Analysis and forecast of China's carbon emission: evidence from generalized group method of data handling (g-GMDH) neural network

Abstract

Achieving carbon emission targets requires an accurate analysis of trends in the rise and fall of carbon emissions. Due to economic growth and energy demand, China's emissions have increased dramatically in the past three decades. To this end, China has pledged to a carbon emissions peak by 2030, which has resulted in a large number of policy initiatives to cut emissions. Therefore, this study aims to present the generalized structure of machine learning (ML) derived group method of data handling (g-GMDH) as an improved alternative for analyzing and predicting China's future CO₂ emission trends. The study used annual data from 1980 to 2019 for gross domestic product, total natural resource rent, industrialization, urbanization, renewable energy, and non-renewable energy consumption to forecast the CO₂ emissions trend from 2020 to 2043. The CO₂ prediction results indicate that China will reach its CO₂ emission peak in 2033. Sensitivity analysis results revealed that industrialization, non-renewable energy, and urbanization significantly impacted the model's output and contributed the most to CO₂ emissions. In contrast, renewable energy consumption contributed the least to CO₂ emissions. The findings of this study can provide valuable guidelines for decision-makers as they set CO₂ reduction goals and implement appropriate energy-saving and emission-reduction initiatives.

Introduction

As the top carbon emitter in the world, the Chinese government on September 2020 announced at the UN General Assembly that China will endeavor to reach its peak CO₂ emissions by 2030 (Normile 2020). To achieve this task, China has implemented numerous policies to reduce carbon emissions, such as the National Key Energy-saving and Low-Carbon Technology Promotion Catalog to enhance the development of low-carbon technologies (NDRC 2017). Also, the nationwide emission trade scheme (ETS) industries in the pilot areas (Hubei, Guangdong, Beijing, Shenzhen, Tianjin, Chongqing, and Shanghai provinces) reduced CO₂ emissions by 15.5% and energy usage by 22.8%, mainly through modifying the industrial structure and technical productivity (Hu et al. 2020). By 2021, the seven pilot provinces had generated an estimated 11.4 billion Yuan in expenses while accumulating 480 million tons of carbon dioxide equivalent (Zhang et al. 2022a). However, because of its detrimental effect on China's economic progress, many researchers continue to dispute the effectiveness of a carbon ETS. For instance, Hübler et al. (2014) evaluated China's carbon ETS using the computable general equilibrium (CGE) model. They found that it reduced gross domestic product (GDP) by around 1% in 2020, and they predict that it may reduce welfare by about 2% by 2030.

A carbon tax was recommended by the National Development and Reform Commission (NDRC), and the Ministry of Finance (MOF) was to be implemented in China by the year 2012; however, the Chinese government did not do so until 2021 (Zhang et al. 2022b). This scheme was proposed to reduce carbon emissions and energy consumption through carbon taxes. Chi et al. (2014) discovered that a carbon tax helped control emissions and energy conservation, and Fu et al. (2021) argued that China should consider implementing carbon taxes that range from 18.37 to 38.25 Yuan per ton. However, establishing a carbon pricing device is difficult due to the rate's unpredictability and its effects on the environment, the energy industry, and the economy.

Apart from ETS, National Key Energy-Saving, and carbon tax, future attempts at realizing the 2030 carbon emissions target in China are centered on the "15^th Five-Year Plan". These include policies aimed at ensuring the share of non-fossil energy will reach around 25% regarding the energy mix; carbon emissions per unit of GDP will drop by more than 65% compared with the 2005 level. Also, the country must strive to achieve an adequate and low-carbon energy infrastructure in the industrial sector by ensuring that renewable sources produce 50% of transmitted electricity. Other policies include urban planning and development geared toward green and low-carbon development. Efforts will be put into developing green cities, towns, and communities and improving resource utilization efficiency by attempting to reduce resource consumption and cut carbon emissions (NDRC 2021).

A growing amount of empirical literature has projected China's CO₂ emissions. However, empirical evidence frequently produces inconsistent results due to variances in methodology, sample periods, and numerous CO₂ indicators. From Table 1, studies that have forecasted a 2030 peak in emissions include the usage of the China-in-Global Energy Model (C-GEM) Zhang et al. (2016), and the logarithmic mean divisia index (LMDI) and decoupling index (LMDI-D) approach (Li and Qin 2019). Both predictions differ from a 2025 peak obtained via the Multi-objective optimization model Yu et al. (2018) and a 2028 to 2030 prediction obtained via the feasible generalized least squares (FGLS) model (Zhang et al. 2021). In addition, Li et al. (2018) used the grey and impacts on ecosystems (I), which are the product of the population size (P), affluence (A), and technology (T) (IPAT) model to assert the feasibility of 2030 peak emissions, provided the GDP in China is lower than 151,426.15 billion. Using the Chinese Integrated MARKAL-EFOM System, Zhang and Chen (2021) also noted that the 2030 peak emission depends on renewable energy sources reaching 60% of total energy consumption by 2050. The conventional predictive models mentioned above are highly accurate, have a small number of parameters, are quite easy to train, and are used for linear sequences. The capacity of these models to fit nonlinear sequences, however, is inferior to that of machine learning techniques.

