A carbon price ensemble prediction model based on secondary decomposition strategies and bidirectional long short‐term memory neural network by an improved particle swarm optimization

To further enhance the precision of carbon trading price forecasting, this research proposes a combined forecasting model, CEEMDAN–VMD–IPSO–BiLSTM, considering the unsatisfactory high‐frequency sequence decomposition and the reliance on unidirectional neural networks in current carbon price‐prediction models. First of all, the original sequence of carbon prices is decomposed into multiple independent subsequences through the completely ensemble empirical mode decomposition with adaptive noise (CEEMDAN) technique. The sample entropy values of each subsequence are calculated to reconstruct them as high‐frequency, low‐frequency, and trend sequences. Second, we employ the variational mode decomposition (VMD) approach to decompose the high‐frequency series. The obtained subsequences, along with the low‐frequency and trend sequences, are separately input into an improved particle swarm optimization (IPSO) optimized bidirectional long short‐term memory neural network (BiLSTM) model for forecasting. Finally, an IPSO–BiLSTM model is used to integrate the forecasting outcomes from the previous step, yielding the ultimate results for predicting carbon prices. The case studies reveal that compared with the benchmark model, this model exhibits superior predictive precision and universality. It offers theoretical support for optimizing carbon market operations and fostering low‐carbon economic growth, holding practical importance.


| INTRODUCTION
With the rapid development of the global economy, environmental and energy issues have become increasingly prominent, with greenhouse gas emissions posing a serious threat to sustainable human development.Historical experience has shown that relying solely on mandatory emission reduction requirements or voluntary emissions reduction by economic entities makes it difficult to achieve emission reduction goals.The emissions trading market for carbon dioxide stands as an efficient avenue to achieve economic emission reduction within a market-driven framework.Since 2011, China has gradually started to develop pilot carbon emission trading systems in multiple regions.In July 2021, the national carbon emission trading market also officially commenced online trading activities.During the 75th United Nations General Assembly, China pledged to reach the peak of its carbon emissions by 2030 and attain carbon neutrality by 2060.As the world's largest emitter of carbon dioxide, China confronts a multitude of challenges in achieving this goal.Therefore, research and forecasting of China's carbon market is particularly important.
As a core element of the carbon market, precisely forecasting the prices levels in the carbon trading market, commonly referred to as "carbon prices," is crucial for policymakers, businesses, and investors.This not only helps investors identify the risks and returns associated with the carbon market, optimize their investment portfolios, and develop targeted investment strategies to maximize the value of carbon assets but also enables businesses to formulate effective risk management strategies and optimize operational decisions, thereby reducing costs and enhancing competitiveness.Accurate carbon price information also plays a significant role in evaluating the effectiveness of existing carbon pricing policies and assists policymakers in adjusting or developing more scientifically sound carbon emission control policies.Furthermore, by enhancing market transparency, accurate predictions of carbon prices contribute to stabilizing market expectations, reducing excessive volatility, and facilitating the smooth operation of the carbon market.Therefore, accurate prediction of carbon prices is crucial for understanding market trends and achieving environmental sustainability.Currently, enhancing the precision of carbon price forecasts and grasping its changing trends has become one of the important issues of concern in both academic research and the industry.
The complexity of carbon trading systems imparts characteristics of nonlinearity and high noise to carbon price time series, thus increasing the difficulty of forecasting carbon prices.In facing this issue, data preprocessing has become a pivotal solution.Consequently, the decomposition-ensemble combined forecasting method, which merges decomposition algorithms beneficial for reducing noise in original data with prediction models, has emerged as a key area of interest in carbon price-prediction research.This method employs decomposition techniques to transform the original series into several subseries of lower complexity, reducing noise while increasing the sample size of the data.In current studies, this method typically involves applying a decomposition algorithm for the primary decomposition of the original series, followed by subsequent forecasting and result integration tasks, which can enhance the precision of carbon price predictions to a certain degree.However, related research indicates that the complexity of high-frequency sequences such as the first intrinsic mode function (IMF1), obtained from the primary decomposition, is significantly greater than that of other subsequences derived from the decomposition, and the decomposition effect is poor, which may adversely affect the overall prediction accuracy. 1,2Liu et al. 3 found that IMF1 contains the primary random components of the data, and the irregularity of the original data set is proportional to the irregularity of IMF1.Experiments by Guo et al. 4 pointed out that excluding IMF1 could slightly enhance forecast accuracy.Therefore, it is essential to research high-frequency sequence decomposition and develop a more effective forecasting model to address the problem of poor decomposition effect.This would further boost the precision of carbon price forecasts, thereby providing a superior support tool for policy formulation and market decision-making in the carbon trading market.
The theoretical basis for predicting future carbon prices based on historical data rests on the autocorrelation present in the carbon price time series.This autocorrelation implies that the value at any given point in time within the same time series is statistically correlated with its historical prices.This indicates that fluctuations in carbon prices are not random; instead, historical carbon price data contain key information that can signal future price changes. 5For instance, if historical carbon price data show an upward trend or cyclical patterns, it may suggest similar price behavior in the future.By identifying and analyzing these significant historical patterns, predictive models are able to capture potential trends and cycles. 6Additionally, market participants typically base their actions on observations and expectations of historical carbon price trends, and such trend-based decision-making patterns may further reinforce existing price trends.Thus, a thorough analysis of historical data can effectively predict future fluctuations in carbon prices. 7Moreover, the carbon market is highly sensitive to changes in policies and regulations.The volatility of carbon prices to some extent reflects the market's expectations and adaptations to policy changes.And these policies usually have clear objectives and implementation paths.By analyzing historical carbon price data, one can identify and extract potential impact patterns of policy changes on prices, thereby providing crucial information for forecasting future carbon prices.
As artificial intelligence (AI) algorithms continue to advance and find widespread application in the field of carbon price forecasting, models for predicting carbon prices are increasingly able to utilize these sophisticated algorithms to extract more valuable information, thus achieving higher prediction accuracy. 8,9This development has introduced many new methods and ideas to carbon price forecasting, enabling the construction of highly accurate predictive models even with the use of historical price data alone. 10,113][14][15][16][17] Consequently, research on carbon price forecasting based on historical price data holds significant scholarly value.
In practical carbon price forecasting research, it is necessary to select an appropriate research path based on forecasting needs and research objectives.The objective of this research is to explore how to effectively eliminate the noise impact of high-frequency sequences in carbon price series, extract as many effective features as possible from historical data, and establish a carbon pricing forecast model with elevated predictive precision, based on a primary decomposition-ensemble method.However, decomposition techniques like empirical mode decomposition (EMD) are more suitable for the decomposition of single price series. 18Furthermore, incorporating other factors that impact carbon prices can enrich input of the model but may increase the complexity of the data in practical applications, resulting in additional noise and the risk of overfitting.Therefore, more complex feature extraction steps are needed to process the data, which will increase the model's running time and complexity.The combination of decompositionensemble methods and AI algorithms for predictive forecasting has been proven to effectively capture key features from historical carbon price data, achieving higher prediction accuracy.Hence, this study chooses to focus on an in-depth analysis of historical carbon price data, striving to find the optimal equilibrium between model intricacy and prediction precision while fulfilling the research purpose.
The subsequent sections of this study are structured as follows: Section 2 mainly introduces previous literature.Section 3 primarily elucidates the foundational principles underpinning the employed methodologies and the establishment process of the proposed combination forecasting model.Sections 4 and 5 primarily presents the data selection, feature analysis, evaluation indicators, empirical research process, and result analysis.Section 6 mainly expounds upon the central discoveries of this paper and outlines avenues for future exploration.

