Forecasting carbon dioxide emission price using a novel mode decomposition machine learning hybrid model of CEEMDAN‐LSTM

Global carbon dioxide emissions have become a great threat to economic sustainability and human health. The carbon market is recognized as the most promising mean to curb carbon emissions, furthermore, carbon price forecasting will promote the role of the carbon market in emissions reduction and achieve reduction targets at lower economic costs for emission entities. However, there are still some technical problems in carbon price prediction, such as mode mixing and larger reconstruction error for the traditional empirical mode decomposition‐type models. Therefore, the innovation of this paper is constructing a novel carbon price prediction model of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)‐long short‐term memory (LSTM), that combines the advantages of CEEMDAN in decomposing the multiscale time‐frequency carbon price signals and the LSTM model in fitting the financial signals. The results show the proposed CEEMDAN‐LSTM model has significant accuracy in predicting the complex carbon price signals. The prediction error and expectation indicators of root mean square error, mean absolute error, mean absolute percentage error, and direction accuracy are 0.638342, 0.448695, 0.015666, and 0.687631, respectively, which is better than other benchmark models. Further evidence convince that the short‐term forecasting performance is superior to the long‐term and medium‐term performance. That evidence concludes that the proposed model is a reliable method to reveal the carbon price‐driving mechanism from the point of multiscale time‐frequency characteristics. Particularly, short‐term forecasting is more accurate and can provide a valuable technical reference for reduction entities and green financial companies to judge the market situation and formulate quantitative transactions.


| INTRODUCTION
With the development of the global economy and industrialization, greenhouse gas emissions have seriously expanded, especially the sharp increase in carbon dioxide emissions has become a key incentive that affects the economy's sustainability and human health. The report of Global Energy Review: CO 2 Emissions in 2021 released by the International Energy Agency (IEA) in March 2022 showed that the global emissions rebounded sharply to the highest level, the carbon emissions from the energy sector reached 36.3 billion tons in 2021, that surpassed the pre-COVID-19 levels. With the improvement of epidemic control, economic recovery has become the primary goal of all countries in 2021, then the energy demand for coal, oil, and electricity is increased. As a result, global carbon emissions have also increased rapidly. Based on the report of the IEA, the global carbon emissions from the coal sector contribute to 1.53 billion tons of carbon emissions. Noteworthy, the significant increase in carbon emissions in 2021 also offset the decline in emissions caused by economic recession since the COVID-19 pandemic. That is, the increased carbon emissions in 2021 are more than the reduction in 2020, which showed a serious "rebound phenomenon" of carbon emissions. Therefore, taking effective measures to curb global carbon emissions, and realizing the economy and environment sustainability have become crucial issues that all countries need to solve.
As a financial innovation, the global carbon emission trading market has become the most constructive measure to solve environmental problems. 1,2 However, compared with other capital markets, the construction of the carbon market is relatively late, and the market efficiency is not high. The market is easily affected by extreme events and energy policies, that make the carbon price have significant characteristics of nonlinear and nonstationary. [3][4][5] Affected by the continuous rise in energy prices, the data show that as of March 2022, the European carbon price has plummeted by more than 35% from 97 euros/ton in early February. Simultaneously, in the last week of February 2022, the carbon price in the UK market also fell below 100 euros/ton, and the price in other carbon markets such as Australia and North America also fell to varying degrees. Recently, concern about the European tension triggered by the Russian and Ukrainian crisis in March 2022, there has been a huge price drop and falling investment expectations in many carbon markets.
Predicting the carbon price and explaining its driving mechanism are important means to promote the maturity of the carbon market price mechanism. 6,7 Consequently, analyzing the impact of market factors on the carbon premium will help the market participants to conduct market judgment, and finally serve the emissions reduction of polluting gas. As for the carbon price forecasting research, the empirical mode decomposition (EMD) technology is the mainstream model to predict carbon prices. The significant advantage of the EMD model is to reveal the multiscale time-frequency characteristics of carbon price signals and reflect the driving process of carbon premium. However, the traditional EMD and adaptive noise ensemble empirical mode decomposition (EEMD) model is prone to the problem of mode mixing. Furthermore, the integrated complementary ensemble empirical mode decomposition (CEEMD) technology has the defects of larger reconstruction error and poor decomposition completeness, which still cannot solve the alignment problem of mode components. Therefore, it is very necessary to employ a more effective model to solve the above problems and put a novel hybrid carbon price forecasting model for improving the accuracy and robustness.
The structure of this paper is designed as follows: the second part introduces the literature, the third part constructs the carbon price forecasting hybrid model, the fourth part is the empirical analysis and the discussion, and the last part is the conclusion and prospects.