Table 1 Studies related to China's projected carbon emissions

In order to overcome the limitations of these prediction models, numerous machine learning methods have been applied which can handle nonlinear sequences. For instance, the Recurrent Neural Network (RNN) has a relationship between the data and time making it suitable for time series analysis. RNN can create sequence-to-sequence mapping between input and output data by adding cyclical connections to neurons. As a result, the input from the time step before influences the output of every subsequent time step. RNN, therefore, accomplishes the "memory" attribute. RNN usage in predicting carbon emissions in China includes Feng et al. (2019) employing the RNN methodology to estimate the regional transport rate of emissions in Hangzhou City. However, because the RNN depends on the data over a lengthy period, gradient disappearance and explosion issues arise during model training (Yin et al. 2019). The problem of minimal gradients is known as the disappearing gradient problem. The disappearing gradient problem mainly affects the lower layers of the network and makes them more challenging to train. Similarly, if the gradient associated with weight becomes extremely large, the updates to the weight will also be extensive. This can cause the gradients to become unstable, preventing the algorithm from converging. This problem of huge gradients is known as the exploding gradients problem.

One means of overcoming RNN limitations concerning sequential data processing is using Long short-term memory (LSTM) developed by (Hochreiter and Schmidhuber 1997). In contrast to RNNs, LSTMs use memory blocks to replace the hidden layer, which can effectively address the issues of gradient disappearance and explosion during RNN training (Sundermeyer et al. 2015). Several studies have employed LSTM to predict carbon emissions in China; case in point, Fan et al. (2022) employed the LSTM technique and confirmed that China will reach a carbon emission peak in 2030. Using sixteen potential input variables to predict carbon emissions (Huang et al. 2019) confirmed that LSTM has higher predictive accuracy than that (Back propaganda neural network) BPNN. But because the LSTM approach contains so many parameters, it is difficult to calculate. Consequently, the LSTM's training period is lengthy (Sorkun et al. 2020).

This study contributes a new perspective to the extant literature by employing the robust generalized structure of the group method of data handling (g-GMDH) as an improved neural network to predict CO₂ emissions in China. The g-GMDH neural network possesses a self-organizing nature to automatically tune model parameters and generate the optimal CO₂ emission model structure while using less computation time. The g-GMDH has a better adaptation and generalization ability with the advantage of not being prone to over fit and can perform excellently even when the predictive variables are irregular. Also, unlike other standard machine learning models trained to forecast CO₂ emissions, the g-GMDH neural network does not require a manual adjustment of learning parameters to generate the best outcome. Based on significant sources of carbon emissions, the g-GMDH uses reproducible models to select predictors and thresholds to deliver carbon emissions intensity prediction from 2020 to 2043. The g-GMDH model performance in predicting carbon emissions was further compared with Long Short-Term Memory (LSTM). Furthermore, this study investigates the impact of influencing factors on CO₂ emissions and then, distinguishes which indicator has the most and least significant effect on CO₂ emission.

Economic output from the secondary industry in China is the primary determinant of CO₂ emissions (Fan et al. 2015), with China's population expansion and urbanization linked to increased environmental degradation, resource consumption, and CO₂ emissions in recent years (Mendonça et al. 2020; Ridzuan et al. 2020; Chai et al. 2020). Natural resources, industrialization, and urbanization are therefore considered in the g-GMDH model employed in this study because they all have an indirect effect on CO₂ emissions due to energy use. Table 1 reports related works done in forecasting CO₂ emissions in China, and Fig. 1 reports the trend of CO₂ emissions in China and globally from 1980 to 2019.

