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1.INTRODUCTIONIn order to actively respond to the carbon peak carbon neutral development strategy, most countries in the world have taken the initiative to commit to carbon peak carbon neutral time node, and China declared in the general debate of the 75th session of the United Nations General Assembly that it will achieve carbon peak by 2030 and carbon neutral by 2060. Since then, provinces, cities and counties have responded positively, and are gradually carrying out research and exploration on the implementation path of carbon peaking, energy conservation and emission reduction path, and the formulation, implementation and implementation of these paths are inseparable from the accurate prediction of carbon emissions, especially the prediction of carbon emissions based on different energy consumption industries. At present, from the perspective of research objects, most studies on carbon dioxide emissions are based on a single prediction of carbon emissions in the terminal energy industry. For example, LMDI method1,2 is used to analyze the influencing factors of carbon emissions in the terminal energy industry. STIRPAT model was used to analyze and predict the influencing factors of carbon emissions from agriculture3, industry4,5 and transportation6, and grey model7 was used to predict carbon emissions from agriculture. The construction industry often uses machine learning8,9 to predict carbon emissions, and the carbon emissions of residential life are predicted by ARIMA model. For example, Xu10 analyzed carbon emissions in Northwest China from the perspective of quantity and spatial pattern based on ARIMA model and standard deviation ellipse, and found that carbon emissions in Northwest China showed a slowly rising trend, and the main area of carbon emission moves to the northwest. There are few studies on the carbon emission prediction of the service industry, especially that of the seven terminal energy consumption industries. The existing studies only comprehensively predict the carbon emission of some industries11. From the perspective of forecasting methods, different terminal energy consumption industries have different data hiding relationships, and there is a single model for carbon emission forecasting. At present, the mainstream models for CO2 emission prediction include STIRPAT model12, LEAP model13, BP neural network model, etc. For example, Pan14 applied STIRPAT model combined with scenario analysis to analyze the time of carbon peaking, and Huang et al.15 applied STIRPAT model to study the positive promoting effect of population, GDP and energy intensity on carbon emissions from energy consumption. LEAP forecasting model has a wide range of applications and can be used to study and analyze the carbon emission trend of various industries in cities, provinces and countries. Ma et al.16,17 applied LEAP model to study the provincial and municipal transportation industry, and She et al.18 studied the key influencing factors of China’s energy carbon emission by combining Markov model and LEAP model. The advantage of BP neural network model for carbon emission prediction is that the prediction accuracy is high, but the disadvantage is that the prediction path is unknown, and other models need to be combined to predict carbon emission. For example, Ji et al.19 combined grey analysis and BP neural network model, and Zhang et al.20 optimized BP neural network model with IPSO model, and found that the error of the BP neural network model optimized by IPSO was lower in predicting carbon emissions in Shandong Province. Some scholars use ARIMA to forecast carbon emissions by combining other models, and find that it has the advantages of simple operation and few variable requirements. For example, Hu et al.21,22 combined BP and ARIMA models for analysis, and found that the prediction error of China’s carbon emission is closer to zero than that of BP and ARIMA separately. Han et al.23 combined the recursive least squares method with the forgetting factor and ARIMA model to make short-term one-step prediction of transportation. In summary, combined with the current research status of carbon emission prediction, focusing on different end-use energy consumption industries, considering that XGBoost model is good at handling small samples and highly volatile data24,25, and the linear and nonlinear prediction relationship of different Sectors should be comprehensively considered. Therefore, ARIMA and XGBoost are complementary to each other to predict the carbon emission time series of different end energy consumption industries (agriculture, industry, construction, service industry and others, energy production and processing, transportation, and residential life), and MAPE (mean absolute percentage error) value is used to evaluate and verify the carbon emission prediction results. To ensure the accuracy and feasibility of the complementarity model, the above research provides more accurate guidance for the study of the two-carbon path. 2.MATHEMATICAL MODELING2.1ARIMA model (differential integrated autoregressive moving average model)The ARIMA model is mainly composed of three parts, namely differential process (I), autoregressive model (AR) and moving average model (MA). In general, the operation process of ARIMA model is to make the time series stable by difference first, and then make specific predictions by AR and MA models. 