It has become an international consensus that carbon emissions (CE) lead to global warming [1]. To this end, many countries have signed the Paris Agreement to work together to reduce CE [2]. Although some measures to reduce CE have been tried, some countries still have a lot of CE [3]. For example, in recent years, the CE of the US and China have been in the top two, and Japan, Russia, and India ranked in the top five in CE [4]. In addition, before 2005, the US ranked first in CE, while China ranked first in CE after 2005. However, the per capita CE of the US is more than twice that of China. And in 2021, the US will account for 23.03% of the world’s total CE [4]. Therefore, the US is still the major CE country in the world.

The CE of the US mainly comes from the consumption of fossil energy, such as natural gas, oil, and coal. To actively respond to the impact of CE on the global climate, the US has put forward a “zero CE action plan”, which draws on and extends the previous two United Nations-led sustainable development solution network reports, and the plan aims to achieve zero CE by 2050. However, for a long time to come, natural gas, oil, and coal are still the main fossil fuels [5]. Among these three energies, different energies have different contributions to CE, and this contribution rate is very important for the government to formulate corresponding policies [6]. In the past few decades, the contribution of coal, oil, and natural gas to CE has gradually changed and is not static. For example, the contribution of natural gas to CE is gradually increasing. Therefore, this fluctuation brings challenges to accurately predicting the contribution of different fossil energy to CE.

In recent years, many studies have focused on forecasting CE [7]. From the perspective of models, it can be roughly divided into three categories [8]. The first type is the traditional model. Meng and Niu [9] think that except for a few countries, the main figures of CE from fossil energy are S-shaped curves. They select the logistic function to simulate the S-shaped curve and to improve the goodness of fit by applying three algorithms. The results indicate that their model is better than the linear model based on simulation risk and goodness of fit. Pérez-Suárez and López-Menéndez [10] employ the environmental Logistic curve and extended environmental Kuznets curve to explain CE in different countries and their result allows them to obtain some ex-ante projections. The second type is the grey model (GM) based on grey theory. Lin et al. [11] proposed the GM (1, 1) to predict CE in Taiwan, and the results show the average residual error of the GM (1, 1) is below 10%. Wang and Ye [12] introduce the power exponential term of the relevant variables as exogenous variables in a multivariable GM. Their results indicate that the model developed by them can reflect the mechanism of the non-linear effects of the gross domestic product on CE from fossil energy consumption. Wang and Li [13] proposed a new method for discussing the relationship between CE and economic growth. This method is based on the derived non-equigap grey Verhulst model optimized by particle swarm optimization. Their results show that the Chinese government should take effective measures to reduce CE. In addition to the above GMs, some researchers also proposed some innovative GMs, such as a new nonlinear discrete GM (1, N) in Ding et al. [14], a hybrid Verhulst-GM (1, N) in Ofosu-Adarkwa et al. [15], a novel grey rolling prediction model in Zhou et al. [16], a novel fractional grey Riccati model in Gao et al. [17], a new cumulative GM in Qiao et al. [18], and a novel continuous fractional nonlinear grey Bernoulli model with grey wolf optimizer in Xie et al. [19]. The third type is machine learning model based on artificial intelligence technology. Sun et al. [20] establish some factors impacting CE based on 22 influencing factors. On this basis, they put forward a coupling model concerning extreme learning machine (ELM) optimized by particle swarm optimization (PSO). The results show that the developed PSO-ELM outperforms the ELM and back propagation neural network in predicting CE. Zhao et al. [21] establish a hybrid of the mixed data sampling regression model and back propagation neural network (BPNN) to forecast CE and their developed model is remarkably better than several benchmark models. In addition, some models have also been developed, such as improved Gaussian regression in [22], a hybrid algorithm based on an improved lion swarm optimizer in [23], and a model according to BPNN based on random forest and PSO in [24].

In addition to simply studying CE prediction, there are also studies on the relationship between CE and other indicators [25]. For example, Pao and Tsai [26] analyzed the relationship between CE, energy consumption, and economic growth. Mohmmed et al. analyzed the relationship between CE and economic growth, development, and human health [27]. Menyah and Wolde-Rufael analyzed the relationship between CE and nuclear energy, renewable energy, and economic growth in the US [28]. Mason et al. discussed the relationship between CE energy demand, and wind generation in Ireland using evolutionary neural networks [29].

