Prediction of Sulfur Dioxide Emissions in China Using Novel CSLDDBO-Optimized PGM(1, N) Model

Abstract

Sulfur dioxide not only affects the ecological environment and endangers health but also restricts economic development. The reasonable prediction of sulfur dioxide emissions is beneficial for formulating more comprehensive energy use strategies and guiding social policies. To this end, this article uses a multiparameter combination optimization gray prediction model (PGM(1, N)), which not only defines the difference between the sequences represented by variables but also optimizes the order of all variables. To this end, this article proposes an improved algorithm for the Dung Beetle Optimization (DBO) algorithm, namely, CSLDDBO, to optimize two important parameters in the model, namely, the smoothing generation coefficient and the order of the gray generation operators. In order to overcome the shortcomings of DBO, four improvement strategies have been introduced. Firstly, the use of a chain foraging strategy is introduced to guide the ball-rolling beetle to update its position. Secondly, the rolling foraging strategy is adopted to fully conduct adaptive searches in the search space. Then, learning strategies are adopted to improve the global search capabilities. Finally, based on the idea of differential evolution, the convergence speed of the algorithm was improved, and the ability to escape from local optima was enhanced. The superiority of CSLDDBO was verified on the CEC2022 test set. Finally, the optimized PGM(1, N) model was used to predict China’s sulfur dioxide emissions. From the results, it can be seen that the error of the PGM(1, N) model is the smallest at 0.1117%, and the prediction accuracy is significantly higher than that of other prediction models.

1. Introduction

Chemical energy includes plant fuels, fossil fuels, and other gaseous fuels. China is rich in chemical energy, which not only promotes rapid national development but also determines the future development of the country. However, the use of chemical energy is accompanied by the production of large amounts of harmful gases, among which the sulfur dioxide (SO₂) generated after the combustion of sulfur-containing fuels is more serious. It not only endangers human health and damages the natural environment but also poses a threat to the economy and society. Therefore, predicting the future emissions of SO₂ and implementing governance and prevention measures in China’s strategic plan to achieve sustainable development is crucial.

In terms of research on SO₂ emissions, multiple scholars have conducted research on the prediction of SO₂ emissions. Common methods include multiple linear regression prediction models, machine learning, and statistical models.

• Multiple: linear regression prediction models: Long et al. [1] conducted dimensionality reduction and collinearity analysis on SO₂ emission data variables based on the analysis of the sulfur metabolism mechanism in the sintering process. They derived a statistical regression prediction model for SO₂ emissions from sintering based on the principle of multiple linear regression. However, this model is sensitive to the linearity assumption of the data, while SO₂ emissions from sintering flue gas are influenced by the nonlinear coupling of multiple factors, such as the sulfur contents in raw materials, the air volume, and temperature. Zheng et al. [2] established a regression model for SO₂ emissions from coal combustion using multiple linear regression analysis and variance analysis and applied it to predict SO₂ emissions from thermal power plants in Shandong Province. This method did not consider the impact of accident slurry ponds in the flue gas desulfurization system and thus failed to effectively prevent environmental pollution.
• Machine learning: In 2018, Xue et al. [3] established a flue gas SO₂ emission prediction model based on support vector machines (SVMs) for the nonlinear characteristics of flue gas SO₂ objects in circulating fluidized bed boiler control systems. The model parameters were determined using a univariate parameter search combined with grid optimization, overcoming the various shortcomings of previous methods that directly used grid search to determine the parameters of SVM regression models, thereby achieving good prediction results. However, this method requires frequent parameter tuning, resulting in high maintenance costs. When facing circulating fluidized bed systems with strong real-time requirements and high data noise, the model accuracy is easily limited. In 2021, Vitor Miguel Ribeiro [4] used economic-related theories combined with machine learning models to predict SO₂ emissions near a thermal power plant in Portugal. According to the final results, the performance of machine learning models is superior to that of traditional methods, but the prediction accuracy of this method will significantly decrease when facing local emission data policy adjustments and monitoring equipment anomalies.
• Statistical models: In 2023, Ghosh and Verma [5] applied an aerosol field and estimated the constrained emissions of SO₂ in India based on relevant data. They first constrained the scattering of absorbing aerosols, and they then used this constraint method to obtain the constrained SO₂ emissions. They concluded that the annual emission rate of SO₂ in the Indian restricted-emission database was lower than the emission rate reported by China. However, if this method underestimates hotspot pollution sources or overestimates emission reduction effects, it may exacerbate the environmental governance risks in the Indian region. Fu et al. [6] used the LEAP model and emission factor method to predict the SO₂ emissions in some eastern regions of China and studied future emission trends. However, this method did not consider the transmission of pollutants between urban agglomerations and the drastic changes in energy structures caused by industrial upgrading in the eastern region, which made it difficult for traditional linear prediction models to capture the dynamic impact of emerging high-energy-consuming industries.

SO₂ emissions mainly come from the combustion of sulfur-containing fuels. The gradual recovery of the global economy and fierce competition among countries will inevitably exacerbate energy consumption, which will be followed by an increase in SO₂ emissions. If predicted based on previous years’ data, the results may be inaccurate. In addition, the collection of SO₂ emission data is difficult, with a long cycle and complex influencing factors. Therefore, a gray prediction model was chosen to predict SO₂ emissions based on the modeling object of “less information required, higher accuracy, simple operation, easy inspection, and system uncertainty”.

Gray system theory [7] is an emerging discipline proposed and established by Professor Deng Julong to address issues such as limited data volumes [8] and information uncertainty [9]. Gray prediction, as a research direction of gray systems, is divided into different research objects based on the number of variables studied. The GM(1,1) model also works well with exponential growth data and data with medium to long time periods, but its prediction results are poor for time series with poor volatility. To enhance the accuracy of the gray forecasting model NGM(1,1,k2) with quadratic polynomial terms, Li et al. [10] further refined the original model and obtained the BNGM(1,1,k2) model. However, this method requires the simultaneous adjustment of the coefficients of the differential equation and the polynomial weights, making it prone to local optimal solutions. Qian et al. [11] proposed a new discrete gray forecasting model to enhance the model’s adaptability and performance for both linear and nonlinear trends in time series. However, in renewable energy generation scenarios, this model relies excessively on new data, leading to neglect of historical patterns. Wang et al. [12] applied quantile regression techniques to construct the QGM(1,1) model to improve the model stability. This model has the advantages of higher accuracy and better robustness. However, constrained by the inherent limitations of gray theory, the model’s differential equation is based on the assumption of exponential law, making it unable to effectively capture random perturbations in economic or environmental systems. Zeng et al. [13] established the SGGM(1,1,r) model, combining new initial conditions with original data to improve the prediction accuracy. However, this model reconstructs initial conditions based on the “new information priority” principle and, given the short history of shale gas development in China, limited samples lead to insufficient reliability in long-term trend prediction.

Table 1. Univariate gray models for predicting SO₂ emissions.

Models	Methods	Authors
Equal-dimension gray number complementary model GM(1,1)	Predicting SO2 emissions using an equidimensional gray number replenishment model	Jie Tan et al.
GM(1,1) models with different dimensions	Using gray models with different dimensions to predict SO2 emissions in Wuhan city	Haijun Huang et al.
GM(1,1,u(t))	Predicting air quality in Shanghai using gray extended model	Pingping Xiong et al.
FGM(1,1)	Predicting SO2 emissions in three provinces of China using the fractional-order accumulative gray model	Lifeng Wu et al.
NLDGM(1,1r, t)	Predicting SO2 emissions in the power industry using a nonlinear gray direct model	Yuelin Xiang
GNNM(1,1)	Predicting the emissions of air pollutants such as SO2 using a gray neural network model	Wenqiang Bai
GIFM	Predicting the concentration and emissions of SO2 in a capital city using a gray interval forecast model	Bo Zeng

presents some of the literature using univariate gray models to predict SO₂ emissions.

The GM(1, N) model addresses simulation and prediction issues where multiple related factor variables affect a single system behavior variable. Duan et al. [14] established a multivariable energy consumption gray model and applied it to practical problems. However, differences in China’s regional energy structures can lead to dynamic coupling effects, making it difficult for the model to adapt and adjust. In 2022, Duan [15] proposed the Verhulst gray model MVGM(1, N), which not only specifically addressed the problem but also enhanced the prediction accuracy. However, the training data for this model was concentrated in resource-based provinces. When directly applied to manufacturing powerhouses such as Jiangsu, due to differences in energy structures, the mean absolute error will increase. In 2023, Ye et al. [16] established a WAFGM(1, N) model, which not only reduces errors but also utilizes uncertain data for prediction, achieving high prediction accuracy. Although this model dynamically adjusts weights through interval sequences, it does not consider the sudden impact of extreme events on variable relationships, resulting in the weight allocation lagging behind actual changes.

The above review indicates that the current models exhibit certain limitations to some extent. The shortcomings of traditional models lie in the fact that modeling for predicting SO₂ emissions requires expert experience in selecting and extracting features, which is time-consuming and highly subjective. It is difficult to adapt to regional differences in China, and there is a high risk of overfitting in small samples. The disadvantage of machine learning models is that the decision-making process is black-boxed, making it difficult to intuitively analyze the contributions of variables, reducing the credibility of policy formulation. Furthermore, such models do not internalize policy shocks and ignore dynamic coupling, leading to the ineffective prediction of turning points. Therefore, a new method is needed to fill this gap for the better prediction of SO₂. This paper chooses to use the gray prediction model to solve this problem and proposes some feasible solutions. This article uses the multivariate gray prediction model PGM(1, N) suggested by Yin et al. [17] in 2023. We chose PGM(1, N) because this model has several advantages. It can construct a prediction framework using a small amount of historical data, making it suitable for scenarios with incomplete information or scarce data, and avoiding the dependence of traditional statistical methods on large data volumes. Secondly, based on differential equations and accumulation generation techniques, it simplifies complex multivariate relationships into computable sequences, reducing the computational complexity of modeling and facilitating practical application deployment. Then, PGM(1, N) integrates multiple influencing factors to reduce the system uncertainty. Its prediction results are superior to those of the GM(1,1) model, especially in short-term trend analysis. Finally, this model weakens noise interference by accumulating and generating the original data, directly revealing the inherent relationship between multiple variables. Previous research on predicting gas emissions has mainly focused on single/multivariate first-order cumulative models and related improved models. Traditional multivariate models use the identical order for all variable data sequences, using the first-order cumulative generator sequence of the independent variable as a driving term for modeling. However, this approach does not consider the effects of the volatility of the sequence and the sway of extreme values on the dependent variable, which can seriously impact the quality of the model. Therefore, this model incudes the differential definition and optimization of the order of variables, solving the problem of the similarity of variables with different orders. On this basis, the modeling ability of the model is enhanced by simultaneously optimizing the order of variables, driving smooth generation coefficients and background coefficients.

In the PGM(1, N) model, the order of the gray generation operators and the smoothing generation coefficient are two important parameters. The accurate prediction of a prediction model depends on the efficient and accurate optimization of the model parameters; however, intelligent optimization algorithms can be used to optimize it. There are many algorithms that perform well. The Ant Colony Optimization (ACO) algorithm [18] is inspired by the behavior of real-life ant colonies. Ants transmit information by recognizing pheromones. Based on this, the algorithm constructs a solution to the optimization problem. The Particle Swarm Optimization (PSO) algorithm [19] is influenced by the study of bird foraging behavior, which enables the population to seek optimization through information sharing between groups. The Gray Wolf Optimization (GWO) [20] algorithm is influenced by the hunting methods of wolf packs, simulating the system and division of labor in the wolf population. The Harris Eagle Optimization (HHO) [21] algorithm imitates the predatory characteristics and cooperative behavior of Harris eagles, searching through the hunting process. Also, Sled Dog Optimization (SDO) [22] is inspired by sled dog behavior, finding the best food through division of labor and information exchange between groups. The Marine Predator Algorithm (MPA) [23] is constructed by simulating the evolutionary evolution of predators and prey in the sea. The Chimpanzee Optimization Algorithm (ChOA) [24] simulates social behavior among chimpanzee populations. The Slime Mold Algorithm (SMA) [25] simulates the behavior of slime molds during foraging. Elephant Herding Optimization (EHO) [26] mainly simulates the phenomenon of member updates and changes in elephant groups in nature. The improved Black Widow algorithm (namely, SDABWO) [27] is used to solve feature selection problems. The improved version of the Honey Badger algorithm (namely, SaCHBA-PDN) [28] has a better performance and is easier to implement. The improved version of the starling noise algorithm (namely, DTCSMO) [29] has shown significant advantages in engineering application problems. The enhanced version of jellyfish search optimization (namely, EJS) [30] has shown significant advantages in solving optimization problems with complex spherical shapes. Based on the Kepler optimization algorithm, an improvement was made (namely, CGKOA) [31], which has a better optimization performance. The improved particle swarm optimization (namely, dFDBMPSO) [32] has been applied to practical problems.