| LITERATURE REVIEW
According to the prediction model differences of previous studies, we divide the existing research into two parts: one is the volatility statistical technology represented by GARCH cluster models; the other is the hybrid model represented by EMD decomposition technology. We also find that, as a classical volatility modeling technology, the GARCH type models are early used to reveal the volatility driving mechanism of the carbon price. Hence, we believe the above literature classification is reasonable and credible.

| Volatility statistical technology for forecasting carbon price
It is found that GARCH cluster models can describe the carbon price better than implied volatility models and knearest neighbor models, 8 the threshold GARCH models can effectively reveal the asymmetric characteristics of the carbon price. 9 Research concluded that the GARCH model based on Markov regime transfer is better than other GARCH cluster models in forecasting the shortterm carbon price. 10,11 Similarly, Zeitlberger and Brauneis 12 put that the AGARCH and GJR-GARCH models subject to the generalized error distribution can accurately forecast the European carbon future price, while the out-of-sample forecasting is more robust. The AR-GARCH model can capture the carbon market uncertainty and predict the carbon price volatility, 13 particularly, the model can reveal the nonlinear impact of policy regulation on carbon price. 14 The GARCH model with the normal distribution is difficult to describe the abnormal volatility feature of the carbon market, Sanin et al. 15 found that the ARMAX-GARCH model with Gaussian time-varying jumping process can explain the changes of the European carbon price. Conducted the DCC-GARCH and ARCH model to study the volatility spillover between fossil energy and carbon market, it is found that the returns of coal, crude oil, and natural gas have a significant shock on the short-term European carbon price. [16][17][18] Carbon market has price fractal characteristics, the fractional Brownian motion model with the volatility parameter determined by the GARCH model showed a stable prediction performance in European carbon option price. 19,20 2.2 | Hybrid mode decomposition model for forecasting carbon price The EMD technology can decompose the carbon price signals into various mode components, and reveal the carbon price-driving mechanism from the perspective of different time scales. 21,22 Combined the advantages of EMD and least squares support vector machine (LSSVM), Zhu et al. 23 proposed a mixed carbon price forecasting model EMD-LSSVM, the results showed that the proposed model has significant prediction accuracy and stability in the European carbon market. Integrated the factor analysis technology into the EMD-LSSVM model, Sun and Huang 24 concluded that it is necessary to detect the relationship between different mode signals so as to improve the prediction accuracy. Decomposed the carbon price signals into different intrinsic mode functions (IMFs), fitted those IMF signals by GARCH and LSSVM model, and further conducted the particle swarm optimization, genetic algorithm (GA), spike neural network (SNN) and deep neural network (DNN) algorithm to optimize the hybrid model, the results put that the carbon price prediction performance of EMD-ARMA-LSSVM, EMD-LSSVM, EMD-GA-BP, EEMD-SNN, and EMD-DNN-BP are significantly better than other single models. [25][26][27] Employed the improved empirical mode decomposition technology (MEEMD), Yang et al. 28 used the improved whale optimization algorithm (IWOA) to optimize the long short-term memory (LSTM) model, the research suggested that the hybrid model of MEEMD-IWOA-LSTM has stability and robustness in predicting the price of emerging carbon markets such as Beijing, Fujian, and Shanghai. Followed the same idea, Huang et al. 29 and Li et al. 30 constructed an integrated carbon price prediction model based on variational mode decomposition (VMD). The results indicated that the integrated models of VMD-GARCH and VMD-LSTM can effectively predict the European carbon price in the rising stage, and also emerging China's carbon market. [31][32][33] For solving the mode mixing problem in the EMD process, Li et al. 34 used the model of CEEMD and VMD to decompose the original carbon price signals, the conclusion proposed that the second mode decomposition technology has obvious advantages in carbon price prediction. To overcome the point prediction defects of traditional EMD techniques, Ji et al. 35 constructed a three-stage vertical carbon price interval prediction model based on the improved complete EEMD technology, the research convinced significant prediction reliability of the proposed model in the European carbon market. In addition, the model of interval discrete wavelet transform, interval empirical mode decomposition, and interval VMD are also integrated into a mixed model for predicting carbon price and showed predicting superiority and robustness. 36,37