Fig. 1

Trend of CO₂ emissions in China and globally from 1980–2019

Materials and methods

Data handling and processing procedures

The input variables used include renewable energy consumption, non-renewable energy consumption, total natural resource rent, industrialization, economic growth, and urbanization. Carbon emission (CO₂), which will serve as our output variable, is the byproduct of CO₂ emissions from burning fossil fuels. As for input variables, total natural resource rent (TNRR) is the accumulation of coal (soft and hard) rents, oil rentals, forest rents, mineral rents, natural gas rents, and other rents. Gross domestic product GDP per capita (constant 2015 US$) is a gauge of economic expansion: The total use of biomass, solar, wind, hydropower, and geothermal energy is called renewable energy (REN). The total consumption of petroleum, gas, coal, and other liquids is known as non-renewable energy (NREN): Urbanization (URB) is indicated by the population of cities. Industrialization (IND) is measured as a value-added percentage of GD. Variables are changed to logarithms to evaluate estimated coefficients of elasticity and smoothness (Riti et al. 2017).

This research periodically employs annual data covering 1980 to 2019. Data for TNRR, IND, URB, and GDP were taken from the World Bank's World Development Indicators online database (WDI, 2021). Data on renewable energy consumption, carbon dioxide (CO₂) emissions, and non-renewable energy consumption were obtained from the US Energy Information Administration (US-EIA, 2021). The summary of the data description and statistical data analysis are reported in (Tables 2 and 3).

Table 2 Summary of data description

Table 3 Statistical parameters of the data used to create the model

During data processing, feature selection (variable selection) was performed to identify and omit irrelevant, unwanted, and redundant features from data that will not contribute to the accuracy of the forecasting model or could reduce the model's precision. Pearson correlation coefficient (R) was used to estimate the relative importance of the input parameters with the output parameter (Eq. 1). The correlation coefficient (R) values always lie between − 1 and + 1. In this case, the values close to positive indicate a similar association between the two variables, while the values close to zero show a weak association between the two-variables pair, and the values close to negative indicate an inverse association between independent variables (Mahmoud et al. 2019).

$$R_{ut} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {u_{i} - \overline{u}} \right)\left( {t_{i} - \overline{t}} \right)} }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {\left( {u_{i} - \overline{u}} \right)^{2} } } \sqrt {\sum\nolimits_{i = 1}^{n} {\left( {t_{i} - \overline{t}} \right)^{2} } } }}$$

(1)

where u_i and t_i denote the individual sample points indexed with i, n represents the size of the samples, $\overline{u}$ denotes the mean sample, and analogously for $\overline{t}$.

The input parameters R-values are REN, NREN, TNRR, IND, GDP, and URB, with the output parameters of CO₂ shown in Fig. 2. The normalization processing was done using Eq. 2:

$$X_{{{\text{NORM}}}} = \frac{{x - x_{\min } }}{{x_{\max } - x_{\min } }}$$

(2)

Fig. 2

Heat map showing the correlation coefficient matrix of the input variables of REN, NREN, TNRR, IND, GDP, and URB with the output parameters of CO₂

Equation 2, $x$ is the actual value, $X_{{{\text{NORM}}}}$ signifies the dataset normalized value, $x_{\min }$ and depicts the minimum value and $x_{\max }$ the maximum value. The selected technique enables the computational learning algorithm to execute faster, improves the model's correctness, reduces over fitting, and also it decreases the complexity of the model (Al-Abadi et al. 2020).

LSTM neural network

Long short-term memory (LSTM) neural network is a recurrent neural network (RNN) designed to handle long-term dependencies in sequential data. LSTM networks use special units, including a "memory cell," that can maintain information over time and regulate the flow of information through the network, which makes it ideal for processing and predicting data with long-term dependencies. The basic idea behind using LSTM neural network is to model the complex relationships between various factors influencing China's carbon emissions. Therefore, the network takes in the historical emissions data and influencing factors suits of REN, NREN, TNRR, IND, GDP, and URB to learn patterns and trends in the data, allowing it to make predictions about future emissions. The advantage of using an LSTM network for this application is its ability to handle long-term dependencies in the data, in which the first stage in the LSTM process is identifying precisely what kind of data can travel through the cell state. Figure 3. shows the LSTM neural network structure.

Fig. 3

Structure of LSTM neural network (Ahmed et al. 2022a)

Equations (3)–(10) demonstrate the update of the LSTM memory cells. From (Fig. 3), the notations for each equation are $x_{t}$ representing the input vector, $\sigma$ representing the sigmoid function $f_{t} ,o_{t} ,{\text{and }}i_{t}$ are values of forget gate, output gate, and input gate, respectively.$W_{f} ,W_{i} ,W_{C} ,{\text{and }}W_{o}$ are the weight matrices of gate and cell states. $h_{t}$ is the hidden state of the LSTM. $\tilde{c}_{t}$ represents the temporary vector for the cell states, $c_{t}$ is the vector for the cell states, and. $b_{f} ,b_{o} ,b_{i} ,{\text{ and }}b_{C}$ are bias vectors of forget gate, output gate, input gate, and temporary cell state.