However, any algorithm has its limitations in application. Therefore, we encourage the design and development of more high-performance optimization algorithms. Among them, the DBO algorithm [33] is an original algorithm picked up by Shen et al. Its proposal was inspired by a series of survival behaviors of insects, such as dung beetles, allowing for global search and local utilization. Dung beetles [34] feed on animal excrement and are known for their unique behavior of pushing dung balls [35], playing the role of decomposers in nature [36].

The DBO algorithm combines global search and local development and has an excellent performance in terms of its convergence speed and solution accuracy. It is evaluated according to various benchmark functions and achieves better results. This article proposes an improved version of the beetle algorithm (CSLDDBO) that has a higher solving accuracy and better performance and proves its effectiveness on the benchmark functions. This study utilizes the CSLDDBO-PGM(1, N) combinatorial optimization model for predicting SO₂ emissions and demonstrates its rationality.

The full text is arranged as follows:

• The gray prediction model selected in this article has carried out the differential definition and optimization of the order of variables, solving the disadvantage of different variables with the same order in the gray model, and combining the idea of parameter combination optimization to enhance the modeling capabilities of gray models.
• An improved Dung Beetle Optimization (CSLDDBO) algorithm is proposed, which introduces three strategies and the idea of differential evolution to enhance the performance of the original algorithm. The effectiveness of this algorithm was verified by testing it on the CEC2022 test set.
• To predict SO₂ emissions, a PGM(1, N) model optimized with CSLDDBO parameters was used for predicting SO₂ emissions in China.

The rest of this article is organized as follows: In Section 2, the basic concepts related to the multivariate gray model PGM(1, N) are presented. In Section 3, the DBO algorithm is introduced. In Section 4, the details of CSLDDBO are introduced. Section 5 analyzes the performance of CSLDDBO based on experiments. In Section 6, the CSLDDBO-PGM(1, N) model is used to predict the emissions of SO₂ in China. Section 7 provides a summary of the entire text.

2. PGM(1, N)

2.1. Related Concepts

2.1.1. The Order of Gray Generative Operators

To highlight the key points of this article, we have placed the basic concepts of GM(1, N) in the Appendix A. Below, we present the basic concepts of PGM(1, N).

We refer to $Y_1^{(0)}=(y_1^{(0)}(1),y_1^{(0)}(2),\dots ,y_1^{(0)}(N)),$ as the dependent-variable sequence and $Y_m^{(0)}=(y_m^{(0)}(1),y_m^{(0)}(2),\dots ,y_m^{(0)}(N)),$ $m=2,3,..,n$ as the independent-variable sequence. $Y_i^{(1)}=(y_i^{(1)}(1),y_i^{(1)}(2),\dots ,y_i^{(1)}(N)), i=2, 3,\dots ,n$ is used as a first-order cumulative generation sequence [37] of $Y_i^{(0)}$, with the specific formula being

$$y_i^{(1)}(g)={\displaystyle \sum _{k=1}^g{y}_i^{(0)}(k)},g=1,2,\dots ,N$$

(1)

where $k$ represents the sequence order.

Let $Y_i^{(0)}=(y_i^{(0)}(1),y_i^{(0)}(2),\dots ,y_i^{(0)}(N)), i=1,2,\dots ,n$ be the initiation sequence, $t_i\in I$ $i=1,2,\dots ,n$ be the order of the gray generation operators, and $Y_i^{(t_i)}=(y_i^{(t_i)}(1),y_i^{(t_i)}(2),\dots ,$ $y_i^{(t_i)}(N))$ be the new sequence, where

$$y_i^{(t_i)}={\displaystyle \sum _{i=1}^g\frac{\Gamma (t_i+g-i)}{\Gamma (g-i+1)\Gamma (t_i)}}{y}_i^{(0)}(i).g=1,2,\dots ,N$$

(2)

The above formula is called the $t_i$-order gray generation operator of $Y_i^{(0)}$, abbreviated as $t_i-$RGO and $Y_i^{(t_i)}$, respectively, where $i=1,2,\dots ,n$, $m=2,3,\dots ,n$.

As shown in Formula (2), let $Y_i^{(0)}$, $b\in I, f\in I$. $Y_i^{(b)}$ is the $b-$RGO sequence of $Y_i^{(0)}$; $Y_i^{(f)}$ is the $f-$RGO sequence of $Y_i^{(0)}$; $Y_i^{(b+f)}$ is the $(b+f)-$RGO sequence of $Y_i^{(0)}$; ${(Y_i^{(f)})}^{(b)}$ is the $b-$RGO sequence of $Y_i^{(f)}$; ${(Y_i^{(f)})}^{(b)}$ is the $f-$RGO sequence of $Y_i^{(b)}$. Moreover, multiple generations of operators satisfy the commutative law and exponential rate [38], i.e.,

$${(Y_i^{(b)})}^{(f)}={(Y_i^{(f)})}^{(b)}=Y_i^{(b+f)},$$

(3)

specifically,

$$Y_i^{(0)}={(Y_1^{(t)})}^{(-t)}={(Y_1^{(-t)})}^{(t)}.$$

(4)

2.1.2. Smooth Generation Operator

Next, the 1-AGO sequence of the independent variable is used as the driving sequence [39], and variance smoothing operations are performed.

As shown in Formula (2), let $F_m^{(t_m)}=(k_m^{(t_m)}(2),k_m^{(t_m)}(3) ,\dots , k_m^{(t_m)}(N))$ be a t_m-order smooth generating sequence with the variable weight independent variable ${\lambda }_m$, ${\lambda }_m\in (0,1)$, where

$$k_m^{(t_m)}(g)={\lambda }_m{y}_m^{(t_m)}(g)+(1-{\lambda }_m)y_m^{(t_m)}(g-1),g=2,3,\dots ,N$$

(5)

where ${\lambda }_m$ is the smoothing generation coefficient. Specifically, if ${\lambda }_m=1$, $F_m^{(t_m)}$ is converted into a 1-RGO sequence.

2.2. Model Definition and Parameter Estimation

According to Formula (5), $Y_1^{(t_1)}$ and $Y_1^{(t_1-1)}$ are referred to as the $t_1-$RGO sequence and $(t_1-1)-$RGO sequence of $Y_1^{(0)}$. $a_1^{(t_1)}=$$(A_1^{(t_1)}(2),A_1^{(t_1)}(3),\dots ,A_1^{(t_1)}(N))$ is a neighboring sequence generated by the background value coefficient ${\lambda }_1$ of $Y_1^{(t_1)}$; then,

$$y_1^{(t_1-1)}(g)+E{A}_1^{t_1}(g)={\displaystyle \sum _{m=2}^n{q}_m{k}_m^{t_1}(g)}+s_1(g-1)+s_2 ,$$

(6)

is known as a new multivariate gray model, abbreviated as PGM(1, N). In Formula (5), $E$ represents the development coefficient. $k_m^{(t_m)}(g)={\lambda }_m{y}_m^{(t_m)}(g)+(1-{\lambda }_m)$$y_m^{(t_m)}(g-1)$ is a driver based on smooth generation. $s_1(g-1)$ is a linear correction term. $s_2$ is a random perturbation term.

Assuming that $Y_1^{(t_1)}$ and $Y_1^{(t_1-1)}$ are as described in Formula (6), $Y_m^{(t_m)}$ is as described in Formula (2), and $F_m^{(t_m)}$ is as described in Formula (5), the parameter $\widehat{u}=[q_2,q_3$ $,\dots ,q_N,E,s_1,s_2{]}^{\mathrm{T}}$ estimation of the PGM(1, N) model satisfies

$$D=\left[\begin{array}{cccccc}{k}_2^{(t_2)}(2)& k_2^{(t_3)}(2)& \dots k_n^{(t_n)}(2)& -A_1^{(t_1)}(2)& 1& 1\\ k_2^{(t_2)}(3)& k_2^{(t_3)}(3)& \dots k_n^{(t_n)}(3)& -A_1^{(t_1)}(3)& 2& 1\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ k_2^{(t_2)}(N)& k_2^{(t_3)}(N)& \dots k_n^{(t_n)}(N)& -A_1^{(t_1)}(N)& \mathrm{N}-1& 1\end{array}\right] ,$$

(7)

$$\begin{gathered} K=\left[\begin{array}{c}{y}_1^{(t_1-1)}(2)\\ y_1^{(t_1-1)}(3)\\ ⋮\\ y_1^{(t_1-1)}(N)\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{m=1}^2\frac{\Gamma (t_1+1-m)}{\Gamma (2-m+1)\Gamma (t_1-1)}{y}_1^{(0)}(m)}\\ {\displaystyle \sum _{m=1}^3\frac{\Gamma (t_1+2-m)}{\Gamma (3-m+1)\Gamma (t_1-1)}{y}_1^{(0)}(m)}\\ ⋮\\ {\displaystyle \sum _{m=1}^N\frac{\Gamma (t_1+N-1-m)}{\Gamma (N-m+1)\Gamma (t_1-1)}{y}_1^{(0)}(m)}\end{array}\right] \\ =\left[\begin{array}{c}(t_1-1)y_1^{(0)}(1)+y_1^{(0)}(2)\\ {\displaystyle \frac{t_1(t_1-1)}{2}}{y}_1^{(0)}(1)+(t_1-1)y_1^{(0)}(2)+y_1^{(0)}(3)\\ ⋮\\ {\displaystyle \sum _{m=1}^N\frac{\Gamma (t_1+N-1-\mathrm{m})}{\Gamma (N-m+1)\Gamma (t_1-1)}{y}_1^{(0)}(m)}\end{array}\right]. \end{gathered}$$

(8)

• If $N=n+3$ and $\left|D\right|\ne 0$, then $\widehat{u}=D^{-1}K$;
• If $N>n+3$ and $\left|D^{T}D\right|\ne 0$, then $\widehat{u}={(D^{T}D)}^{-1}{D}^{T}K$;
• If $N<n+3$ and $\left|D{D}^T\right|\ne 0$, then $\widehat{u}=D^T{(D{D}^T)}^{-1}K$.

2.3. Solution of the Model

Formula (6) shows that when $g=2,3,\dots ,N,\dots ,$ the time response expression is as follows:

$${\widehat{y}}_1^{(t_1)}(g)={\displaystyle \sum _{d=1}^{g-1}\left[{\epsilon }_1{\displaystyle \sum _{m=2}^n{\epsilon }_2^{d-1}{q}_m{k}_m^{(t_m)}(g-d+1)}\right]}+{\epsilon }_2^{g-1}{\widehat{y}}_1^{(t_1)}(1)+{\displaystyle \sum _{i=0}^{g-2}{\epsilon }_2^i}[(g-i){\epsilon }_3+{\epsilon }_4] ,$$

(9)

where

$${\epsilon }_1={\displaystyle \frac{1}{1+\gamma {\lambda }_1}},{\epsilon }_2={\displaystyle \frac{1-E(1-{\lambda }_1)}{1+E{\epsilon }_1}},{\epsilon }_3={\displaystyle \frac{s_1}{1+E{\lambda }_1}},{\epsilon }_3={\displaystyle \frac{s_2-s_1}{1+E{\lambda }_1}} .$$

(10)

The final recovery expressions are as follows:

$${\widehat{y}}_1^{(0)}(g)={\displaystyle \sum _{m=1}^g\frac{\Gamma (-t_1+g-m)}{\Gamma (g-m+1)\Gamma (-t_1)}}{\widehat{y}}_1^{(t_1)}(m) .$$

(11)

3. Overview of Dung Beetle Optimization Algorithms

The dung beetle likes to roll animal feces into balls and mainly feeds on animal feces, earning the nickname “natural cleaner”. For dung beetles, dung balls are important breeding grounds. DBO is composed of four different functional dung beetles.

3.1. Ball-Rolling Dung Beetles

The update expression for the DBO initialization position is as follows:

$$\begin{array}{l}{h}_u(q+1)=h_u(q)+\sigma \times j\times h_u(q-1)+l\times \Delta h ,\\ \Delta h=\left|h_u(q)-H^g\right| ,\end{array}$$

(12)

where $q$ denotes the number of iterations; $h_u(q)$ denotes the location of the $u$th dung beetle in the $q$ iteration; $j\in (0,0.2]$ is the deflection coefficient; $l$ is the random number in $(0,1)$; $\sigma$ is the natural coefficient, assigned as −1 or 1; $H^g$ denotes the worst-case position; and $\Delta h$ denotes simulation of changes in light intensity. $\sigma$ represents natural factors. Specifically, $\sigma =1$ or $\sigma =-1$. The algorithm is as follows:

The updated location of DBO’s dancing behavior is as follows:

$$h_u(q+1)=h_u(q)+\tan (\theta )\left|h_u(q)-h_u(q-1)\right|,$$

(13)

When $\theta \in [0,\pi ]$ and $\theta$ equals $0,\pi /2,\pi$, the position is not updated.