| Comment on previous literature
Those pieces of literature provide a valuable reference for this study, but there are still the following two deficiencies: first, the GARCH cluster models usually request strict tail distribution assumption, which makes the prediction performance of GARCH cluster models questionable. More importantly, the time scale heterogeneity and time-frequency characteristics of the carbon price have been ignored. Second, the traditional EMD technology is prone to the problems of mode mixing and matrix alignment obstacles, that increase the reconstruction error and residual noise. Therefore, to overcome the theoretical defects of the above models, this paper conducts the complete ensemble empirical mode decomposition with adaptive noise (CEEM-DAN) to realize the multiscale decomposition of the original carbon price signals. The model has two significant advantages compared with other EMD-type models: one is the CEEMDAN model can reduce the mode reconstruction error with a small average number of iterations, can solve the mode mixing problem, and improve the noise reduction performance. The other one is that it can depict the multiscale time-frequency characteristics of carbon price series, and reveal the driving mechanism of carbon premium from different time frequencies point. For the latest study, Zhou et al. 38 constructed the CEEMDAN-LSTM model to predict the Guangzhou carbon price, and a novel VMD-LSTM model is suggested for improving prediction accuracy. Although the CEEMDAN and VMD model can solve the mode mixing problem, and the LSTM model has good prediction performance, the robustness verification of the LSTM model for forecasting carbon price in different time scales cannot be neglected. Actually, due to its special gate structure, the prediction performance of the LSTM model has a certain correlation with the time scale of the prediction period. Consequentially, revealing the carbon price prediction robustness of the LSTM model in different time periods is a non-negligible task in this paper.
The innovation of this paper is constructing a new mode decomposition machine learning CEEMDAN-LSTM model for predicting the nonlinear and nonstationary carbon price, so as to provide new evidence explanation for the driving process of carbon premium. Furthermore, this paper also tests the prediction stability of the proposed CEEMDAN-LSTM model in different prediction periods to support a more robust performance.
There are two basic logic steps of the proposed model, on the one hand, based on the strengths of CEEMDAN, the proposed model decomposes the original carbon price signals into several modes, and reveals the multiscale time-frequency characteristics of carbon price; on the other hand, the LSTM network with the advantage of time series fitting is used to conduct the multistep prediction of the multiscale mode information, then the final prediction value can be obtained after summarizing the multiscale mode prediction results.