| LITERATURE REVIEW
To improve the prediction accuracy, academics have conducted extensive research on carbon priceprediction models, and many data-driven models have been proposed and applied to carbon price-prediction research.According to the differences of prediction models, the existing research on carbon price prediction can be divided into three directions: traditional statistical prediction methods, AI methods and combined prediction models.Moreover, due to the significant position of the Chinese carbon market in global carbon emissions, accurately predicting its carbon prices is particularly essential.

| Traditional statistical methods
Research on carbon price prediction using traditional statistical methods mainly involves constructing forecast models with various techniques, such as multivariate regression, autoregressive integrated moving average (AR-IMA), generalized autoregressive conditional heteroskedasticity (GARCH), dynamic model averaging (DMA), and Markov models, based on historical price data to predict carbon prices in different regions.Byun and Cho 19 have modeled European Union (EU) carbon futures prices, demonstrating the applicability of the GARCH model for carbon price forecasting.Çanakoğlu et al. 20 have enriched carbon price forecasting methods by establishing econometric time series, regime-switching models, and multivariate vector autoregression models based on historical carbon price data.Benz and Trück 21 effectively modeled and predicted short-term spot carbon prices in the European Union Emissions Trading System (EU ETS) by combining Markov models with AR-GARCH models.Facing the complexity of the carbon market, Koop and Tole 22 effectively forecasted the EU carbon market with the DMA method, accurately capturing carbon price dynamics and achieving precise forecast outcomes.
Despite the computational simplicity and speed of traditional statistical methods, carbon price data often exhibit strong volatility and significant nonstationary and nonlinear characteristics.Traditional statistical methods, mostly based on linear assumptions, may not adequately capture the nonlinear relationships in carbon prices, potentially leading to biased forecasting outcomes.Furthermore, traditional statistical methods typically require data stationarity, necessitating the transformation of carbon price series into stationary series through differencing and other methods, which may result in the loss of valuable information.Consequently, in recent years, single statistical methods have been less commonly used for prediction in related research.

| AI methods
To surmount the limitations of traditional statistical methods, AI algorithms have gradually developed, providing new directions for carbon price forecasting.They deduce complex nonlinear mapping relationships related to the series by statistical analysis of historical data, demonstrating a strong capability to seize nonlinear trends concealed within carbon market prices, and thus are widely applied in carbon price forecasting. 23Zhu and Wei 24 employed the least squares support vector machine to forecast four different carbon trading prices, thereby improving the model's robustness.Li et al. 25 chose the long short-term memory (LSTM) method to build a multivariate LSTM prediction model for forecasting China's carbon prices, to improve the capability to capture and extract data features, and the results show that this model is well-suited for carbon price forecasting.AI methods demonstrate greater robustness and generalization capabilities when compared with statistical approaches. 26However, given that carbon prices, as financial data, are highly responsive to market information changes and characterized by strong randomness and complexity, and considering the drawbacks and limitations of each intelligent method, using a single AI model may not effectively capture the complex patterns of data changes, and hence may not achieve the desired prediction effect.

| Combined forecasting models
In the realm of carbon price forecasting, combined models have become a key tool for improving forecasting performance, especially in capturing nonlinear features, due to their ability to integrate the advantages of various single models while avoiding their shortcomings.Given that carbon price data typically presents as complex, noisy time series, researchers have developed combined forecasting models based on decomposition-ensemble methods.These methods generally include a fusion of data decomposition techniques, intelligent optimization algorithms, and AI forecasting methods.On the one hand, decomposition methods like EMD can effectively reduce data noise and isolate crucial characteristics, making it easier for the forecasting model to learn. 27On the other hand, bioinspired optimization methods like particle swarm optimization (PSO) mimic natural evolutionary mechanisms, automatically adjusting and optimizing model parameters to more effectively manage fluctuations and uncertainties in carbon price data, thus improving the model's forecasting accuracy and stability.As a result, decomposition-ensemble based combined models are highly capable in carbon price forecasting, delivering precise and stable predictions. 28ianwei et al. 29 constructed a new combined forecasting method, which first decomposes the carbon price series into multiple IMFs using extreme symmetric mode decomposition and then integrates predictions using a least squares support vector regression model optimized synchronously with the PSO algorithm.The study indicates that in comparison to other benchmark models, this model exhibits superior predictive precision and demonstrates robustness and effectiveness in carbon price forecasting.Zhu et al. 30 employed EMD and hierarchical clustering methods to decompose and reconstruct the EU ETS carbon futures prices.While simplifying the complex series, this approach revealed the short, medium, and long-term fluctuations and trends in the carbon market.Chai et al. 31 suggested a hybrid model that employs an extreme learning machine (ELM) optimized by PSO to forecast various components obtained through decomposition.The study found that the PSO algorithm effectively reduces the prediction error of the model.In an example examining the Hubei carbon market in China, compared with models not utilizing the PSO algorithm, this model showed significant improvements in its predictive accuracy, and the root mean square error (RMSE) and other evaluation indicators have an optimization rate of more than 50%.][34][35][36][37] In recent research, Zhou et al. 38 created a carbon price-prediction approach that combines the completely ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method and LSTM.The research outcomes reveal that when contrasted with conventional decomposition approaches like EMD, the CEEMDAN method effectively reduces mode mixing by introducing adaptive noise, thereby more accurately capturing the ZOU and ZHANG | 2571 complex volatility of carbon price data and exhibiting superior performance in decomposing carbon price time series.However, the study found that CEEMDAN's performance in processing high-frequency components like IMF1 was suboptimal, with the decomposition results showing strong volatility and irregularity, potentially adversely affecting overall forecasting accuracy.Through a series of improvements, they discovered that using the variational mode decomposition (VMD) method for a secondary decomposition of highfrequency sequences, such as IMF1, characterized by strong volatility, could reduce prediction errors.Similarly, Zhou and Wang 1 also proposed a combined forecasting model that uses VMD for a secondary decomposition of the most volatile IMF1 after the original sequence is decomposed with CEEMDAN.Compared with models without secondary decomposition, this model improved RMSE by 55.2%, greatly enhancing the model's predictive accuracy.It demonstrates that the secondary decomposition approach founded on CEEMDAN-VMD can effectively enhance the accuracy of the decomposition-ensemble prediction model.

