 under the impact of COVID-19 will decrease by approximately 3.15% from 2019 to 9893 million tons. The results bear important policy implications for planners in investment in clean energy and infrastructure to achieving the goal of a low-carbon transition.

Introduction

The continuous spread of COVID-19 worldwide has caused substantial economic losses to all countries (Ozili & Arun, 2020) and seriously threatened people’s lives. Until the pandemic is completely resolved, the shadow of the virus will linger in the future. The widespread and profound impact of COVID-19 has dampened the otherwise optimistic global economic outlook. Major economies in the world are facing considerable challenges. In contrast to its impact on the economy, the COVID-19 pandemic had a positive effect on improving global environmental issues (Saadat et al., 2020; Zambrano-Monserrate et al., 2020). During the epidemic, China and most other countries imposed a severe nationwide lockdown, and most economic activities were restricted, resulting in a short-term reduction in air pollution levels, a slowdown in greenhouse gas emissions such as CO₂, and a significant improvement in air quality (Arora et al., 2020; Mahato et al., 2020).

CO₂ emission is an essential manifestation of human economic activities. Owing to its massive impact on the climate and environment (Manabe & Wetherald, 1980), it has become a major concern in the international community. However, in modern times, due to human social productivity and various economic activities, CO₂ emission remains high, leading to frequent extreme weather and continuous climate disasters. To address this ordeal facing human society, taking resolute measures to reduce CO₂ emissions is necessary. The 2010 Chinese Government Work Report pointed out that 30 years of extensive economic development has caused an enormous toll on resources and the environment. Therefore, changing the economic development mode, promoting the transformation of China’s economy, and adopting sustainable development and low-carbon environmental protection are urgent matters. In the 13th Five-Year Plan, the Chinese government proposed controlling the increase in CO₂ emissions and reducing the atmosphere’s carbon intensity. Since September 2020, when China’s leaders proposed at the United Nations General Assembly that “carbon dioxide emissions should strive to peak by 2030, and strive to achieve carbon neutrality by 2060”, top-level design has been made frequently and intensive deployment has been made around CO₂ emissions. Based on these backgrounds, in order to better serve the overall situation of China’s carbon peak and carbon neutrality, and to propose positive and reasonable measures and countermeasures, it is especially crucial to study and forecast the situation of China’s CO₂ emissions, and to estimate the impact of the COVID-19 epidemic on China’s CO₂ emissions.

The COVID-19 pandemic has changed the global energy supply and demand pattern, resulting in a significant drop in global daily CO₂ emissions (Le Quéré et al., 2020). The prediction of CO₂ emissions in the context of the pandemic is crucial for studying the impact of the pandemic on the environment. Moreover, such predictions help formulate policies that contribute to environmental protection and green economic growth. At present, many models and methods have been applied to the prediction of CO₂ emissions. These models can be divided into three categories, statistical analysis models, intelligent models, and grey prediction models. Table 1 summarizes some methods that use statistical analysis models and intelligent models to predict CO₂ emissions. They are applied to predict CO₂ emissions in different regions and different sectors. The advantages of statistical analysis models are simplicity and ease of implementation. Given that most of the parameters have practical significance on different occasions and studies, the model has strong interpretability. These advantages make a wide range of applications possible. However, statistical analysis models inevitably have shortcomings. The simplicity of their models is often based on the premise of sufficient statistical assumptions (Wu et al., 2014). They are too idealistic, and their prediction accuracy is usually limited by the degree of satisfaction with these statistical assumptions. The idealized model is challenging to use in areas with high accuracy requirements. The development of intelligent models based on machine learning in recent years provides a possible solution to this problem. Intelligent models can achieve high prediction accuracy without relying on strict mathematical and statistical assumptions. Moreover, intelligent modes have strong generalization capabilities, showing significant advantages in prediction problems in many fields. However, an intelligent model’s dependence on computers and program packages makes it difficult to be applied without these tools. Besides, an intelligent model is challenging to perform well in cases with small samples of poor information.

Table 1.

Recent research on CO₂ emission forecasting using statistical analysis models and intelligent models

Model type	References	Model	Application
Statistical analysis model	Fang et al. (2018)	Gaussian process regression (GPR)	Total CO2 emissions of US, China, and Japan
	Xu and Lin (2015)	Vector autoregression (VAR)	CO2 emission reduction in China’s transport sector
	Nyoni and Bonga (2019)	ARIMA	CO2 emissions in India
	Aydin (2015)	Multiple linear regression (MLR)	Energy-related CO2 emissions in Turkey
	Meng and Niu (2011)	Logistic equation	CO2 emissions from fossil fuel combustion
	Hosseini et al. (2019)	Multiple polynomial regression (MPR)	CO2 emissions in Iran
Intelligent model	Xu et al., (2019a, 2019b)	Artificial neural network (ANN) and scenario analysis	Determining China’s ${\text{CO}}_2$ emissions peak
	Sun and Liu (2016)	Least squares support vector machine (LSSVM)	Three major industries and CO2 emissions from residential consumption
	Qiao et al. (2020)	Lion swarm optimizer	CO2 emission forecasting
	Wen and Cao (2020a)	Support vector machine (SVM) and improved chicken swarm optimization (ICSO)	Influencing factor analysis and forecasting of residential energy-related CO2 emissions
	Wen and Yuan (2020)	BP neural network (BPNN) based on random forest and PSO	CO2 emissions in China’s commercial industry
	Wen and Cao (2020b)	LSSVM and butterfly optimization algorithm (BOA)	Household CO2 emissions

Grey prediction theory is a mathematical method used to solve systems with incomplete information. It was first proposed by Deng (1982, 1989) in 1982. This theory provides a reliable solution to the prediction problems of small samples and inadequate information systems. In the past 30 years, grey prediction theory has been widely used in various fields such as industry, agriculture, transportation, and energy (Liu & Lin, 2006), which show strong vitality. At present, many grey prediction models have been applied to the CO₂ emission prediction problem. Table 2 summarizes these models.

Table 2.

Recent research on CO₂ emission forecasting using grey forecasting model

Author	Model	Application
Lu et al. (2009)	GM(1, 1)	CO2 emission from Taiwan’s road transportation sector
Meng et al. (2014)	Hybrid model of GM(1, 1)	Energy-related CO2 emissions
Xu et al., (2019a, 2019b)	Adaptive grey model with buffered rolling mechanism (BR-AGM(1, 1))	Chinese greenhouse gas emissions from energy consumption
Wu et al. (2020)	Conformable fractional non-homogeneous grey model (CFNGM(1, 1, k, c))	CO2 emissions of BRICS countries
Gao et al. (2020)	Fractional grey Riccati model FGRM(1, 1)	Carbon emission prediction
Wu et al. (2015)	GM(1, N) model with opposite-direction accumulated generating operator (GOM(1, N))	CO2 emissions in BRICS
Ding et al. (2017)	Grey multivariable model based on the trends of driving variables	Chinese CO2 emissions from fuel combustion
Wang and Ye (2017)	Nonlinear grey multivariable models TNGM(1, N)	Chinese carbon emissions from fossil energy consumption
Ofosu-Adarkwa et al. (2020)	A hybrid Verhulst–GM(1, N) model (V-GM(1, 4))	Forecasting CO2 emissions of China’s cement industry
Ye et al. (2021)	Time-delay multivariate grey model (ATGM(1, N))	CO2 emissions from China’s transportation sector
Wang and Wang (2022)	Fractional grey Bernoulli model (FGBM(1, 1, ta))	Predict the CO2 emissions of the USA, India, Asia Pacific

As shown in Table 2, grey prediction models are mainly divided into univariate and multivariate grey models. The univariate grey model GM(1, 1) and its enhanced model mainly study the continuous development and change process of a single system variable without involving other variables. In complex systems, the system characteristic variables are often closely related to other factors, and the univariate grey model will have difficulty in playing its role. Deng proposed the GM(1, N) model to study the influence of multiple influencing factors on characteristic variables. This model is a factor state model, which is often used to analyse the state of the system at a particular moment, and it will cause unacceptable errors when used to predict problems. To overcome the above defects, scholars have continued to study and improve the model. These improvements are mainly divided into parameter estimation optimization and model structure optimization, which are summarized in Fig. 1 Scholars have continued to study and improve the model (Ding, 2019; Zeng & Li, 2018). Tien (2012) adds control coefficients to the right side of the GM(1, N) grey differential equation to solve the model’s prediction problem. To further study the nonlinear relationship between system characteristic variables and system influencing factors, Wang (2014) proposed a grey multivariable power exponent model.