The forget gate controls the decision using sigmoid, which produces a 0 to 1 value ft based on the prior output h_t−1 and the recent input x_t. It determines whether or not the previous information C_t−1 learned at the last minute has been transferred entirely or partially. Equation 3 shows the ft formula.

$$f_{t} = \sigma \left( {W^{fx} x_{t} + W^{fh} h_{t - 1} { + }b_{f} } \right)$$

(3)

The second step is determining what data should be added to the cell state. To produce i_t and decide which values should be changed, the input gate uses a sigmoid. After that, a tanh layer is employed to construct a new candidate value $\tilde{c}_{t}$, as reported in Eqs. 4 and 5. The following are lists of computing processes:

$$i_{t} = \sigma \left( {W_{i} \cdot [h_{t - 1} ,x_{t} ] + b_{i} } \right)$$

(4)

$$\tilde{c}_{t} = \tanh \left( {W_{c} \cdot [h_{t - 1} ,x_{t} ]{ + }b_{c} } \right)$$

(5)

The process of forgetting undesired information and adding new information is the third step. The old cell state C_t-1 is multiplied by f_t which ignores all unrequired information, and then, it is added to $i_{t} \tilde{c}_{t}$ that will give the new cell state C_t as shown in Eq. 6

$$c_{t} = f_{t} c_{t - 1} + i_{t} \tilde{c}_{t}$$

(6)

The output of the LSTM model is generated last. The tan h layer compresses the internal state C_t, which is then multiplied by the output gate $o_{t}$ as illustrated in Eqs. 7 and 8:

$$o_{t} = \sigma \left( {W^{{{\text{ox}}}} x_{t} + W^{{{\text{oh}}}} h_{t - 1} { + }b_{o} } \right)$$

(7)

$$h_{t} = o_{t} \tan \,h(c_{t} )$$

(8)

The equations above $W^{x}$ depict the weights between the inputted layer and inputted node x; the weights between the inputted node and hidden layers are characterized by $W^{h}$; b portrays the bias term, $\sigma$ which denotes the sigmoid function. Also, all three gates are sigmoid units $\sigma$, as defined in the equation

$$\sigma (x) = {\text{sigmoid}} = \frac{1}{{1 + e^{ - x} }}$$

(9)

And tanh is expressed in Eq. (10):

$$\tanh (x) = \frac{{e^{x} - e^{ - x} }}{{e^{x} + e^{ - x} }}$$

(10)

Generalized structure of GMDH (g-GMDH) model

The g-GMDH neural network is an advanced data analysis, prediction, and modeling tool, offering numerous advantages in a wide range of applications. Firstly, its robustness and adaptability allow it to effectively handle complex, nonlinear, and noisy data, making it suitable for diverse domains (Shen et al. 2019). Secondly, the automatic model selection feature eliminates the need for predefined structures, iteratively generating and evaluating models to find the optimal fit while reducing overfitting risk. Additionally, g-GMDH's interpretability fosters a better understanding of relationships between input and output variables, aiding informed decision-making. Its scalability enables the efficient processing of large datasets through parallel processing capabilities, making it an excellent choice for big data applications and real-time analysis. Lastly, the network's reduced training time, attributed to its inductive nature and iterative model generation process, allows for faster model development and deployment, enhancing overall performance and efficiency.

The g-GMDH neural network is well-suited for predicting carbon emissions because it can handle large and complex datasets, and it can automatically identify and extract relevant features from the data. By analyzing historical data on carbon emissions, the g-GMDH neural network can learn patterns and relationships that can then be used to make accurate predictions about future emissions. The GMDH model's goal is to identify a function $\hat{f}$ that is used as an estimation rather than an actual function, f, to evaluate the output (CO₂), y, in the presence of a given input vector U = (u₁, u₂, u₃, …, u_n), as close to its actual output as possible, p (CO₂,). As a result, if you have a single output and n data pairs with numerous inputs, you will get:

$$y_{i} = f\left( {u_{i1} ,u_{i2} ,u_{i3} ,....,u_{in} } \right) \, \left( {i = 1,2,3,....,M} \right)$$

(11)

The GMDH network may be trained to evaluate output values t using any given input vector $U = u_{i1} ,u_{i2} ,u_{i3} ,....,u_{in}$, implying:

$$y_{i} = \hat{f}\left( {u_{i1} ,u_{i2} ,u_{i3} ,....,u_{in} } \right) \, \left( {i = 1,2,3,....,M} \right)$$