| Research on price prediction of carbon market in China
As the largest CO 2 emitter globally, China not only possesses a vast potential carbon market but also plays a pivotal role in the global efforts to reduce carbon emissions. 39The volatility of its carbon market prices significantly impacts both domestic and international carbon reduction strategies and promotes the transition of the global economy towards a low-carbon model. 40Therefore, employing scientific methods to accurately forecast carbon prices in China is crucial for formulating effective global carbon reduction strategies and advancing the economy's green transformation.However, compared with the more established international carbon markets like the EU, which was initiated in 2005, China's carbon market started later and is still in a phase of development and improvement, with comparatively limited research on carbon price forecasting available. 32There is a need for more research on carbon price forecasting in China's carbon market to provide a scientific basis for optimizing its market mechanism, evaluating policy effects, and facilitating effective integration with international markets, thereby ensuring its maximum contribution to global carbon reduction efforts.Currently, research on carbon price forecasting in China's carbon market primarily treats carbon price prediction as a time series forecasting issue, widely adopting composite prediction models based on AI algorithms to model historical price data.Sun and Huang 2 developed a novel hybrid model for carbon price forecasting based on a quadratic decomposition-integration approach combined with a genetic algorithm-optimized backpropagation (BP) neural network model, using historical prices as input for forecasting studies on China's carbon pilot.The results show that the proposed forecasting model demonstrated effectiveness and robustness in several carbon pilots, including Hubei.Similarly, Zhou et al. 41 established a carbon price composite prediction model based on a decomposition-integration forecasting method, which, after being optimized with a Gray Wolf optimization algorithm enhanced ELM, predicted carbon price data for four pilot carbon markets, including Hubei, exhibiting excellent predictive performance in experiments.
Furthermore, numerous scholars, building on their research of China's carbon market, have expanded their scope to include the globally leading EU carbon market.This expansion aims to validate the generalization capabilities of carbon price-prediction models across different market environments through comparative cross-market studies, that is, the models' capacity to accurately predict new, unseen data.Specifically, the carbon markets in China and the EU showcase the differences between developing powerhouses and developed economies in terms of stages of economic development, industrial structures, and patterns of energy consumption.These differences enable scholars to evaluate more scientifically and rigorously the predictive performance of models when processing data with varying characteristics.For example, Niu et al. 42 designed a hybrid forecasting system that incorporates error correction and divide-and-conquer strategies to enhance the accuracy of carbon price predictions.The effectiveness and reliability of the proposed forecasting system in carbon price prediction were demonstrated using data from both the Chinese and EU carbon markets.Similarly, Hao et al. 43 developed a composite model based on feature selection and multiobjective optimization algorithms, which, in simulation experiments using daily price data sets from the Chinese and EU carbon markets, showed superior predictive performance compared with other models.Such cross-market studies not only deepen the understanding of the models' generalization capabilities but also promote the application and development of forecasting techniques in the global carbon market.

| Comment on previous literature
Drawing from the aforementioned research, it can be concluded that: First, the decomposition-integration forecasting method exhibits excellent predictive capabilities within the domain of carbon price prediction.However, the primary decomposition is less effective for highly volatile IMFs.Studies have shown that a secondary decomposition strategy rooted in the CEEMDAN-VMD method can effectively address this issue, achieving better decomposition results and enhancing the precision of carbon price forecasts.Second, the field of carbon price forecasting currently relies on unidirectional neural networks like LSTM for forecasting.However, in forecasting complex time series, such as carbon prices, data from both previous and subsequent time points can affect the current moment.Additionally, carbon price data typically exhibit long-term trends and seasonal fluctuations, and LSTM may face problems of gradient vanishing or exploding when processing long sequences.5][46] Compared with LSTM, BiLSTM can increase the accuracy of time series prediction by an average of 37.78%, 47 and demonstrates good predictive effectiveness in forecasting carbon prices. 48Furthermore, the PSO algorithm can enhance the precision and performance of models for predicting carbon prices.However, it is prone to getting trapped in local optima.Lastly, China's carbon market plays a crucial role in advancing global climate governance and promoting the development of international carbon markets.However, due to its later start compared with more mature carbon markets such as the EU's, China's carbon market is still in a phase of continuous improvement and development, leaving ample room for research.Furthermore, joint research of China's carbon market and the experienced EU carbon market can scientifically and rigorously validate the generalization capabilities of prediction models.
Therefore, to address the aforementioned issues, based on the decomposition-ensemble forecasting framework, this research introduces a carbon price combination forecast model, named CEEMDAN-VMD-IPSO-BiLSTM, which incorporates the CEEMDAN method, VMD method, improved particle swarm optimization algorithm (IPSO), and BiLSTM model.It primarily focuses on the Chinese carbon market, while also examining the EU carbon market to verify the model's generalization capability.The innovations of this research can be briefly outlined as follows: (1) This paper introduces an innovative model for forecasting carbon prices, known as CEEMDAN-VMD -IPSO-BiLSTM.The model combines the second-order decomposition method, an improved PSO algorithm, the BiLSTM model, and the Decomposition-Ensemble forecasting framework, aiming to maximize the strengths of individual models while minimizing the shortcomings of each single model.
(2) An IPSO algorithm is proposed to determine five crucial hyperparameters for BiLSTM.This optimization algorithm possesses strong global search capabilities, enabling the rapid establishment of a highprecision prediction system.(3) The simulation research was conducted based on several carbon markets, including the highly representative three Chinese pilot carbon markets and the EU's Intercontinental Exchange (ICE), to assess the model's predictive performance across different data sets.The outcomes showed that the proposed model exhibited higher predictive accuracy and universality.

| VMD
VMD is a nonrecursive technique for signal processing developed by Dragomiretskiy and Zosso.It is capable of breaking down time-series data into several IMFs that each have a finite bandwidth, and it adaptively updates the optimal central frequency and bandwidth for each IMF.This method exhibits excellent robustness against noise and effectively avoids endpoint effects and mode mixing problems.The essence of the VMD decomposition process lies in constructing a constrained variational model and achieving signal decomposition through iterative seeking the best solution for this model.For detailed steps of the VMD process, please refer to the paper by Dragomiretskiy and Zosso. 50This paper does not elaborate further.