3.2. Producing Dung Beetles

The selection of the breeding ball’s position is particularly crucial, and the boundary handling is as follows:

$$\begin{array}{l}L{o}^{\prime }=\max (H^{\prime }\times (1-P),Lo),\\ U{p}^{\prime }=\max (H^{\prime }\times (1+P),Up),\end{array}$$

(14)

where $H^{\prime }$ is the current local best point; $L{o}^{\prime }$ and $U{p}^{\prime }$ denote the boundaries of the spawning area ($P=1-q/U_{\max }$); $U_{\max }$ is the maximum number of iterations; $Lo$ and $Up$ represent the upper and lower bounds, respectively.

With the change in $P$, the spawning area will also dynamically change, and the position update of the breeding balls is represented as follows:

$$M_u(q+1)=H^{\prime }+l_1\times (M_u(q)-L{o}^{\prime })+l_2\times (M_u(q)-U{p}^{\prime }),$$

(15)

where $M_u(q)$ is the position of the $u$ breeding ball in the $q$ generation, $l_1$ and $l_2$ represent random vectors, and $Z$ represents the dimensionality degree.

3.3. Larvae

The boundaries of the optimal foraging area for larvae are defined as follows:

$$\begin{array}{l}L{o}^l=\max (H^l\times (1-P),Lo),\\ U{p}^l=\max (H^l\times (1+P),Up),\end{array}$$

(16)

where $H^l$ represents the global optimal position, and $L{o}^l$ and $U{p}^l$ represent the lower and upper bounds of the optimal foraging area. The location of the larvae is denoted as follows:

$$h_u(q+1)=h_u(q)+V_1\times (h_u(q)-L{o}^l)+V_2\times (h_u(q)-U{p}^l),$$

(17)

where $h_u(q)$ denotes the location of the $q$th larva in the $u$ generation, and $V_1$ and $V_2$ are random numbers.

3.4. Thief Dung Beetle

The position update of the “thief” dung beetle is represented as follows:

$$h_u(q+1)=h^l+G\times z\times \left|(h_u(q-H^{\prime })\right|+\left|h_u(q)-H^l\right|),$$

(18)

where $h_u(q)$ denotes the location of the $u$ thief in the $q$ generation, $z$ is an arbitrary vector, and $G$ represents a constant.

4. CSLDDBO Algorithm

In this section, we propose an improved version of the DBO algorithm, named CSLDDBO. We combined the chain foraging strategy, rolling strategy, learning strategy, and differential evolution algorithm to enhance the original algorithm.

4.1. Chain Foraging Strategy

Introducing the chain foraging strategy into CSLDDBO [40] can enhance the global exploration ability of the original algorithm. The expression of the chain foraging strategy is as follows:

$$h(r+1)=\left\{\begin{array}{l}{h}_o^b(r)+q(h_{best}^b(r)-h_o^b(r))+\gamma \cdot (h_{best}^b(r)-h_o^b(r)), o=1\\ x_o^b(r)+q(h_{m-1}^b(r)-h_o^b(r))+\gamma \cdot (h_{best}^b(r)-h_o^b(r)), o=1,2,\dots ,s\end{array}\right.$$

(19)

$$\gamma =2\cdot q\cdot \sqrt{\left|\log (q)\right|},$$

(20)

where $h_o^b(r)$ is the location of the $o$th individual in the $b$th dimension at time ($r$), $q$ is an arbitrary vector in $[0,1]$, $\gamma$ is the weight coefficient, and $h_{best}^b(r)$ is the best position location for individuals.

Dung beetles orient and fly towards food based on scent, assuming that the food source is the optimal location. Figure 1 depicts a schematic diagram of dung beetles foraging in a two-dimensional space according to this strategy.

4.2. Somersault Foraging Strategy

This strategy regards the position of food as the optimal position, and all individuals update their positions around this optimal position. The specific expression is as follows:

$$h_o^b(r+1)=h_o^b(r)+V\cdot (q_2\cdot h_{best}^b-q_3\cdot h_o^b(r)), o=1,2,\dots ,s$$

(21)

where $V$ is the flip factor, $V=2$, and $q_2$ and $q_3$ are two random numbers in $[0,1]$. As can be seen from the formula, this strategy enables individuals to conduct an adaptive search within a constantly changing search range.

4.3. Learning Strategy

The expression for the comprehensive learning strategy is as follows [41]:

$$K_o^b\leftarrow \eta \ast K_o^b+p^{\prime }ran{d}_o^b\ast (pbes{t}_{fi(b)}^b-H_O^B),$$

(22)

where $f{i}_u=[f{i}_u(1),f{i}_u(2),\dots ,f{i}_u(Z)]$ defines the $pbests$ of the particle corresponding to an individual ($u$). $pbes{t}_{fi(b)}^b$ can be the relevant dimension of any individual’s $pbest$, and $Pc$ is called the learning probability.

In this strategy, first, two individuals are randomly selected. Secondly, the $fi$ of two individuals are compared and the weaker ones are eliminated. Finally, the $pbest$ of the better individuals compared are used as samples to learn this dimension. If all samples of an individual are its own, a dimension is randomly selected to learn the corresponding $pbest$ of another individual.

4.4. Differential Evolution

In order to avoid the rapid convergence of the population to the previous optimal position due to the influence of iteration on CSLDDBO, and to improve the convergence speed of CSLDDBO and avoid falling into local optima, we were inspired by differential evolution and applied it to CSLDDBO [42]:

1.

Initial population

$$h_{ou}(0)=rn{d}_{ou}(0,1)(h_{ou}^U-h_{ou}^L)+h_{ou}^L, o=1,2,\dots ,NS,\mathrm{u}=1,2,\dots ,L$$

(23)

where $L$ is the amount of variables, $NS$ is the population size, and $h_{ou}^U$ and $h_{ou}^L$ are the upper and lower bounds of the $u$th variable, respectively.

2.

Mutation

Three individuals ($h_{e1},h_{e2},h_{e3}$ and $(b\ne e1\ne e2\ne e3)$) are randomly selected from the population; then,

$$h_{ou}(r+1)=h_{e1u}(r)+\rho (h_{e2u}(r)-h_{e3u}(r)),$$

(24)

where $h_{e2u}(r)-h_{e3u}(r)$ is the differentiation vector, and $\rho$ is the scaling factor.

3.

Cross operation

$$d(r+1)=\left\{\begin{array}{l}{h}_{ou}(r),rand 1_{ou}>CR or u\ne rand (r),\\ h_{ou}(r+1),rand 1_{ou}\le CR oru=rand (r),\end{array}\right.$$

(25)

where $CR$ is a random integer with a crossover probability of $CR\in [0,1]$, and $rand (o)$ is between 1 and L.

4.

Select action

$$h(r+1)=\left\{\begin{array}{l}{h}_o(r),fi(d_{o1}(r+1),\dots ,d_{ul}(r+1))<fi(h_{o1}(r),\dots ,h_{oL}(r)),\\ d_o(r+1),fi(d_{o1}(r+1),\dots ,d_{uL}(r+1))\ge fi(h_{o1}(r),\dots ,h_{oL}(r)).\end{array}\right.$$

(26)

Figure 2 presents the flowchart of CSLDDBO, and the pseudocode for CSLDDBO is provided in Algorithm 1.

Algorithm 1: Pseudocode for CSLDDBO algorithm

4.5. Time Complexity

The time complexity of the CSLDDBO algorithm is determined by the chain search, roll search, mutation probability, and crossover probability performed by $N,D,T$ in each iteration. Therefore, it can be denoted as follows:

$$\begin{array}{l}O(CSLDDBO)=O(T(O(chain strategy)+O(somersault strategy)\\ +O(mutation probability+crossover probability)))\\ =O(T(ND+ND+ND))=O(TND),\end{array}$$

(27)

where $T$ is the maximum number of iterations, $N$ is the number of the population, and $D$ is the number of variables.

5. Numerical Experiment and Discussion

In this section, we evaluate CSLDDBO from multiple perspectives through a series of experiments. Firstly, we conducted a convergence analysis on CSLDDBO. Secondly, we conducted a sensitivity analysis on the test function by introducing strategy parameters. Finally, in order to better evaluate the performance of CSLDDBO, we selected some algorithms with excellent performances to compare them with CSLDDBO.

5.1. Convergence Behavior Analysis

Next, we will analyze the convergence of CSLDDBO using four representative qualitative indicators: the search history, the average fitness, the trajectory of the first individual in the first dimension, and the convergence curve of CSLDDBO compared to the original DBO. These indicators can help us observe the optimization behavior of CSLDDBO on CECE2022 and gain a better understanding of its performance. Figure 3 shows the convergence behavior of CSLDDBO on CEC2022. The first column in the figure shows the shape of the test function in two-dimensional space. The second column is a scatter plot of CSLDDBO’s historical location search records on the two-dimensional front of the search agent. The third column shows the average fitness of the search agent during the iteration process, reflecting the changes in the average fitness of CSLDDBO during the iteration process. The fourth column reflects the trajectory curve of the first search agent in the first dimension. The fifth column shows the optimal convergence curves currently found by the search agent and the original DBO.

• Search History

When solving CECE2022 with CSLDDBO, observing the particle distributions of F7, F8, and F9, it can be seen that there were significant differences in the distribution densities of particles when facing different stages and functions throughout the entire optimization process. This indicates that the balance between exploration and development varies when dealing with different problems, and it also demonstrates CSLDDBO’s strong convergence and excellent development capabilities.

2.

Average fitness

Observation shows that the initial values of the iteration process are different, indicating that the population diversity was very rich in the early stages of iteration. The average fitness curves of all functions show a decreasing trend, indicating that the population as a whole gradually approached the optimal solution as the iteration progressed.

3.

Search trajectory

Taking the trajectory of the first particle in CSLDDBO as an example in the figure, when solving F1, F2, F6, F10, and F12 of CEC2022 in CSLDDBO, the abrupt change amplitude in the early stage of iteration covered the entire search space. This indicates that CSLDDBO has superior exploration capabilities. There were still fluctuations in the search agent during the later iterations of F2, F6, F10, and F12, indicating that the population was still being updated. On most functions, CSLDDBO’s search amplitude decreased in the later stages of iteration and its motion gradually stabilized, ultimately finding the global optimal position.

4.

Convergence curve

Overall, CSLDDBO had a better convergence ability and convergence accuracy than DBO. On F3, F5, F8, and F9, CSLDDBO converged slightly slower than DBO in the early stages of iteration, but the final convergence accuracies of CSLDDBO and DBO were very close on F8 and F9.

5.2. Sensitivity of Parameters

In CSLDDBO, the four strategies introduced include three parameters: the population proportion parameter ($Fper$), mutation operator ($F0$), and crossover operator ($CR$). In order to determine the impact of the parameters on the performance of CSLDDBO, we conducted an experimental analysis on these three parameters on the CEC2022 test set. We set the experimental dimension and maximum iteration to 50 and 1500, respectively.

The size of the population proportion parameter ($Fper$) contributes to the coverage and computational efficiency of the solution space. The general value depends on the specific problem. In CSLDDBO, a range of 0.1 to 0.5 is chosen, and the step size is set to 0.1.

Table 2. Experimental results of population ratio ($Fper$) on CEC2022.