| CEEMDAN model
The CEEMDAN model is a special form of mode decomposition technology based on EMD. It is a model innovation derived from the improvement of EMD-type models. Actually, the EMD method is an effective adaptive data processing method, that can decompose the price signals with multiscale time-frequency characteristics. 21 The main contribution of the EMD model is to decompose the complex time series into several IMFs and a residual sequence. The IMFs represent the local signals with timefrequency characteristic at different time scales, that is used to reveal the spectral volatility of the price signals. While the residual indicates the long-term market trend. However, the biggest defect of the EMD model is the mode mixing problem during the decomposition process when the price signals do not the complete white noise process, or there are abnormal events in the signals. Although the EEMD technology proposed by Wu and Huang can reduce the signal's noise, the decomposition process of EEMD may lead to mode components cannot be orthogonal, and matrix alignment problems due to the difference of the added-noise signals. 39 Based on the above analysis, to overcome the problems of mode mixing and matrix alignment, this paper uses the CEEMDAN method carried out by Torres et al. 40 to decompose the nonlinear and nonstationary carbon price signals. Different from the noise addition process of the EEMD model proposed by Wu and Huang and CEEMD model put by Yeh et al., 41 the core of the CEEMDAN method is adding a new white noise into the residual after the first-order IMF is calculated, then a new IMF component is obtained. In particular, the white noise added into the CEEMDAN decomposition process is the IMF component obtained by EMD decomposition. The added noise is reduced step by step, and the residual noise in the mode component is less, so the reconstruction error can finally be reduced. The flowchart of the CEEMDAN model for decomposing the carbon price signals can be shown in Figure 1. The algorithm steps of the CEEMDAN model are as follows: Step 1: Add the positive and negative pairs of white noise to the original carbon price x(t), then get a new signal sequence: x t a n t represents the added noise after the ith decomposition (i = 1, 2, …, n); a 0 denotes the noise value, and q = 1 or q = 2 is used to ensure pairwise noise addition; r t ( ) i 1 means the first residual after the decomposition; the final first-order component can be obtained by taking the average of t IMF ( ) Compare with Formulae (1) and (2), it can be seen that after calculating the component mean, the positive and negative pairs of white noise can offset each other, which can optimize the denoising performance of the E t (IMF ( )) i 1 sequence.
Step 2: Calculate the first-order residual of the final Continue to add the IMFs with positive and negative white noise to the residual series r t ( ) 1 , and construct a new signal sequence r t a E n t ( ) + (−1) ( ( )) q i 1 11 . Then, decompose the obtained new series n times to obtain the second-order component: According to the same idea, the second-order residual of the final component r t ( ) 2 can be calculated: The flowchart of complete ensemble empirical mode decomposition (EMD) with adaptive noise model for decomposing the carbon price signals.
Step 3: Repeat the process of Steps 1 and 2 to acquire And then the final K + 1 order IMF component of Similarly, the final K + 1 order component residual of r t ( ) 2 is expressed as: Step 4: Repeat the Step 3, when the extreme number of the remaining components is less than 2, the K IMF component can be acquired from the whole CEEMDAN model. The original signal can be showed as: In the above steps, the CEEMDAN algorithm continues to add new adaptive noise to the residual series after EMD decomposition, reduces the residual noise of the mode component, and serves to solve the mode mixing problem to the maximum extent.

| LSTM model
The carbon price series have nonlinear and nonstationary characteristics, especially the time-frequency IMFs have the attribute of time scale heterogeneity. Therefore, this paper conducts the LSTM model, which has the advantages of financial data fitting to capture the timefrequency characteristic and fit the nonlinear signals. As an improved style of the traditional recurrent neural network, the special cell structure of the LSTM model is the key to fitting the complex financial time series. 42 Where, the long memory ability of cell structure is mainly decided by the designed gate structure, namely, forget gate, input gate, and output gate. The designation of those gate structures is used to update the carbon pricing factors. The training process of each gate structure can be described as follows: First, the forget gate is used to decide the forgotten information and filter information from the previous hidden layer network, and employs the sigmoid function to map the current input xt and previous hidden union Ct-1 into the value between 0 and 1. Then, the forgotten output of f t is obtained. Second, the input gate determines the information be saved and updated in the current memory union Ct. That is, the saved information in a current layer can be calculated by the sigmoid function by the original information xt and previous hidden layer memory Ht-1. Thirdly, the output gate is designed to acquire the filtered information of the current cell, the network output of the current layer ht can be finally calculated by the tanh function. Furthermore, the multilayer LSTM model is a network stack of a single LSTM network. Generally, the multi-layer LSTM model can extract the financial data signals, and improve the performance of carbon price prediction more than a single-layer network.
The forget gate is used to filter the input carbon price IFM signals and previously hidden layer characteristics, the forget gate output is shown as: The input gate can saves the filtered carbon price signals, and determines the output of the input gate data: The output gate determines the memorized characteristic in the hidden cell unit and obtains the network output by the activation function: Specifically, the above weight functions need to be calculated separately during the learning process, that is  C t mean the information update vector of the input gate, C t denotes the update vector of the output gate. ht means the network output of the LSTM model. b f , b i , b C , and b o represent the bias information. σ is the sigmoid activation function.