| BiLSTM
The LSTM network is a derivation of the recurrent neural network (RNN), addressing the limitation of RNNs in handling long-term dependencies. 51The core idea is to employ memory cells to retain long-term historical information and utilize specially designed gate mechanisms to select whether to retain or forget information.The unit architecture of LSTM is depicted in Figure 1.
When dealing with time-series data, the unidirectional LSTM method demonstrates excellent predictive performance for nonlinear time series.However, it can only access information from the previous node and cannot utilize the influence of the next node's information on the current node.BiLSTM is an improved model developed on the foundation of LSTM.It consists of two LSTM layers oriented in opposite directions, processing data along the time series in both forward and reverse directions.During the training phase, both directional networks can access the entire sequence data, allowing the model to concurrently learn the forward and backward dependencies of each data point within the time series.This bidirectional learning mechanism enables the BiLSTM model to fully utilize the global information of the time series data during the training phase, thereby deeply learning the patterns and dependencies of the data.This comprehensive understanding of data dynamics enhances the BiLSTM's ability to predict future data points.
Although BiLSTM cannot access future data points during the actual prediction phase, it can still utilize the historical data patterns learned during the training process for efficient prediction.Compared with RNN and unidirectional LSTM, BiLSTM demonstrates superior performance and predictive capability when dealing with complex time series data.
The BiLSTM computation encompasses both forward and reverse computations.In this, the horizontal axis delineates the bidirectional progression of the time series, while the vertical axis signifies the unidirectional passage of information from the input to the hidden layer and from the hidden layer to the output layer.The basic architecture of the BiLSTM is depicted in Figure 2. The core computational procedure of BiLSTM can be represented by the formula (3).
Where h t + and h t − represent the outputs of the hidden layer from both the forward and backward LSTM networks at time t, respectively.LSTM + and LSTM − denote the operations of the forward and backward LSTM, while W + and W − are the matrices containing the Unit structure of the long short-term memory network.
weights for the forward and backward LSTM layers.b y is the bias term of the output layer.

| PSO
The PSO algorithm was introduced by R. Eberhart and J. Kennedy in 1995 and subsequently enhanced with the incorporation of inertia weight by Y. Shi and R. Eberhart in 1998.It is an intelligent optimization technique inspired by the collective behavior of bird flocks or fish schools as they search for food.
When solving optimization problems, the PSO algorithm updates particle velocity and position by tracking both the individual best particle and the best particle across the entire population.This process can be delineated as follows: In a D-dimensional search space, there is a particle swarm consisting of m randomly initialized particles.At the tth iteration, the position and velocity of the ith particle in the jth dimension are denoted as X i j t , and V i j t , , respectively.Each particle continuously updates its position and velocity to explore the entire state space by tracking its individual best solution p best t i and the global best solution g best t found by the entire population.The ultimate goal is to gradually search for the optimal position, that is, the optimal solution.The updates of velocity and position follow formulas (4) and (5).
where ω represents the inertia weight, c 1 and c 2 are the learning factors, r 1 and r 2 are the uniformly distributed random numbers within the range [0, 1].

| IPSO
While PSO has shown significant effectiveness in solving complex optimization problems, it lacks effective parameter control and is prone to get stuck in local optima, resulting in a decline in the PSO algorithm's capacity for comprehensive global exploration and search efficiency.Therefore, this study makes improvements to the inertia weight and learning factors within the PSO method to ensure the efficiency and accuracy in carbon priceprediction optimization.

| Improvement of inertial weights
When initializing the neural network parameters within the PSO algorithm, the choice of a fitting inertia weight value can strike a harmony between particles' global exploration and local exploration prowess, thereby augmenting the efficiency of the PSO technique in optimization.During the algorithm's initial search phase, where the particles have higher fitness values, a linear decrease in the initial inertia weight should be adopted to weaken the influence of these particles during particle updates.This allows the algorithm to have stronger global search capability at this stage and quickly transition into the local search.In the later optimization phase, where particle fitness values are smaller, a nonlinear inertia weight should be used to increase the impact of individual particles during updates, enabling the algorithm to converge quickly.However, in the standard PSO algorithm, the inertia weight remains constant and does not change with the number of iterations, which limits the flexibility and generalization ability of the model.Therefore, this article suggests an adaptive inertia weight that combines linear and nonlinear value strategies and uses the sine function to control the nonlinear variation of the inertia weight, making the value of ω random.The designed inertia weight is shown in formula (6).
where ω max and ω min denote the maximum and minimum inertia weights, respectively, f i represents the fitness value of the current particle, f min is the minimum fitness value among the particles, t stands for the current iteration number, and t max signifies the maximum iteration number of the particle swarm.

| Improvement of learning factors
The learning factors c 1 and c 2 dictate how much weight the individual and other particle experiences carry in shaping the optimization trajectory, reflecting the exchange of information between particles.Suitable learning factors can accelerate the search speed of particles and diminish the possibility of the algorithm getting trapped in local optima.
In practical applications, as the iteration progresses, it is usually necessary for the value of c 1 to gradually decrease to expedite the search speed during initial iterations and improve global search capability.In the later iterations, the value of c 2 needs to gradually increase to facilitate local refinement search while enhancing precision.However, the standard PSO algorithm typically sets c 1 equal to c 2 and a fixed value, which falls short in satisfying the needs of realworld applications.Therefore, this study proposes an improved learning factor method based on the cosine function, as shown in Equations ( 7) and (8).

| Sample entropy (SE)
SE constitutes a technique devised by Richman and Moorman 52 to gauge the intricacy of time series, and it represents an improvement over approximate entropy.In contrast to the latter, the computation of SE remains unaffected by data length and exhibits better robustness and consistency.In practical applications, the definition of sample entropy can be expressed as follows: where N represents the length of the input signal, m represents the dimension, and r represents the tolerance of similarity.

| The construction process of the proposed model
The carbon price predictive model presented in this research refers to the combined forecasting framework based on VMD secondary decomposition proposed by Zhou et al. 38 The modeling process is depicted in Figure 3, with the detailed steps outlined as follows: (1) The original carbon price series is processed using the CEEMDAN method, resulting in its decomposition into multiple IMFs and a residual sequence.