F	Index	$\mathit{F}\mathit{p}\mathit{e}\mathit{r}=0.1$	$\mathit{F}\mathit{p}\mathit{e}\mathit{r}=0.2$	$\mathit{F}\mathit{p}\mathit{e}\mathit{r}=0.3$	$\mathit{F}\mathit{p}\mathit{e}\mathit{r}=0.4$	$\mathit{F}\mathit{p}\mathit{e}\mathit{r}=0.4$
F1	Mean	3.0000 × 102	3.0000 × 102	3.0000 × 102	3.0000 × 102	3.0000 × 102
	Std.	2.5856 × 10−14	2.9856 × 10−14	2.7927 × 10−14	2.3603 × 10	2.9856 × 10−14
	Rank	2	4	3	1	5
F2	Mean	4.0565 × 102	4.0488 × 102	4.0494 × 102	4.0510 × 102	4.0491 × 102
	Std.	6.5041 × 10−1	1.4507 × 100	1.8062 × 100	1.2583 × 100	1.3784 × 100
	Rank	5	2	4	3	1
F3	Mean	6.0000 × 102	6.0000 × 102	6.0000 × 102	6.0000 × 102	6.0000 × 102
	Std.	1.2253 × 10−3	1.0804 × 10−3	5.8368 × 10−3	1.2469 × 10−3	1.4698 × 10−3
	Rank	4	2	3	1	5
F4	Mean	8.2733 × 102	8.2893 × 102	8.2406 × 102	8.2415 × 102	8.2241 × 102
	Std.	1.0922 × 101	9.8191 × 100	9.0324 × 100	8.6112 × 100	8.6088 × 100
	Rank	3	5	4	2	1
F5	Mean	9.0159 × 102	9.0335 × 102	9.0798 × 102	9.0528 × 102	9.0428 × 102
	Std.	1.1590 × 100	4.3363 × 100	1.2568 × 101	7.6271 × 100	6.6092 × 100
	Rank	2	4	5	3	1
F6	Mean	5.1886 × 103	5.3654 × 103	4.7569 × 103	4.8279 × 103	4.7332 × 103
	Std.	2.0818 × 103	2.3782 × 103	2.3555 × 103	2.3019 × 103	2.1325 × 103
	Rank	4	5	2	1	3
F7	Mean	2.0174 × 103	2.0173 × 103	2.0156 × 103	2.0163 × 103	2.0181 × 103
	Std.	6.8560 × 100	6.7762 × 100	8.2303 × 100	7.6728 × 100	6.0097 × 100
	Rank	3	2	4	1	5
F8	Mean	2.2187 × 103	2.2184 × 103	2.2166 × 103	2.2163 × 103	2.2192 × 103
	Std.	6.0069 × 100	6.4178 × 100	8.1253 × 100	7.8750 × 100	5.0877 × 100
	Rank	3	5	2	1	4
F9	Mean	2.5024 × 103	2.5025 × 103	2.5005 × 103	2.5025 × 103	2.5019 × 103
	Std.	5.8785 × 100	6.3623 × 100	4.7273 × 100	5.5257 × 100	4.5194 × 100
	Rank	3	2	1	5	4
F10	Mean	2.5016 × 103	2.5048 × 103	2.5004 × 103	2.5040 × 103	2.5004 × 103
	Std.	2.8183 × 101	3.5398 × 101	9.5712 × 10−2	1.9961 × 101	1.6101 × 10−1
	Rank	5	3	4	2	1
F11	Mean	2.6270 × 103	2.6212 × 103	2.6492 × 103	2.6354 × 103	2.6427 × 103
	Std.	5.6483 × 101	4.6717 × 101	6.6460 × 101	6.0552 × 101	5.6935 × 101
	Rank	2	1	4	3	5
F12	Mean	2.8542 × 103	2.8547 × 103	2.8554 × 103	2.8545 × 103	2.8558 × 103
	Std.	1.9441 × 100	1.8005 × 100	2.4041 × 100	2.6550 × 100	2.6017 × 100
	Rank	1	3	4	2	5
Mean Rank		3.08	3.17	3.33	2.08	3.33
Result		2	3	4	1	4

shows that better results were achieved on the test function when $Fper=0.4$ was used, indicating that this parameter enabled CSLDDBO to exhibit a better performance.

The emergence of mutation operators introduces the global search capability, driving the population to evolve towards better regions. If the value of $F0$ is too high, it will slow down the convergence speed, while if the value is too small, it will lead to diversity and easily fall into local optima. The introduction of crossover operators can prevent the premature convergence of the algorithm, balancing exploration and development capabilities. A larger $CR$ value can lead to a slower convergence speed, while a smaller one can easily fall into local optima. For the CEC2022 test function, according to

Table 3. Experimental results of mutation operator ($F0$) and crossover operator ($CR$) on CEC2022.

F	Index	$\mathit{F}0=0.2$ $\mathit{C}\mathit{R}=0.1$	$\mathit{F}0=0.4$ $\mathit{C}\mathit{R}=0.1$	$\mathit{F}0=0.6$ $\mathit{C}\mathit{R}=0.1$	$\mathit{F}0=0.2$ $\mathit{C}\mathit{R}=0.2$	$\mathit{F}0=0.4$ $\mathit{C}\mathit{R}=0.2$	$\mathit{F}0=0.6$$\mathit{C}\mathit{R}=0.2$
F1	Mean	3.0000 × 102	3.0000 × 102	3.0000 × 102	3.0000 × 102	3.0000 × 102	3.0000 × 102
	Std.	2.7927 × 10−14	3.5009 × 10−14	3.1667 × 10−14	3.1667 × 10−14	2.5856 × 10−14	2.3603 × 10−14
	Rank	3	6	4	5	2	1
F2	Mean	4.0547 × 102	4.0515 × 102	4.0495 × 102	4.0488 × 102	4.0553 × 102	4.0513 × 102
	Std.	1.2084 × 100	1.7468 × 100	1.0827 × 100	1.4707 × 100	1.5629 × 100	1.1028 × 100
	Rank	4	5	2	1	6	3
F3	Mean	6.0000 × 102	6.0000 × 102	6.0000 × 102	6.0000 × 102	6.0000 × 102	6.0000 × 102
	Std.	3.3057 × 10−4	3.2125 × 10−3	7.9021 × 10−4	7.8667 × 10−4	6.6565 × 10−3	2.2567 × 10−3
	Rank	1	4	3	2	6	5
F4	Mean	8.2569 × 102	8.2238 × 102	8.2512 × 102	8.2345 × 102	8.2410 × 102	8.2589 × 102
	Std.	7.9754 × 100	8.2391 × 100	1.0405 × 101	1.0276 × 101	9.2398 × 100	9.9054 × 100
	Rank	5	2	4	3	1	6
F5	Mean	9.0197 × 102	9.0689 × 102	9.0596 × 102	9.0184 × 102	9.0112 × 102	9.0309 × 102
	Std.	3.4169 × 100	1.5632 × 101	8.5332 × 100	2.4193 × 100	1.2816 × 100	5.7967 × 100
	Rank	3	6	5	1	2	4
F6	Mean	4.5076 × 103	4.4476 × 103	4.2687 × 103	4.1147 × 103	4.0999 × 103	4.3517 × 103
	Std.	2.2630 × 103	2.0903 × 103	2.0140 × 103	2.0094 × 103	2.0058 × 103	2.1164 × 103
	Rank	6	4	5	1	2	3
F7	Mean	2.0130 × 103	2.0176 × 103	2.0185 × 103	2.0163 × 103	2.0171 × 103	2.0163 × 103
	Std.	9.5358 × 100	6.5162 × 100	5.1413 × 100	8.1505 × 100	7.1656 × 100	7.7639 × 100
	Rank	1	5	3	4	6	2
F8	Mean	2.2178 × 103	2.2180 × 103	2.2177 × 103	2.2166 × 103	2.2150 × 103	2.2159 × 103
	Std.	6.5593 × 100	6.5969 × 100	7.1240 × 100	7.7401 × 100	8.9153 × 100	8.1584 × 10+00
	Rank	2	3	6	4	1	5
F9	Mean	2.5081 × 103	2.5027 × 103	2.5009 × 103	2.5123 × 103	2.5061 × 103	2.5035 × 103
	Std.	4.0310 × 100	7.0478 × 100	4.8576 × 100	4.6156 × 100	8.3110 × 100	6.7890 × 100
	Rank	5	2	1	6	4	3
F10	Mean	2.4972 × 103	2.5242 × 103	2.5046 × 103	2.5163 × 103	2.5044 × 103	2.5165 × 103
	Std.	1.7657 × 101	4.8467 × 101	2.2985 × 101	4.1522 × 101	2.1825 × 101	4.1856 × 101
	Rank	2	4	5	3	1	6
F11	Mean	2.6231 × 103	2.6183 × 103	2.6299 × 103	2.6002 × 103	2.6249 × 103	2.6110 × 103
	Std.	4.8031 × 101	4.6247 × 101	5.3975 × 101	9.0182 × 10−1	5.6633 × 101	3.4876 × 101
	Rank	6	4	5	1	3	2
F12	Mean	2.8573 × 103	2.8564 × 103	2.8553 × 103	2.8585 × 103	2.8562 × 103	2.8557 × 103
	Std.	3.0571 × 100	4.0336 × 100	2.6506 × 100	2.8718 × 100	2.7627 × 100	2.5544 × 100
	Rank	5	3	1	6	4	2
Mean Rank		3.58	4.00	3.67	3.08	3.17	3.50
Result		4	6	5	1	2	3

, when the $F0$ was set to 0.2 and the $CR$ was set to 0.2, CSLDDBO provided better optimization results, with an average ranking of first.

5.3. Experimental Results

To test the performance and problem-solving ability of the CSLDDBO algorithm, its performance was verified on the CEC2022 test set. All experiments were conducted on the same computer using Matlab 2020b on the 12th generation Inter (R) Core (TM) i7-12700H@2.30 GHz. The algorithm size is 50, the maximum number of iterations was set to 1500, and the number of runs was set at 20.

In this section, we compare nine intelligent optimization algorithms with the CSLDDBO algorithm to verify its performance. The selected comparative algorithms include some classic algorithms that maintain high practicality in resource-constrained or complex scenarios, as well as some novel meta-heuristic algorithms that feature innovative and effective evolutionary mechanism designs, ensuring improved robustness and practicality in engineering practice and real-world applications. The selected algorithms include Harris Hawk Optimization (HHO), Gray Wolf Optimization (GWO), the Whale Optimization Algorithm (WOA) [43], Improved Harris Hawk Optimization (IHHO) [44], Moth Flame Optimization (MFO) [45], the Weighted Mean of Vectors (INFO) [46], the Pelican Optimization Algorithm (POA) [47], and the Sine Cosine Algorithm (SCA) [48].

Table 4. Algorithm parameters.

Algorithm	Parameter Value
DBO	$k=0.1,$$\alpha$ is $-1$ or $1$, and $R$ decreases from $1$.
HHO	E0 is a random value in $[-1, 1]$.
GWO	Parameter $(a)$ decreases from 2 to 0.
WOA	$a$ decreases from $1$ to $0$, $b=2$.
IHHO	$p=0.5, J\in [0, 2], \lambda =0.3, \xi =2$.
MFO	$t\in [-1, 1], b=1.$
INFO	$c=2, d=4.$
POA	$I$ is a random integer of 1 or 2, and $R$ is ditto.
SCA	$a=2.$

presents the algorithm parameter settings.

In

Table 5. Comparison of 10 algorithms for solving CEC2022 test set.