| The proposed CEEMDAN-LSTM model
Based on the nonlinear, nonstationary, and time-frequency heterogeneity characteristics of the carbon price series, this paper constructs a novel mode decomposition machine learning hybrid model of CEEMDAN-LSTM to predict the carbon price. The proposed model combines the strengths of the CEEMDAN and LSTM model, that is, the CEEMDAN model is used to fit the nonlinear and nonstationary carbon price signals, and decompose the original carbon price into different IMF modes. However, the LSTM model is designed to improve the forecasting ability of IMF information obtained by the CEEMDAN model.
The empirical processing follows the idea of "decomposition-prediction-integration." First, during the decomposition stage, the CEEMDAN model is used to decompose the original carbon price to obtain the mode signal IMFs and a residual. These mode signals reflect the time-frequency volatility of carbon prices. Secondly, in the prediction stage, the LSTM model is conducted to fit the nonlinear trend of the mode signals and residual, respectively, so as to reveal the prediction performance of each time scale signal. Finally, in the integration stage, the final predicting value of the proposed model can be calculated by summing the above modes predicting results and residual prediction value.
Furthermore, to prove the prediction superiority of the proposed model, this paper also compares the prediction performance with other benchmark models, and finally supports convincing evidence by the robustness test for different predicting periods. The logic framework of the proposed CEEMDAN-LSTM model is designed as shown in Figure 2.

| Evaluation criteria
This article uses the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) as error indicators to evaluate the deviation between the predicted price and the real price. The smaller the error value, the deviation error from the predicted value to the real one is lower, which denotes the model predicting performance is perfect. Furthermore, we also use the direction accuracy (DA) indicator to test the consistency probability between the investor's prediction direction and the real market trend. The greater the DA value, the predicted performance is more consistent with the real price trend and investors' expectation. The evaluation criteria are as follows: where Y y y y = { , , …, }

| Data and statistical analysis
This paper chooses the settlement price of continuous future contracts of European Union Allowance (EUA) products traded in the European Union Emission Trading Scheme to measure the market price of carbon emissions. The reason is that, as the largest carbon emissions trading market in the world, the EUA's future products have significant characteristics of larger trading volume and stronger liquidity, that can better reflect the market trading price of global carbon dioxide compared with other certification emission reduction (CER) products and EUA spot products. 43,44 The data samples range from June 2, 2009 to November 23, 2021 excluding discontinuous data and holiday nontransaction data, a F I G U R E 2 The logic framework of the proposed complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)long short-term memory (LSTM) model. total of 3195 data are obtained. All data are from the European energy exchange.
Based on the statistical results shown in Table 1, we find that: firstly, the average carbon price is 14.65, the skewness is 1.999 and the kurtosis is 7.152, indicating that the carbon price has an obvious spike and thick tail distribution. A positive skewness suggests that there is an "outlier" phenomenon on the right side of the carbon price distribution. While a high kurtosis means that the big variance is caused by the extreme value of low frequency greater or less than the price data. Furthermore, the significance of JB statistics shows the carbon price does not obey the normal distribution, and LB statistics denotes the carbon price has obvious long memory characteristics. Second, as for the stability test, the ADF critical value is 3.618, which is significant at the 1% level, the result indicates the original carbon price has the characteristic of nonstationary. Third, as for the nonlinear test, when the embedding dimensions are 5 and 10, the critical value are 136.64 and 299.94, which are significant at the 1% level, and the results prove the nonlinear characteristic of the carbon price signals.
The above results conclude that the original carbon price signals have the characteristics of spike and thick tail, nonnormal, nonstationary and nonlinear. So, we believe it is suitable to use the CEEMDAN model to perform timefrequency decomposition of the price signals.

