1.1.1. Forecasting methods in the carbon market.

As carbon trading systems are complex mechanisms, a common theme of discussion in the wider literature is how to enhance the precision of carbon market forecasts and the interpretability of the results. In recent years, carbon market forecasting methods have been classified into three main categories: econometric models, artificial intelligence models, and hybrid models. Conventional statistical and econometric methods are extensively utilized for time series forecasting. For example, Liu et al. [8] found that the GARCH-MIDAS model combining economic policy uncertainty (EPU) and realized volatility (RV) is more effective in predicting China’s carbon volatility than the model based on RV only. Niu and Liu [9] studied the volatility of EU quota futures using the mixed data sampling method based on the GJR-GARCH model. They found that accounting for the asymmetric effect of volatility and the influence of macroeconomic variables can substantially enhance the model’s predictive capability. In addition, Zhang et al. [10] extended the classical autoregressive model and found that the EPU index possesses substantial long-term predictive power for the volatility of EU carbon allowance (EUA) futures.

Although traditional econometric models can intuitively measure the importance of characteristics, they falter in capturing the non-linear aspects of carbon prices. Artificial intelligence models not only have the arithmetic calculation capabilities of traditional econometric models, but also can capture historical information, volatility and nonlinear characteristics. Therefore, they significantly improve the accuracy and performance of predictive analysis. Their advantages in predicting financial time series have now been widely proven [11]. Xu et al. [12] used two traditional econometric models alongside three machine learning algorithms to predict carbon returns in China’s carbon trading market, and found that the random forest model had higher accuracy in predicting carbon returns in China than traditional econometric model. Gong et al. [13] chose extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM) and deep neural network (DNN) models to study how climate change impacts carbon futures returns. The findings indicate that the prediction accuracy markedly enhances when climate factors are incorporated into machine learning or deep learning models. Li et al. [14] used multivariate LSTM, multi-layer perceptron, support vector regression and recurrent neural network (RNN) for predicting Hubei emissions allowances (HBEA) and Guangdong emissions allowances (GDEA), and compare the prediction results with each other, which show that multivariate LSTM performs better for carbon price forecasting. Pan et al. [15] constructed an LSTM model that introduced particle swarm optimization to predict carbon prices and extracted effective keywords from Google search volume indices as inputs, which improved the prediction accuracy. Wang et al. [16] proposed a new hybrid prediction model based on a stacked ensemble learning algorithm. which integrates the echo state network, stacked autoencoder, support vector machine (SVM), and LSTM models as base learners, and a convolutional neural network (CNN) – bidirectional LSTM hybrid model as the meta-learner. This model effectively extracts time series features, and achieves high prediction accuracy and good generalisation ability.

AI models outperform traditional econometrics models in predicting nonlinear and non-stationary data. However carbon price data usually contain a large amount of noise and volatility, posing challenges for single models to handle complex, multilevel variations while ensuring predictive stability. For this reason, some scholars have proposed hybrid models and gradually improved them. Huang et al. [17] introduced a new decomposition-ensemble paradigm VMD-GARCH/LSTM-LSTM model, and found that the prediction errors of all decomposition-ensemble models are smaller than that of a single model, which confirms that decomposition-ensemble methods are effective in carbon price forecasting, and the study also shows that nonlinear integration algorithms can boost the prediction performance of multi-scale models. Wang et al. [18] developed a deep learning-based framework for forecasting carbon prices in Hubei province’s carbon trading pilot, and the study showed that principal component analysis and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) methods significantly improve the accuracy of carbon price prediction.