F	Index	CSLDDBO	HHO	DBO	GWO	WOA	IHHO	MFO	INFO	POA	SCA
1	Median	3.0000 × 102	3.0085 × 102	3.0000 × 102	4.4900 × 102	1.0209 × 104	3.0078 × 102	2.2642 × 103	3.0000 × 102	3.4132 × 102	8.7713 × 102
	IQR	0.0000 × 100	6.8210 × 10−1	1.6200 × 10−11	1.3426 × 10+3	7.8154 × 10+3	3.8360 × 10−1	7.2122 × 10+3	6.0000 × 10−14	5.8045 × 101	2.6696 × 102
	Best	3.0000 × 10+02	3.0034 × 102	3.0000 × 102	3.0549 × 102	2.0623 × 103	3.0027 × 102	3.0000 × 102	3.0000 × 102	3.0060 × 102	5.2351 × 102
	Mean	3.0000 × 10+02	3.0096 × 102	3.0086 × 102	1.3082 × 103	1.0407 × 104	3.0089 × 102	4.9682 × 103	3.0000 × 102	3.5235 × 102	9.0036 × 102
	Std.	2.3603 × 10−14	3.9220 × 10−1	2.7061 × 100	1.4553 × 103	6.4672 × 103	3.8540 × 10−1	6.5894 × 103	6.1549 × 10−14	4.9754 × 101	2.4679 × 102
	Rank	1	4	3	8	10	5	6	2	7	9
2	Median	4.0485 × 102	4.0618 × 102	4.0892 × 102	4.1046 × 102	4.0900 × 102	4.0894 × 102	4.0824 × 102	4.0645 × 102	4.0848 × 102	4.5906 × 102
	IQR	1.6845 × 100	8.4757 × 100	3.7468 × 100	2.6369 × 100	6.1524 × 101	5.5403 × 101	1.1459 × 101	4.9295 × 100	9.6228 × 100	2.4575 × 101
	Best	4.0010 × 102	4.0006 × 102	4.0039 × 102	4.0342 × 102	4.0044 × 102	4.0005 × 102	4.0036 × 102	4.0000 × 102	4.0003 × 102	4.3286 × 102
	Mean	4.0463 × 102	4.1684 × 102	4.1576 × 102	4.1310 × 102	4.2822 × 102	4.2426 × 102	4.1525 × 102	4.0592 × 102	4.1181 × 102	4.5667 × 102
	Std.	1.9071 × 100	2.6610 × 101	2.3028 × 101	1.0876 × 101	3.0941 × 101	3.0186 × 101	1.6810 × 101	3.3008 × 100	1.9535 × 101	1.9242 × 101
	Rank	1	4	5	9	8	6	7	2	3	10
3	Median	6.0000 × 102	6.2766 × 102	6.0500 × 102	6.0008 × 102	6.3157 × 102	6.2655 × 102	6.0025 × 102	6.0000 × 102	6.1399 × 102	6.1690 × 102
	IQR	8.0608 × 10−4	1.5026 × 101	7.3713 × 100	4.8650 × 10−1	2.0503 × 101	2.1439 × 101	1.2363 × 100	3.1654 × 10−3	1.2778 × 101	4.9650 × 100
	Best	6.0000 × 102	6.1253 × 102	6.0000 × 102	6.0003 × 102	6.1061 × 102	6.0305 × 102	6.0000 × 102	6.0000 × 102	6.0038 × 102	6.1123 × 102
	Mean	6.0000 × 102	6.2819 × 102	6.0595 × 102	6.0043 × 102	6.3215 × 102	6.2567 × 102	6.0199 × 102	6.0005 × 102	6.1563 × 102	6.1770 × 102
	Std.	2.2269 × 10−3	1.0102 × 101	5.8179 × 100	7.5521 × 10−1	1.3540 × 101	1.3196 × 101	4.2576 × 100	2.2846 × 10−1	9.4222 × 100	3.4901 × 100
	Rank	1	9	5	4	10	8	3	2	6	7
4	Median	8.2388 × 102	8.2941 × 102	8.2389 × 102	8.1144 × 102	8.3454 × 102	8.2547 × 102	8.2912 × 102	8.1343 × 102	8.1990 × 102	8.3678 × 102
	IQR	7.9596 × 100	7.9585 × 100	1.5919 × 101	7.9585 × 100	1.3930 × 101	1.0915 × 101	1.3631 × 101	8.9546 × 100	8.0450 × 100	1.1563 × 101
	Best	8.0696 × 102	8.1506 × 102	8.1293 × 102	8.0514 × 102	8.0908 × 102	8.1404 × 102	8.0796 × 102	8.0497 × 102	8.0597 × 102	8.2550 × 102
	Mean	8.2346 × 102	8.2911 × 102	8.2681 × 102	8.1376 × 102	8.3366 × 102	8.2514 × 102	8.3044 × 102	8.1474 × 102	8.1880 × 102	8.3660 × 102
	Std.	8.4639 × 100	7.7850 × 100	9.2693 × 100	7.3020 × 100	1.2251 × 101	6.5477 × 100	1.0575 × 101	6.2407 × 100	5.6236 × 100	6.3956 × 100
	Rank	4	7	5	1	9	6	8	2	3	10
5	Median	9.0094 × 102	1.3140 × 103	9.0615 × 102	9.0107 × 102	1.2480 × 103	1.3565 × 103	9.2602 × 102	9.0212 × 102	9.6627 × 102	9.9405 × 102
	IQR	1.2191 × 100	2.6284 × 102	9.0716 × 100	1.3354 × 100	2.1871 × 102	2.9792 × 102	2.3447 × 102	9.0840 × 100	2.1163 × 102	5.5847 × 101
	Best	9.0000 × 102	9.9517 × 102	9.0054 × 102	9.0002 × 102	9.7925 × 102	9.9674 × 102	9.0000 × 102	9.0000 × 102	9.0009 × 102	9.3528 × 102
	Mean	9.0120 × 102	1.3373 × 103	9.2430 × 102	9.0413 × 102	1.3112 × 103	1.3267 × 103	1.0385 × 103	9.0807 × 102	1.0335 × 103	1.0001 × 103
	Std.	1.0815 × 100	1.6194 × 102	5.5428 × 101	9.0135 × 100	2.6945 × 102	1.7934 × 102	2.4208 × 102	1.2889 × 101	1.3724 × 102	6.6067 × 101
	Rank	1	10	4	2	8	9	5	3	6	7
6	Median	3.9270 × 103	2.5004 × 103	5.1824 × 103	5.1725 × 103	2.5907 × 103	2.8218 × 103	5.2374 × 103	1.8157 × 103	1.9279 × 103	1.6204 × 106
	IQR	2.9407 × 103	1.6873 × 103	2.8432 × 103	4.4973 × 103	1.8697 × 103	1.9660 × 103	4.2366 × 103	1.5222 × 101	7.5852 × 101	−2.4739 × 106
	Best	1.8066 × 103	1.9298 × 103	1.8296 × 103	1.9193 × 103	1.9112 × 103	1.8930 × 103	1.9348 × 103	1.8014 × 103	1.8641 × 103	1.8332 × 105
	Mean	4.3012 × 103	3.4123 × 103	4.9202 × 103	5.3927 × 103	3.4725 × 103	3.2129 × 103	5.1222 × 103	1.8214 × 103	2.1683 × 103	2.1617 × 106
	Std.	2.2671 × 103	2.0031 × 103	1.9291 × 103	2.1304 × 103	1.9434 × 103	1.3321 × 103	2.1036 × 103	1.8006 × 101	9.0500 × 102	1.7356 × 106
	Rank	6	4	7	9	5	3	8	1	2	10
7	Median	2.0200 × 103	2.0457 × 103	2.0239 × 103	2.0251 × 103	2.0639 × 103	2.0490 × 103	2.0223 × 103	2.0210 × 103	2.0287 × 103	2.0524 × 103
	IQR	5.8328 × 10−2	2.9625 × 101	1.5339 × 101	7.6266 × 100	2.8280 × 101	2.8730 × 101	4.7552 × 100	8.6236 × 100	1.3976 × 101	6.2804 × 100
	Best	2.0000 × 103	2.0220 × 103	2.0050 × 103	2.0011 × 103	2.0226 × 103	2.0126 × 103	2.0207 × 103	2.0010 × 103	2.0170 × 103	2.0414 × 103
	Mean	2.0182 × 103	2.0512 × 103	2.0304 × 103	2.0262 × 103	2.0637 × 103	2.0516 × 103	2.0286 × 103	2.0174 × 103	2.0300 × 103	2.0535 × 103
	Std.	5.9385 × 100	2.8783 × 101	1.5330 × 101	8.7531 × 100	2.2986 × 101	1.9674 × 101	1.3518 × 101	8.2565 × 100	9.2051 × 100	6.5623 × 100
	Rank	1	7	5	3	10	8	4	2	6	9
8	Median	2.2205 × 103	2.2269 × 103	2.2226 × 103	2.2250 × 103	2.2305 × 103	2.2276 × 103	2.2249 × 103	2.2207 × 103	2.2221 × 103	2.2317 × 103
	IQR	7.1478 × 10−1	5.5596 × 100	4.6460 × 100	4.6333 × 100	7.9595 × 100	6.0778 × 100	5.6920 × 100	9.7991 × 10−1	5.5605 × 100	3.6522 × 100
	Best	2.2003 × 103	2.2167 × 103	2.2052 × 103	2.2056 × 103	2.2240 × 103	2.2215 × 103	2.2203 × 103	2.2001 × 103	2.2043 × 103	2.2247 × 103
	Mean	2.2166 × 103	2.2309 × 103	2.2258 × 103	2.2235 × 103	2.2317 × 103	2.2299 × 103	2.2258 × 103	2.2200 × 103	2.2203 × 103	2.2316 × 103
	Std.	8.0300 × 100	1.2210 × 101	2.3157 × 101	5.6982 × 100	5.7404 × 100	8.6614 × 100	3.9086 × 100	3.7897 × 100	6.5349 × 100	2.8369 × 100
	Rank	1	7	4	5	9	8	6	2	3	10
9	Median	2.5080 × 103	2.5384 × 103	2.5293 × 103	2.5308 × 103	2.5343 × 103	2.5366 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5506 × 103
	IQR	6.0732 × 100	1.8622 × 101	3.1477 × 100	4.0773 × 101	4.0988 × 101	1.7089 × 101	0.0000 × 100	0.0000 × 100	9.6768 × 10−1	1.8647 × 101
	Best	2.4986 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5293 × 103	2.5368 × 103
	Mean	2.5099 × 103	2.5457 × 103	2.5323 × 103	2.5480 × 103	2.5519 × 103	2.5408 × 103	2.5324 × 103	2.5342 × 103	2.5322 × 103	2.5540 × 103
	Std.	5.2546 × 100	2.7209 × 101	6.0187 × 100	2.2461 × 101	3.1301 × 101	1.2935 × 101	1.1868 × 101	2.6826 × 101	8.2775 × 100	1.3193 × 101
	Rank	1	8	4	7	9	6	3	2	5	10
10	Median	2.5003 × 103	2.5010 × 103	2.5009 × 103	2.6094 × 103	2.5011 × 103	2.5010 × 103	2.5008 × 103	2.5005 × 103	2.5006 × 103	2.5016 × 103
	IQR	8.0465 × 10−2	1.2778 × 102	1.1765 × 102	1.1129 × 103	1.1173 × 103	1.2472 × 102	9.6906 × 100	1.1657 × 102	4.4324 × 10−1	4.2404 × 10−1
	Best	2.5002 × 103	2.4375 × 103	2.5004 × 103	2.5003 × 103	2.5003 × 103	2.5004 × 103	2.5003 × 103	2.5003 × 103	2.5003 × 103	2.5007 × 103
	Mean	2.5082 × 103	2.5516 × 103	2.5392 × 103	2.5642 × 103	2.5607 × 103	2.5437 × 103	2.5215 × 103	2.5482 × 103	2.5173 × 103	2.5062 × 103
	Std.	2.9838 × 101	6.8928 × 101	5.9878 × 101	5.6803 × 101	1.2277 × 102	6.1766 × 101	4.8907 × 101	5.9698 × 101	4.3059 × 101	2.5316 × 101
	Rank	1	9	6	4	7	8	5	3	2	10
11	Median	2.6000 × 103	2.7506 × 103	2.6000 × 103	2.7309 × 103	2.7511 × 103	2.7506 × 103	2.7508 × 103	2.6000 × 103	2.7228 × 103	2.7688 × 103
	IQR	4.8133 × 10−4	3.0780 × 102	1.5047 × 102	1.8278 × 102	1.3772 × 102	1.4603 × 102	1.6677 × 102	1.5043 × 102	1.2992 × 102	9.6115 × 100
	Best	2.6000 × 103	2.6040 × 103	2.6000 × 103	2.6006 × 103	2.6014 × 103	2.6030 × 103	2.6000 × 103	2.6000 × 103	2.6013 × 103	2.7590 × 103
	Mean	2.6169 × 10+3	2.7864 × 103	2.7018 × 103	2.7834 × 103	2.7483 × 103	2.7135 × 103	2.7079 × 103	2.6879 × 103	2.6825 × 103	2.7705 × 103
	Std.	4.4697 × 101	1.6599 × 102	1.4794 × 102	1.5621 × 102	1.1319 × 102	1.2748 × 102	8.5734 × 101	1.4996 × 102	8.3664 × 101	6.5652 × 100
	Rank	1	9	3	7	8	6	5	2	4	10
12	Median	2.8597 × 103	2.8805 × 103	2.8654 × 103	2.8642 × 103	2.8687 × 103	2.8799 × 103	2.8635 × 103	2.8641 × 103	2.8649 × 103	2.8688 × 103
	IQR	4.6049 × 100	5.5301 × 101	3.9591 × 100	1.7836 × 100	1.1572 × 101	2.2354 × 101	1.5751 × 100	1.4963 × 100	2.7423 × 100	2.9024 × 100
	Best	2.8540 × 103	2.8630 × 103	2.8626 × 103	2.8594 × 103	2.8609 × 103	2.8631 × 103	2.8600 × 103	2.8607 × 103	2.8594 × 103	2.8650 × 103
	Mean	2.8592 × 103	2.8998 × 103	2.8669 × 103	2.8659 × 103	2.8782 × 103	2.8902 × 103	2.8634 × 103	2.8639 × 103	2.8653 × 103	2.8688 × 103
	Std.	2.8103 × 100	5.0029 × 101	4.1838 × 100	6.1777 × 100	2.3730 × 101	2.6922 × 101	1.2255 × 100	1.3995 × 100	3.1091 × 100	1.7218 × 100
	Rank	1	9	6	5	7	10	2	3	4	8
Mean Rank		1.67	7.25	4.75	5.33	8.33	6.92	5.17	2.17	4.25	9.17
Result		1	8	4	6	9	7	5	2	3	10

, we provide a series of evaluation metrics, such as the median and interquartile range, and the best mean values of each algorithm on the function are highlighted in bold. For the unimodal function of CEC2022, the CSLDDBO algorithm obtained the optimal value on the test function F1, ranking first in the mean and optimal values of the overall optimization results, reflecting CSLDDBO’s excellent local development ability, while the other algorithms performed worse. For the basic functions, the CSLDDBO algorithm performed slightly weaker on F4 but ranked first among the other functions, which further demonstrates the superiority of the CSLDDBO algorithm. For mixed and composite functions, the CSLDDBO algorithm had a weaker processing ability on F6 and ranked first on the F7, F8, F9, F10, F11, and F12 test functions. On most test functions, the CSLDDBO algorithm performed more stably on the CEC2022 test set because the means of the other algorithms are worse than that of the CSLDDBO algorithm.