(12)

To tackle this difficulty, GMDH creates a general relationship in the framework of a mathematical description of output and input parameters termed the reference. The goal is to find the GMDH network that minimizes the square difference between the expected and actual output, as follows:

$$\sum\limits_{i = 1}^{M} {\left[ {\hat{f}\left( {u_{i1} ,u_{i2} ,u_{i3} ,....,u_{in} } \right) - p_{i} } \right]}^{2} \to \min$$

(13)

The Kolmogorov–Gabor polynomial, also identified as the polynomial series, which is the Volterra functions complex discrete form (Anastasakis and Mort 2001; Najafzadeh and Azamathulla 2013), can represent the general relationship between input and output parameters in the mode of:

$$p = a_{o} + \sum\limits_{i = 1}^{N} {a_{i} u_{i} + } \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {a_{ij} u_{i} u_{j} } + } \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {\sum\limits_{k = 1}^{N} {a_{ijk} u_{i} u_{j} u_{k} + ...} } }$$

(14)

Equation 14 can be simplified by the simplified GMDH network's partial quadratic polynomial equation (Shen et al. 2019)

$$\, y = W\left( {u_{i} ,u_{j} } \right) = a_{o} + a_{1} u_{i} + a_{2} u_{j} + a_{3} u_{i}^{2} + a_{4} u_{j}^{2} + a_{5} u_{i} u_{j}$$

(15)

This network of associated neurons generates the mathematical association between the input–output variables stated in Eq. 13. The weighting Eq. 14 coefficients are calculated using regression techniques to reduce the variation between an actual (p) and anticipated (y) output as each variable pair of u_i and u_j input is minimized (Armaghani et al. 2020). Figure 4. depicts the GMDH network design in a schematic form.

Fig. 4

Type of the GMDH network design

A tree of polynomials is formed using the quadratic equation from the provided Eq. 14, in which the weighting coefficients can be calculated using the least square approach method. The quadratic function of weighting coefficients W_i is obtained to optimally match the output in the entire set of output-input data pairs as follows:

$$E = \frac{{\sum\nolimits_{i = 1}^{M} {(p_{i} - W_{i} ())^{2} } }}{M} \to min$$

(16)

To fit the general form of GMDH algorithms, the possibilities for dual independent variables among the overall n input parameters are drawn. To construct the regression polynomial described in Eq. 16, which suits the dependent observations better in the least-square sense. Consequently $C_{n}^{2} = {{n\left( {n - 1} \right)} \mathord{\left/ {\vphantom {{n\left( {n - 1} \right)} 2}} \right. \kern-0pt} 2}$, observations can create quadratic polynomial neurons $\left\{ {(p_{i} ;u_{xi} ,u_{yi} ); \, (i = 1,2,3....M)} \right\}$ for various $x, \, y \in \left\{ {1, \, 2,3, \, . \, . \, . \, .n} \right\}$ in the feed-forward network's first layer. M data triples can now be created $\left\{ {(p_{i} ;u_{xi} ,u_{yi} ); \, (i = 1,2,3....M)} \right\}$, making use of $x, \, y \in \left\{ {1, \, 2,3, \, . \, . \, . \, .n} \right\}$ the form of:

$$\left[ {\begin{array}{*{20}c} {u_{1x} } & {u_{1y} } & : & {p_{1} } \\ {u_{2x} } & {x_{2y} } & : & {p_{2} } \\ {...} & {...} & : & {...} \\ {u_{mx} } & {u_{my} } & : & {p_{m} } \\ \end{array} } \right]$$

(17)

The following matrix expression can be expressed directly for each row of the m data triples using the quadratic sub-expression indicated in Eq. 14 as follows:

$$Aa = P$$

(18)

a illustrates an unknown vector of quadratic polynomial weighting factors in Eq. 12:

$$a = \left[ {a_{0} ,a,_{1} a_{2} ,a_{3} ,a_{4} ,a_{5} } \right]^{T}$$

(19)

Superscript, T, indicating the matrix transposition:

$$P = \left[ {p_{1} ,p_{2} ,p_{3} ,...p_{M} } \right]^{T}$$

(20)

Standard equations are solved using the least-square approach, which is developed from the concept of multiple regression analysis, which is in the form of:

$$a = \left( {A^{TA} } \right)^{ - 1} A^{T} P$$

(21)