| Data collection
This research takes the carbon trading price data from the Chinese and EU carbon markets, which hold important positions in the global carbon market, as the empirical research objects.Starting from 2011, China has progressively launched trial programs for carbon emissions trading in eight provinces and cities.Due to the different economic structures, government policies, and corporate participation willingness in each pilot market, there are significant differences in carbon trading prices among other markets (Figure 4 shows the trading situation of various carbon pilots in China in 2022).Therefore, this study chooses the daily trading prices of three carbon pilot projects located in the central, southern, and northern regions of China, which have representative and typical regional development characteristics, as the research objects, to comprehensively cover the price characteristics of the Chinese carbon market.The Guangdong Carbon Exchange, which has continuously led the country in terms of trading scale, is selected as the primary research object for the Chinese carbon market.In addition, the Hubei carbon pilot, which has a high market share and a mature carbon market, and the Beijing carbon pilot, which experiences relatively intense carbon price fluctuations, are chosen as supplementary cases for the carbon market in China.
The data sources are from the official websites of each carbon exchange.
In addition, the closing prices of carbon emission allowances continuous futures from the largest European carbon emission futures exchange, the ICE, are used as representative data for the EU carbon market to further validate the model's accuracy and universality.The data source is from the investment platform (https://cn.investing.com/). The

| Data characteristics
Figures 5 and 6 illustrate the trends of the carbon price sequences, showcasing that the Beijing carbon pilot exhibits the highest frequency of fluctuations.Table 2 presents the statistical description of the data sets on carbon prices of each carbon market calculated using Python 3.9.0,displaying the discrete distribution and volatility of their data sequences.From the statistical values, it can be observed that the Skewness values for all carbon market price data exceed 0, indicating a right-skewed distribution with a long right tail; the Kurtosis values are all less than 3, meaning a platykurtic distribution; the Jarque-Bera values are much larger than 0, indicating that the carbon price sequences in each market do not follow a normal distribution.The P-values obtained from the augmented Dickey-Fuller test for Guangdong, Hubei carbon pilot, and the EU ICE are all above 0.05, suggesting the nonstationary nature of the carbon price data sequences in these three carbon markets.Additionally, Figure 7

| Parameter setting
In this study, we configure the VMD method with a decomposition layer of 10, and IPSO has a population size of 20 with 100 iterations and particle velocities in the range of [−5, 5].Through IPSO, five parameters of the BiLSTM neural network, including the number of layers, the number of neurons per layer, the number of samples per training, learning rate, and Dropout rate, are optimized within the ranges of [1, 4], [20, 200], [10, 20], [1 −3 , 1 −2 ], and [0.03, 0.19], respectively.The experiments were conducted on a central processing unit with an R7-5800H processor, 16.0 GB of RAM, running on the Windows 11-64-bit operating system, with TensorFlow = 2.5.0 installed, and using Python version 3.9.0.To avoid errors as much as possible and obtain higher prediction accuracy, the prediction strategy of 1 day in advance is adopted in this study.To alleviate the algorithm's randomness, each model's average of 10 runs was taken as its final prediction result.

| Model evaluation
For an accurate evaluation of each model's predictive performance, this study uses five metrics as evaluation criteria: the coefficient of determination (R 2 ), RMSE, mean absolute error (MAE), mean absolute percentage error (MAPE), and the SIGN method (SIGN).Let y t represent the actual carbon price at time t, y ˆt be the predicted carbon price at time t, y ¯t be the average of actual carbon prices at time t, and n be the number of prediction samples.The calculation formulas for the first four indicators are as follows:  R 2 represents the squared percentage of the correlation between the forecasted and factual values of the variable, with its numerical range generally from 0 to 1.The larger the R 2 value of a model, the better its fit.The smaller the RMSE, MAE, and MAPE values of a model, the smaller the error between its predicted values and actual values, demonstrating a better predictive accuracy of the model.The SIGN method can be used to assess the ability of various models to capture the trend changes in carbon prices.In the SIGN method, the original carbon price series and model prediction results are first classified and labeled: a label of "1" indicates an upward trend in prices, "0" indicates prices remain unchanged, and "−1" indicates a downward trend in prices.Subsequently, the model's predictive performance on trends is evaluated by calculating the directional accuracy between the labels of the original series and the predicted results.Generally speaking, a directional accuracy above 50% is considered an effective prediction, between 60% and 80% indicates the model has a good predictive performance, and accuracy exceeding 80% shows that the model excels in predicting the direction of carbon price trends.Through the comprehensive application of these evaluation metrics, the model's performance in prediction accuracy and trend capturing, among other dimensions, can be thoroughly assessed, thereby ensuring the accuracy and reliability of the evaluation results.

| Prediction process
This paper conducts a detailed analysis of the Guangdong carbon market as its primary research subject.First, the CEEMDAN method is employed to decompose the original time series of Guangdong ETS carbon prices, resulting in seven IMFs and one residual component.It is assumed that the closing prices of each trading day are continuous and uninterrupted.From the decomposition results of CEEMDAN in Figure 5, it can be observed that the frequency of fluctuations decreases gradually from IMF1 to IMF8, and the variability trends become clearer.The SE values are calculated for all IMFs and their complexities are analyzed.Subsequences with similar entropy values are added to each other for reconstruction to enhance computational efficiency and avoid excessive decomposition leading to information loss. 31A higher SE value indicates a greater complexity of the component.As shown in Table 3, under the two-parameter combinations, the SE values of IMF1 and IMF2 are significantly higher than those of other components, while the SE values of IMF6-IMF8 are all below 0.1, indicating relatively low complexities and distinct variability trends.Therefore, IMF1 and IMF2, IMF3-IMF5, and IMF6-IMF8 are reconstructed, respectively, as highfrequency components, low-frequency components, and trend components, as shown in Table 1.These three component sequences can, to some extent, demonstrate the short-term fluctuations of the carbon market, the cyclical nature of the market economy and significant events, as well as the long-term trends of the carbon market. 30ubsequently, to reduce the complexity of the highfrequency components and enhance predictive accuracy, the VMD method is employed to perform a secondary decomposition on the high-frequency components, resulting in 10 new IMFs.The subsequences obtained from this decomposition exhibit greater regularity compared with the original high-frequency components and are more amenable to prediction.The specific reconstruction and secondary decomposition processes are illustrated in Figure 8.
Lastly, three BiLSTM models are constructed for synchronized hyperparameter optimization through IPSO.These models are utilized to predict the 10 components (V-IMFs) obtained from the secondary decomposition of the high-frequency component, as well as the low-frequency and trend sequences.All prediction outcomes are fed into an IPSO-BiLSTM for integration, resulting in the final carbon price-prediction results.This completes the entire process of decomposition-reconstruction-prediction-integration.The fitted prediction results are depicted in Figure 9.
In the two stages of individual prediction and integrated prediction of various frequency components, the application of BiLSTM exhibits significant differences.In the first stage, when dealing with relatively simple decomposed subsequences, the IPSO-optimized BiLSTM model typically has a simpler structure with fewer layers and neurons, focusing on capturing patterns and dynamics within the component.In the second stage, which involves processing complex information from multiple components, BiLSTM demonstrates a more complex structure with additional layers and neurons, aiming to effectively integrate these components to provide a comprehensive composite forecast.This reflects the flexibility and effectiveness of BiLSTM in responding to varying data complexities and forecasting objectives.