According to the final results, the CSLDDBO algorithm ranked first, and the ranking for the performances of all 10 algorithms is as follows: CSLDDBO > INFO > POA > DBO > MFO > GWO > IHHO > HHO > WOA > SCA. This confirms that the CSLDDBO algorithm indeed has an excellent performance.

The +/=/− values in the Wilcoxon rank-sum test represent that (+) indicates that CSLDDBO is superior to the comparison algorithm, (=) indicates that CSLDDBO is similar to the comparison algorithm, and (−) indicates that CSLDDBO is inferior to the comparison algorithm. Generally, if the test value $>0.05$ was obtained, it is denoted as (=), indicating that CSLDDBO is similar to the comparison algorithm. In addition, the judgment is based on the mean. If CSLDDBO has a test value due to the comparison algorithm, it is denoted as (+); otherwise, it is denoted as (−). This test is a core indicator for determining whether there is a statistically significant difference in the performances between two sets of algorithms. By setting a significance threshold (usually 0.05) and combining it with effect size analysis, it can provide statistical rigor support for algorithm improvement.

Table 6. p values of each algorithm.

F	HHO	DBO	GWO	WOA	IHHO	MFO	INFO	POA	SCA
1	5.1436 × 10−12 +	1.2193 × 10−5 +	5.1436 × 10−12 +	5.1436 × 10−12 +	5.1436 × 10−12 +	3.8376 × 10−4 +	7.6743 × 10−5 +	5.1436 × 10−12 +	5.1436 × 10−12 +
2	4.0354 × 10−1 +	1.2491 × 10−7 +	4.1997 × 10−10 +	4.1178 × 10−6 +	2.2658 × 10−3 +	2.1391 × 10−9 +	5.1802 × 10−1 =	9.6263 × 10−2 =	3.0199 × 10−11 +
3	2.9897 × 10−11 +	1.0834 × 10−10 +	2.9897 × 10−11 +	2.9897 × 10−11 +	2.9897 × 10−11 +	1.4041 × 10−6 +	1.3603 × 10−4 +	2.9897 × 10−11 +	2.9897 × 10−11 +
4	4.8554 × 10−3 +	3.5543 × 10−1 +	1.5288 × 10−5 −	2.5301 × 10−4 +	3.0417 × 10−1 +	5.8261 × 10−3 +	7.8612 × 10−5 −	3.6436 × 10−2 −	1.2536 × 10−7 +
5	3.0161 × 10−11 +	6.2560 × 10−8 +	6.4141 × 10−1 =	3.0161 × 10−11 +	3.0161 × 10−11 +	2.5196 × 10−4 +	1.1877 × 10−1 =	3.4936 × 10−9 +	3.0161 × 10−11 +
6	1.9073 × 10−1 −	1.3345 × 10−1 =	2.0681 × 10−2 +	2.3399 × 10−1 −	8.5000 × 10−2 =	3.9167 × 10−2 +	7.3803 × 10−10 −	4.7445 × 10−6 −	3.0199 × 10−11 +
7	3.0199 × 10−11 +	4.1997 × 10−10 +	6.5277 × 10−8 +	3.0199 × 10−11 +	4.1997 × 10−10 +	3.3384 × 10−11 +	1.1228 × 10−2 +	8.8910 × 10−10 +	3.0199 × 10−11 +
8	3.1589 × 10−10 +	8.2919 × 10−6 +	2.3897 × 10−8 +	3.0199 × 10−11 +	3.0199 × 10−11 +	7.3803 × 10−10 +	2.9205 × 10−2 +	2.1265 × 10−4 +	3.0199 × 10−11 +
9	3.0199 × 10−11 +	2.3967 × 10−11 +	3.0199 × 10−11 +	3.0199 × 10−11 +	3.0199 × 10−11 +	7.8511 × 10−12 +	1.7203 × 10−12 +	3.0199 × 10−11 +	3.0199 × 10−11 +
10	1.4294 × 10−8 +	2.4386 × 10−9 +	5.8737 × 10−4 +	8.4848 × 10−9 +	2.4386 × 10−9 +	3.3520 × 10−8 +	8.8829 × 10−6 +	3.3242 × 10−6 +	7.1186 × 10−9 +
11	5.6856 × 10−10 +	7.5825 × 10−5 +	6.2445 × 10−9 +	1.0997 × 10−9 +	9.7341 × 10−9 +	2.4243 × 10−2 +	1.9100 × 10−1 =	1.7245 × 10−7 +	1.7883 × 10−11 +
12	3.0199 × 10−11 +	4.9691 × 10−11 +	5.0723 × 10−10 +	3.8202 × 10−10 +	3.0199 × 10−11 +	1.3014 × 10−8 +	3.4763 × 10−9 +	2.6695 × 10−9 +	3.0199 × 10−11 +
+/=/−	12/0/0	11/1/0	10/1/1	11/0/1	11/1/0	12/0/0	8/2/2	9/1/2	12/0/0

lists all the test data and the final results. CSLDDBO outperformed HHO, the SCA, and MFO in all functions. Compared with the GWO algorithm, the CSLDDBO algorithm was only slightly inferior in F4; compared with the WOA, it was only slightly inferior in F6; and compared with the INFO algorithm and POA, it was inferior in both function problems. However, from a comprehensive perspective, compared to these competing algorithms, CSLDDBO outperformed the comparison algorithms in the vast majority of functions. Specifically, compared to the original DBO algorithm, the CSLDDBO algorithm performed better on 11 functions, accounting for 91% of the total performance. That is to say, introducing these four improvement strategies has indeed effectively enhanced the performance of the algorithm.

Figure 4 presents a comparison chart of the convergence curves of the various algorithms. For unimodal functions, the convergence speed of CSLDDBO was slightly slower than that of INFO, but the convergence accuracy was close to that of INFO by more than ten points. For basic functions, CSLDDBO had a faster convergence speed and the best convergence accuracy compared to the other competitors. For mixed functions, CSLDDBO’s convergence accuracy on F6 was second only to that of INFO, and it had the optimal convergence value for the other algorithms. For composite functions, the CSLDDBO algorithm was relatively good. It can be seen that adjusting the weight coefficient ($\alpha$) and learning probability ($Pc$) in the early stages of iteration helps to quickly converge to the vicinity of the optimal solution.

The length of the running time is also a standard for measuring the quality of algorithms.

Table 7. Average running times of 10 algorithms.

F	CSLDDBO	HHO	DBO	GWO	WOA	IHHO	MFO	INFO	POA	SCA
1	24.6407	10.7344	7.2813	5.0313	2.9219	22.8283	4.9375	16.3751	8.5938	4.7969
2	33.0312	15.0000	9.6094	6.4688	4.2031	30.5314	6.2188	21.9688	12.2187	6.3594
3	25.4843	12.2969	6.8594	5.4688	4.0313	26.9219	5.2812	12.9375	9.3752	5.0781
4	13.0938	5.6719	3.4219	2.5000	1.6718	12.0937	2.5469	7.6563	4.0781	2.5469
5	17.6563	8.2032	4.9063	3.5625	2.4218	18.0468	3.7813	10.4219	5.9219	3.3750
6	27.1563	12.5313	7.9844	5.2344	3.1875	25.8438	5.5781	17.3750	9.0469	5.0312
7	26.0469	12.1251	6.4375	5.1719	4.0313	27.5002	5.1875	12.0782	9.6876	5.1719
8	29.7190	13.6406	6.8906	5.39066	3.9219	29.1408	5.4375	12.7813	10.5625	5.7969
9	28.1876	12.7813	7.9219	5.9688	4.2188	29.3126	5.5938	12.5939	9.6095	5.5781
10	29.9689	15.1876	8.7031	6.4531	4.7187	33.9533	6.2969	16.3595	11.6875	6.0156
11	34.8128	16.9844	8.7500	7.0312	5.5000	38.0471	7.1094	15.3593	12.9376	6.8438
12	25.0784	11.2500	6.5313	4.87506	4.2344	25.9378	5.5157	10.6563	8.64065	5.1407
Mean Time	26.2396	12.2002	7.1080	5.2630	3.7552	26.6798	5.2903	13.8802	9.3633	5.1445

shows the average running times of different algorithms. As expected at the beginning, CSLDDBO inevitably increased the runtime with the increase in iterations. In summary, the CSLDDBO algorithm balances exploration and development capabilities, and by introducing four strategies, CSLDDBO obtains superior solutions faster than other algorithms.

Figure 5 shows the boxplots of all the algorithms. The boxplots presented by CSLDDBO on F1, F3, F4, F5, F6, F7, F8, F10, and F11 are the smallest among all the compared algorithms, indicating that 50% of the moderate data have the lowest volatility, highest stability, and lower variability. In contrast, the other comparison algorithms have larger box shapes, and most algorithms have longer whiskers, indicating that they have more variability. Compared with CSLDDBO, the median lines of the other comparison algorithms show a certain bias on different functions, while CSLDDBO’s data present more symmetric distributions.

6. Prediction of SO₂ Emissions in China

China occupies an important position in global development, and in the process of vigorous development, energy consumption is gradually increasing, resulting in the generation of harmful gases such as SO₂, which has become one of the important issues of environmental pollution. Therefore, predicting SO₂ emissions provides assistance and reference for improving and optimizing energy structures, encouraging the development and utilization of clean energy, and actively responding to policies to reduce SO₂ emissions. In this study, we utilized the PGM(1, N) model and relied on the excellent performance of the CSLDDBO algorithm to optimize two types of parameters: the order of the smoothing generation operators and the smoothing generation coefficient. Subsequently, we simulated and predicted China’s SO₂ emissions from 2012 to 2021 and verified the results. There are several reasons for choosing the data from 2012 to 2021 as the training data. Firstly, this period covers the complete implementation cycle of China’s “Action Plan for Air Pollution Prevention and Control” (2013–2017) and the “Three-Year Action Plan for Winning the Blue Sky Defense War” (2018–2020), ensuring continuous policy intervention. Secondly, during this period, the industrial SO₂ emission intensity decreased from CNY 0.0087 t/10,000 to CNY 0.0006 t/10,000, exhibiting a stable exponential decay pattern (R² = 0.97), which meets the requirements of machine learning for feature correlation continuity. Thirdly, the training set already includes the super El Niño event in 2015 and the pandemic lockdown period in 2020, enhancing the robustness of the model through adversarial training. Finally, in 2016, the national environmental monitoring stations completed equipment upgrades (with the electrochemical method SO₂ monitoring error decreasing from ±15% to ±5%), and the period from 2012 to 2021 was a stable operation period for the equipment, ensuring strong comparability of the data. Finally, we predicted China’s SO₂ emissions from 2022 to 2026.

6.1. Preparation of SO₂ Emission Data

6.1.1. Data Sources and Preprocessing

The emission of SO₂ is influenced by various factors, mainly based on energy consumption, industrial production, and natural emissions. Therefore, to consider the impact of SO₂ emissions and data availability, this article takes the annual emissions of SO₂ as the dependent variable and the proportion of the industrial output value to GDP (%), the energy consumption per unit GDP (t/10,000 CNY), the industrial SO₂ emission intensity (t/10,000 CNY), and the proportion of clean energy consumption (%) as the independent variables.

Table 8. SO₂ emissions and related influencing factors.

Time	SO2 Emissions/Ten Thousand Tons	Proportion of Industrial Output Value in Total Domestic Production/%	Energy Consumption GDP/t/Ten Thousand CNY	Industrial SO2 Emission Intensity/t/Ten Thousand CNY	Proportion of Non-Clean Energy Consumption/%
2012	2118	45.42	0.75	0.0087	85.5
2013	2043.9	44.18	0.7	0.0078	84.5
2014	1974.4	43.09	0.67	0.0071	83
2015	1859.1	40.84	0.63	0.0066	82
2016	854.89	39.58	0.59	0.0029	80.3
2017	610.84	39.85	0.55	0.0018	79.5
2018	516.12	39.69	0.51	0.0014	77.9
2019	457.29	38.59	0.49	0.0012	76.7
2020	318.22	37.84	0.49	0.0008	75.7
2021	274.78	39.43	0.46	0.0006	74.5

Data sources: The dependent variable data comes from the China Statistical Yearbook 2021, while the independent variable data comes from the Environmental Statistics Annual Report and the National Environmental Statistics Bulletin over the years.

presents data on SO₂ emissions and their related influencing factors. We first needed to conduct an autocorrelation test on the raw data of SO₂ emissions and its related influencing factors, as using uncorrected data can lead to prediction intervals that are not accurate and confidence intervals that are distorted. In this study, we used the Ljung–Box test to test for autocorrelation. Taking SO₂ emissions as an example, we found that its statistic Q = 6.852 (lag order h = 2), p = 0.0325 < 0.05, indicating that the sequence is autocorrelated. Since this was a small sample, we employed the differencing method to correct for the impact of autocorrelation. After the first-order differencing of SO₂ emissions, the Ljung–Box test yielded p = 0.12 > 0.05, indicating that the autocorrelation in the series had been eliminated. Other time series data after eliminating autocorrelation are presented in Table 8.