Fig. 1.

Optimization approaches of the GM(1, N) model

In response to the problem that the first-order accumulation generation preprocessing of the original data in the traditional grey model may make the accumulated series too steep and unfit to the modelling conditions of the grey index rate, Wu et al. (2013) innovatively proposed the fractional accumulation generation operator and established the fractional grey model, which greatly enhanced the prediction accuracy of the grey model. Scholars have since extended it to different applications and derived numerous grey models, showing good prediction accuracy in different practical problems. Wu et al. (2020) further proposed the conformable fractional accumulation (CFA) and established the CFA non-singular grey model and applied it to the prediction of CO₂ emissions in BRICS countries.

The above and most current grey models are based on first-order derivatives or integer-order derivatives, which are ideal memory models and are not suitable for describing systems with historical dependence. Most systems have fractional-order properties; hence, fractional derivatives can better describe and characterize them. Researchers have found a close connection between fractional-order calculus and power-law phenomena, fractal geometry, and memory processes. Fractional derivatives has been widely used in climate research (Liu et al., 2007) and control theory (Kang et al., 2021; Matušů, 2011). Mao et al. (2016) introduced fractional derivatives into the traditional GM(1, 1) model to establish a new grey model of fractional derivatives and proved its validity. Mao et al. (2020) further developed a fractional grey model based on a non-singular exponential kernel. They successfully applied it to the prediction of precious metal content in e-waste. Kang et al. (2022) developed a time-varying grey model with fractional derivatives to predict traffic flow series.

CO₂ emissions are influenced not only by many objective factors such as GDP, population, and energy consumption (Li et al., 2011), but also by other subjective factors such as government macro-control and energy conservation and emission reduction targets set by enterprises. These factors make historical CO₂ emissions show historical dependence. Although they cannot be visually reflected in the data, ignoring their influence is bound to lead to incalculable errors. Therefore, this paper uses the method of fractional-order derivatives for modelling. The main innovations of this paper are as follows.

1. On the basis of the intrinsic linkage of system characteristic variables and their complicated nonlinear relationship with system influence variables, this paper proposes a fractional multivariate nonlinear grey model (FMNGM[q, N]) based on the traditional GM(1, N) model. The proposed model uses the fractional derivative method to portray and describe the endogenous relationship between system characteristic variables data. In addition, the model assigns different power exponent responses to different system influence variables to simulate their relationship with system characteristic variables.

2. To improve the prediction accuracy of the model, the QPSO algorithm, which is improved on the basis of PSO, is used to optimize the fractional derivative order, the fractional cumulative order, and the power exponent of the system influence variables in the model. The QPSO algorithm shows strong stability in dealing with complex optimization problems with many parameters and improves the generalization ability of the model.

3. CO₂ emissions tend to show different trends at different stages of social and economic development. To improve the prediction accuracy, this paper adopts a segmental modelling approach to compare and analyse the differences between different stages. This paper also uses the models developed in recent periods to predict CO₂ emissions in China in 2019 and 2020, to analyse the impact of COVID-19 on CO₂ emissions in China.

The structure of the rest of this article is as follows. In the second part, we introduced the traditional GM(1, N) model and its existing flaws, and made targeted improvements on this basis and proposed the FMNGM(q, N) model. We use the QPSO algorithm to optimize the parameters of the model. In the third part, we verified the model’s validity and applied it to forecast China’s CO₂ emissions in 2019 and 2020. We provide relevant analysis and suggestions on the basis of the results. In the fourth part, we summarized the paper.

Methodology

In this section, we first present the research hypothesis, then the traditional GM(1, N) model and its parameter estimation are first introduced, and some flaws of the model are explained. The modelling steps and parameter estimation methods of its improved model FMNGM(q, N) are then provided. The parameter optimization method of this model uses the QPSO algorithm. Finally, the process of predicting China’s CO₂ emissions using the proposed model is given.

Research hypotheses

Based on research objectives, several research hypotheses were tested. Firstly, the pandemic will cause a significant decrease in China’s total CO₂ emissions. During the pandemic, the Chinese government takes strong blockade measures, restricts the movement of people, and stops work and production in the first half of 2020, when the pandemic is at its peak. These measures will have a more significant impact on CO₂ emissions, and starting from the second half of 2020, the pandemic in China is effectively controlled and the production and living order is gradually restored, but the level of CO₂ emissions still cannot reach the level before the pandemic.

Secondly, economic show a strong positive correlation with CO₂ emissions. Although the government has been increasing its policies for environmental protection enterprises in recent years, the share of environmental protection industry in China’s industrial structure is still very small, and the main industries driving China’s economic growth, such as manufacturing and real estate, still have high CO₂ emission levels. So CO₂ emissions will continue to rise for a long time.

Thirdly, another important factor is energy consumption, which has a strong positive correlation with CO₂ emissions. The burning of fossil fuels emits large amounts of CO₂, and China's rapid economic development since the reform and opening up has come at the cost of high energy consumption. Although China’s energy efficiency has been improving with the advancement of science and technology, the total energy consumption is still rising, which will inevitably lead to a continuous rise in CO₂ emissions.

Fourthly, in addition, the influence of population factors on CO₂ cannot be ignored, and we assume that it still shows a positive correlation with CO₂ emissions. In fact, different aspects such as population size, population structure, and urbanization of population have different effects on CO₂, while this paper focuses on population size as the research and modelling object.

Traditional GM (1, N) model and its gaps

The GM (1, N) model is a grey forecasting system with N variables and a first-order equation (Tien, 2012). The influence of system-related factors is taken into account; hence, this model can better simulate the system’s operation pattern and development trend than the univariate grey forecasting model such as the GM (1, 1). The definition of the GM (1, N) model is given below.

Definition 1

Assume $x_1^{\left(0\right)}$ is the system characteristics sequence,

$$x_1^{\left(0\right)}=\left(x_1^{\left(0\right)}\left(1\right),x_1^{\left(0\right)}\left(2\right),\dots ,x_1^{\left(0\right)}\left(n\right)\right)$$

and assume $x_i^{\left(0\right)}\left(i=2,3,\dots N\right)$ is the sequence of systematic influences, which are all highly correlated with $x_1^{\left(0\right)}$,

$$x_i^{\left(0\right)}=\left(x_1^{\left(0\right)}\left(1\right),x_1^{\left(0\right)}\left(2\right),\dots ,x_1^{\left(0\right)}\left(n\right)\right)$$

$x_j^{\left(1\right)}$ is the first-order accumulated generating sequence of $x_j^{\left(0\right)}$,

$$x_j^{\left(1\right)}=\left(x_j^{\left(1\right)}\left(1\right),x_j^{\left(1\right)}\left(2\right),\dots ,x_j^{\left(1\right)}\left(n\right)\right)$$

where $x_j^{\left(1\right)}\left(k\right)={\sum }_{m=1}^n{x}_j^{\left(0\right)}\left(m\right),k=1,2,\dots ,n$.

Assume $Z_1^{\left(1\right)}$ is the sequence of immediately adjacent mean generation of $Z_1^{\left(1\right)}$, also called the sequence of background values,

$$Z_1^{\left(1\right)}=\left(z_1^{\left(1\right)}\left(2\right),z_1^{\left(1\right)}\left(3\right),\dots ,z_1^{\left(1\right)}\left(n\right)\right),$$

where $z_1^{\left(1\right)}\left(k\right)=0.5\times \left(x_1^{\left(1\right)}\left(k\right)+x_1^{\left(1\right)}\left(k-1\right)\right)\left(k=2,3,\cdots n\right)$, then the following equation

$$x_1^{\left(0\right)}\left(k\right)+a{z}_1^{\left(1\right)}\left(k\right)=\sum _{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k\right).$$ 1

is called the GM(1, N) model, where $-a$ is called the development coefficient and $b_i$ is called the grey action quantity, $P={\left[a,b_2,b_3\dots b_N\right]}^{\text{T}}$ is called the parameter vector of the model.