In hybrid modeling, secondary decomposition has the obvious advantage of capturing the details that cannot be identified by primary decomposition and can better separate the noise from the useful signals. Sun and Huang [19] first decomposed the time series data into multiple intrinsic mode functions (IMFs) using empirical mode decomposition (EMD), and then applied variational mode decomposition (VMD) to the first intrinsic mode function (IMF1) for further decomposition. Their findings showed that model incorporating the quadratic decomposition algorithm achieved significantly higher prediction accuracy compared to those using undecomposed data or solely EMD methods. Compared to the EMD decomposition method, in order to extract the features of the signal more efficiently, Ozan [20] used multiple machine learning models optimized by CEEMDAN, VMD, and genetic algorithms, and found that the secondary decomposition markedly enhances model robustness and prediction accuracy by comparing the key performance metrics of the primary decomposition and secondary decomposition models. This outcome aligns with the findings of Sun and Huang. Zhou et al. [21] proposed a hybrid framework that combines CEEMDAN, sample entropy, VMD and LSTM. The study showed that the integrated VMD-LSTM method can effectively predict high-frequency time series extracted by CEEMDAN. In addition, through verification and comparison with different datasets, the study further proves that LSTM outperforms gated recurrent unit and DNN in carbon price forecasting. In order to further address the modal mixing issue in decomposition, Yang et al. [22] first performed a preliminary decomposition of carbon prices using ICEEMDAN, and then used multiscale fuzzy entropy to divide the decomposition results into high, medium, and low complexity components. After merging the high complexity components, they performed a second decomposition on the merged components using complete ensemble empirical mode decomposition. It is shown that ICEEMDAN surpasses other decomposition methods when dealing with data from Shenzhen and Hubei, and the secondary decomposition markedly diminishes data complexity, establishing a basis for subsequent precise predictions. Xu et al. [23] also employed an ICEEMDAN method, combined with a wavelet decomposition (WD) to perform a secondary decomposition of the original monthly runoff series. Subsequently, they utilised a seagull optimisation algorithm (SOA) optimised SVM to predict each component. This model framework effectively improved the prediction accuracy.

1.1.2. Factors influencing the carbon market.

There is a wide range of literature that explores the factors affecting carbon prices, and these factors can be analyzed from both internal and external perspectives. In terms of internal factors, Sadefo and Njong [24] used a self-refinement to coarse reconstruction algorithm to decompose the subsequences and residuals extracted from the original carbon price into high-frequency, low-frequency, and trend components, aiming to deeply study the characteristics and basic laws of carbon prices in different frequency bands. Similarly, Zhu et al. [25] used improved random forest using particle swarm optimisation, extreme learning machine and LSTM to predict sequences of differing complexity, following the extraction of diverse features from the original carbon price series, and then combined and reconstructed them to better capture the characteristics of each component.

Although decomposition based on the historical data series of carbon price can effectively improve the prediction accuracy and generalization ability, it is limited by its disregard for external influences such as energy price and economic factors. To compensate for this deficiency, Li and Liu [26] classified external factors into structural and non-structural factors, and used Baidu search index as a non-structural data source. They highlighted that incorporating non-structural data can enhance the informational scope and consequently augment predictive accuracy. Similarly, Huang and He [27] refined the Baidu search index in feature selection by selecting the Baidu search index corresponding to low carbon economy, carbon trading, carbon emissions, carbon exchange rate, emission reduction, and carbon neutrality as unstructured influencing factors. Zhang et al. [28] also considered 20 factors related to carbon prices, including Baidu search index, when constructing a mixed model for forecasting carbon price. Finally, the Lasso method was used to extract variables strongly linked to carbon prices. The experimental results showed that the model performed well in terms of prediction accuracy and stability. Li et al. [29] selected text data from three sources: news reports, policy documents, and social commentary. They then integrated China’s daily time-series data on carbon emissions and the text data into a multimodal feature set using a convolutional neural network inceptionv2 and a text convolutional neural network optimised with an attention mechanism. This study comprehensively considered the factors influencing carbon emissions.

As for the structural factors, using external influences such as macroeconomics, energy prices, and international carbon markets as model inputs can substantially enhance the precision of carbon price forecasts [30–31]. Mao and Yu [32] used the SHAP method to analyze the key indicators of carbon price forecasting, and found that the carbon price was affected by the multiple influences of macroeconomic conditions, international carbon market prices, exchange rates, and domestic and international major energy prices. Zhang et al. [33] on the other hand, constructed a dynamic self-learning integrated forecasting framework, choosing Brent crude oil futures price and natural gas futures price as fuel and energy factors, and stoxx 600 index and goldman sachs commodity index (GSCI) as economic factors. The study demonstrates that the model achieves both high forecasting accuracy and stability. In addition, Hu and Cheng [34] selected the main external factors with the largest information coefficient based on the maximum information coefficient (MIC) method, revealing that the Guangdong carbon market price is heavily affected by the EUA price. Wang et al. [35] also selected features using the MIC, covering five aspects: fossil energy, macroeconomics, exchange rates, industrial development, and climate. The study showed that variables reflecting the overall economic situation, such as stock indices, exchange rates, and industrial development indices, have a significant positive contribution to the medium-term and long-term trend of carbon prices, while recent fossil fuel prices have a negative impact on the short-term trend of carbon prices due to supply and demand. It is worth noting that Wang et al. [36] also selected internal carbon market data such as opening price, maximum price and minimum price, and weather pollution such as air quality index (AQI) and PM2.5 as factors affecting the carbon price, and proposed an intelligent optimization decomposition and integration model based on multi-feature fusion, which achieves excellent prediction performance. Zheng et al. [37] also divided the factors affecting carbon emissions into three aspects. Among them, the carbon market aspect is reflected by the EU allowance futures carbon price (EUAP), EU allowances futures carbon trading volume and the Hubei carbon trading volume. The economic environment aspect is reflected by the dollar against the RMB exchange rate and CSl 300 index, etc., and the daily AQI of Wuhan City to reflect the natural environment. The study found that Brent crude oil futures settlement price and EUAP have a positive impact on carbon prices, while CSl 300 index has a negative impact on carbon prices.