The WT [31, 32] can realize the time subdivision of the signal at high frequency and the detail subdivision at low frequency, complete the multi-scale refinement of the signal, and overcome the shortcomings of the Fourier transform (FT).

The wavelet reconstruction process is the inverse process of wavelet decomposition, which can be referred to in references [33–35] for details.

The LSTM network was presented by [36, 37], which avoids the phenomenon of gradient disappearance and gradient explosion in the traditional RNN. The LSTM network includes three gates, namely input, forgetting, and output gates. The controlling equation in the LSTM network is displayed in the following Eqs. (1) ~ (6):

where f_t means the output data of the forgetting gate at the current moment; x_t means the current input value; g_t and i_t means the value handled by the Sigmoid and tanh function in the input gate; W_o, W_i, W_f, W_c, and b_i, b_c, b_o, b_f mean the weight matrix and bias vectors of the corresponding threshold; o_t means the output data of output gate at the current moment; S_t and S_t-1 mean the output cell state at the current moment and the last moment; h_t and h_t-1 mean the data output information at the current moment and the last moment.

The SSA was studied by [38] according to the anti-predation and predation behavior of sparrows during the foraging process. The SSA has divided the entire sparrow population into discoverers, joiners, and scouts. The finder is responsible for providing foraging areas and directions for the entire sparrow population, and the scout is responsible for alerting the sparrow population, the optimal position is fixed by the fitness value. It is found that the finder location update and initial population play an important role in the property of SSA. Therefore, we will improve SSA in two aspects proposed by [39]. These two aspects are improvement of Sobol initial population and improvement of finder location update, respectively. For improvement of Sobol initial population, the initial value can be generated through Sobol sequence. For improvement of finder location update, the location update of the finder will to some extent affect the sparrow’s search ability. Therefore, improvements should be made to the public display of the finder’s location update. Moreover, the T distribution will be introduced to SSA to improve the local search performance in the later stage proposed by [40].

Since 1965, the US has ranked first in CE in the world. Until 2005, China’s CE exceeded the US, and the US ranked second. However, the per capita CE of the US still ranked first in the world. The reason for the high CE in the US is the massive use of coal, natural gas, and petroleum. Due to the increasingly serious problem of global warming, reducing CE has become imminent. Finding the main sources of CE and formulating corresponding policies are the main ways to reduce CE. Because of the large amount of CE and the collectability of data in the US, it is very representative, and it is very reasonable to study it as an example.

The raw data for the average contribution rate forecasting of CE generated by coal, natural gas, and petroleum in the US are taken from EIA (https://www.eia.gov/). A total of 1836 data are selected, and the data of CE generated by coal, natural gas, and petroleum are 612, 612, and 612, and the starting and ending time of selection is from January 1973 to December 2023. For the CE data generated by coal, its max, min, mean and SD are 203, 49, 138, and 34. For CE data generated by natural gas, its max, min, mean and SD are 195, 50, 101, and 27. For CE data generated by petroleum, its max, min, mean and SD are 241, 133, 191, and 16.

In this study, the prediction process consists of five parts, which are described in detail as follows:

In this section, the CE produced by the coal is decomposed by using Haar wavelet, Daubechies wavelet, and Symflets wavelet into 231 high-frequency components and 231 low-frequency components. Then ISSA-LSTM is utilized to forecast the 462 components respectively, and the final prediction results are obtained through reconstruction. Three parameters of LSTM, which are learning_rate, maxEpochs, and numHiddenUnits, are optimized by ISSA.

Fig. 1 gives the comparison of prediction results of the test set based on different layers of different wavelet’s orders and ISSA-LSTM for the coal. In Fig. 1 (a), the prediction results obtained by OH, T¹H, T²H, F¹H, F²H, S¹H, and S²H (from one layer to seven layers of Haar wavelet) can closely follow the changing trend of the raw data at many points. However, the difference between the predicted value obtained by F²H and S¹H and the observed value at many points is a little larger than that obtained by other layers, especially for F²H.