| Carbon price-prediction results for the Guangdong market in China
This article selected 17 models for comparative experiments to thoroughly validate the viability and efficacy of the proposed composite forecasting model CEEMDAN-VMD-IPSO-BiLSTM. Table 4 presents the values of various prediction performance evaluation indicators for each model, while Figure 10 depicts the comparison results of fitted predictions and actual data for multiple models in the Guangdong carbon market.Several noteworthy discoveries extracted are as follows: As shown in Figures 9 and 10, the fitted prediction results of the proposed CEEMDAN-VMD-IPSO-BiLSTM model align with the actual data trends and outperform other comparative models, demonstrating a significant superiority in carbon price forecasting for the Guangdong carbon market.According to Table 4, compared with other benchmark models, this model yields the optimal prediction results, with R 2 , RMSE, MAE, MAPE, and SIGN values of 0.9906, 1.3639, 1.0419, 1.3639%, and 78.39%, respectively, proving that this model can serve as an effective tool for carbon price prediction and analysis in the Guangdong carbon market.
In the comparative experiments of six individual models, it can be observed that the predictive performance of BiLSTM stands notably superior to LSTM and other models.Compared with the LSTM model, the R 2 and SIGN of the BiLSTM model increased by 6.8% and 10.7%, while RMSE, MAE, and MAPE decreased by 37.0%, 40.8%, and 39.3%, respectively.This indicates that BiLSTM exhibits excellent predictive performance in the Guangdong carbon market.This superiority arises from BiLSTM's ability to simultaneously consider both forward and backward information in the time series, T A B L E 3 The calculation results of the sample entropy of each intrinsic mode function (IMF) in Guangdong.allowing it to better capture the long-term dependency relationships in carbon price-prediction tasks and thus enhance prediction accuracy.The comparative experimental results between the LSTM model and CEEMDAN-LSTM, as well as between the BiLSTM model and CEEMDAN-BiLSTM, demonstrate that employing the CEEMDAN method for decomposing carbon price sequences can improve the precision of model predictions.Compared with the BiLSTM model, CEEMDAN-BiLSTM exhibits optimization rates of 1.5% for R 2 , 27.2% for RMSE, 39.1% for MAE, 31.9% for MAPE, and 1.6% for SIGN.Similarly, compared with the LSTM model, CEEMDAN-LSTM showcases optimization rates of 1.9% for R 2 , 9.0% for RMSE, 7.6% for MAE, 7.2% for MAPE, and 1.0% for SIGN, all demonstrating significant improvement.
By comparing the predictive results of the CEEMDAN-BiLSTM model with those of the EMD-BiLSTM and EEMD-BiLSTM models, it is demonstrated that, compared with other decomposition methods, CEEMDAN is more suitable for the decomposition of carbon price sequences.Its RMSE, MAE, MAPE, SIGN, and R 2 values are 2.6633, 1.9527, 2.6077%, 70.35%, and 0.9644, respectively.This is because CEEMDAN can overcome mode mixing and endpoint effect issues, yielding more stable decomposed sequences.
When comparing the BiLSTM model with the PSO-BiLSTM model, the latter demonstrates higher predictive accuracy, with optimization rates of 0.4%, 7.7%, 7.9%, 12.6%, and 2.8% for R 2 , RMSE, MAE, MAPE, and SIGN, respectively.This indicates that utilizing PSO for hyperparameter optimization of BiLSTM can enhance its predictive capability.Moreover, the predictive accuracy of the IPSO-BiLSTM model is significantly superior to that of the PSO-BiLSTM model, with optimization rates of 0.9%, 9.8%, 19.8%, 12.6%, and 2.4% for R 2 , RMSE, MAE, MAPE, and SIGN, respectively.This suggests that the proposed improvements to the inertia weight and learning factor of the PSO algorithm in this study can enhance the efficiency and accuracy of parameter optimization.
In the comparative experiments investigating the impact of PSO and IPSO algorithms on the predictive performance of models, such as CEEMDAN-BiLSTM and CEEMDAN-VMD-BiLSTM, it was found that models utilizing PSO or IPSO algorithms achieved higher predictive accuracy.Furthermore, the predictive results of models using the IPSO algorithm consistently outperformed those using the PSO algorithm.For instance, the model proposed in this paper, CEEMDAN-VMD-IPSO-BiLSTM, demonstrated the highest predictive accuracy.Compared with CEEMDAN-VMD-BiLSTM, it showed improvements of 1.2% in R 2 , 33.8% in RMSE, 39.5% in MAE, 42.2% in MAPE, and 7.1% in SIGN.In comparison with CEEMDAN-VMD-PSO-BiLSTM, improvements of 0.8%, 27.9%, 14.1%, 20.1%, and 3.3% were achieved.This confirms that the addition of PSO and IPSO algorithms to the CEEMDAN-BiLSTM, CEEMDAN-VMD-BiLSTM, and other decompositionintegration predictive frameworks can effectively enhance the accuracy of carbon price prediction.At the same time, it further provides evidence that the improved strategy of the PSO proposed in this paper is effective and can enhance the optimization ability of PSO.