| Mode characteristic analysis of carbon price signals
This article conducts the CEEMDAN model to decompose the original carbon price signals, and then 11 modes and 1 residual represented different time scale information are obtained. Figure 3 shows the volatility trend of each mode component. As we can see from Figure 3, with the signals change from IMF1 to IMF11, the frequency change from strong to weak, the market signals impact on carbon price is becoming more and more stable, as a result, the fluctuation range of each mode is getting smaller and smaller.
According to the frequency period of each mode showed in Table 2, we conclude that: firstly, the signal period of IMF1, IMF2, IMF3, IMF4, IMF5, and IMF6 are basically within 2 months, the average period is relatively short, and the signal shock is high, indicates that those signals have an obvious impact on the short term carbon price. For example, the short-term carbon quota supply and demand and other irregular events are important external drivers of short-term shocks in the carbon price. These explanations are consistent with the conclusion of Ji et al. 45 and Shi et al. 46 that the supply and demand capacity of carbon quotas affected the short-term carbon price significantly. Second, the signal period of the residual term is the longest, and the signal shock frequency is low. The residual means the price change caused by the market macro factors that mainly reflect the long-term market equilibrium. Thirdly the remaining modes can be regarded as the medium-term factors that affect the carbon price signals. Those modes represent the market response of investors to policy adjustment, extreme event impact, and other factors.

| Parameter optimization
Based on the model designation mentioned above, we employ the proposed CEEMDAN-LSTM model to realize the multistep prediction of the original carbon price signals.
To improve the prediction stability, it is necessary to preset and optimize the neural network parameters of the proposed model. As we know, the iterations and neuron nodes are important training parameters, the iterations represent the updated times of data training, and the appropriate number of iterations can accelerate the convergence and training process of the neural network. Hence, under the constraint of the definite loss function, the optimal iteration is the training times that correspond to the lowest objective loss. As a chain network, each hidden layer of LSTM has similar neuron nodes. More neuron nodes can improve the training and generalization ability of the proposed model, but it may also cost more training time and produce an overfitting phenomenon. 47 Therefore, referring to the experience of Shen et al. 48 and Yun et al., 49 this paper uses the step-by-step experimental method to determine the optimal network parameters, specifically, we calculate the model training error when the iteration are 50, 100, 200, 300 and the neuron nodes of the proposed model are 4,8,16,32,64, and 128 separately. The results conclude that when the iteration number is 200 and the neuron node is 128, the error indicators of RMSE, MAE, and RMSE are 0.851783, 0.601614, and 0.021578, respectively (as shown in Table 3), that is the minimum value of the whole test sample. So, we use those parameters to predict the carbon price.  Mode  IMF1  IMF2  IMF3  IMF4  IMF5  IMF6  IMF7  IMF8  IMF9  IMF10

| Predicting results analysis
According to the defined network parameters, we conduct the CEEMDAN-LSTM model to perform the multi-step prediction of carbon price mode signals, the prediction performance can be shown in Figure 4. Furthermore, to evaluate the superiority and relative advantages of the proposed model, we also choose the back propagation (BP) neural network and the gated recurrent unit (GRU) network as the comparison models, then construct a cluster of benchmark models based on other decomposition technologies such as CEEMD, EEMD, and EMD. The benchmark models designed with the same network F I G U R E 4 Intrinsic mode function (IMF) prediction performance of the carbon price signals based on complete ensemble empirical mode decomposition with adaptive noise-long short-term memory model. parameters, and the research results put in Table 4 conclude that: First, compared with other benchmark models, the prediction performance of the hybrid model with LSTM show high prediction accuracy in both error indicators and expected indicator. In particular, the prediction superiority of the CEEMDAN-LSTM model is the best, specifically, the error values of RMSE, MAE, and MAPE are 0.638342, 0.448695, and 0.015666, respectively. The error is the smallest of all models, the market expectation indicator DA is 0.687631, which is the largest of all models. These pieces of evidence reveal that the proposed CEEMDAN-LSTM model has significant prediction accuracy and stability, the model can not only reveal the multiscale time-frequency driving mechanism of carbon price signals but also has good market potential, it can provide technical support for investors to take market judgment and predict market prospects.
Second, the hybrid prediction models constructed based on the CEEMDAN model suggest perfect prediction accuracy. That is, CEEMDAN-LSTM, CEEMDAN-GRU, and CEEMDAN-BP models have the smallest error and the largest market expectation value in various hybrid models. For example, the error indicators RMSE, MAE, and MAPE of the CEEMDAN-GRU model are 0.650083, 0.472725, and 0.01731, respectively, the market expected indicator DA is 0.542066. The error indicators RMSE, MAE, and MAPE of the CEEMDAN-BP model are 4.264262, 2.585904, and 0.066193, respectively, the market expected indicator DA is 0.542066. According to Figures 5 and 6, we can also clearly observe that the carbon price prediction performance of the hybrid models based on CEEMDAN technology has significant advantages, the deviation between the predicted value and the real one is the smallest, and the dynamic error is relatively stable. These pieces of evidence argue that the CEEMDAN model has obvious strengths in carbon price signals decomposition, it can describe the multiscale time-frequency characteristics of carbon price more accurately. The result indirectly proves that the CEEMDAN model can reduce the carbon price signal noise to the greatest extent, the proposed predicting model can provide a valuable reference for investors to judge the market situation and formulate investment strategies.