Fig. 1 (b)–(f) gives the comparison of forecasting results of the test set based on one-seven layers of Daubechies wavelet’s two-six orders (ODBT¹, T¹DBT¹, T²DBT¹, F¹DBT¹, F²DBT¹, S¹DBT¹, S²DBT¹, ODBT², T¹DBT², T²DBT², F¹DBT², F²DBT², S¹DBT², S²DBT², ODBF¹, T¹DBF¹, T²DBF¹, F¹DBF¹, F²DBF¹, S²DBF¹, S²DBF¹, ODBF², T¹DBF², T²DBF², F¹DBF², F²DBF², S¹DBF², S²DBF², ODBS, T¹DBS, T²DBS, F¹DBS, F²DBS, S¹DBS, S²DBS) and ISSA-LSTM for the coal. From Fig. 1 (b), it can be seen that the prediction results obtained by ODBT¹, S¹DBT¹, and S²DBT¹-ISSA-LSTM greatly deviate from the raw data, the forecasting results obtained by T¹DBT¹, T²DBT¹, F¹DBT¹, and F²DBT¹ can closely follow the changing trend of the raw data, and the difference with the observed value is small. Moreover, from Fig. 1 (c), the prediction results obtained by F²DBT²-ISSA-LSTM greatly deviate from the raw data, the prediction results gotten by the other layers can closely follow the changing trend of the raw data, and the difference with the raw data is small. In Fig. 1 (d)–(f), the predicted values obtained from ODBF¹, T¹DBF¹, T¹DBF¹, F¹DBF¹, F²DBF¹, S¹DBF¹, S²DBF¹, ODBF², T¹DBF², T¹DBF², F¹DBF², F²DBF², S¹DBF², S²DBF², ODBS, T¹DBS, T²DBS, F¹DBS, F²DBS, S¹DBS, S²DBS can follow the trend of the raw data, and the difference is not large.

Fig. 1 (b)–(f) gives the comparison of forecasting results of the test set based on one-seven layers of Symflets wavelet’s two-six orders (OSYMT¹, T¹SYMT¹, T²SYMT¹, F¹SYMT¹, F²SYMT¹, S¹SYMT¹, S²SYMT¹, OSYMT², T¹SYMT², T²SYMT², F¹SYMT², F²SYMT², S¹SYMT², S²SYMT², OSYMF¹, T¹SYMF¹, T²SYMF¹, F¹SYMF¹, F²SYMF¹, S¹SYMF¹, S²SYMF¹, OSYMF², T¹SYMF², T²SYMF², F¹SYMF², F²SYMF², S¹SYMF², S²SYMF², OSYMF, T¹SYMS, T²SYMS, F¹SYMS, F²SYMS, S¹SYMS, S²SYMS) and ISSA-LSTM for the coal. For Fig. 1 (g), (h), and (j), the deviation between the forecasting data obtained by different layers of Symflets’s two-, three-, and five-orders and the raw data is small. Furthermore, in Fig. 1 (i) and (k), although the deviation between the predicted value obtained by F²SYMF¹ and S²SYMS and the observed value is large, the deviation between the forecasting data obtained by other layers (e.g. T¹SYMF¹ and F¹SYMF¹ et al.) and the observed value is still small.

Table 1 displays the prediction error based on seven different layers of different orders of different wavelet and ISSA-LSTM for the coal.

In Table 1, the prediction error of the S²H is the smallest compared with the prediction error of the one to six layers of the Haar wavelet according to six error evaluation indicators including U1, U2, RMSPE, RMSE, MAPE, and MAE. Therefore, based on the prediction error comparison results of different layers of the Haar wavelet, using the S²H to predict CE generated by coal is the best.