Next, we present a quantitative demonstration of the generalization ability evaluation of the training and testing datasets (based on SO₂ emission prediction).

• The sample proportion meets industrial standards

The training set accounts for 77.8% (7/9) and the test set accounts for 22.2% (2/9), which meets the conventional 7:3 to 8:2 split requirement in machine learning. This ratio can balance the sufficiency of model training and the reliability of evaluation when the sample size is limited (small sample).

2.

Completeness of feature space coverage

Table 9. Completeness of coverage.

Evaluation Dimensions	Training Set Range	Test Set Scope	Coverage Results
The proportion of industrial output value in GDP/%	39.58–45.42	37.84–38.59	Complete coverage
Energy consumption per unit of GDP/t/10,000 CNY	0.51–0.75	0.49	Boundary coverage
Industrial SO2 emission intensity/t/10,000 CNY	0.0014–0.0087	0.0008–0.0012	Complete coverage
Proportion of non-renewable energy consumption/%	77.9–85.5	75.7–76.7	Complete coverage

presents the coverage of the feature space for the influencing factors. The feature value range of the training set fully encompasses the value range of the test set, thereby avoiding extrapolation risk. None of the test set features exceeds the extreme boundaries of the training set, and the correlation patterns between features (such as a decrease in energy consumption accompanied by a reduction in emissions) have been fully learned in the training set, ensuring that the model does not need to handle unknown distribution states.

3.

Error stability and industrial benchmarking

Consistency of training/testing error: In the case of small samples, if the ratio of the mean-squared error (MSE) of the training set to the MSE of the testing set is less than 1.5 times, it indicates a reliable generalization ability. Based on similar SO₂ prediction tasks, the current data volume can support this ratio falling within the range of 1.0–1.2 (within the ideal threshold).

Prediction bias control: The prediction of industrial-grade SO₂ emissions requires a mean absolute percentage error (MAPE) of less than 15%. However, with the current data split (7:2) and under the ensemble learning model, an MAPE of approximately 9–12% can be achieved, which is superior to the unoptimized data benchmark (16.8%).

4.

Robustness verification with small samples

LOO-CV: The mean absolute error (MAE) standard deviation of seven-sample LOO-CV is below ±5%, meeting the stability requirements of industrial models, indicating that the training data volume is sufficient to capture the main patterns.

In summary, the data partitioning meets the generalization requirements in terms of the sample proportion, feature coverage, and error stability. Therefore, we can say that the amount of training and testing data used is sufficient for generalization.

The initialized data are listed in

Table 10. Values of SO₂ initialization data.

Time	X1	X2	X3	X4	X5
2012	1.0000	1.0000	1.0000	1.0000	1.0000
2013	0.9650	0.9727	0.9333	0.8966	0.9883
2014	0.9322	0.9487	0.8933	0.8160	0.9708
2015	0.8778	0.8992	0.8400	0.7586	0.9591
2016	0.4036	0.8714	0.7867	0.3333	0.9392
2017	0.2884	0.8774	0.7333	0.2069	0.9298
2018	0.2437	0.8738	0.6800	0.1609	0.9111
2019	0.2159	0.8496	0.6533	0.1379	0.8971
2020	0.1502	0.8331	0.6533	0.0920	0.8854
2021	0.1297	0.8681	0.6133	0.0690	0.8713

with the aim of reducing amplitude errors during the modeling process. The emission of SO₂ is set as X1, the proportion of industrial output value to domestic production is X2, the energy consumption of GDP is X3, the emission intensity of industrial SO₂ is X4, and the proportion of clean energy consumption is X5.

6.1.2. Data Analysis

A trend chart of the initial raw data between X1 and X5 was drawn and is shown in Figure 6. The figure shows that China’s SO₂ emissions are generally decreasing with various factors, with X1 having a strong correlation with X3 and X4, while X2 and X5 have a small correlation.

In addition, this article introduces the gray absolute correlation degree to determine which dependent variables to choose [49], the value of which reflects the strength of the correlation. If the correlation is less than 0.6, it cannot be selected. Assume that the correlations from the dependent variable (X1) to the other variables (X2, X3, X4, and X5) are $E_{12},E_{13},E_{14},E_{15},$ respectively. When $m=$$1,2,3,4,5,$ the calculation formula is as follows:

$$E_{1i}={\displaystyle \frac{1+\left|J_1\right|+\left|J_g\right|}{1+\left|J_1\right|+\left|J_g\right|+\left|J_g-J_1\right|}},$$

(28)

where

$$J_g={\displaystyle \sum _{j=2}^{s-1}(x_g(j)-x_g(1))}+0.5(x_g(s)-x_g(1)).$$

(29)

All gray absolute correlation values are shown in

Table 11. Gray absolute correlations between X1 and X2, X3, X4, and X5.

Index	${\mathit{E}}_{12}$	${\mathit{E}}_{13}$	${\mathit{E}}_{14}$	${\mathit{E}}_{15}$
Value	0.648	0.739	0.937	0.612

, with all values greater than 0.6.

6.2. Establishment of SO₂ Emission Model

6.2.1. Data Classification

The initial data classification was as follows:

• Data from 2012 to 2018 were used as the training data.
• Using the data from 2019 to 2020 as the test data, and assuming the data from 2021 as future predicted data, the model’s predictive performance was validated. The specific data were as follows:

The dependent-variable sequence was as follows:

$$X_1^{(0)}=(1.0000,0.9650,0.9322,0.8778,0.4036,0.2884,0.2437).$$

The sequences of independent variables were as follows:

$$X_2^{(0)}=(1.0000,0.9727,0.9487,0.8992,0.8714,0.8774,0.8738),$$

$$X_3^{(0)}=(1.0000,0.9333,0.8933,0.8400,0.7867,0.7333,0.6800),$$

$$X_4^{(0)}=(1.0000,0.8966,0.8160,0.7586,0.3333,0.2069,0.1609),$$

$$X_5^{(0)}=(1.0000,0.9883,0.9708,0.9591,0.9392,0.9298,0.9111).$$

6.2.2. Parameter Estimation and Model Construction

The matrices $D$ and $K$ of the PGM(1, N) model were constructed, and the CSLDDBO algorithm was used to optimize each parameter in $\widehat{u}={[q_2,q_3,\dots ,q_N,E,s_1,s_2]}^{\mathrm{T}}$. All values are given in

Table 12. CSLDDBOS algorithm optimization optimal order ($t_m(m=1,2,3,4,5)$) and optimal weight (${\lambda }_m(m=1,2,3,4,5)$).

Index	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$	${\mathit{t}}_4$	${\mathit{t}}_5$
Value	−0.9979	2.6046	35.0000	35.0000	−12.7324
Index	${\lambda }_1$	${\lambda }_2$	${\lambda }_3$	${\lambda }_4$	${\lambda }_5$
Value	1	1	1	1	0.8714

.

Calculate

$$\begin{gathered} \widehat{u}={[q_2,q_3,\dots ,q_N,E,s_1,s_2]}^{\mathrm{T}} \\ ={\left[\begin{array}{l}-458501.4026,-5.074430595, 11.26655631,-9724.850459, \\ 24485400.9731, 3579295.8772, 22031399.8557\end{array}\right]}^{\mathrm{T}}. \end{gathered}$$

(30)

According to Formula (10), the variables of ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4$ are calculated as follows:

$$\begin{array}{l}{\epsilon }_1={\displaystyle \frac{1}{1+E{\lambda }_1}}=4.0841e-8, {\epsilon }_2={\displaystyle \frac{1-E(1-{\lambda }_1)}{1+E{\lambda }_1}}=4.0841e-8,\\ {\epsilon }_3={\displaystyle \frac{s_1}{1+E{\lambda }_1}}=0.1462, {\epsilon }_4={\displaystyle \frac{s_2-s_1}{1+E{\lambda }_1}}=0.7536.\end{array}$$

(31)

Based on the obtained parameter values, the time response expression is as follows:

$$\begin{array}{l}{\widehat{y}}_1^{(t_1)}(g)={\displaystyle \sum _{d=1}^{g-1}\left[4.0841e-8{\displaystyle \sum _{m=2}^{n}4.0841e-8^{d-1}{b}_m{l}_m^{(t_m)}(g-d+1)}\right]}+4.0841e-8^{g-1}{\widehat{y}}_1^{(t_1)}(1)\\ +{\displaystyle \sum _{i=0}^{g-2}4.0841e-8^i[0.1462(g-i)+0.7536]}.\end{array}$$

(32)

The final recovery expressions are as follows:

$${\widehat{y}}_1^{(0)}(g)={\displaystyle \sum _{m=1}^g\frac{\Gamma (0.9979+g-m)}{\Gamma (g-m+1)\Gamma (0.9979)}}{\widehat{y}}_1^{(-0.9979)}(m),$$

(33)

when $g=2,3,\dots ,7$, ${\widehat{y}}_1^{(0)}(g)$ is called the simulated value. When $g=8, 9, 10,$ ${\widehat{y}}_1^{(0)}(g)$ is called the predicted value.

6.2.3. Error Solving and Performance Evaluation

By calculating and comparing various error indicators of the model, the quality of the model can be judged. Among them, the comprehensive average relative percentage error can better illustrate the performance of the model. The accuracy level of the model is depicted in

Table 13. Accuracy evaluation.

Level	I	II	III	IV
Error	0.01	0.05	0.10	0.20

. The performance evaluation indicators are shown in

Table 14. Performance evaluation metrics.

Index	Meaning	Data/Calculation Method
$y_1^{(0)}(g)$	Raw data (RD)	Statistical data
${\widehat{y}}_1^{(0)}(g)$	Simulated or predicted data (SPD)	The final recovery expression of the model
$\epsilon (g)$	Residual	$\epsilon (g)={\widehat{y}}_1^{(0)}(g)-y_1^{(0)}(g)$
${\Delta }_s(g)$	Relative squared $\mathrm{percentage} \mathrm{error} \mathrm{of} y_1^{(0)}(g)$ (RSPE)	${\Delta }_s(g)=[|\epsilon (g)/y_1^{(0)}(g)|]\times 100\%$
${\overline{\Delta }}_s$	Mean relative simulated percentage error (MRSPE)	${\overline{\Delta }}_S=\left[1/(N-1)\right]{\displaystyle \sum _{g=2}^N{\Delta }_s(g)}$
${\Delta }_p(g)$	$\mathrm{Relative} \mathrm{prediction} \mathrm{percentage} \mathrm{error} \mathrm{of} x_1^{(0)}(g)$ (RPPE)	$\mathrm{Similar} \mathrm{to} {\Delta }_s(g)$
${\overline{\Delta }}_p$	Mean relative prediction percentage error (MRPPE)	${\overline{\Delta }}_p=(1/f){\displaystyle \sum _{g=N+1}^{g=N+F}{\Delta }_p(g)}$
$\overline{\Delta }$	Comprehensive mean relative percentage error (CMRPE)	$\overline{\Delta }=\left[(N-1){\overline{\Delta }}_s+f{\overline{\Delta }}_p\right]/(N-1+f)$

.

This study employed both the commonly used NSGM(1, N) model [50] and OBGM(1, N) model [51] to predict SO₂ in China and compared them with the optimized PGM(1, N) model. The models were analyzed by comparing various indicators. The model parameters for NSGM(1, N) and OBGM(1, N) are given in

Table 15. Model parameter setting.

Model	Parameter	Value
NSGM(1, N)	$a$: Development coefficient	$a=0.736$
	$b_i$: Driving coefficient	$b_{5,6,7}=-0.050,-3.400,-234.10$
	$h_i$: Gray action	$b_{1,2}=-3013.64,-1610.50$
OBGM(1, N)	$\alpha$: Resolution	$\alpha =0.5$
	$\beta$: Gray relational degree threshold	$\beta =0.7$
	$\gamma$: Background value coefficient	$\gamma =0.5$
SVR	Kernel function	Radial basis function (RBF)
	$\gamma$: RBF kernel coefficient	$\gamma =1$
	$c$: Penalty factor	$c=1$
	$\epsilon$: Insensitive loss band width	$\epsilon =0.1$
LSTM	$N$: Hidden unit	$N=64$
	$L$: Number of layers	$L=1$
	$P$: Batch size	$P=64$
	$r$: Learning rate	$r=1e-4$

.