2.2.1. LightGBM based Boruta feature selection method.

In the Boruta framework, Random Forests are usually used to compute the importance of features, for example, Ghosh et al. [39] proposed an ensemble feature selection (EFS) method incorporating the Boruta and regularized random forest algorithms for filtering explanatory features. The effectiveness of combining the EFS process with advanced AI-based prediction models is well demonstrated. In this paper, we introduce the LightGBM model to be used as a base model in the Boruta algorithm to evaluate the relative significance of original and shadow features.

Let the closing price of the three carbon markets be the target variable, denoted as , where denotes the target value of the ith sample, and denotes the value of the feature over all samples. Boruta’s algorithm introduces shadow features, defined as:

(3)

Among them, the shadow feature are obtained by randomly permuting the values of the original feature and do not contain any predictive information, so the shadow feature can be used as a benchmark for the uninformative features.

For a given feature , its importance score is defined as the cumulative split gain of the feature across all decision trees.

(4)

Where denotes the total number of decision trees in the LightGBM model, denotes the number of nodes in the jth tree, is an indicator function. When is used as the splitting feature at the kth node of the jth tree, Otherwise . Gain denotes the gain of the split at the kth node of the jth tree for the feature .

By using the above formula, the importance scores of all shadow features are calculated and the maximum value is taken as the upper limit of the importance of the shadow feature, denoted as:

(5)

Boruta’s algorithm performs multiple rounds of iterations, regenerating the shadow features and calculating the feature importance scores in each round, and then comparing them with the importance scores of the original features until all features are labeled as “confirmed” or “rejected”, or until the maximum number of preset iterations is reached. The stability and reliability of the filtering results are ensured through multiple rounds of iterations. The final retained feature set is defined as:

(6)

2.2.2. feature selection method based on Lasso regression.

The main reason for using linear models for feature selection in this paper is that financial data is noisy and nonlinear models may be too flexible and easy to extract irrelevant patterns from the noise, leading to poor feature selection [40].

Lasso regression controls the sparsity of features by adding a regularization term to the objective function of ordinary least squares regression. The optimization objective function for Lasso regression is defined as:

(7)

Where is the number of samples, is the target variable, is the target value of the ith sample, and is the value of the jth feature of the ith sample. is the intercept term of the model. is the regression coefficient corresponding to the feature. is the regularisation parameter, which controls the strength of regularisation.

In the optimization process, Lasso regression imposes sparsity constraints on some of the characteristic coefficients, resulting in a sparse vector of regression coefficients . Ultimately, the model retains the set of features corresponding to non-zero coefficients as valid features. Let the solution of the feature coefficients be , then the final selected feature set is defined as:

(8)

2.3.1. ICEEMDAN decomposition.

ICEEMDAN is further optimized on the basis of CEEMDAN algorithm, and its core idea is to add noise to the original signal and extract the IMF at each step. it can effectively capture the essence of the sequence and reduce the volatility of the data to improve the prediction accuracy [41], the steps of ICEEMDAN can be summarized as follows.

Step 1. Introduce a set of white noise signals for each sample (where denotes the number of the noise sample, and (), and superimpose it with the original signal superimposed. Form the added noise signal.

(9)

Where is the noise amplitude.

(10)

For each noise-added signal, EMD is used to extract, and calculate the average value of the first mode in all the added noise signals.

Step 2. For each subsequent modal function , ICEEMDAN is extracted by recursion.

