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As the population of the earth is being exponentially increasing, the exhaustion of carbon dioxide is increasing day by day resulting in the extreme overheating of the environment, which has become a significant reason for climate change. Global efforts to mitigate climate change were focused on the reduction of future days with extreme overheating of the environment. There are many research and surveys that have been conducted by various scientists, students, and other officials which were about the reasons for the high emission of CO₂ in different countries. Most of the empirical studies took the parametric modeling approach to analyze the factors that initiate and support the emission of CO₂. However, the traditional parametric approach optimizes a function to a known form with a set of finite and pre-determined parameters. This rigidity limits the predictive power of the parametric models. In recent years, nonparametric machine learning techniques have played dominantly with the enhancement of the forecast.

In this paper, a Bayesian nonparametric kernel prediction algorithm in machine learning is applied to predict CO₂ emissions. A literature review has been conducted so that the proper independent variables have been identified. Classical least squares, robust least squares, and algorithms of the GPR were introduced and their prediction performance, including the evaluation criteria that are effective in the measurements for model performance, were summarized. The reliability and efficiency of the proposed algorithms were then demonstrated through the comparison between the actual data and the predicted results. It is found that GPR can give the most accurate predictions on CO₂ emissions.

The growth of the economy, energy utilization, and CO₂ emissions are deeply related to each other. Kolstad and Krautkraemer (1993) point out that while the use of resources like the energy has a bright side on growth, it has negative environmental impacts. Traditional growth theories like the Solow growth model failed to consider the environmental impacts of growth (Solow, 1956). More modern growth theories study the interrelationship among energy, the environment, and economic growth (see, for example, Kolstad and Krautkraemer (1993), Jorgenson and Wilcoxen (1993) or Xepapadeas (2005) for a brief review).

Empirical studies depict that the growth of the economy and the ingestion of energy incorporates with the process of CO₂ emission. Recently, Hu et al. (2020) study the dominant reasons for carbon emission among the Belt and Road countries and find that CO₂ emissions have increased significantly due to economic growth. Similarly, Shabaz et al. (2013) found that in Indonesia, the emission of CO₂ increased for the extreme boost of the economic zone, while Shahbaz et al. (2016) found that economic growth led to CO₂ emissions in Bangladesh and Egypt. Meanwhile, other studies discovered a bilateral causal relationship among the three variables. Munir et al. (2020) prove the fact that there is a relation of aftermaths and economy between GDP and energy ingestion in the major countries of the ASEAN (Association of Southeast Asian Nations), while Liu and Hao (2018) find that in energy-exporting countries, there is a bilateral relationship which may be a full-duplex connection between CO₂ emissions, energy utilization, and GDP per capita. Similarly, a repeating loop effect is observed between energy ingestion, CO₂ emission, and the advancement of the economy by Kahouli (2018). Accordingly, Mohmannd et al. (2020) observed the working principle of the causal relationship among transportation infrastructure, economic growth, and transportation emissions from 1971 to 2017 in Pakistan. The results show short-term causality from transportation infrastructure, economic growth, fuel consumption to CO₂ emissions, and the long-run relationship between economic advancement and infrastructure.

Apart from the growth and energy consumption, industrialization, population growth, and income level also contributed a great share in global carbon emissions. Minx et al. (2011) found that “industrialization” can be taken into consideration for the rapid increase of carbon dioxide emission in China from 2002 to 2007 while Zhang et al. (2014) found that the growth of the tertiary industry can decrease the CO₂ exhalation intensity. Nasir et al. (2021) examined the connection between the factors which are the exhalation of CO₂, industrialization, growth of the economy, energy ingestion, and several connecting factors from 1980 to 2014 in Australia. The observations of those involved say that all variables affect CO₂ emissions. Li et al. (2021) discussed the effect of the growth and structure of the economy on per capita CO₂ emissions in 147 countries from 1990 to 2015. The results show that at the global level, economic growth and economic structure are the most significant positive and positive effects, respectively.