The parameters of the model can be estimated by using the least squares method,

$$\widehat{P}={\left(B^{\text{T}}\text{B}\right)}^{-1}{B}^{\text{T}}Y.$$ 2

where

$$B=\left[\begin{array}{cccc}-z_1^{\left(1\right)}\left(2\right)& x_2^{\left(1\right)}\left(2\right)& \cdots & x_N^{\left(1\right)}\left(2\right)\\ -z_1^{\left(1\right)}\left(3\right)& x_2^{\left(1\right)}3)& \cdots & x_N^{\left(1\right)}\left(3\right)\\ \cdots & \cdots & \cdots & \cdots \\ -z_1^{\left(1\right)}\left(n\right)& x_2^{\left(1\right)}\left(n\right)& \ddots & x_N^{\left(1\right)}\left(n\right)\end{array}\right],Y=\left[\begin{array}{c}{x}_1^{\left(0\right)}\left(2\right)\\ x_1^{\left(0\right)}\left(3\right)\\ ⋮\\ x_1^{\left(0\right)}\left(n\right)\end{array}\right].$$

Definition 2

The whitening equation of GM (1, N) model is expressed as.

$$\frac{\text{d}{x}_1^{\left(1\right)}}{\text{d}t}+a{x}_1^{\left(1\right)}=\sum _{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k\right).$$ 3

Assuming that $x_i^{\left(0\right)}\left(i=2,3,\dots N\right)$ varies negligibly, then ${\sum }_{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k\right)$ can be considered a grey constant, and solving the whitening differential equation in Definition 2 yields the estimated time response equation,

$${\widehat{x}}_1^{\left(1\right)}\left(k+1\right)=\left[x_1^{\left(1\right)}\left(0\right)-\frac{1}{a}\sum _{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k+1\right)\right]e^{-ak}+\frac{1}{a}\sum _{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k+1\right).$$ 4

where $x_1^{\left(1\right)}\left(0\right)$ can be replaced by the initial value $x_1^{\left(0\right)}\left(1\right)$ of the GM(1, N) model. Then doing the cumulative reduction yields the prediction of the GM(1, N),

$${\widehat{x}}_1^{\left(0\right)}\left(k+1\right)={\widehat{x}}_1^{\left(1\right)}\left(k+1\right)-{\widehat{x}}_1^{\left(1\right)}\left(k\right).$$ 5

This modelling process shows that the GM(1, N) model inevitably has shortcomings.

First, in terms of structure, as a factorial state model, the structure of the GM(1, N) model appears to be simple. As a grey prediction model with N variables and a first-order equation, the GM(1, N) model does not transform into the traditional GM(1, 1) model when N = 1. At the right end of the GM(1, N) model, a missing control coefficient $\text{u}$ exists. As a result, the stability and prediction accuracy of the model are affected to a great extent. This limitation reflects that the GM(1, 1) model has structural problems.

Second, the setting of the model itself is not reasonable. Most of the system characteristic variables in nature have a complex relationship with their influencing factors rather than a simple linear correlation. Hence, the setting of the right-hand term ${\sum }_{i=2}^N{b}_i{x}_i^{\left(1\right)}\left(k\right)$ model is too simple to represent its complex relationship with system characteristics variable $x_1^{\left(0\right)}$, which will affect the prediction accuracy of the model.

In terms of data and preprocessing, in many cases, several of the system characteristic data that we study have shown a large upward trend, and the integer-order accumulation of the original data used in the GM(1, N) model will make the series too steep. As a result, the grey exponential rate is no longer satisfied, which is unfavourable for grey modelling. Moreover, integer-order accumulation will enlarge the model’s perturbation bound, thus affecting the model accuracy. For these reasons, making targeted improvements to the GM(1, N) model is necessary.

The proposed FMNGM (q_,N) model

In this section, an FMNGM(q, N) is developed to address the previously discussed flaws of the GM(1, N) model with targeted improvements.

Definition 3

Assume $r$ as a positive real number $(r>0)$ and $n$ as a positive integer, then

$${\mathrm{\nabla }}^{-r}x\left(n\right)=\left[\begin{array}{c}r\\ n\end{array}\right]\ast x\left(n\right)=\sum _{i=1}^n\left[\begin{array}{c}r\\ n-i\end{array}\right]x\left(n\right).$$ 6

is called the r-order accumulation of $x\left(n\right)$, where $\left[\begin{array}{c}r\\ n\end{array}\right]=\frac{r\left(r+1\right)\cdots \left(r+n-1\right)}{n!}$ and * is the discrete convolution operator ($x\left(n\right)\ast y\left(n\right)={\sum }_{r=1}^{n}x\left(n-r\right)y\left(r\right)$). When $m=\left[r\right]$, ${\mathrm{\nabla }}^{r}x\left(n\right)={\mathrm{\nabla }}^m{\mathrm{\nabla }}^{-\left(m-r\right)}x\left(n\right)$ is called the r-order difference of $x\left(n\right)$ (Cheng, 2011).

Let $n$ be a positive integer and $u$ and $v$ real numbers, then

$$\left[\begin{array}{c}u\\ n\end{array}\right]\ast \left[\begin{array}{c}v\\ n\end{array}\right]=\left[\begin{array}{c}u+v\\ n\end{array}\right].$$ 7

Definition 4

Assume $x^{\left(0\right)}=\left(x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\dots x^{\left(0\right)}\left(n\right)\right)$ is a non-negative original sequence, $r$ is a positive real number, and $k=1,2,\dots n$. Define $A^r(r>0)$ as an r-order accumulative generator, and $A^r$ has the following matrix form:

$$A^r=\left(\begin{array}{ccccc}\left[\begin{array}{c}r\\ 0\end{array}\right]& 0& 0& \cdots & 0\\ \left[\begin{array}{c}r\\ 1\end{array}\right]& \left[\begin{array}{c}r\\ 0\end{array}\right]& 0& \cdots & 0\\ \left[\begin{array}{c}r\\ 2\end{array}\right]& \left[\begin{array}{c}r\\ 1\end{array}\right]& \left[\begin{array}{c}r\\ 0\end{array}\right]& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \left[\begin{array}{c}r\\ n-1\end{array}\right]& \left[\begin{array}{c}r\\ n-2\end{array}\right]& \left[\begin{array}{c}r\\ n-3\end{array}\right]& \cdots & \left[\begin{array}{c}r\\ 0\end{array}\right]\end{array}\right).$$ 8

Then we have $x^{\left(r\right)}\left(k\right)={\mathrm{\nabla }}^{-r}{x}^{\left(0\right)}\left(k\right),x^{\left(r\right)}=A^r{x}^{\left(0\right)}$, and $x^{\left(r\right)}$ is the r-order accumulated generating sequence of $x^{\left(0\right)}$. To be formally uniform with ${\mathrm{\nabla }}^{-r}$, $r>0$ is used. In fact, for the definition of $A^r$, the values of $r$ can be extended to the range of real numbers.

Theorem 1

$x^{\left(0\right)}$ is the inverse accumulated generating sequence of $x^{\left(r\right)}$. Then,

$$x^{\left(0\right)}\left(k\right)={\mathrm{\nabla }}^r{x}^{\left(r\right)}\left(k\right),x^{\left(0\right)}=A^{-r}{x}^{\left(r\right)}.$$ 9

Definition 5

Positive real number $\mathrm{\alpha }$ satisfies $n-1<\alpha \le n$, n is a positive integer, and $x\left(t\right)$ is a function defined in interval $\left[a,b\right]$. Then,

$${}_0^C{D}_t^{\alpha }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\int }_0^t{(t-\tau )}^{n-\alpha -1}{x}^{\left(n\right)}\left(\tau \right)\text{d}\tau .$$ 10

is called the Caputo fractional derivative (Caputo & Fabrizio, 2015).

Definition 6

If $x_1^{\left(0\right)}$ is the system characteristic data, $x_i^{\left(0\right)}\left(i=2,3,\dots k\right)$ is the system influence data, then $x_1^{\left(r\right)}$ and $x_i^{\left(r\right)}$ are $x_1^{\left(0\right)}$ and $x_i^{\left(0\right)}$ rth-order accumulated generating sequences. $Z_1^{\left(r\right)}$ is the background value sequence of $x_1^{\left(r\right)}$, also known as the sequence of adjacent mean values. Then, the equation

$${}_0^C{D}_t^q{x}_1^{\left(r\right)}+a{x}_1^{\left(r\right)}=\sum _{i=2}^N{b}_i{\left(x_i^{\left(r\right)}\right)}^{m_i}+u.$$ 11

is called the whitenization equation of FMNGM(q, N) model, where $0<q\le 1$, and ${}_0^C{D}_t^q{x}_1^{\left(r\right)}$ is the Caputo fractional derivative of $x_1^{\left(r\right)}$.