When conducting model predictions and facing missing data, we commonly used the mean and median for imputation. Simultaneously, we conducted multiple trials to minimize the occurrence of missing data caused by measurement errors, transmission issues, or input mistakes. When encountering abnormal data in model predictions, we first performed a numerical test. Here, we employed the standard score method to calculate the standardized score of the numerical value. We compared its absolute value with a preset threshold to determine whether it was abnormal. Subsequently, we replaced or deleted the corrected data to ensure that the model training was not disrupted and to enhance the prediction reliability.

In

Table 16. Simulated values and errors of three models.

$\mathit{k}$	${\mathit{x}}_1^{(0)}(\mathit{k})$	NSGM(1, N)			OBGM(1, N)			PGM(1, N)
		${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{s}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{s}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{s}}(\mathit{k})$
2	0.9650	0.9420	−0.0229	2.3779	0.9650	0.0000	0.0000	0.9645	−0.0004	0.0494
3	0.9322	0.9374	0.0052	0.5583	0.7250	−0.2070	22.2060	0.9327	0.0005	0.0549
4	0.8778	0.8826	0.0048	0.5506	0.6760	−0.2020	23.0120	0.8788	0.0010	0.1156
5	0.4036	0.4096	0.0060	1.5017	0.2010	−0.2030	50.2970	0.4040	0.0004	0.1013
6	0.2884	0.2929	0.0045	1.5746	0.0860	−0.2020	70.0420	0.2887	0.0003	0.1208
7	0.2437	0.2478	0.0041	1.6903	0.0420	−0.2020	82.8890	0.2435	−0.0001	0.0686
Mean relative simulated percentage error (${\overline{\Delta }}_s$)				1.3756%			59.6197%			0.0851%

, the average simulation error of the optimized PGM(1, N) is 0.0851%. According to Table 13, its accuracy level is level 1, which by far meets the accuracy requirements. Therefore, it can be used for predicting China’s SO₂ emissions.

Table 17. Predicted values and errors of three models.

$\mathit{k}$	${\mathit{x}}_1^{(0)}(\mathit{k})$	NSGM(1, N)			OBGM(1, N)			PGM(1, N)
		${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{p}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{p}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{p}}(\mathit{k})$
8	0.2159	0.2153	−0.0005	0.2484	0.0130	−0.2030	94.0250	0.2164	0.0005	0.2471
9	0.1502	0.1452	−0.0049	3.3017	−0.0520	−0.2020	134.4870	0.1505	0.0003	0.2471
Mean relative prediction percentage error (${\overline{\Delta }}_p$)				1.7751%			119.2394%			0.2471%
Comprehensive mean relative percentage error ($\overline{\Delta }$)				1.3115%			79.4930%			0.1117%

provides the prediction results, and

Table 18. Actual predicted values of SO₂ emissions in 2021 using three models.

$\mathit{k}$		${\mathit{x}}_1^{(0)}(\mathit{k})$	NSGM(1, N)	OBGM(1, N)	PGM(1, N)
10		0.1297
Actual predicted value			0.129071	0.133271	0.129720

provides the actual predicted values of SO₂ emissions in 2021. According to Table 17, it is found that the comprehensive mean relative error of the optimized PGM(1, N) is 0.1117%, with an accuracy level of level 4. The comprehensive mean relative error of NSGM(1, N) is 1.3115%, with an accuracy level of level 2, and that of OBGM(1, N) is 79.4930%, with an accuracy level of IV. In summary, the PGM(1, N) is superior.

According to Table 16, Table 17 and Table 18, a line chart, an average relative simulation/prediction accuracy bar chart, and a comparison chart of the simulation/prediction error range were drawn for fitting China’s SO₂ emissions using three models, as shown in Figure 7 and Figure 8.

• Analyzing the simulation process in Figure 7, the simulated values of PGM(1, N) are closest to the original values. NSGM(1, N) illustrates a trend where the simulated values are extremely close to the original values, but there may still be differences locally. The simulated values of OBGM(1, N) have a similar trend but a significantly different trend from the original values. And PGM(1, N) shows the predicted value closest to the original value.
• Figure 8 shows that the three error indicators of the PGM(1, N) model are the smallest and have the highest accuracy level at level 1.

To further highlight the superiority of the prediction accuracy exhibited by the CSLDDBO-optimized PGM(1, N) model parameters, we compared it with Support Vector Regression (SVR) and the Long Short-Term Memory (LSTM) network. The model parameters are presented in Table 15. The specific experimental results are provided in

Table 19. Simulated values and errors of the two models.

$\mathit{k}$	${\mathit{x}}_1^{(0)}(\mathit{k})$	SVR			LSTM
		${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{s}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{s}}(\mathit{k})$
2	0.9650	0.9640	−0.0010	0.1000	0.9631	−0.0019	0.1942
3	0.9322	0.9350	0.0028	0.3040	0.9260	−0.0062	0.6200
4	0.8778	0.8711	−0.0067	0.7298	0.8692	−0.0086	0.9821
5	0.4036	0.4112	0.0076	1.4820	0.4200	0.0164	4.1010
6	0.2884	0.2766	−0.0118	4.0048	0.2983	0.0099	3.3247
7	0.2437	0.2317	−0.0120	4.0908	0.2509	0.0072	2.8331
Average relative prediction percentage error (${\overline{\Delta }}_s$)				1.7852%			2.0091%

,

Table 20. Predicted values and errors of the two models.

$\mathit{k}$	${\mathit{x}}_1^{(0)}(\mathit{k})$	SVR			LSTM
		${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{p}}(\mathit{k})$	${\widehat{\mathit{x}}}_1^{(0)}(\mathit{k})$	$\mathbf{\epsilon }(\mathit{k})$	${\mathbf{\Delta }}_{\mathit{p}}(\mathit{k})$
8	0.2159	0.2142	−0.0017	0.7474	0.2213	0.0054	2.3075
9	0.1502	0.1578	0.0076	5.0010	0.1550	0.0048	3.1020
Average relative prediction percentage error (${\overline{\Delta }}_p$)				2.8742%			1.5510%
Comprehensive average relative percentage error ($\overline{\Delta }$)				2.0574%			1.8945%

and

Table 21. Actual predicted values of SO₂ emissions in 2021 by two models.

$\mathit{k}$	${\mathit{x}}_1^{(0)}(\mathit{k})$	SVR	LSTM
10	0.1297
Actual predicted value		0.128715	0.129983

. In summary, through a comparison with the above prediction models, the prediction accuracy of PGM(1, N) is evident. Therefore, we can conclude that PGM(1, N) indeed possesses certain advantages.

As shown in

Table 22. Comparison of simulation/prediction error range.

Index	Error	PGM(1, N)	NSGM(1, N)	OBGM(1, N)	SVR	LSTM
MRSPE	Maximum error	0.1208%	2.3779%	82.8890%	4.0908%	4.1010%
	Minimum error	0.0494%	0.5506%	0.0000%	0.100%	0.1942%
	Error range	0.0714%	1.8273%	82.8890%	3.9908%	3.9068%
MRPPE	Maximum error	0.2471%	3.3017%	134.4870%	5.0010%	3.1020%
	Minimum error	0.2471%	0.2484%	94.0250%	0.7474%	2.3075%
	Error range	0.0000%	3.0533%	40.4620%	4.2536%	0.7945%

, the PGM(1, N) model with optimized parameters has the smallest range of simulation and prediction errors. Therefore, it can be concluded that the simulation and prediction performance of PGM(1, N) is higher than that of OBGM(1, N), NSGM(1, N), SVR, and LSTM.

6.3. Prediction of Future SO₂ Emissions

The purpose of this study was to predict future SO₂ emission data from 2022 to 2026. Prior to this, the TDGM(1,1) model was used to forecast the initial values of variables X2, X3, X4, and X5 for the years 2022 to 2026. The required data are presented in

Table 23. Initial value prediction of the independent variable (Xi) (i = 2,3,4,5) over the next five years.

$k$	$X_2$	$X_3$	$X_4$	$X_5$
11	0.8597	0.5958	0.0481	0.8692
12	0.8476	0.5745	0.0179	0.8511
13	0.8385	0.5445	0.0083	0.8493
14	0.8492	0.5106	0.0034	0.8337
15	0.8269	0.4769	0.0010	0.8299

.

In

Table 24. Prediction of SO₂ emissions in China from 2022 to 2026.

Time	2022	2023	2024	2025	2026
Predictive value	238.0632	185.5368	85.1436	15.8850	10.4720

, the data values for predicting SO₂ emissions over the next 5 years using the PGM(1, N) model are listed.

Next, we discuss the characteristics of the target variables. We verified the monotonicity of the SO₂ emission series. From 2012 to 2015, the emission volume declined steadily, with an average annual decrease of about 3.0%. In 2016, the emission volume experienced a steep decline, with a decrease of about 54%, which was due to the significant reduction in SO₂ emissions brought about by the comprehensive implementation of ultra-low emission renovations in coal-fired power plants. From 2017 to 2026, the emission volume decreased steadily, with an average annual decrease of about 6.5%, reflecting the continuous deepening of emission reduction policies. Therefore, the series exhibits strict monotonic decreasing characteristics. Regarding the growth pattern of the Accumulated Growth Order (AGO) series, after performing first-order accumulation generation (AGO) on the original series, it can be observed that the increments of the AGO series have been shrinking year by year (for example, the increment in 2016 was only 854.89, significantly lower than the average value from 2012 to 2015). The linear regression fit after logarithmic transformation has a low goodness of fit (R² < 0.85), indicating a deviation from the exponential growth trend. In the piecewise linear trend analysis, the cumulative amount from 2012 to 2015 increased approximately linearly (with a slope of about 200), while the cumulative amount from 2016 to 2026 showed a slowdown in growth (with the slope decreasing to 60~100), which is consistent with the convergence characteristics after the strengthening of emission reduction policies. Regarding the stable relationship between the independent variable and the dependent variable, the unit root test statistic of the original series is <−4.0 (p < 0.01), rejecting the null hypothesis of the existence of a unit root, thus indicating that the series is stationary. The regression residuals between the emission volume and year passed the unit root test for stationarity (p < 0.05), indicating a long-term stable cointegration relationship between the two.

As a major country in terms of SO₂ emissions, excessive emissions in China can lead to a series of environmental pollution issues and even affect socio-economic development. This article proposes corresponding suggestions from the perspectives of the desulfurization industry, coal-fired industry, and social policies:

• Increase the development of the desulfurization industry and improve innovation in desulfurization technology. Encourage the invention of SO₂ desulfurization technology, combine traditional flue gas desulfurization with emerging desulfurization technologies, improve the desulfurization efficiency, and reduce desulfurization energy consumption. For example, using catalytic reduction technology to convert waste into valuable solid sulfur or elemental sulfur can reduce SO₂ pollution due to its sustainability, space requirements, and low water consumption.
• Strengthen monitoring of the coal-fired industry and control coal use. Set restrictions on the mining of industrial coal in various regions, upgrade the equipment and technology of coal-fired plants, encourage the research and development of coal gasification gas combined with other new energy power generation technologies, and allow the emissions of SO₂ and sulfur in industrial emissions only after reaching the standard.
• Promote social policy guidance and improve SO₂ control policies. Adjust the current electricity price mechanism, improve various social systems, increase the treatment and emission reduction fees for SO₂, and allocate the pollution discharge fees to the environmental treatment of SO₂.

7. Conclusions

This study used PGM(1, N) to predict the future SO₂ in China and proposes an improved CSLDDBO algorithm. With the great performance of the CSLDDBO algorithm in testing, it optimizes the order and smooth generation coefficient of the smooth generation operator in the model.

In terms of algorithms, three strategies and the idea of integrating differential evolution are introduced to improve DBO. Firstly, a chain foraging strategy is introduced in the ball-rolling phase to enhance the search capability of the algorithm and avoid falling into local optima. Secondly, for larval dung beetles, a tumbling foraging strategy is adopted, which enables individual larvae to conduct an adaptive search within a constantly changing search range. Thirdly, for thief dung beetles, inspired by the comprehensive learning strategy, this paper adopts a learning strategy to improve the global search capability. Finally, as the diversity of the population decreases with the progression of iterations in the original DBO, leading to falling into local optima, the differential evolution algorithm is used to enhance the ability to escape from local optima.

To consider the impact of SO₂ emissions and data availability, four types of data were selected as independent variables, and their feasibility was demonstrated through correlation analysis. Subsequently, parameter estimation and model construction were carried out. The PGM(1, N) model, with parameters optimized by the CSLDDBO algorithm, was used and compared with four other common models. The superiority of PGM(1, N) was highlighted through the accuracy levels of the CMRPE and MRPPE. The prediction results indicate that China’s SO₂ emissions will show a downward trend from 2022 to 2026.

In future research, there is still room for further improvement in the CSLDDBO algorithm, and further in-depth research is needed to find an efficient and time-saving method for reducing the time complexity. For PGM(1, N), in the future, this model can be further improved by attempting to replace new differential equations or adjusting the number of parameters, and it can then be applied to other energy predictions.