For the entire set of m triples data, Eq. 21 signifies the optimum quadratic weighting coefficients given the vector in Eq. 1. To tackle the issue of linear dependency and equation complexity, the generalized Group Method of Data Handling (GMDH) can build a higher-order polynomial (Ivakhnenko, 1968). By minimizing the fitness function in Eq. 22, the training sub-samples are used to construct the optimal model structure:

$$AR(s) = \frac{1}{{N_{B} }}\sum\limits_{i = 1}^{{N_{B} }} {(z_{i} } - z_{i} (B))^{2} \to \min$$

(22)

The fitness function is based on the d-fold cross-validation criteria, which randomly selects training and testing subsamples considering all information in data samples. A comprehensive search is conducted on models classified of similar complexity, allowing the entire search termination rule to be planned. The models are compared to the measured samples, and the process is repeated till the criterion is reached. Because of the constraint imposed by the computation period time, it is suggested to increase parameters because of the criterion value after specific iterations' layer and calculation time while putting the models together. Then, for the chosen set of best variables, complete arranging approaches are used until progress is minimal. This gives the option of including more information components in the input and saving successful elements between layers to get the best model. In the first phase, the user defines the data sample for the model. In the second level, several layers are used to express model complexity. In the third step, the best models are created, and in the fourth step, the best model is chosen. During the fifth stage, discriminating criteria are used to complete the extra model definition, as shown in Fig. 5.

Fig. 5

Flowchart of the generalized structure of combinatorial GMDH algorithm

Model validation

The g-GMDH neural network is a powerful and versatile tool for forecasting carbon emissions, as it can effectively model and predict complex relationships between various influencing factors. The g-GMDH network relies on inductive modeling and self-organization principles, making it capable of handling nonlinear data and adapting to changing patterns. As a result, it is well-suited for capturing intricate connections between factors such as gross domestic product, total natural resource rent, industrialization, urbanization, renewable energy, and non-renewable energy, which contribute to carbon emissions.

When choosing influencing factors as input variables, it is vital to consider several factors to set up an accurate and trustworthy carbon emission forecast model. There are several criteria, including data availability, interpretability, correctness, relevance, and consistency. The influencing elements should be relevant to carbon emission prediction, and the data used should be accurate, reliable, and easily accessible over a specific time. The data should also be consistent across chosen time and cover a broad range of factors that contribute to carbon emissions. When applied to carbon emission forecasting, the g-GMDH neural network iteratively generated CO₂ models, automatically selecting the optimal fit based on predefined criteria. This process helped to reduce overfitting, improving the accuracy and reliability of emission forecasts. Furthermore, the interpretability of the CO₂ g-GMDH model enabled a better understanding of emission trends and the underlying relationships between contributing factors.

The (g-GMDH) model and Long Short-Term Memory (LSTM) algorithms were implemented in MATLAB R2021a with Windows 10 operating system. The root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (R) were utilized as statistical indicators to analyze the results of the predicted CO₂ models. A correlation coefficient (R) near 1 indicates optimal model performance. On the other hand, if the MAE and RMSE values approach zero when comparing models, the model is a reliable predictor. Equations 23, 24, and 25 provide mathematical expressions for RMSE, MAE, and R (Asante-Okyere et al. 2018; Okon et al. 2021)

$$R = \frac{{\sum\nolimits_{i = 1}^{N} {\left( {y_{i} - \overline{y}} \right)} \left( {Y_{i} - \overline{Y}_{i} } \right)}}{{\left( {\sqrt {\sum\nolimits_{i = 1}^{N} {\left( {y_{i} - \overline{y}_{i} } \right)}^{2} } } \right)\left( {\sqrt {\sum\limits_{i = 1}^{N} {\left( {Y_{i} - \overline{Y}_{i} } \right)}^{2} } } \right)}}$$

(23)

$${\text{MAE}} = \frac{1}{N}\sum\limits_{i = 1}^{n} {\left| {y_{i} - Y_{i} } \right|}$$

(24)

$${\text{RMSE}} = \sqrt {\left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {(\mathop y\nolimits_{i} - \mathop Y\nolimits_{i} )^{2} } } \right)}$$

(25)

whereas N stands for the number of data points, $\overline{y}_{i}$ signifies the mean number for the measured variables,$y_{i}$ stands for the actual carbon emission values, the expected carbon emission is characterized by $Y_{i}$, and the mean values of the predicted variable are $\overline{Y}_{i}$ (Adeniran et al. 2019; Ahmadi and Chen 2019). If the MAE and RMSE values approach 0 during training, the model has performed well. The trained model's ability to function effectively when validated using withheld testing data, on the other hand, is given greater weight. As a result, if the test results are favorable, the best-performing model with improved generalization capacity can be chosen if (MAE and RMSE) approach zero (Asante-Okyere et al. 2018).