Assume $m=\left[q\right]+1$, then

$${}_0^C{\mathrm{\nabla }}_t^{q}x\left(t\right){\triangleq }_0{\mathrm{\nabla }}_t^{-m+q}\left[{\mathrm{\nabla }}^{m}x\left(t\right)\right].$$ 12

is the $q$-order fractional difference of $x\left(t\right)$.

Given that ${}_0^C{D}_t^q{x}_1^{\left(r\right)}$ is a Caputo fractional derivative, we can consider replacing it with a Caputo fractional difference. According to the above formula, the Caputo fractional difference of $x_1^{\left(r\right)}\left(k\right)$ is ${}_1^C{\mathrm{\nabla }}_k^q{x}_1^{\left(r\right)}\left(k\right){\triangleq }_1{\mathrm{\nabla }}_k^{-m+q}\left[{\mathrm{\nabla }}^m{x}_1^{\left(r\right)}\left(k\right)\right]$. When $0<q\le 1$, $m=1$; hence, ${}_0^C{D}_t^q{x}_1^{\left(r\right)}{\approx }_1^c{\mathrm{\nabla }}_k^q{x}_1^{\left(r\right)}\left(k\right){\triangleq }_1{\mathrm{\nabla }}_k^{-1+q}\left[\mathrm{\nabla }{x}_1^{\left(r\right)}\left(k\right)\right]$. According to the definition of integer-order difference,${}_0^C{D}_t^q{x}_1^{\left(r\right)}{\approx }_1^c{\mathrm{\nabla }}_k^q{x}_1^{\left(r\right)}\left(k\right){\triangleq }_1{\mathrm{\nabla }}_k^{-1+q}\left[\mathrm{\nabla }{x}_1^{\left(r\right)}\left(k\right)\right]$.

Given that $x_1^{\left(r-1\right)}=A^{\left(r-1\right)}{x}_1^{\left(0\right)}$, we have:

$$\begin{array}{cc}& {}_1{\mathrm{\nabla }}_k^{-\left(1-q\right)}{x}_1^{\left(r-1\right)}=A^{\left(r-1\right)}{\mathrm{\nabla }}_k^{-\left(1-q\right)}{x}_1^{\left(0\right)}{=}_1{\mathrm{\nabla }}_k^{-\left(1-q\right)}{x}_1^{\left(r-1\right)}\hfill \\ \hfill & =A^{\left(r-1\right)}{\mathrm{\nabla }}_k^{-\left(1-q\right)}{x}_1^{\left(0\right)}\left(\begin{array}{ccccc}\left[\begin{array}{c}r-1\\ 0\end{array}\right]& 0& 0& \cdots & 0\\ \left[\begin{array}{c}r-1\\ 1\end{array}\right]& \left[\begin{array}{c}r-1\\ 0\end{array}\right]& 0& \cdots & 0\\ \left[\begin{array}{c}r-1\\ 2\end{array}\right]& \left[\begin{array}{c}r-1\\ 1\end{array}\right]& \left[\begin{array}{c}r-1\\ 0\end{array}\right]& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \left[\begin{array}{c}r-1\\ n-1\end{array}\right]& \left[\begin{array}{c}r-1\\ n-2\end{array}\right]& \left[\begin{array}{c}r-1\\ n-3\end{array}\right]& \cdots & \left[\begin{array}{c}r-1\\ 0\end{array}\right]\end{array}\right)\hfill \\ \hfill & ·\left(\begin{array}{c}\left[\begin{array}{c}1-q\\ 0\end{array}\right]x^{\left(0\right)}\left(1\right)\\ \begin{array}{c}\left[\begin{array}{c}1-q\\ 1\end{array}\right]x^{\left(0\right)}\left(2\right)+\left[\begin{array}{c}1-q\\ 0\end{array}\right]x^{\left(0\right)}\left(2\right)\\ {\sum }_{i=1}^3\left[\begin{array}{c}1-q\\ n-i\end{array}\right]x^{\left(0\right)}\left(3\right)\\ ⋮\\ {\sum }_{i=1}^n\left[\begin{array}{c}1-q\\ n-i\end{array}\right]x^{\left(0\right)}\left(n\right)\end{array}\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{ccccc}\left[\begin{array}{c}r-q\\ 0\end{array}\right]& 0& 0& \cdots & 0\\ \left[\begin{array}{c}r-q\\ 1\end{array}\right]& \left[\begin{array}{c}r-q\\ 0\end{array}\right]& 0& \cdots & 0\\ \left[\begin{array}{c}r-q\\ 2\end{array}\right]& \left[\begin{array}{c}r-q\\ 1\end{array}\right]& \left[\begin{array}{c}r-q\\ 0\end{array}\right]& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \left[\begin{array}{c}r-q\\ n-1\end{array}\right]& \left[\begin{array}{c}r-q\\ n-2\end{array}\right]& \left[\begin{array}{c}r-q\\ n-3\end{array}\right]& \cdots & \left[\begin{array}{c}r-q\\ 0\end{array}\right]\end{array}\right)x_1^{\left(0\right)}\hfill \\ \hfill & =A^{\left(r-q\right)}{x}_1^{\left(0\right)}\hfill \\ \hfill \end{array}$$

Then, we can obtain

$${}_0^C{D}_t^q{x}_1^{\left(r\right)}{\approx }_1^c{\mathrm{\nabla }}_k^q{x}_1^{\left(r\right)}\left(k\right)\triangleq x_1^{\left(r-q\right)}\left(k\right).$$ 13

Definition 7

Equation

$$x_1^{\left(r-q\right)}\left(k\right)+a{Z}_1^{\left(r\right)}\left(k\right)=\sum _{i=2}^N{b}_i{(x_i^{\left(r\right)}\left(k\right))}^{m_i}+u.$$ 14

is called FMNGM(q, N) model, where $Z_1^{\left(r\right)}\left(k\right)=\frac{1}{2}\left(x_1^{\left(r\right)}\left(k-1\right)+x_1^{\left(r\right)}\left(k\right)\right)$. The parameter vector of the model is $P={\left[a,b_2,b_3\dots b_N,u\right]}^{\text{T}}$, which can be estimated by using the least squares method. $\widehat{P}={\left(B^{T}B\right)}^{-1}{B}^{\text{T}}Y$, where

$$\begin{array}{cc}\hfill Y& ={\left[\begin{array}{c}{x}_1^{\left(r-q\right)}\left(2\right)\\ x_1^{\left(r-q\right)}\left(3\right)\\ ⋮\\ x_1^{\left(r-q\right)}\left(n\right)\end{array}\right]}_{n-1\times 1},\widehat{P}={\left[\begin{array}{c}a\\ b_2\\ b_3\\ ⋮\\ b_N\\ u\end{array}\right]}_{N+1\times 1},\hfill \\ \hfill B& ={\left[\begin{array}{ccccc}-Z_1^{\left(r\right)}\left(2\right)& {(x_2^r\left(2\right))}^{m_2}& \cdots & {\left(x_N^r\left(2\right)\right)}^{m_N}& 1\\ -Z_1^{\left(r\right)}\left(3\right)& {(x_2^r\left(3\right))}^{m_2}& \cdots & {(x_N^r\left(3\right))}^{m_N}& 1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -Z_1^{\left(r\right)}\left(n\right)& {(x_2^r\left(n\right))}^{m_2}& \cdots & {\left(x_N^r\left(n\right)\right)}^{m_N}& 1\end{array}\right]}_{n-1\times N+1}\hfill \\ \hfill \end{array}$$

We use the Laplace transform to solve Eq. 14. The Laplace transform formula of the Caputo fractional derivative is

$$\left(L({}_0^C{D}_t^{q}x\left(t\right)\right)=S^{q}X\left(S\right)-\sum _{k=0}^{m-1}{S}^{q-k-1}{x}^{\left(k\right)}\left(0\right).$$ 15

where $m-1<q\le m$, and $m$ is a positive integer.