Studies on population have thus far concentrated on the relationship between population growth and emission increase. The effect of population growth on CO₂ emissions can be summarized as follows (Birdsall, 1992): On one side, the energy demand was increased for power generation, industry, and transport. On the contrary, it increased deforestation emissions due to population growth. Empirically, Knapp and Mookerjee (1996) conducted a Granger causality test on annual data from 1880 to 1989 to determine the connecting clauses between global population expansion and carbon dioxide exhaustion. The results show there is a short-term dynamic relationship between the exhaustion of carbon dioxide and population growth. Very recently, Zhang et al. (2020) analyzed the knot between CO₂ emissions, GDP, and fuel ingestion in China and ASEAN countries. It was found that carbon density, energy intensity, GDP, and population are positively correlated with CO₂ emissions. Empirical findings also show that the developing countries are facing the effect of overpopulation and that’s why, they are facing more of a carbon emissions record per year other than the developed countries (Shi, 2003).

In the past decade, the theory and methodology of the Environmental Kuznets Curve (EKC) have been used to analyze the relationship between the net income and exhaustion of carbon of an area (Dinda, 2004; Williams and Rasmussen, 1996). According to the EKC, at relatively low-income levels, emissions increase as income increases. After a certain point, emissions will decline with income. Thus, the emission of CO₂ varies concerning the level of income. Luo et al. (2021) investigated the influencing factors of Shanghai’s CO₂ emissions from 1995 to 2017. They found that personal disposable income is one of the top drivers of CO₂ emissions. Yuan et al. (2014) examined the long-term relationship between China’s per capita income, ingestion of energy, and the emission of CO₂ from 1953 to 2008. They found out, there is a unilateral Granger inter-relation between the gross national income and the emission of CO₂.

Based on the literature above, it concludes that economic up-gradation, energy utilization, manpower density, industrialization, and income can be classified as the predominant factors affecting CO₂ emissions. Other factors might also affect CO₂ emissions in China. For example, R&D (Nguyen et al., 2020; Jones, 1995), financial development (Bhattacharya et al., 2017; Zaidi et al., 2019; Wang et al., 2020), the degree of foreign direct investment (Essandoh et al., 2020; Le et al., 2020; Khan and Rana, 2021. etc). This paper limits the focus on how well the different prediction models perform based on the information set which includes only the most predominant driver of CO₂ emissions and excludes those unimportant ones to be captured by the stochastic terms in the models.

Gaussian Process Regression (GPR) method can be introduced as a non-parametric Bayesian regression method (Gershman and Blei, 2012). It captures a wide variety of relations between inputs and outputs and lets the data determine the complexity of the underlying functions through the means of Bayesian inference (Williams, 1998). Considering the output   y of a function w at input x with independent and identically distributed random noise ε ∼ N ( 0 , σ n 2 ) . The function accompanied with the distributed random noise can be presented as: y = w ( x ) + ε   (1)

In classical linear regression, w ( x ) is deterministic whereas the noise term is random. In Gaussian process regression, however, w ( x ) is assumed to be random and follows a Gaussian process. A Gaussian process is an extension of multivariant Gaussian distribution to infinite dimensions; any finite subset sampled from the Gaussian process follows multiple Gaussian distributions (MacKay, 1998). The distribution over functions can be described with the help of the Gaussian process, w ( x ) = GP ( m ( x ) , k ( x , x ' ) ) (2) where x is applied as the input variable, m ( x ) is denoted as the mean function, finally, k ( x , x ' ) is known as covariance function. These two functions are defined respectively as: m ( x ) = E [ w ( x ) ] k ( x , x ' ) = c o v [ w ( x ) , w ( x ' ) ] . (3)

A finite collection of function values sampled from the Gaussian process follows multiple Gaussian distributions: [ w ( x ( 1 ) ) , w ( x ( 2 ) ) , … , w ( x ( n ) ) ] T ∼ N ( μ , K ) (4) where K is a n × n (n by n) matrix with the entries K i j = k ( x ( i ) , x ( j ) ) and μ has entries μ i = m ( x ( i ) ) . Given a training set that contains observation points y = [ y ( x ( 1 ) ) , y ( x ( 2 ) ) , … , y ( x ( n ) ) ] T and function values w = [ w ( x ( 1 ) ) , w ( x ( 2 ) ) , … , w ( x ( n ) ) ] T , it follows that the conditional distribution p ( y | w ) and the Gaussian prior p ( w ) are N ( w , σ ε 2 I ) and N ( μ , K ) , respectively. By definition, the set of observations y and the set of function values w follow a joint multivariate Gaussian distribution. The join distribution p ( w ( x ∗ ) ,   y ) is defined as [ w ( x ∗ ) y ] ∼ N ( [ μ ∗ μ ] , [ k ( x ∗ , x ∗ ) k ∗ T k ∗ K + σ ε 2 I ] ) (5)