(11)

The sum of the extracted modal functions is removed from the original signal to obtain the k-1th-order residual signal . Then, a new white noise is added to each to form a new noise-added residual signal .

(12)

Then, for each , the kth-order modal function is extracted using EMD, and the kth-order modal mean of all the noise-added residuals is calculated to obtain the kth-order intrinsic modal function.

(13)

Step 3. When the residual becomes a monotonic function or meets a preset energy threshold, the decomposition process is terminated and the final residual part no longer has a significant oscillatory component. At this point, the signal is represented as:

(14)

Where is the total number of modes obtained from the decomposition, and is the residual term.

2.3.2. SDAE.

Stacked denoising autoencoder is a deep learning model for unsupervised learning. In this paper, stacked autoencoder is used for noise reduction of high frequency components after ICEEMDAN decomposition, the model consists of two layers of encoder and decoder, and the signal noise reduction is achieved by layer-by-layer dimensionality reduction and reconstruction.

Step 1. Input the matrix consisting of high frequency components IMF1 through IMF4 of size .

Layer 1 encoder: input 2340 dimensional data, dimensionality reduction to 1170, then to 585. Where is the weight matrix, and is the bias vector is the activation function Tanh.

(15)

Layer 1 decoder: reconstructs the 585-dimensional representation to a 2340-dimensional input size. Where is the transpose of the encoder’s weight matrix, and is the bias vector of the decoder, and is the activation function Tanh.

(16)

Layer 2 encoder: receives the output of the layer 1 decoder and once again reduces the 2340 dimensional data to 1170, and finally to 585. Where is the weight matrix of the second layer encoder, and is the corresponding bias.

(17)

Layer 2 decoder: restore the 585-dimensional representation to 2340 dimensions to generate the final denoised signal.

(18)

Step 2. The input data is reduced in dimension by two layer encoder and restored by two layer decoder to extract the signal characteristics and remove noise. The forward propagation process is as follows:

(19)

Where denotes the input high frequency component containing noise, and denotes the denoised signal after reconstruction.

step3. To realize the denoising effect, the model uses MSE as a loss function for calculating the error between the reconstructed signal and the target noise-free signal.

(20)

Where is the ith sample value in the target noise-free signal, is the ith sample value in the denoised reconstructed signal, and is the number of samples. During training, the loss value is dynamically calculated for each iteration, and the Adam optimizer is used to minimize and gradually update the network parameters and to reduce the reconstruction error.

2.3.3. VMD.

VMD is an adaptive signal processing method designed to decompose a complex signal into a set of IMFs with different center frequencies and narrowband characteristics, thereby effectively extracting features from the original time series [42]. Different from the traditional EMD, the VMD is based on the principle of bandwidth minimization and adopts a variational approach to decompose the signal, ensuring that the decomposed modes have good localization properties in the frequency domain.

The process of VMD is as follows:

(21)

Where is the kth modal function, is the center frequency of the modal function , and is the composite function of the Dirac function and the Hilbert transform, which is used to obtain the analytic signal of the signal, indicates the convolution operation, and is the frequency shift factor, which shifts the modal function to near the center frequency .

(22)

Where is a balance parameter that controls the trade-off between bandwidth minimization and reconstruction accuracy, and is the Lagrange multiplier for imposing constraints on the decomposition.

VMD employs the alternating direction multiplier method (ADMM) to address the above optimization problem. ADMM gradually approaches the global optimal solution by separately optimizing the modal function and the center frequency , as well as the Lagrange multiplier , in each iteration. Fixing and , the update formula for the modal function is solved by minimizing the partial derivative of the Lagrangian function with respect to each .

(23)

Where denotes the inverse Fourier transform and is the frequency domain representation of the modal function .

Then, fixing and , minimizing the partial derivative of the Lagrangian function with respect to , we obtain an updated formula for the central frequency.

(24)

Finally the Lagrange multipliers are updated to progressively satisfy the constraints. Where is the step parameter, which is used to control the speed of convergence of the Lagrange multipliers.

(25)

The VMD algorithm, by iterating the above steps repeatedly, makes the modal function and center frequency gradually converge to a steady state, thus obtaining a modal decomposition result that satisfies the bandwidth minimization condition.