Theorem 2

The solution of the FMNGM(q, N) model is.

$${\widehat{x}}_1^{\left(r\right)}\left(k\right)=\left[\frac{x_1^{\left(r\right)}\left(1\right)-M\left(k\right)H_{k=1}}{G_{k=1}}\right]H\left(k\right)+M\left(k\right)k^{q}Q\left(k\right).$$ 16

where $M\left(k\right)={\sum }_{i=2}^N{b}_i{(x_i^{\left(r\right)}\left(k\right))}^{m_i}+u,H\left(k\right)={\sum }_{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+1\right)}$ and $Q\left(k\right)={\sum }_{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}.$

Theorem 3

By using Eq. (16), we obtain the predicted value of the FMNGM (q, N) model, and the predicted value of the original sequence can be obtained according to the following equation:

$${\widehat{x}}^{\left(0\right)}={\mathbf{A}}^{-r}{\widehat{x}}^{\left(r\right)}.$$ 17

Parameter optimization of QPSO algorithm for FMNGM (q, N)

PSO is an evolutionary computational technique that originated from the study of the predatory behaviour of birds (Shi and Eberhart, 1999). It is widely used in many fields such as function optimization, neural network training, and image recognition owing to its simplicity, ease of implementation, and excellent optimization performance. However, as the study of PSO progressed, researchers found that the algorithm has limitations. For example, the loss of search diversity and the lack of randomness of the population in space make the algorithm easily fall into local optimality (Bai, 2010). Moreover, the stability of the algorithm is poor due to the number of parameters that need to be set in the model (inertia factor $\omega$, local learning factor $c_1$, and global learning factor $c_2$), which are not conducive to finding optimal parameters. Therefore, Sun et al. (2004) designed the QPSO algorithm in 2004 on the basis of PSO. QPSO cancels the moving direction property of particles and increases the randomness of particle removal. It has only one set of innovation parameter α, which, to some extent, compensates for the shortcomings of the PSO algorithm. As a result, the ability of the algorithm to find the global optimization is greatly enhanced, thus improving the performance of the algorithm. In this paper, the QPSO algorithm is applied to the FMNGM (q, N) model’s parameter optimization. The parameters to be optimized in this paper are fractional-order derivatives ($q$), fractional cumulative order ($r$), and power exponent of the system influence variable ($m_i$). The fitness function established with the objective of minimizing MAPE is: $\text{Fitness}=\frac{1}{n-1}{\sum }_{k=1}^{n-1}\frac{\left|{\widehat{x}}_1^{\left(0\right)}\left(k+1\right)-x_1^{\left(0\right)}\left(k+1\right)\right|}{x_1^{\left(0\right)}\left(k+1\right)}$. Algorithm 1 shows the specific steps for optimizing the above parameters using QPSO.

Modelling methods

Figure 2 shows the process of predicting Chinese ${\text{CO}}_2$ emissions using the proposed FMNGM(q, N) model.

Fig. 2.

Flowchart of proposed FMNGM(q, N) model to forecast China’s CO₂ emissions

Forecasting China’s CO₂ emissions under the influence of the COVID-19 pandemic

In this section, the essential information and sources of the data are first introduced, and then, segmental modelling is performed on the basis of data characteristics. The first segment of the data are used for model validation. The proposed model’s results are compared with those of GM(1, 1), GM(1, N), ARIMA, and SVR according to MAPE, RMSE, MAE, and other indicators. The second segment of the data is used to build a new model to predict Chinese CO₂ emissions under the influence of the COVID-19 pandemic.

Data collection

According to the literature, the economy (GDP), population (P), and energy consumption (EC) are key factors positively correlated with CO₂ emissions. However, the complexity and uncertainty in this relationship present challenges to applying the model. Table 3 shows the collected and organized data.

Table 3.

Original data of Chinese CO₂ emissions and its influencing factors

Year	× 1	× 2	× 3	× 4
	CO2 emissions (million tons)	GDP (100 million yuan)	Population (million)	Total energy consumption (million tons of coal)
2001	3297.4	110,863.1	127,627	155,547
2002	3551.9	121,717.4	128,453	169,577
2003	4110.9	137,422	129,227	197,083
2004	4782.7	161,840.2	129,988	230,281
2005	5448.9	187,318.9	130,756	261,369
2006	6004.1	219,438.5	131,448	286,467
2007	6517.2	270,092.3	132,129	311,442
2008	6712	319,244.6	132,802	320,611
2009	7177.8	348,517.7	133,450	336,126
2010	7873	412,119.3	134,091	360,648
2011	8615.3	487,940.2	134,735	387,043
2012	8863.5	538,580	135,404	402,138
2013	9234.4	592,963.2	136,072	416,913
2014	9164.2	643,563.1	136,782	425,806
2015	9137.2	688,858.2	137,462	429,905
2016	9099.2	746,395.1	138,271	435,819
2017	9289.6	832,035.9	139,008	448,529
2018	9570.8	919,281.1	139,538	464,000

Table 3 and Fig. 3 indicate that China’s CO₂ emissions showed a significant upward trend from 2001 to 2009. This increase may be due to China’s rapid economic growth after it joined the WTO in 2001. Traffic CO₂ emissions have also continued to rise. After 2009, under the leadership of the USA, Europe, China, Japan, and other countries, a new technological revolution championing artificial intelligence, quantum information technology, and clean energy was launched worldwide. This phase is known as the fourth industrial revolution or the green industrial revolution. Taking this shift as an opportunity, China began to carry out large-scale industrial changes and upgrades and gradually transformed the country’s economic development mode. This change is intuitively reflected in CO₂ emissions. After 2010, the growth momentum of CO₂ emissions in China has gradually slowed down and increased steadily. Figure 4 shows that CO₂ emissions have a strong nonlinear relationship with GDP, population, and total energy consumption.

Fig. 3.

Chinese CO₂ emissions over the years

Fig. 4.

Relationship between Chinese CO₂ emissions and its influencing factors

The above data were divided into two stages: 2001–2009 and 2010–2018. They were modelled and compared separately to demonstrate the validity and usefulness of the proposed model. The comparison serves to verify the significant differences in economic development patterns and other aspects between the two stages. Since the COVID-19 outbreak at the end of 2019 is included in the 2010–2018 time period and the increase in the level of economic development in a short period of time is negligible, we will use the model developed in the latter stage to predict 2019 and 2020 Chinese CO₂ emissions to analyse and illustrate the impact of COVID-19 on Chinese CO₂ emissions.

Validation of FMNGM(q, N)

Parameter estimation of the proposed model

In this section, the model was modelled using data from 2001 to 2009, and the model was tested by predicting 2010 and 2011 CO₂ emissions. QPSO was used to optimize the model parameter fractional derivatives ($q$), fractional cumulative order ($r$), and power exponents of the system influence variables ($m_2$, $m_3$, $m_4$). The algorithm parameters were set as the number of iterations $N=100$ and the innovation parameter $\alpha =0.5$. The number of particles is set to $\text{num}=100$, and the dimension of particles is set to dim = 5. Table 4 shows the optimization results.

Table 4.

Parameter optimization results of FMNGM(q, N) using 2001–2009 data

Parameter	$q$	$r$	$m_2$	$m_3$	$m_4$
Value	0.8890	0.1800	1.7074	1.6697	0.6097

The parameters in Table 4 are used for modelling, and the least squares method is used to obtain the parameter vector $P={\left[a,b_2,b_3\dots b_N,u\right]}^{\text{T}}$ of the FMNGM(q, N) model with the estimated values shown in Table 5.

Table 5.

Parameter estimation results of FMNGM(q, N) using 2001–2009 data

Parameter	$a$	$b_2$	$b_3$	$b_4$	$u$
Value	1.33	$2.78\times {10}^{-7}$	$-9.94\times {10}^{-6}$	8.13	$-3.38\times {10}^{-3}$

Comparisons and analysis

In this part of the analysis, to better reflect the practicality and effectiveness of the proposed model, we used three indicators—MAPE, RMSE, and MAE—to evaluate the different model fitting and prediction performance. They are defined as follows:

$$\text{MAPE}=\frac{1}{n-1}\sum _{k=2}^n\frac{\left|{\widehat{x}}^{\left(0\right)}k-x^{\left(0\right)}k\right|}{x^{\left(0\right)}k}\times 100\%.$$ 18

$$\text{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left|{\widehat{x}}^{\left(0\right)}k-x^{\left(0\right)}k\right|}^2}.$$ 19

$$\text{MAE}=\frac{1}{n}\sum _{i=1}^n\left|{\widehat{x}}^{\left(0\right)}k-x^{\left(0\right)}k\right|$$ 20

Table 6 shows the results of the FMNGM (0.889, 4) model developed in Sect. 3.3.1 and the competing models GM(1, 1), GM(1, N), ARIMA, and SVR.