Here, I will be considered as the identity matrix, σ ε 2 is the unknown variance of the random noise and ( k ∗ ) i = k ( x ∗ , x ( i ) ) for i = 1,2 , … , N . Using the Bayesian rule, the predictive posterior, p ( w ( x ∗ ) | y ) ∼ N ( w ¯ ∗ , Σ ∗ ) , can be obtained, and the mean w ¯ ∗ and variance Σ ∗ are defined by w ¯ ∗ = μ ∗ + ( k ∗ ) T [ K + σ ε 2 I ] − 1 ( y − μ ) Σ ∗ = k ( x ∗ , x ∗ ) − ( k ∗ ) T [ K + σ ε 2 I ] − 1 ( k ∗ ) (6)

The covariance function determines the characteristics of the Gaussian method that can be expressed as k ( x ( i ) , x ( j ) ) . The covariance function models the dependence between the function values at different input points x ( i ) and x ( j ) . The covariance function is often called the kernel of the Gaussian process. There are many possible options for the prior covariance function. A popular kernel is the exponential covariance function which allows the model to general a non-negative definite covariance matrix for any set of input points (Williams and Rasmussen, 1996). The exponential covariance function is defined as k ( x ( i ) , x ( j ) ) = σ f 2 e x p ( − ∥ x ( i ) − x ( j ) ∥ 2 2 I 2 ) + σ ε 2 δ i j (7) where I is the characteristic length scale, σ f 2 is the signal variance, and δ i j is a Kronecker delta. The Gaussian process regression employs a set of hyperparameters θ including I , σ f 2 and σ n 2 to increase or reduce the priority correlation between points and consequentially the variability of the resulting function. The hyperparameters θ can be optimized based on the log-likelihood framework: L = log ⁡ p ( y | w , θ ) = − 1 2   y T C − 1 y − 1 2 log | C | − n 2 log ⁡ 2 ⁡ π ,     C = K + σ ε 2 I (8)

More details about the regression process of Gaussian can be researched and acknowledged in the book of Williams and Rasmussen (2006), available free online and is accessible via the link: www.GaussianProcess.org/gpml.

A literature review has been conducted so that five independent variables; namely: economic growth, energy consumption, population, industrialization, and income, have been identified. In this study, the GPR method and the other proposed algorithms are applied to study carbon emissions in China. Economic growth is approximated by GDP (100 million RMB), energy consumption is approximated by per capita energy consumption (tons of standard coal), the population is approximated by population size (10,000 people), industrialization is weighted by the percentage of secondary industry in China, and income is measure by the average annual salary (RMB).

The data of GDP, population size, energy consumption, percentage of secondary industry, and average annual salary are collected from the China City Statistical Yearbook. CO₂ emissions data come from four main sources of energy consumption. These are electricity, fuel, heating, and transportation. Those data can be obtained and calculated through the China Urban Construction Statistical Yearbook, the China City Statistical Yearbook, and the submerged government Panel on the change of weather and climate. Since some of those data is not available after 2014, the data in this paper range from the year 2002–2014.

In this paper, the Gaussian process regression method is proposed for CO₂ emissions analysis in China. The traditional linear regression approach is limited by its rigid functional form and the approach often encounters an over-fitting problem. The Gaussian progress regression approach relaxes the parametric assumption by applying the Bayesian nonparametric inference approach. The preciseness and exactitude of the prediction of the exponential GPR were compared and discussed with the classical least squares and the robust least-squares model. Based on the outcome of the whole study, it is proved that the Gaussian progress regression algorithms can give the most accurate predictions on CO₂ emissions compared with the other two traditional models and thus is applicable for CO₂ emissions prediction analysis to enhance forecast performance.

The prediction performances of the selected methods discussed only focus on the six predominant factors affecting carbon emissions. Future research should focus on further reviewing the completeness of the set of driving factors and the effectiveness of model predictions, compared them with other commonly used models.