2.3.4. CPO.

CPO is a biomimetic optimization algorithm that simulates the group defense behavior of the crown porcupines to achieve a global optimal solution search in the balance of exploration and development. The defense mechanism of the crown porcupines includes vision, sound, smell and physical attacks, among which vision and sound correspond to the exploration operator, and smell and physical attacks correspond to the development operator. In addition, CPO also introduces the cyclic population reduction technique to preserve population diversity and enhance the convergence speed [43].

Step1. In the initialization phase, the positions of the individuals are randomly distributed within the given upper and lower bounds, and the position of the th individual in the population is initialized as follows:

(26)

Where and are the lower and upper bounds of the jth dimension respectively, and rand is a random number in the range [0,1].

Step2. In each iteration, the CPO updates the position of an individual through two mechanisms: exploration and exploitation.

The exploration phase simulates the defensive extension behavior of the crown porcupine with the individual position update equation.

(27)

Where is a coefficient that controls the exploration range, is a random number used to increase the diversity of individuals, is a random individual in the population, and is the average of the current population position.

The development phase corresponds to the clustering behavior between individuals, and its position update formula is:

(28)

Where is the individual with optimal fitness in the current population, and is the control parameter for the convergence step size, and is a [0,1] random number.

Step 3. After updating the position, if the individual exceeds the boundary, the boundary reflection mechanism is used to keep it within the reasonable range.

(29)

Step 4. At the end of each iteration, the fitness of every individual in the population is assessed, and the prevailing optimal solution is retained as the global optimal solution of the population. If the new solution surpasses the global optimal solution, the global optimal solution is updated.

2.4.1. LSTM.

LSTM is an improved model of RNN. Compared with traditional RNN, LSTM significantly enhances the learning ability of long-term dependencies through a structured network design, and shows stronger predictive accuracy in tasks involving long sequence data [44]. The LSTM effectively controls the flow of information at each time step by introducing a structure with input gates, forget gates and output gates, as shown in Fig 1, enabling the model to retain important information in long sequence data and suppress cumulative errors. The computational process of the LSTM unit consists of four steps:

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Fig 1. Structure of LSTM.

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(1) Oblivion gate: based on the hidden state of the previous time step and the current input , the output of the forgetful gate is calculated to determine the part of the previous momentary cell state that needs to be forgotten.

(30)

(2) Input gate: determines the importance of the current input . The input gate and the candidate memory state are calculated as follows:

(31)

(32)

(3) Updating the cell state: combining the outputs of the forgetting gate and the input gate, the cell state at the current moment is updated.

(33)

(4) Output gate: calculate the output gate , and based on the current cell state update the hidden state . Where denotes the sigmoid activation function, and denotes the element level multiplication.

(34)

(35)

2.4.2. Attention mechanism.

The Attention mechanism allocates weights to each component of the input sequence, allowing the model to concentrate on critical elements and enhancing its capacity to capture temporal or spatial relationships. The mechanism obtains weighted output by calculating the similarity between the vector , vector , and vector . The calculation formula is:

(36)

Where is the query matrix, and is the key matrix, and the value matrix, and denotes the dimension of the key vector, which is used as a scaling factor to stabilize the range of values.

2.4.3. Optuna.

Optuna is an automated hyperparameter optimization framework. Its “define-by-run” feature allows users to flexibly define the hyperparameter space and search for the optimal hyperparameter combination in a parallelized manner. In addition, the framework enables users to query and document the hyperparameter values throughout the optimization process [45].

Optuna defines the hyperparameter optimization problem as the minimization of an objective function. In this framework, the goal is to identify a set of hyperparameters that minimizes the objective function .

(37)

Where denotes the model performance metric determined by the set of hyperparameters , and denotes the combination of hyperparameters sampled from the defined parameter space. Optuna uses the Tree-structured Parzen Estimator algorithm to quickly converge to the optimal solution by dynamically adjusting the posterior probability distribution to guide the sampling of the next set of hyperparameters.

4.1.1. Closing price data.

The study used daily carbon prices in Hubei, Guangdong, and Shanghai from April 4, 2014 to September 30, 2024 as the experimental data, and the first 85% of the dataset was chosen as the training set and the last 15% as the test set. Fig 3 shows the trend of the original price series across the three carbon markets.

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Fig 3. Trends of the original price series of the three carbon markets.

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4.1.2. Influence factor data.

The model in this paper takes into consideration the factors influencing the carbon price, including structural and non-structural factors.