Results and discussion

Performance indicators and error analysis of CO₂ emissions in China

CO₂ emissions models were produced from the primary data set by gathering and identifying numerous factors and dynamic development characteristics affecting carbon emission in China before 2019. The generalized structure of the GMDH (g-GMDH) model had the best overall performance compared to the most used machine learning algorithm of Long Short-Term Memory (LSTM). From the training results, the g-GMDH had the lowest error margin of 0.019 and 0.0105 for RMSE and MAE, respectively, which was less than 27% to 35% of computation time better than the LSTM techniques during CO₂ emissions prediction. On the other hand, the LSTM CO₂ model archived RMSE and MAE values of 0.0440 and 0.09035, respectively, during training Table 3. To practically assess the ability of the trained models, they were tested on withheld data distributed from 2008 to 2019. These data did not contribute to the development of the CO₂ models; therefore, they can provide an unbiased assessment of any new dataset.

It was discovered that the g-GMDH CO₂ model was the best-performing model, which generated predictions close to the actual CO₂ emissions values. This was seen from the results shown in Fig. 6 that g-GMDH obtained the least RMSE and MAE values of 0.1162 and 0.3722, respectively. The LSTM produced CO₂ prediction having error margins scores of 0.2257 and 0.4815 for RMSE and MAE, respectively, as reported in Table 4. The least RMSE and MAE score from g-GMDH indicates that the CO₂ prediction results do not deviate much from the measured CO₂ value. Therefore, the least RMSE value of 0.1162 during testing makes the proposed CO₂ model the best and most stable CO₂ emissions model compared to the LSTM technique.

Fig. 6

RMSE and MAE error analysis for g-GMDH and LSTM during CO₂ emissions forecast

Table 4 CO₂ predictive models statistical measures

The g-GMDH model is a self-organizing model in which the model's topology optimizes itself based on the data input of influencing elements for carbon emission and actual CO₂. The g-GMDH avoids the issue of overfitting that plagues ANN and standard ML by objectively picking the CO₂ model with optimum complexity, which proves the noise resistance ability exhibited by g-GMDH. The performance of the developed predictive CO₂ models for g-GMDH and LSTM, as compared to the actual CO₂ data, is presented using the Taylor diagram (Fig. 6). The Taylor diagram was drawn to show and interpret the correlation coefficient, root means square error, and standard deviation of the models. The results during CO₂ forecasting reveal that the best-performing CO₂ predictive model is g-GMDH, as it performed perfectly compared to the LSTM CO₂ model (Fig. 7).

Fig. 7

Evaluation of g-GMDH and LSTM CO₂ models using Taylor diagram

Forecasting China's carbon dioxide emissions

The g-GMDH is a proper self-organizing technique and tool for categorizing data collection and presenting a suitable output (CO₂) based on the data observed for appropriate forecasting problems, including time series (Yahya et al. 2019). The g-GMDH algorithm produced excellent results in estimating greenhouse gas emissions and was used to examine the CO₂ emissions trend from 2020 to 2043. The dataset was divided into two consecutive segments of training and testing datasets to forecast carbon emissions. The training set was used for estimating parameters and learning the models, which were distributed from 1980 to 2007. The testing set was used to measure the predictive performance of the CO₂ emissions model and was distributed from 2008 to 2019, and then, carbon emissions were forecasted from 2020 to 2043. From 1980 to 2019, two parts were used in this approach to predict more accurate results. Following the above-mentioned technique, the algorithm predicted China's CO₂ emissions yearly from 2020 to 2043. The deviation from the measured CO₂ value can be examined visually (Figs. 8 and 9).

Fig. 8

CO₂ emission intensity forecast for g-GMDH

Fig. 9

CO₂ emission intensity forecast for LSTM

The historical pattern of CO₂ emission in China, as shown in Fig. 8, demonstrates a constant increase in CO₂ emission over the sampled period. From this study's projection, the Chinese nation will achieve its carbon emission peak in 2033. This finding is in-line with Zhou et al. (2022), who found that under the current policy implementations, China will reach a carbon emissions peak in 2035. This finding contradicts the commitment made at the UN general assembly by the Chinese government to reach its peak by 2030. The reported result may be primarily because the high GDP expansion cancels out the emission-reduction advantages of energy-related technology advancements and energy structure optimization. This finding demonstrates that the peak goals cannot be successfully attained under the current energy policy objectives. Comparison of actual CO₂ and predicted CO₂ models performance of g-GMDH and LSTM can be examined visually from Fig. 10.