Table 6.

Comparison of the results of each model

Year	CO2 emissions million tons)	FMNGM(0.889, 4)		GM(1, 1)
	Actual values	Fitted values	APE (%)	Fitted values	APE (%)
2001	3297.40	3297.40		3297.40
2002	3551.90	3467.70	2.37	3932.97	13.27
2003	4110.90	4058.38	1.28	4311.76	5.07
2004	4782.70	4801.81	0.40	4727.03	0.77
2005	5448.90	5471.32	0.41	5182.30	1.89
2006	6004.10	5999.28	0.08	5681.41	0.67
2007	6517.20	6568.36	0.79	6228.59	0.52
2008	6712.00	6805.54	1.39	6828.47	4.15
2009	7177.80	7132.13	0.64	7486.13	7.44
MAPE (%)		0.92		4.69
RMSE		54.99		263.58
MAE		46.68		242.54

	Actual values	Predicted values	APE (%)	Predicted values	APE (%)
2010	7873.00	7787.00	1.092	8207.13	4.244
2011	8615.30	8560.93	0.631	8997.56	4.437
MAPE (%)		0.86		4.34
RMSE		71.94		359.00
MAE		70.18		358.19

GM(1, 4)		ARIMA		SVR
Fitted values	APE (%)	Fitted values	APE (%)	Fitted values	APE (%)
3297.40		3377.97	2.44	3431.84	4.08
3041.66	14.37	3621.95	1.97	3775.18	6.29
4400.35	7.04	4157.85	1.14	4238.23	3.10
4868.87	1.80	4801.89	0.40	4787.40	0.10
5445.32	0.07	5440.56	0.15	5327.47	2.23
5966.60	0.62	5972.81	0.52	5825.13	2.98
6519.30	0.03	6464.71	0.81	6382.76	2.06
6758.33	0.69	6651.46	0.90	6793.73	1.22
7100.77	1.07	7098.01	1.11	7148.66	0.41
3.21		1.05		2.50
212.44		55.58		131.88
131.55		49.91		115.05

Predicted values	APE (%)	Predicted values	APE (%)	Predicted values	APE (%)
7662.23	2.677	7098.01	9.84	7648.91	2.85
8275.13	3.948	7021.51	18.50	8126.84	5.67
3.31		13.41		4.258
282.97		1192.58		380.00
275.47		1114.20		356.276

First, we compare the FMNGM (0.889, 4) model with the other two grey models, GM (1, 1) and GM (1, N). Table 6 shows that in terms of model fitting and prediction, the MAPE, RMSE, and MAE values ​​of the proposed model are all the smallest. GM(1, 1) only considers the changing trend of the system characteristic variables; hence, better prediction accuracy is not easy to obtain. It has the worst prediction performance among the three grey models. GM(1, N) introduces system influence variables based on GM(1, 1) to reflect their relationship with the system characteristic variables. The increase in the amount of information improves the model effect. The proposed model entirely takes into account the nonlinearity and complexity of this relationship. The introduction of the power exponent of the system influence variable greatly enhances the interpretability of the model. Besides, the introduction of control coefficients dramatically improves the prediction performance of the model.

Then we compare our model with two non-grey models, the classic ARIMA model for time series forecasting and the SVR method in machine learning. Table 6 shows the results, which indicate that the proposed model is optimal in terms of fitting and prediction. The fitted MAPE value of ARIMA reached 1.05%, but the predicted MAPE was 13.41%, which indicates that overfitting has occurred. SVR is currently one of the popular machine learning models. Through constant adjustment of parameters, it can also achieve good prediction results. For problems with a large amount of data, SVR is the right choice, but the proposed model has certain advantages for small sample data.

To make the conclusions adequate, Fig. 5 further visualizes the difference between our model and other grey and non-grey models. Figure 5 shows that the proposed model better captures the trend of the original data series and is almost synchronized with the actual value curve. Despite matching the actual value curve in the general trend, the other models show significant deviations in some years. This gap is pronounced in the 2010–2011 forecast years, given that the FMNGM(0.889,4) model fits the original data more closely. This result is clear evidence that the proposed model has better forecasting performance than the other four models.

Fig. 5.

Comparison between the FMNGM(0.889, 4) model and other models

Figure 6 intuitively shows the performance of the proposed model and other models in terms of RMSE and MAE metrics. The smaller the value, the closer to the centre of the radar plot, the better the prediction effect. It can be seen that the proposed model has the smallest RMSE and MAE values compared to the other models, both in the fitting and prediction stages, which indicates that the proposed model outperforms the other competing models in the comparison of these two evaluation metrics.

Fig. 6.

RMSE and MAE performance of the proposed model and the other models

In addition, we used the APE criterion to evaluate the error between the forecast data and the real data for each year, which is defined as

$$\text{APE}=\frac{\left|{\widehat{x}}^{\left(0\right)}\left(k\right)-x^{\left(0\right)}\left(k\right)\right|}{x^{\left(0\right)}\left(k\right)}\times 100\%.$$ 21

Figure 7 shows the APE values of CO₂ emissions from 2001 to 2011 for the above five models. The APEs are close to zero, which indicates that the model has high precision. The difference between the models is noticeable in Fig. 7 The two grey models GM(1, 1) and GM(1, 4) are significantly worse and less stable than the other three models, whereas FMNGM(0.889, 4) and ARIMA perform better. For the forecast years 2010–2011, FMNGM(0.976, 4) has the smallest APE. Overall, the proposed model performs best in terms of this metric.

Fig. 7.

Comparison of fitted and predicted errors from various models

Forecasting China’s CO₂ emissions in 2019 and 2020

The above model results demonstrate the applicability of the proposed model to the problem of predicting China’s CO₂ emissions. This section will model the data from 2010 to 2018 and compare them with the above model to analyse the differences between the two different economic development stages and then forecast China’s CO₂ emissions for 2019 and 2020 under the influence of the COVID-19 pandemic.

Building a new FMNGM(q, N) model using 2010–2018 data

The new prediction model is built following the same steps as in Sect. 3.2.1. Table 7 and Table 8 show the parameter estimation results.

Table 7.

Parameter optimization results of FMNGM(q, N) using 2010–2018 data

Parameter	$q$	$r$	$m_2$	$m_3$	$m_4$
Value	0.5096	0.0862	0.0842	1.2298	1.3019

Table 8.

Parameter estimation results of FMNGM(q, N) using 2010–2018 data

Parameter	$a$	$b_2$	$b_3$	$b_4$	$u$
Value	$8\text{.82}\times {10}^{-1}$	$-7.04\times {10}^{-3}$	$-1\text{.12}\times {10}^{-2}$	$7\text{.40}\times {10}^{-4}$	$4\text{.41}\times {10}^4$

The structure and modelling process of the FMNGM(q, N) model shows that its parameters have practical significance. Comparing the parameters of the two models built using 2001–2009 and 2010–2018 data, we can analyse the differences in the development trends of China’s CO₂ emissions at different stages and derive the changes in the relationship between CO₂ emissions and its influencing factors. The $a$ in the model is defined as the development coefficient, and the value of a reflects the development trend of the data. The larger the value of $a$, the more upward the trend of the data. The values of a for the two models mentioned above are 1.330 and 0.882, and respectively. These results indicate a more pronounced upward trend of China’s CO₂ emissions in the 2001–2009 time period than in the 2010–2018 period, which is consistent with the analysis in Fig. 4 and Sect. 3.1.

The modelling results for the 2001–2009 data show that $m_2$, $m_3$, and $m_4$ are 1.7074, 1.6697, and 0.6097, respectively; whereas the above parameters of the model built using the 2010–2018 data are 0.0842, 1.2298, and 1.3019, respectively. These results reflect that the complex nonlinear relationship between CO₂ emissions and GDP, population, and energy consumption is changing. The value of the power exponents corresponding to the variable GDP changes the most from 1.7074 to 0.0842. Coefficient ${\text{b}}_2$ has the opposite sign between the two modelling results, indicating that the relationship between GDP and CO₂ emissions has changed from a positive correlation to a slight negative correlation.