1. Structural factors

The structural factors covered in this paper include environmental and climatic factors, energy factors, carbon market factors, exchange rate and monetary factors, domestic industry factors, and economic factors.

1. (1). Environmental and climatic factors

In this paper, daily AQI, Daily maximum temperature and Daily minimum temperature are used to reflect the impact of environment and climate on carbon price. In particular, the relevant data of Guangdong province and Hubei province are replaced by AQI, maximum air temperature and minimum air temperature data of Guangzhou city and Wuhan city, respectively.

1. (2). Energy factors

In this paper, the Brent crude oil futures settlement price, Daqing crude oil spot price, Thermal coal futures closing price, China coking coal composite index, and NYMEX natural gas closing price are selected to represent the impact of energy on the carbon price, in which the Thermal coal futures closing price comes from Zhengzhou commodity exchange.

1. (3). Carbon market factors

In this paper, EUA futures closing price, EUA futures volume, GDEA volume, HBEA volume, and Shanghai emission allowance (SHEA) volume are chosen to represent the EU carbon market and the impact of internal factors of the carbon market on carbon price.

1. (4). Exchange rates and monetary factors

In the paper, the USD to CNY exchange rate, the USD to EUR exchange rate, and the RMB index are chosen to reflect the influence of exchange rate and monetary factors on carbon prices in the three markets.

1. (5). Domestic industry factors

In this paper, the Chemicals index, Textile index, Steel index, Energy index, Agricultural and sideline products index, Rubber and plastic index, Non-ferrous metals index, and Commodity price index are selected to reflect the impact of domestic industries on the carbon prices of the three markets.

1. (6). Economic factors

In the paper, NASDAQ index, S&P 500 index, SSE composite index volume, SSE composite index closing price, CSI 300 index volume, CSI 300 index closing price, SHFE: Gold futures closing price, SHFE: Silver futures closing price, and BDI index are selected to react to the influence of economic factors on the price of carbon market.

1. Unstructural factors

Unstructured indicators can capture market sentiment, public attention and behavioral changes that are difficult to be reflected in traditional structured data, making the information more comprehensive. This study uses Baidu search index as unstructured data, collecting daily search volumes for relevant keywords in Guangdong, Hubei, and Shanghai. Keywords include carbon, low carbon, low carbon economy, energy conservation, emission reduction, carbon sink, carbon emission, carbon trading, carbon emission trading, carbon footprint, carbon neutrality. Finally, the search volume of all keywords in the region is summed as the unstructured data for the region. This study selects data from the same period to match all variables.

4.1.3. Data preprocessing.

This paper first employs the XGBoost method to impute all missing values, using the complete features as input. By learning the complex nonlinear relationships among features, it effectively predicts missing data and significantly improves the accuracy of imputation. Subsequently, to eliminate the impact of different feature dimensions, normalisation is applied to scale all data to the [0, 1] range, standardising the feature scale. Next, to construct training samples suitable for predicting carbon emission allowance closing prices, a sliding window length of 10 is set, corresponding to two weeks of trading days. Finally, the processed data set is divided into a training set comprising the first 85% and a test set comprising the remaining 15% for model training and performance evaluation. During parameter tuning, we used a time series cross-validation method to divide the training set into five folds in chronological order. In each fold, the training set was always earlier than the validation set to match the actual scenario of time series modelling and avoid information leakage.

4.3.1. ICEEMDAN decomposition and denoising.

This study employs the ICEEMDAN method to decompose the original carbon price series from three carbon markets into modal components of differing frequencies. As illustrated in Fig 4, the IMF1 sequence has the highest frequency, and the subsequent IMF frequency gradually decreases, and the fluctuations gradually level off. From the figure, it can be seen that the trend of the low-frequency components and the trend of the original series as a whole are roughly consistent, and the trend of the low-frequency components is more gentle.

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Fig 4. Primary decomposition of carbon price ICEEMDAN.