| Robustness test of the CEEMDAN-LSTM model
The proposed CEEMDAN model can capture the multiscale time-frequency characteristics of carbon price signals, and reveal the special carbon premium driving mechanism. However, the LSTM model has significant advantages in dealing with financial time series. Therefore, to prove the stability and robustness of the proposed CEEMDAN-LSTM model in different time scales, this paper readjusts the time scale of the carbon price series and tests the model's robustness in the long-term, medium-term, and short-term prediction periods. Where, the length of the last 3 years, 2 years, and 1 year of the carbon price series are intercepted into the long-term, medium-term, and short-term, respectively. That is, the last 700, 460, and 230 trading days of the sample are taken as the testing set, and the other samples are regarded as the training set. The network structure of the test model is consistent with the previous design. The empirical results in Table 5 showed that, firstly, in terms of the long-term, medium-term, and short-term carbon price prediction performance, the CEEMDAN-LSTM model has a lower error indicators value than that of other benchmark models. For example, the long-term prediction error indicators of RMSE, MAE, and MAPE are 2.955937, 1.369345, and 0.024355, which are significantly lower than other benchmark models of CEEMDAN-GRU and CEEMDAN-BP. Similarly, the medium and shortterm prediction errors are also lower than in other models. Compared with Figures 7-9, it can be found that the predicted trend of the CEEMDAN-LSTM model is basically consistent with the real carbon price, the dynamic error deviation is much smaller than that of other models. This shows that the prediction superiority of the proposed model has reliable robustness on different time scales. The CEEMDAN has been proved to be an effective technology to reduce data decomposition noise and reconstruction error, the LSTM model can also nonlinear map the complex modes with time scale differences.
Second, the CEEMDAN-LSTM model has the smallest error performance in the short-term carbon price prediction, the dynamic error is also small F I G U R E 6 Dynamic predicting error of proposed complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)long short-term memory (LSTM) model and its benchmark models. BP, back propagation; CEEMD, complementary ensemble empirical mode decomposition; EEMD, ensemble empirical mode decomposition; EMD, empirical mode decomposition; GRU, gated recurrent unit; MAE, mean absolute error; MAPE, mean absolute percentage error; RMSE, root mean square error.
T A B L E 5 Predicting performance of the proposed and benchmark models in the long-medium-short term of carbon price series. compared with other long-term and medium-term prediction performance. That is, the short-term prediction error values of RMSE, MAE, and MAPE are 1.013417, 0.703062, and 0.011943, respectively, which are significantly lower than the long-term and medium-term error values. With the prediction period reducing from the long-term to the short-term, the carbon price prediction error of the proposed model decreases, as a result, the prediction accuracy is gradually improving. As shown in Figures 10 and  11, the out-of-sample prediction error of the proposed CEEMDAN-LSTM model gradually decreases as the reducing of prediction period from the long term to the short period. The reasonable explanation is that the reducing of the prediction period means the samples used for model training are increasing so that the network structure of the proposed model is improved to the greatest extent. Therefore, the out-of-sample prediction performance shows strong accuracy and stability. These conclusions are basically similar to Lesort's idea, the better fitting performance always benefits from the optimized machine learning structure with more training samples. 50 The short term predicting performance of the proposed complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)-long short-term memory (LSTM) model and other benchmark models. BP, back propagation; GRU, gated recurrent unit.
F I G U R E 10 Dynamic root mean square error (RMSE) error of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)-long short-term memory (LSTM) and its benchmark models. BP, back propagation; GRU, gated recurrent unit. PROSPECTS