With the changes in China’s economic development model, the continuous development of low-carbon industries and clean energy technologies has brought huge economic benefits to society and reduced CO₂ emissions to a certain extent. Such changes have resulted in a negative correlation between GDP and CO₂ emissions. The total population has a negative correlation with the impact of CO₂ emissions. CO₂ emissions are related to the number of people as well as population growth, age structure, and population urbanization rate. The slowdown of population and the intensification of population ageing have led to the above results. China is the world’s largest energy consumer (Swartz & Oster, 2010). As the primary source of CO₂ emissions, the increasing consumption of fossil fuels such as coal, oil, and natural gas has led to the continuous rise of China’s CO₂ emissions.

Forecasting China’s CO₂ emissions in 2019 and 2020 using the FMNGM(q, N) model

The COVID-19 pandemic has affected China’s GDP and EC to a certain extent, but the impact on the huge number of P can be ignored. The relationship between these three and ${\text{CO}}_2$ emissions largely depends on the level of economic and social development. The impact of the pandemic on GDP, EC, and P is intuitively reflected in the numerical value, and changes in this relationship are not significant. Our model mainly reflects the interdependence between system characteristic variables and system influencing factors. Therefore, selecting the 2010–2018 time period with a relatively close economic development level and social development to predict CO₂ emissions in 2019 and 2020, a period greatly affected by the pandemic, is reasonable.

A one- to two-year lag exists in the data statistics of CO₂ emissions; hence, no agency has yet published data for China’s CO₂ emissions beyond 2018. The structure of the FMNGM (q, N) model shows that system characteristic variables’ values can be predicted after obtaining the values of future system influence variables. The Chinese National Bureau of Statistics has released data on GDP, population, and energy consumption for 2019. 2020 Chinese GDP data are derived from International Monetary Fund (IMF) projections, P data are extrapolated from the 2019 natural growth rate, and EC data are derived from the Chinese government’s 2020 primary energy consumption planning target. Table 9 presents the data for the system influence variables.

Table 9.

Value of GDP, P, and EC in 2019 and 2020

Year	GDP (100 million yuan)	Population (million)	Total energy consumption (million tons of coal)
2019	990,865.1	140,005	486,000
2020	100,0774	140,505	480,000

Table 10 shows the results obtained by using the model established in Sect. 3.3.1 to predict the data of 2019 and 2020.

Table 10.

Forecast of Chinese CO₂ emissions in 2019 and 2020 (unit: million tons)

Year	2019		2020
Values	10,214.36		9,893.38

Figure 8 shows the results of using the FMNGM (0.51, 4) model to predict China’s CO₂ emissions from 2010 to 2020, in which 2010–2018 is the fitting part, and 2019 and 2020 are the backward prediction parts. During this period, the trend of China’s CO₂ emissions was flatter than that during the 2001–2009 time period; some years even showed a downward trend. Nonetheless, the proposed model performs well from the modelling effect, which further validates the model effectiveness and practicality. According to the forecast, CO₂ emissions in 2019 will increase slightly to 10.214 billion tons. Moreover, the outbreak of the pandemic at the end of 2019 has reduced China’s CO₂ emissions in 2020 by approximately 320 million tons lower than in 2019, to 9.893 billion tons.

Fig. 8.

Forecasted results of Chinese CO₂ emissions during 2010–2020

According to the authoritative data site Our World in Data (https://ourworldindata.org/co2-emissions), China’s CO₂ emissions in 2019 are estimated at 10,170 Mt, and our model's result of 10,214 Mt differs by only 44 Mt. Due to the lag in CO₂ statistics, the website does not yet show China’s CO₂ emissions in 2020. BP World Energy Yearbook 2020 estimates China’s CO₂ emissions under the impact of the epidemic and concludes that China’s total CO₂ emissions in 2020 will be 9,899.3 Mt, while our model's prediction is 9,893.4 Mt, a difference of only 5.9 Mt. The comparison with these authoritative websites and institutions proves the accuracy of our model in the problem of CO₂ emission prediction.

The model results are largely consistent with the hypotheses of our previous studies. First, we assumed a significant decline in carbon emissions in 2020 due to the epidemic, unlike the upward trend in previous years, which is confirmed by the forecast results. Second, based on literature studies and historical experience, we hypothesized a complex nonlinear relationship between economic, demographic, and energy factors and CO₂, respectively, and that they all contribute positively to CO₂ emissions. The results of segmental modelling show that the relationship between economy, population, energy consumption, and CO₂ emissions exhibits dynamic changes. Contrary to our empirical assumptions, the relationship between population and CO₂ emissions is negative at all stages. Consistent with our hypothesis, energy consumption is positively correlated with CO₂ emissions at different stages. For economic factors, the effect on CO₂ emissions is reversed at different stages, with a positive contribution to CO₂ emissions in the 2001–2010 period and a hindering effect in the 2011–2020 period, which may be related to the level of economic development.

Discussion and suggestions

On the basis of the results of the above model, we present the following conclusions and recommendations.

1. In terms of model results, China’s CO₂ emissions will fall in 2020 to 9,893 million tons after a rise in 2019. After the outbreak of the COVID-19 pandemic in late 2019, the Chinese government took decisive quarantine measures to stop the further spread of the virus. Some production and business activities were thus halted. China paid a substantial economic cost to control the epidemic. The new model results also reflect that China has achieved good results in controlling CO₂ emissions. Compared with the rapid growth of CO₂ emissions in 2001–2009, the growth of CO₂ emissions in China has been stable in recent years. The stable growth is related to the change of economic development and the gradual replacement of traditional fossil energy with clean energy.

2. In a practical sense, to achieve the vision of carbon neutrality, China must continue to insist on developing a low-carbon economy on three levels. First, the government should strengthen the demonstration and promotion of low-carbon technologies. The International Energy Agency (IEA) states that the vast majority of global CO₂ emission reductions on the path to net zero emissions between now and 2030 will come from currently available technologies. Second, promote the innovative application of green supply chain. Enterprises are the main body of emission reduction, and green supply chain is mainly through the environmental management of the leading enterprises to drive the actions of enterprises in their supply chain to achieve the lowest environmental impact, highest energy efficiency and lowest carbon emissions throughout the product life cycle. Third, give full play to the role of market mechanisms in regional emission reduction. Making full use of the national carbon market, which is a major institutional innovation to control and reduce greenhouse gas emissions and promote green and low-carbon development using market mechanisms.

Conclusion and future works

In this paper, a novel FMNGM model is proposed to investigate the relationship between GDP, P, EC, and CO₂ emissions and to forecast China’s CO₂ emissions in 2019 and 2020 under the influence of the COVID-19 pandemic. The main work of this paper is summarized as follows:

1. The fractional accumulation of the raw data makes the data suitable for the grey modelling conditions, and the Caputo fractional derivative method is used to describe the endogenous relationship between CO₂ emission data over the years. These methods improve the accuracy of the model.

2. Different power exponents were assigned to different influencing factor variables of CO₂ emissions to portray their complex nonlinear relationship with CO₂ emissions. These power exponents varied at different stages of economic development, reflecting that this nonlinear relationship was changing.

3. The FMNGM(q, N) model shows good adaptability in predicting CO₂ emissions in China for this problem. The results also demonstrate the validity and practicality of the model, which has better prediction accuracy in this problem than other competing models.

4. The predictions of China’s CO₂ emissions for 2019 and 2020 have a small increase in 2019 compared with previous years, and a certain drop in 2020 due to the impact of the COVID-19 pandemic.

In conclusion, the proposed FMNGM(q, N) model is a feasible and effective method for predicting China’s CO₂ emissions.

The research results of this paper mainly include two aspects. Firstly, for the CO₂ forecasting problem, this paper only studies the total carbon emission in China, but the model and the methodology of this paper are not only developed for the carbon emission problem in China, but also applicable to the forecasting and analysis of carbon emission in other regions or a single sector. Secondly, the new model focuses on the complex nonlinear relationship between the characteristic variables and the influencing variables within the system. Although the model is developed for CO₂ emissions, it is actually a factorial state model, which is theoretically applicable to other multivariate problems with multiple influencing factors. This implies that the new proposed model has the potential for wider application.