In Table 1, the prediction error of F²DBT¹ is the smallest compared with the prediction error of the other layers of Daubechies wavelet’s two orders. For the different layers of Daubechies wavelet’s three orders, the prediction error of the two layers is smallest. Concerning the different layers of Daubechies wavelet’s four, five, and six orders, the prediction error of five, six, and seven layers are smallest, according to six different error evaluation indicators. Compared with F²DBT¹, T¹DBT², S¹DBF², and S²DBS, the prediction error of F²DBF¹ is the smallest. And U1, U2, RMSE, MAPE, MAE, and RMSPE of F²DBF¹ reduced by 0.2912, 0.6088, 1.6227, 0.3125, 2.4501, 0.2876, 0.4163, 0.4384, 0.0905, 1.1624, 0.0829, 2.9960, 0.0091, 0.0136, 0.0210, 0.0053, 0.0370, 0.0060, and 0.0036, 0.0033, 0.0018, 0.0171, 0.0016, 0.0282 compared with that of the optimal of Daubechies wavelet’s other orders.

In Table 1, the prediction error of seven, seven, two, and three layers of Symflets wavelet’s two-, three-, four-, and five-orders is the smallest compared with the prediction error of the different layers of Symflets wavelet’s two-, three-, four-, and five-orders. For the prediction error of different layers of Symflets wavelet’s six orders, MAE and MAPE of two layers are the smallest, however, RMSE, RMSPE, U1, and U2 of six layers are smallest, that is to say, the six prediction error indicators are not uniform. So, it is uncertain whether the prediction performance of two layers is good or the prediction performance of six layers is good based on the Symflets wavelet’s six orders. In a word, compared with the optimal layers of Symflets wavelet’s two-, three-, four-, and six-orders, the prediction error evaluation index of T²SYMF² is the smallest.

By comparing the prediction error of S²H, F²DBF¹, and T²SYMF², it is found that the prediction error of F²DBF¹ is the smallest based on six indexes. Therefore, F²DBF¹ is suitable for decomposing CE generated by coal, and using ISSA-LSTM to predict its components can obtain lower prediction error indicators.

Similar to section 4.1, in this Subsection, the optimal layers of wavelet’s orders based on ISSA-LSTM for CE produced by the natural gas are determined. Fig. 2 represents the comparison of prediction results of the test set based on different layers of different wavelet’s orders and ISSA-LSTM for the natural gas including the comparison of different layers of Haar wavelet, Daubechies wavelet’s different orders, and Symflets’s different orders. To better show the coincidence between the forecasting data and the observed data, some predicted values are hidden.

In Fig. 2 (a), for a total of 59 predicted values, some predicted values deviate greatly from the observed data, for example, the second and 52nd points obtained by OH, and the fourth point obtained by F¹H, F²H, S¹H, and S²H. The rest of the predicted values obtained from OH et al. have a small deviation from the observed values, such as the 57th, 58th, and 59th points.

In Fig. 2 (b) – (f), different layers of Daubechies wavelet’s different orders can follow the trend of the raw data. However, at some points, the deviation between the predicted value and the observed value is very large, such as the prediction values obtained by ODBT¹ at the second, third, fourth, 51st, 52nd, and 53rd points, by F²DBT¹ at 51st, 52nd and 53rd, 55th, 56th, and 57th, by ODBT² at second, third, 6th, 7th, 8th, 55th points, by ODBF¹ at fourth point, by F²DBF² at 51 st, 52 nd th, 53rd, 54 th, 55th, 56th, 57th, and 58th points, and by F²DBS at 51st, 52nd, 53rd, 54th, 55th, 56th, and 57th points. In general, the predicted value obtained from different layers of Daubechies wavelet’s four orders is in good agreement with the observed value at most points.

In Fig. 2 (g), at the front of the breakpoint, except for the large difference between the forecasting data obtained from OSYMT¹ and the observed data, the predicted value obtained from T¹SYMT¹, T²SYMT¹, F¹SYMT¹, F²SYMT¹, S¹SYMT¹, and S²SYMT¹ is in good agreement with the observed value. At the back of the breakpoint, the predicted value obtained from different layers of Symflets’s different orders is very different from the raw data. In Fig. 2 (h), the predicted value obtained from OSYMT² is quite different from the raw data at most points. In Fig. 2 (i), behind the breakpoint, the predicted value obtained from S¹SYMF¹ is quite different from the raw data. In Fig. 2 (j), the difference between the predicted value obtained from different layers of Symflets’s different orders and the observed value is very small. Similar to the prediction result in Fig. 2 (h), the predicted value obtained from OSYMS is quite different from the observed value at most points in Fig. 2 (k).