| Conclusion
Global carbon dioxide emissions from energy combustion and industrialization have reached their highest level in 2021. The sharp increase in carbon emissions not only weak the emissions reduction effort over the years, but also affect the long-term sustainability of economic growth and human health. Therefore, it is of great significance to study the emissions reduction mechanism of carbon dioxide, especially exploring the price-driving mechanism of the carbon emission trading market. For existing carbon price studies, the EMD-type models have become the mainstream method to decompose carbon price signals into multiscale modes. However, these models have some theoretical defects in practice, such as mode mixing problems and lower noise reduction performance during the data decomposition process. The innovation of this paper is constructing a new mode decomposition hybrid carbon price prediction model CEEMDAN-LSTM, and testing the prediction robustness of the proposed model in different time scales and prediction periods. Based on the idea of "decomposition-prediction-integration," this model reveals the driving mechanism of carbon premium from the perspective of a multidimensional time scale. The main conclusions are summarized as follows: First, the CEEMDAN model has advantages in describing the multi-scale time-frequency characteristics of carbon price signals. The proposed CEEMDAN-LSTM model can effectively map the nonlinear carbon price. Compared with other benchmark models, the prediction accuracy and stability of the proposed model have been convinced. The results show that this model can provide new evidence for revealing the carbon premium from the point of multiscale time-frequency characteristics. The results can also provide valuable reference for investors, emissions reduction entities, and carbon market regulators to judge the market situation, and formulate investment strategies and other market transactions. Furthermore, the carbon market price will be more efficient with the use of the proposed model and guide the polluting entities to achieve the emission reduction goals and sustainable development.
Second, the carbon price prediction results of the CEEMDAN-LSTM model have significant robustness and stability. That is, no matter for the long-term, medium-term, and short-term period, the prediction error of the proposed model is the smallest compared with other benchmark models, and the prediction accuracy is higher. Particularly, the short-term prediction performance of the CEEMDAN-LSTM model is outstanding. That is, with the reducing of the prediction period, the samples used for model training and parameter optimization increasing, and the out-ofsample prediction accuracy and stability are improving. This conclusion can provide technical reference for market investors, financial companies, and emissions reduction entities to predict the market trend of the carbon price, contribute to more valuable market decisions. F I G U R E 11 Dynamic mean absolute percentage error (MAPE) error of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)-long short-term memory (LSTM) and its benchmark models. BP, back propagation; GRU, gated recurrent unit.

| Prospects
In this paper, a novel mode decomposition machine learning hybrid model of CEEMDAN-LSTM is constructed for forecasting carbon price. The conclusions convince the superiority and robustness of the proposed model in fitting and predicting the European carbon price.
Then, the EMD-type of decomposition technologies are suitable for the signal decomposition of a single price series, which makes the proposed model of this paper can only predict the price based on the carbon price lag series, the result may neglect the nonlinear impact of structural pricing factors such as coal, crude oil, and capital market factors on the carbon price. Actually, apart from the structured factors, unstructured factors including investor sentiment and policy factors are also important factors that cannot be ignored. Therefore, for improving the prediction accuracy, future research needs to employ the hybrid models of machine learning, text mining, and other technologies to reveal the dynamic impact of the structured and unstructured factors on carbon price.