However, the model has limitations. For example, when we applied Laplace transform method to solve the whitenization equation of the model, we treated the right end of the model as a grey constant M(k) for simplicity. In fact, in many occasions, the system influences are changing, and the magnitude of change is not small. Hence, the simplification we made may have a negative impact on the final prediction results of the model. Besides, we adopted the fractional derivative method in the model, which has improved prediction accuracy. However, its specific impact on the model and its practical significance need to rely on powerful mathematical tools to give a convincing proof in theory. In addition, the COVID-19 pandemic persists, and its long-term impact cannot be accurately predicted, which brings challenges to the prediction of future CO₂ emissions. To solve these problems, further research needs to be conducted.

In terms of empirical research, the work confirms the validity of the proposed new model for the regional CO₂ emission prediction problem, but this paper only considers three influencing factors, namely, economy, population, and energy consumption. In fact, CO₂ emission as a complex system has many influencing factors, such as urbanization level, industrial structure, energy structure, and energy intensity, which can also have a significant impact on CO₂ emission. Therefore, in the subsequent study, for carbon emissions in different regions, the variables should be screened to select the influencing factors with higher correlation. Second, in order to make the study more practical and instructive, our next study will focus on the carbon emissions of a particular sector or industry in a region.

Appendix 1: The Proof of Theorem 1

Proof.

Assume $m$ to be the smallest positive integer greater than $r$. Then,

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}^r{x}^{\left(r\right)}\left(k\right)& ={\mathrm{\nabla }}^m{\mathrm{\nabla }}^{-\left(m-r\right)}{\mathrm{\nabla }}^r{x}^{\left(0\right)}\left(k\right)\hfill \\ \hfill & ={\mathrm{\nabla }}^m\left[\begin{array}{c}-m+r\\ n\end{array}\right]\ast \left[\begin{array}{c}r\\ n\end{array}\right]x^{\left(0\right)}\left(k\right)\hfill \\ \hfill & ={\mathrm{\nabla }}^m\left[\begin{array}{c}m\\ n\end{array}\right]x^{\left(0\right)}\left(k\right)\hfill \\ \hfill ={\mathrm{\nabla }}^m{\mathrm{\nabla }}^{-m}{x}^{\left(0\right)}\left(k\right)=x^{\left(0\right)}\left(k\right).\\ \hfill \end{array}$$

According to Definitions 3 and 4,

$$A^{-r}{x}^{\left(r\right)}=\left(\begin{array}{ccccc}\left[\begin{array}{c}-r\\ 0\end{array}\right]& 0& 0& \cdots & 0\\ \left[\begin{array}{c}-r\\ 1\end{array}\right]& \left[\begin{array}{c}-r\\ 0\end{array}\right]& 0& \cdots & 0\\ \left[\begin{array}{c}-r\\ 2\end{array}\right]& \left[\begin{array}{c}-r\\ 1\end{array}\right]& \left[\begin{array}{c}-r\\ 0\end{array}\right]& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \left[\begin{array}{c}-r\\ n-1\end{array}\right]& \left[\begin{array}{c}-r\\ n-2\end{array}\right]& \left[\begin{array}{c}-r\\ n-3\end{array}\right]& \cdots & \left[\begin{array}{c}-r\\ 0\end{array}\right]\end{array}\right)·\left(\begin{array}{c}\left[\begin{array}{c}r\\ 0\end{array}\right]x^{\left(0\right)}\left(1\right)\\ \begin{array}{c}\left[\begin{array}{c}r\\ 1\end{array}\right]x^{\left(0\right)}\left(2\right)+\left[\begin{array}{c}r\\ 0\end{array}\right]x^{\left(0\right)}\left(2\right)\\ {\sum }_{i=1}^3\left[\begin{array}{c}r\\ n-i\end{array}\right]x^{\left(0\right)}\left(3\right)\\ ⋮\\ {\sum }_{i=1}^n\left[\begin{array}{c}r\\ n-i\end{array}\right]x^{\left(0\right)}\left(n\right)\end{array}\end{array}\right)=x^{\left(0\right)}$$

Therefore, $x^{\left(0\right)}\left(k\right)={\mathrm{\nabla }}^r{x}^{\left(r\right)}\left(k\right),x^{\left(0\right)}=A^{-r}{x}^{\left(r\right)}$.

The Proof of Theorem 2

Proof.

Given that ${\sum }_{i=2}^N{b}_i{(x_i^{\left(r\right)}\left(k\right))}^{m_i}+u$ does not contain unknown variable $x_1^{\left(r\right)}$, it can be considered a grey constant $M\left(k\right)$. Let $X\left(s\right)=L\left(x_1^{\left(r\right)}\left(t\right)\right)$, then

$$S^{q}X\left(S\right)-S^{q-1}{x}_1^{\left(r\right)}\left(0\right)+aX\left(S\right)=\frac{M\left(k\right)}{S}.$$ 22

The following results can be obtained by solving Eq. (16):

$$X\left(S\right)=\frac{\frac{M\left(k\right)}{S}+S^{q-1}{x}_1^{\left(r\right)}\left(0\right)}{S^q+a}.$$ 23

Finding the inverse Laplace transform of $X\left(S\right)$ can obtained by the following results:

$$\begin{array}{cc}\hfill x_1^{\left(r\right)}\left(k\right)& =L^{-1}\left[X\left(S\right)\right]=L^{-1}\left[\frac{\frac{M\left(k\right)}{S}+S^{q-1}{x}_1^{\left(r\right)}\left(0\right)}{S^q+a}\right]\hfill \\ \hfill & =M\left(k\right)L^{-1}\left[\frac{1}{S\left(S^q+a\right)}\right]+x_1^{\left(r\right)}\left(0\right)L^{-1}\left[\frac{S^{q-1}}{S^q+a}\right].\hfill \\ \hfill \end{array}$$ 24

According to the relationship between Laplace inversion and Mittag-Leffler equation, we can obtain:

$$L^{-1}\left[\frac{1}{S\left(S^q+a\right)}\right]=L^{-1}\left(\frac{S^{-1}}{S^q+a};S\right)=k^q{E}_{q,q+1}\left(-a{k}^q\right)=k^q\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)};$$ 25

$$L^{-1}\left(\frac{x_1^{\left(r\right)}\left(0\right)S^{q-1}}{S^q+a};S\right)=x_1^{\left(r\right)}\left(0\right)E_{q,1}\left(-a{k}^q\right)=x_1^{\left(r\right)}\left(0\right)\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+1\right)}.$$ 26

Therefore,

$${\widehat{x}}_1^{\left(r\right)}\left(k\right)=x_1^{\left(r\right)}\left(0\right)\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+1\right)}+M\left(k\right)k^q\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}.$$ 27

Let ${\widehat{x}}_1^{\left(r\right)}\left(1\right)=x_1^{\left(r\right)}\left(1\right)$, then:

$$x_1^{\left(r\right)}\left(0\right)=\frac{x_1^{\left(r\right)}\left(1\right)-M\left(k\right){\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}}{{\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+1\right)}}.$$ 28

Bringing Eq. (28) into Eq. (27) yields:

$${\widehat{x}}_1^{\left(r\right)}\left(k\right)=\left[\frac{x_1^{\left(r\right)}\left(1\right)-M\left(k\right){\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}}{{\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+1\right)}}\right]\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+1\right)}+M\left(k\right)k^q\sum _{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}.$$ 29

Let ${\sum }_{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+1\right)}=H\left(k\right),{\sum }_{h=0}^{\infty }\frac{{\left(-a{k}^q\right)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}=Q\left(k\right),{\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+1\right)}=H_{k=1}$, and ${\sum }_{h=0}^{\infty }\frac{{(-a)}^h}{\mathrm{\Gamma }\left(qh+q+1\right)}=Q_{k=1}$. Hence,

${\widehat{x}}_1^{\left(r\right)}\left(k\right)=\left[\frac{x_1^{\left(r\right)}\left(1\right)-M\left(k\right)H_{k=1}}{G_{k=1}}\right]H\left(k\right)+M\left(k\right)k^{q}Q\left(k\right)$.

Authors contributions

Shuhua Mao was invovled in formal analysis, funding acquisition, writing of review, editing, and project administration. Wen Lei was responsible for data collection and curation, conceptualization, writing of original draft, visualization, and formal analysis. Yonghong Zhang contributed to language editing, validation, and methodology.

Declarations

Conflict of interest

The authors declare no conflict of interest regarding the publication of this paper. This research was partly supported by the National Natural Science Foundation of China (Project No. 51479151 and No. 71871174).