Table 2 gives the prediction error based on seven different layers of different orders of different wavelet and ISSA-LSTM for the natural gas. It can be seen from Table 2 that the forecasting error of two layers of Haar wavelet is the smallest, which is different from other layers of Haar wavelet based on RMSE, MAE, MAPE, RMSPE, U1, and U2.

As can be seen from Table 2, for two, three, four, and six orders of Daubechies wavelet, the prediction error obtained by two, two, two, and two layers is the smallest. For five orders of Daubechies wavelet, the prediction error obtained by six layers is the smallest. In addition, compared with two, two, two, and two layers of Daubechies wavelet’s two, three, four, and six orders, six layers of Daubechies wavelet’s five orders has the smallest prediction error, and the U1, U2, RMSE, MAPE, MAE, and RMSPE of six layers of Daubechies wavelet’s five orders are 0.0009, 0.0018, 0.2580, 0.2277, 0.0934, 0.0030, 0.0009, 0.0018, 0.2529, 0.1508, 0.0847, 0.0017, 0.0009, 0.0017,0.2466, 0.1413, 0.0582, 0.0022, and 0.0003, 0.0005, 0.0772, 0.0711, 0.0459, 0.0003, lower than those of two, two, two, and two layers of Daubechies wavelet’s two, three, four, and six orders.

As can be seen from Table 2, for two, three, and four orders of Symflets wavelet, the prediction error obtained by two, two, and two layers is the smallest. For five orders of Symflets wavelet, the prediction error obtained by six layers is the smallest. For six orders of Symflets wavelet, the prediction error obtained by four layers is the smallest. For optimal layers of Symflets wavelet’s different orders, the prediction error obtained by S¹SYMF² is the smallest.

According to the prediction error results in Table 2, the prediction error of S¹DBF² is the smallest, and compared with T¹H and S¹DBF², the U1, U2, RMSE, MAPE, MAE, and RMSPE of S¹SYMF² is reduced by 0.0097, 0.0194, 2.7225, 1.0423, 0.4981, 0.0182, and 0.0019, 0.0038, 0.5368, 0.3274, 0.1744, 0.00368. Therefore, it is most appropriate to use S¹SYMF² to decompose CE generated by natural gas, which can obtain the minimum forecasting error and more accurate prediction results.

In this subsection, the optimal layers of wavelet’s orders based on ISSA-LSTM for CE produced by petroleum are determined. The wavelet basis function used is still Haar, Daubechies, and Symflets.

Fig. 3 (a) demonstrates the comparison of forecasting results of the test set according to different layers of the Haar wavelet. It can be seen from Fig. 3 (a) that the predicted value obtained from OH, T¹H, T²H, F¹H, F²H, S¹H, and S²H is in good agreement with the observed value on the whole, but there are still some points where the predicted value and the observed value differ greatly, for example, the predicted value obtained from OH and T¹H et al. is at point 4. Fig. 3 (b)–(f) demonstrates the comparison of forecasting results of test sets based on different layers of Daubechies wavelet’s different orders. For Fig. 3 (b)–(f), the forecasting data and the raw data are in good agreement, except for the large deviation between the forecasting data and the raw data obtained from ODBS, at some points.

Fig. 3 (g)–(k) demonstrates the comparison of forecasting results of test sets based on different layers of Symflets wavelet’s different orders. It can be seen from Fig. 3 that the forecasting data obtained from different layers of Symflets wavelet’s different orders is in good agreement with the observed data, especially the predicted value obtained from OSYMF², T¹SYMF², T²SYMF², F¹SYMF², F²SYMF² S¹SYMF², and S²SYMF² is in the best agreement with the observed value. However, the forecasting data gotten from OSYMT¹ deviates greatly from the raw data, at individual points, such as points 4 and 5 in Fig. 3 (g). The forecasting data obtained from OSYMS deviates greatly from the raw data, at individual points, such as points 4 and 5 in Fig. 3 (k).

Table 3 displays the prediction error based on seven different layers of different orders of different wavelet and ISSA-LSTM for the petroleum. In Table 3, it can be seen that the prediction error of F¹H is the smallest, based on U1, U2, RMSE, MAPE, MAE, and RMSPE, but it is not much different from that of S¹H and S²H.