| Carbon price-prediction results for other carbon markets in China
To comprehensively evaluate the accuracy and effectiveness of the proposed model, this study also conducted simulations in the carbon markets of Hubei and Beijing, which have different trading volumes, volatility trends, and geographical regions.The structures and parameters of the models used in the comparative experiments are consistent with those mentioned earlier.Tables 5 and 6 showcase the prediction assessment outcomes for each model.
Looking at the predictive evaluation results from the two carbon trading markets, the proposed combined forecasting model CEEMDAN-VMD-IPSO-BiLSTM demonstrates superior predictive performance compared with the other comparative models.For example, in the prediction of carbon prices in Hubei, the proposed model achieves R 2 , RMSE, MAE, MAPE, and SIGN values of 0.9836, 0.9719, 0.7751, 1.7108%, and 78.91%, respectively.In comparison to the individual BiLSTM model, its optimization rates are 20.2%, 48.6%, 46.8%, 46.2%, and 18.5%, respectively.
Specifically, in the Beijing carbon market, the use of CEEMDAN for data decomposition resulted in a decrease in the predictive accuracy of the model.This is due to the high-frequency volatility present in the Beijing carbon market, as illustrated in Figure 5.In comparison to other carbon markets, the difficulty of decomposing high-frequency sequences increases, making it challenging for the CEEMDAN method to achieve optimal decomposition results.However, with the addition of VMD for secondary decomposition, the model's predictive accuracy improves significantly.For instance, compared with CEEMDAN -BiLSTM, the CEEMDAN-VMD-BiLSTM model saw improvements of 25.5% in R 2 and 24.3% in SIGN, while RMSE, MAE, and MAPE decreased by 53.2%, 52.3%, and 54.8%, respectively.This once again demonstrates that using VMD for secondary decomposition can effectively address the issue of poor decomposition performance for high-frequency sequences by CEEMDAN, highlighting the superiority of the secondary decomposition strategy utilized in this research.
In general, the results from validating the efficacy of each submodel in the combination model selected in this paper, aiming to enhance predictive precision and the test of the proposed PSO improvement strategy, are consistent with those obtained in the Guangdong carbon market, which reflects the preeminence and universality of the advanced prediction model in forecasting carbon prices.

| Carbon price forecast results for the EU carbon market
To further validate the forecast precision and efficacy of the composite predictive model introduced in this study in different countries' carbon markets, the carbon prices from the EU's ICE were used as the predictive data set.utilizing the PSO algorithm effectively enhances the predictive capabilities of models.(5) When contrasting the predictive outcomes derived from models employing the IPSO algorithm with those using the PSO algorithm, the former demonstrates higher predictive accuracy.Among them, the CEEMDAN-VMD-IPSO-BiLSTM model put forth in this paper yields the most predictive results.In comparison to the CEEMDAN-VMD-PSO-BiLSTM model, it shows improvements of 3.7%, 46.6%, 22.8%, 20.4%, and 6.0% in terms of R 2 , RMSE, MAE, MAPE, and SIGN, respectively.Additionally, compared with the predictive outcomes of PSO-BiLSTM, the R 2 and SIGN of the IPSO-BiLSTM improved by 3.7% and 1.3%, while RMSE, MAE, and MAPE decreased by 10.6%, 6.8%, and 4.4%, respectively.This confirms that the proposed improving strategy of the inertia weight and learning factor in this study can improve the PSO algorithm's search capacity.

| CONCLUSION AND FUTURE WORK
Greenhouse gas emissions gravely threaten the sustainable development of humanity.As the most cost-effective tool for reducing emissions, the accurate forecasting of carbon trading prices is not only crucial for market stability and participant decisionmaking but also significantly impacts the achievement of long-term environmental goals.To address the limitations of the primary decomposition method and the problem of relying on unidirectional neural networks in the current forecasting of carbon prices methods, this study introduces a composite forecasting model named CEEMDAN-VMD-IPSO-BiLSTM, based on a decomposition-integration forecasting framework.The model, validated using highly representative carbon price data from three pilot markets in China and the EU's emissions trading market, achieved favorable predictive results, effectively enhancing the accuracy and stability of forecasts.The findings of this paper provide substantial theoretical support for the establishment and development of carbon markets and offer empirical guidance for formulating more effective carbon reduction strategies.The principal conclusions drawn from this paper are as follows: (1) In comparison to models such ARIMA, BP, deep neural network, and LSTM, the BiLSTM model has demonstrated significant advantages in single-model forecasting, proving its potential within the domain of carbon price forecasting.This is primarily attributed to BiLSTM's bidirectional network structure and gated mechanism, which enable it to effectively avoid the problem of gradient vanishing and more efficiently capture complex features.This conclusion also suggests that research into bidirectional neural networks is of significant value and offers a new direction for future research on forecasting carbon prices.
(2) The study has significantly enhanced the predictive performance of the BiLSTM model by optimizing its five key hyperparameters using the IPSO algorithm.This improvement is attributed to the proposed adjustments in the PSO algorithm's inertia weight and learning factors, which increased the algorithm's flexibility and reduced the risk of converging to local optima, thereby enhancing search efficiency and accuracy.This demonstrates that refining the PSO algorithm and adjusting the hyperparameters of the BiLSTM model is meaningful, offering valuable guidance for future research in PSO algorithms and BiLSTM hyperparameter optimization.(3) The integration of the decomposition-integration forecasting framework with the secondary decomposition approach significantly improves the model's predictive performance.This method not only effectively addresses the limitations of single decomposition but also thoroughly extracts highcomplexity component features, transforming data into more stable sequential modules, significantly enhances the overall accuracy of carbon price forecasting.Practically, this approach has potential applications in several areas, including carbon market analysis, policy formulation, and investment decision-making.From a theoretical perspective, this conclusion not only enriches the research on carbon price and time series forecasting but also offers a new direction for developing more efficient and precise predictive models.(4) The proposed CEEMDAN-VMD-IPSO-BiLSTM composite prediction model has demonstrated its significant effectiveness and general applicability in carbon price forecasting across multiple data sets, providing more accurate and stable predictions compared with other benchmark models.This model not only enriches the theoretical foundation of time series prediction but also provides a powerful decision-support tool for participants in the carbon market.Policymakers can leverage the accurate market insights offered by the prediction model to assess the effectiveness of existing and proposed carbon pricing policies, such as carbon taxes or capand-trade systems.This allows for timely adjustments or the formulation of more scientific and rational carbon emission control policies, ensuring the healthy operation of the carbon market and the achievement of carbon reduction targets.For investors, this model, by providing precise carbon price forecasts, can help them evaluate the risks and returns associated with the carbon market and identify potential investment opportunities.This supports investors in crafting effective investment strategies, such as asset diversification, determining the timing for market entry and exit, and hedging with financial derivatives, to effectively reduce investment risks and minimize potential losses, optimize the investment portfolio, and maximize returns.For businesses, this predictive model can assist in identifying financial risks associated with carbon pricing, enabling the timely formulation of relevant risk management strategies to mitigate the adverse effects of carbon price volatility, such as by locking in future costs through hedging strategies like futures contracts.Additionally, it can provide decision support for optimizing operational strategies and strategic planning.For instance, when the carbon price forecast indicates an upward trend, businesses can adjust their production processes and invest in low-carbon technologies to achieve reductions in carbon emissions and cost savings.
Although the proposed prediction model exhibits high precision, its training time is extensive, making it suitable for scenarios with high accuracy demands.Furthermore, given that decomposition techniques like CEEMDAN are better suited for decomposing signals in single price sequences, and to maintain a balance in the model's complexity, this study focuses on historical carbon price data.The predictive model is based on an in-depth analysis of the characteristics of carbon price data.Within the framework of the decomposition-integration forecasting method, it implements a secondary decomposition strategy that includes both CEEMDAN and VMD methods to effectively break down the complex carbon price data set into several simpler and more predictable subsequences.This process not only enhances the model's capability for feature extraction but also maximizes noise separation and optimizes model input.Furthermore, by incorporating the IPSO-BiLSTM method, it fully leverages its strengths in nonlinear pattern recognition, bidirectional information processing, adaptability, and efficient management of information.Building on the effective utilization of the autocorrelation of time series data, this composite forecasting model deeply mines historical carbon price data, deducing complex nonlinear mapping relationships related to the carbon price series, thereby achieving satisfactory fitting and predictive results based on historical price data.Nevertheless, there is still room for optimizing the error values of the model.
Future research should approach from different perspectives, considering various factors affecting carbon prices, such as national policies, regional economic development, and energy prices.A deeper exploration of the complex relationships between these factors and carbon price fluctuations will better capture the trends in carbon prices, further reduce model errors, and enhance the model's predictive performance.