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Next, the g test is applied to differentiate the high-frequency and low-frequency components in the subsequence to improve the plausibility of sequence reconstruction. Table 2 shows the g values of the sub-sequences after the ICEEMDAN decomposition of the carbon prices in the three markets. For the Guangdong carbon market, it can be seen that the g-values of IMF1 to IMF3 are low, and the percentages are all below 10%, which are typical high-frequency components. While IMF4 belongs to a turning point and no longer behaves as a typical high-frequency component, the percentage is significantly higher than that of IMF3 but still lower than that of IMF6, which is a transition component and may contain part of the high-frequency signals, so it is also considered within the high-frequency component, and the rest are low-frequency components. For the Hubei carbon market, it can be seen that the g-values from IMF1 to IMF3 are lower, and IMF4 belongs to a turning point, which is a transition component, so it is also considered in the high-frequency component, and the rest are low-frequency components. For Shanghai carbon market, it can be seen that the g-values from IMF1 to IMF4 are low, and the percentages are all below 3%, then IMF1, IMF2, IMF3 and IMF4 are high-frequency components, and the rest are low-frequency components.

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Table 2. High-frequency and low-frequency IMF components in the three carbon markets.

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Next, a stacked auto-encoder is used to perform noise reduction on the high-frequency components IMF1 to IMF4, after which the noise-reduced IMF1, IMF2, IMF3, IMF4, and the low-frequency components are summed up to obtain the reconstructed signals of the primary decomposition.

4.3.2. Optimized VMD decomposition.

In order to effectively eliminate mode aliasing. The VMD decomposition is started after the primary decomposition, and CPO is introduced to optimize the alpha, tau, and K parameters in the decomposition process to determine the best decomposition parameters. Fig 5 shows the convergence curves of the three carbon markets, which all show the effectiveness of the CPO algorithm in application. The detailed parameter settings for CPO algorithm are shown in Table 3.

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Table 3. Specific details of CPO optimisation of VMD decomposition.

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Fig 5. Convergence curve of CPO optimization algorithm for carbon market.

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Finally, the optimal K value of Guangdong carbon market and Shanghai carbon market is determined to be 9, which is decomposed into 9 modal components, and the optimal K value of Hubei carbon market is 8, which is decomposed into 8 modal components. Fig 6 specifically shows the time domain waveforms of different modal components.The sensitivity analysis of VMD parameters for the three carbon markets is detailed in Tables 4–5, and Table 6.

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Table 4. Parameter sensitivity analysis of VMD decomposition in the Guangdong carbon market.

https://doi.org/10.1371/journal.pone.0326926.t004

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Table 5. Parameter sensitivity analysis of VMD decomposition in the Hubei carbon market.

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Table 6. Parameter sensitivity analysis of VMD decomposition in the Shanghai carbon market.

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Fig 6. VMD decomposition results for three carbon markets.

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Through the VMD decomposition, the raw signal of Hubei carbon market is decomposed into VIMF1 to VIMF8, where VIMF1 and VIMF2 correspond to the low-frequency trend of the signal, VIMF3 to VIMF5 capture the mid-frequency dynamics of the signal, and VIMF6 to VIMF8 separate the high-frequency oscillations and noise in the signal. The original signals of Guangdong carbon market and Shanghai carbon market are decomposed into VIMF1 to VIMF9, where VIMF1 to VIMF3 show the low-frequency trend components, VIMF3 to VIMF6 gradually capture the mid-frequency characteristics, and VIMF7 to VIMF9 mainly reflect the high-frequency components. Overall, the optimized VMD decomposition achieves an accurate separation of the signal in the time and frequency domains.

4.4.1. Comprehensive feature screening.

By selecting the top and stable elements, the Boruta algorithm identified 13 key factors in the Guangdong carbon market, including the CSI 300 index closing price, NASDAQ index, etc. By Lasso regression method, SSE composite index volume, SSE composite index closing price and other factors were screened in Guangdong carbon market. Finally, through the comprehensive feature screening by Boruta algorithm and Lasso regression, we determined the intersection of the selected feature variables of the two methods as shown in Fig 7. These commonly contained features are finally used as inputs to the forecasting model, including USD to CNY exchange rate, Daqing crude oil spot price, etc.

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Fig 7. Comprehensive characteristic screening results of Guangdong carbon market.

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Hubei carbon market and Shanghai carbon market also screened the final input external factors through the above process as shown in Figs 8 and 9. The results of the final screening for the three carbon markets are detailed in Table 7.

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Table 7. Screening results for the characteristics of the three carbon markets.

https://doi.org/10.1371/journal.pone.0326926.t007

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Fig 8. Comprehensive characteristic screening results of Hubei carbon market.

https://doi.org/10.1371/journal.pone.0326926.g008

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Fig 9. Comprehensive characteristic screening results of Shanghai carbon market.

https://doi.org/10.1371/journal.pone.0326926.g009