In Table 3, compared with the prediction error of other layers of Daubechies wavelet’s two orders, the prediction error of S¹DBT¹ is the smallest. Among different layers of Daubechies wavelet’s three orders, the prediction error of T²DBT² is the smallest. For different layers of Daubechies wavelet’s four, five, and six orders, the prediction error of three layers of Daubechies wavelet’s five, six, and seven orders is the smallest. Furthermore, the prediction error of S²DBS is 1.8623, 0.3367, 0.6033, 0.0109, 0.0050, 0.0099, 0.2356, 0.1054, 0.1684, 0.0018, 0.0006, 0.0013, 0.7782, 0.0622, 0.1270, 0.0060, 0.0020, 0.0041, and 0.7836, 0.0574, 0.1307, 0.0066, 0.0021, 0.0042 lower than that of six, three, five, and six layers of Daubechies wavelet’s two, three, four, and five orders. So, for different layers of Daubechies wavelet’s different orders, the prediction performance of S²DBS is the best, from the perspective of the prediction error evaluation index.

It can be seen from Table 3 that four, seven, four, seven, and four layers of symflets wavelet’s two, three, four, five, and six orders have the smallest prediction error and the best prediction performance, compared with other layers of symflets wavelet’s two, three, four, five, and six orders. Moreover, S²SYMF² has the smallest prediction error, compared with four, seven, four, and four layers of symflets wavelet’s two, three, four, and six orders., the prediction error of S²SYMF² is 2.6265, 0.4120, 0.7792, 0.0155, 0.0070, 0.0139, 0.4456, 0.1807, 0.3024, 0.0030, 0.0012, 0.0024, 1.3517, 0.2814, 0.4887, 0.0078, 0.0036, 0.0072, and 0.2607, 0.0813, 0.1362, 0.0014, 0.0007, 0.0014 lower than that of four, seven, four, and four layers of symflets wavelet’s two, three, four, and six orders.

Based on the comparison results of prediction errors in Table 3, compared with F¹H and S²DBS, S²SYMF² has smaller prediction errors, that is, it has better prediction performance, and is more suitable for processing CE generated by the petroleum, to obtain higher prediction accuracy.

To verify the impact of different components of the proposed model on the prediction results, taking the prediction of carbon emissions from coal as an example, this study investigates the effects of separately using LSTM and LSTM-ISSA on the prediction of carbon emissions from coal. The RMSE, MAE, MAPE, RMSPE, U1 and U2 in predicting carbon emissions from coal using LSTM alone are 8.8167, 2.6239, 7.8382, 0.1002, 0.0501, and 0.0988, while the errors in predicting carbon emissions from coal using LSTM-ISSA are 40.9320, 6.1432, 30.7916, 0.3348, 0.1949, and 0.3327, respectively. Compared with using LSTM alone for prediction, the prediction error of LSTM-ISSA is reduced, indicating that adding the ISSA module reduces the prediction error. The prediction error obtained by using LSTM-ISSA to predict the time series after wavelet processing has been improved compared to using LSTM-ISSA. This indicates that adding a wavelet processing module further improves the prediction performance.

Fig. 4 demonstrates the comparison of forecasting results according to different preprocessing algorithms including F²DBF¹, S¹SYMF², S²SYMF², empirical mode decomposition (EMD) [46], ensemble empirical mode decomposition (EEMD) [47], and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)-ISSA-LSTM for the coal, natural gas, and petroleum [48]. Fig. 4 (a) demonstrates that the forecasting data obtained by other preprocessing methods can well match the raw data, except for the large deviation between the predicted values obtained by EMD-ISSA-LSTM and the actual values. The predicted value obtained by different pretreatment methods is in good agreement with the actual value, and there is no significant deviation between the predicted value and the actual value at any point in Fig. 4 (b). The predicted values obtained by different pretreatment methods are basically in good agreement with the actual values, however, several forecasting data gotten by EMD-ISSA-LSTM deviate greatly from the raw data, for example at points 33, 34, and 35 in Fig. 4 ©.