Fig. 10

Comparison of CO₂ emission prediction performance of g-GMDH and LSTM CO₂ models

The energy and industrial sector, predominantly fueled by coal, is China's key source of carbon emissions. Industrialization, fossil fuel energy, and urbanization contributed the most to environmental distortion. They accounted for 42.4%, 41.1%, and 39.95%, respectively, as seen in Fig. 11. Findings regarding fossil fuel energy conform with that of (Wang et al. 2019). The more significant the proportion of coal used, the greater the region's dependency on fossil fuels in overall energy consumption. Also, in line with the findings of Zhou et al. (2019), the energy consumption structure is thought to have detrimental effects on emissions reduction. Between 2012 and 2014, China was said to have used over half of the world's coal in power generation, thus considerably increasing CO₂ emissions. After 2014, there was a pause in CO₂ emissions rise as the Chinese government implemented several initiatives to reduce CO₂ emissions (Ahmed et al. 2022b). Furthermore, carbon emissions began an upward trend again in 2016. China burnt roughly 64% of coal to meet its energy mandate in 2019 (IEA 2020) and accounting for 52% of worldwide coal consumption, according to Enerdata (2019).

Fig. 11

The percentages of contributing variables in predicting CO₂ emissions

Industrialization in developing countries degrades the environment and harms people (Valadkhani et al. 2019). Increasingly we have seen a significant amount of industrialization in recent decades regarding social and economic development. However, legislated laws to limit fossil fuel use from industrialization activities have not been successfully applied. As a result, a rise in fossil fuel usage fulfills the energy mandate, considered the most severe environmental hazard that causes global warming. Furthermore, in China, urbanization is connected to industrial development, characterized by high energy consumption and low energy efficiency, resulting in environmental damage. Consistent with our findings, other workers have concluded that a higher level of urbanization increases carbon intensity (Rehman and Rehman 2022).

Because China is the world's fastest-growing economy, and its GDP has expanded dramatically over the last 40 years, the beneficial influence of economic expansion on its expanding footprint seems plausible. In every economic sector, the rise in income has boosted resource consumption. As a result, currently, China is the largest consumer of energy and has the highest overall environmental footprint globally. Subsequently, as the economy grows, energy issues become more prevalent in meeting the population's energy needs. For example, large-scale population migration results in consumption and production activities, implying that migration has an ecological impact (Gao et al. 2021). Natural resource richness reduces reliance on imported energy sources. It favors using domestic energy sources that are less polluting, such as natural gas, which helps to alleviate environmental degradation (Joshua et al. 2020). However, in China, this is not the case, as coal, a high-pollutant energy source, is widely used to meet energy demands. Because the Chinese use of natural resources is unsustainable, energy plans should be developed and executed to minimize reliance on traditional, high-polluting energy sources, such as fossil fuel energy sources which are the primary cause of environmental distortion. According to the Global Energy Statistical Yearbook 2021, China has met 28 percent of its energy needs with renewable energy. Our findings suggest that China can effectively cut CO₂ emissions with increased investments in renewable energy sources Li (2020), essential for meeting CO₂ emission reduction goals (Magazzino et al. 2021).

Utilizing inductive modeling and self-organization principles showed the effectiveness of the g-GMDH neural network when forecasting carbon emissions in China. It showed its ability to model complex relationships between various influencing factors. The network iteratively generated and evaluated the CO₂ model automatically, selecting the optimal model to reduce overfitting and enhance accuracy. The interpretability of the CO₂ emission g-GMDH model supports a better understanding of emission trends for China and its contributing factors, enabling the development of targeted policies and strategies for carbon emission reduction and climate change mitigation which can be adapted worldwide.

Conclusion

This study adopted the Group Method of Data Handling (GMDH) technique to anticipate carbon emissions in China from 2020 to 2043 via input elements which included economic development, total natural resource rents, renewable and non-renewable energy, industrialization, and urbanization. To attain this goal, the study initially validated the GMDH model's prediction capacity by examining which input variable has the most significant impact on carbon emissions using actual data. Reported findings revealed that the most critical factors influencing carbon emissions were industrialization, fossil fuel energy use, and urbanization.

In contrast, renewable energy had the lowest impact on carbon emissions across the period studied. The GMDH model's predicted carbon emission values were close to the actual values. We also confirmed from the GMDH model that carbon emissions in China will likely peak around 2033 and fall slowly but steadily until 2043. To achieve the 2030 carbon emissions peaking target, a careful implementation of the pertinent energy economic structure and clean energy architecture by the Chinese government while upholding a sustainable economy is outlined in the "14th and 15th Five-Year Planned" policies.