F I G U R E 2
Basic structure of BiLSTM network.BiLSTM, bidirectional long short-term memory; LSTM, long short-term memory.

( 2 )
Determine SE values for each component sequence derived from the decomposition.Based on these calculation results, all components are classified into high-frequency, low-frequency, and trend components, which are then separately aggregated and reconstructed, resulting in three new component sequences.(3)Apply the VMD method to perform a second decomposition on the high-frequency components, obtaining several new modal components (V-IMFs).(4) All mode components (V-IMFs) acquired from the secondary decomposition are input into an IPSO-BiLSTM model in matrix form for prediction, resulting in the forecast of the high-frequency component.(5)The low-frequency component and the trend component are input separately as vectors into two IPSO-BiLSTM models for prediction, resulting in the forecast of the low-frequency component and the trend component.(6)In the final step, the predictive outcomes of highfrequency components, low-frequency components, and trend components were input into the IPSO-BiLSTM model for integration to yield the ultimate carbon price forecast results.

F I G U R E 3
Flowchart of the mode.BiLSTM, bidirectional long short-term memory; CEEMDAN, completely ensemble empirical mode decomposition with adaptive noise; IMF, intrinsic mode function; IPSO, improved particle swarm optimization; SE, sample entropy; VMD, variational mode decomposition.F I G U R E 4 Volume and turnover of China's carbon pilot carbon trading in 2022.(The inner circle represents the transaction amount, while the outer circle represents the trading volume.) Figures5 and 6illustrate the trends of the carbon price sequences, showcasing that the Beijing carbon pilot exhibits the highest frequency of fluctuations.Table2presents the statistical description of the data sets on carbon prices of each carbon market calculated using Python 3.9.0,displaying the discrete distribution and volatility of their data sequences.From the statistical values, it can be observed that the Skewness values for all carbon market price data exceed 0, indicating a right-skewed distribution with a long right tail; the Kurtosis values are all less than 3, meaning a platykurtic distribution; the Jarque-Bera values are much larger than 0, indicating that the carbon price sequences in each market do not follow a normal distribution.The P-values obtained from the augmented Dickey-Fuller test for Guangdong, Hubei carbon pilot, and the EU ICE are all above 0.05, suggesting the nonstationary nature of the carbon price data sequences in these three carbon markets.Additionally, Figure7's autocorrelation function and partial autocorrelation function results indicate the presence of autocorrelation within the carbon price sequences.

F I G U R E 7
Autocorrelation function and partial autocorrelation function of carbon markets: (A) Beijing, (B) Guangdong, (C) Hubei, and (D) European.

F I G U R E 8
Component reconstruction and quadratic decomposition process.CEEMDAN, completely ensemble empirical mode decomposition with adaptive noise; IMF, intrinsic mode function; SE, sample entropy; VMD, variational mode decomposition.F I G U R E 9 Carbon price forecast results for Guangdong Emissions Trading System.

F
I G U R E 10 Prediction results of numerous models in the Guangdong ETS.BiLSTM, bidirectional long short-term memory; CEEMDAN, completely ensemble empirical mode decomposition with adaptive noise; ETS, Emissions Trading System; IPSO, improved particle swarm optimization; LSTM, long short-term memory; MAE, mean absolute error; MAPE, mean absolute percentage error; PSO, particle swarm optimization; RMSE, root mean square error; VMD, variational mode decomposition.

F
I G U R E 11 Prediction results of numerous models in the European market.BiLSTM, bidirectional long short-term memory; CEEMDAN, completely ensemble empirical mode decomposition with adaptive noise; IPSO, improved particle swarm optimization; LSTM, long short-term memory; MAE, mean absolute error; MAPE, mean absolute percentage error; PSO, particle swarm optimization; RMSE, root mean square error; VMD, variational mode decomposition.
four data sets on carbon prices are partitioned into training sets (80%) and test sets (20%) at a ratio of 4:1 for the purposes of model development and performance assessment.Additionally, cross-validation is employed, allocating 10% of the training set as a validation set for cross-training purposes.The primary research focuses on 1245 sample data of closing prices from the Guangdong Carbon Pilot from July 4, 2017, to September 30, 2020.Of these, samples from July 4, 2017, to September 28, 2021, totaling 996, are designated as the training set, while samples from September 29, 2021, to September 30, 2022, totaling 249, serve as the test set for evaluating model prediction accuracy.All forecast results are based on this test set data.The selection of data from other carbon markets is detailed in Table 1.
The variation trend of carbon price carbon prices in European ETS.ETS, Emissions Trading System; EUA, European.Statistical description of carbon prices in carbon markets used.
T A B L E 2Abbreviations: ADF, augmented Dickey-Fuller; ETS, Emissions Trading System.ZOU and ZHANG| 2579 T A B L E 4 Prediction evaluation indicators of multiple models in the Guangdong market.
T A B L E 5 Prediction evaluation indicators of various models in the Hubei market.: ARIMA, autoregressive integrated moving average; BiLSTM, bidirectional long short-term memory; CEEMDAN, completely ensemble empirical mode decomposition with adaptive noise; IPSO, improved particle swarm optimization; LSTM, long short-term memory; MAE, mean absolute error; MAPE, mean absolute percentage error; PSO, particle swarm optimization; RMSE, root mean square error; VMD, variational mode decomposition.Prediction evaluation indicators of various models in the Beijing market.
AbbreviationsT A B L E 6

Table 7
Prediction evaluation indicators of various models in the EU.