2.1.1DifferentialDifference is the calculation of the difference between adjacent data points in a set of numerical sequences. In time series, difference is used to transform a non-stationary series into a stationary series, reducing or eliminating trends and seasonal variations in time series. A new time series is obtained by first-order difference, where each value is the difference between two adjacent values in the original time series. For a time series Yt, the first difference of the series is defined as: where, Yt is the current value, Yt–1 is the previous value. New sequence is obtained by second-order difference, each value of which is subtracted from two observations separated by one value in the original time series. The second-order difference of the sequence is defined as: where, ΔYt was the first-order difference of the current moment, ΔYt–1 was the first-order difference from the previous moment. For the ARIMA model, multiple differences will lead to overfitting, so the difference is controlled within two times. If the time series is still non-stationary after two differences, it indicates that the problem studied is not suitable for ARIMA model. 2.1.2ARMA modelsThe ARMA model is a combination of autoregressive model (AR) and moving average model (MA). The AR model takes into account the effects of past observations on the current values, and is a linear combination of the current value of the time series and the past p values (Order of autoregressive lag and determines the number of backtracking observations of the model). The mathematical expression of AR model is as follows: where, εt is the prediction error, conforming to normal distribution; φi is the regression coefficient, which automatically fits the optimal parameters when the model is trained. The moving average model (MA) is a linear combination of past residuals or the prediction errors of the AR model. By combining residuals to observe the vibration of residuals, the prediction of random fluctuations can be effectively eliminated. MA is a linear combination of the current value of the time series and q lag errors in the past (the order of sliding average lag determines the amount of white noise in the model backtracking). Mathematical expression of MA model: In equation (4), εt–1, εt–2, ……εt–p is the random disturbance term of the first q period (zero mean white noise sequence), θi is the smoothing coefficient, and the optimal parameters will be automatically fitted during model training. Therefore, the expression of the ARMA model is: 2.2XGBoost modelsXGBoost, the full name of “Extreme Gradient Boosting”, was a gradient boosting model proposed by Chen26. The operation principle of XGBoost is to continuously add decision trees, and learn a new function to fit the residual predicted by all decision trees in the past. The whole process is similar to using gradient descent method to continuously optimize the objective function to achieve the best classification effect. Each decision tree in the process acts as a “weak learner” and can be integrated into a powerful classifier. The XGBoost model first scores the structure of the tree through the objective function to judge the quality of the tree structure, and then determines the shape and depth of each decision tree through node splitting to determine the structure of the tree. The prediction of the XGBoost model for each sample can be expressed as: where, fk(xi) is the value corresponding to the sample in the kth decision tree, and K is the number of decision trees. 2.2.1Objective functionEach decision tree corresponds to an objective function, and the objective function of XGBoost includes the loss function between the predicted value and the real value and the regular term. Be defined as: where, After considering the residual of past predictions, the newly generated prediction value of Kth tree can be expressed as: where, Therefore, the objective function can be rewritten as: XGBoost approximates the loss function by Taylor second-order expansion, so the objective function is approximated as: where, gi is the first derivative, hi is the second derivative: The regular term is used to reduce the complexity of the model. When XGBoost trains the model, the complexity can be controlled by the number of leaf nodes, the depth of the tree and the value of leaf nodes, and the regular term can be concretized into the following formula: Where T represents the number of leaf nodes of the Kth decision tree; ωj is the output value of the leaf node; Since the loss function of the first k-1 tree has been determined, the objective function is a constant term except for the about part of fk(xi), which does not affect the optimal solution of fk(xi). The objective function can be converted into the following expression: In the Kth decision tree, there is and can only be one corresponding leaf node, so the sum of the values of all samples in the Kth tree and the sum of the values of all samples corresponding to leaf nodes is the same. The sample set on node j is defined as Ij = {xi|q(xi) = j},where q(xi) is the index function that maps the sample to the leaf node, and the regression value on leaf node j is ωj = fk(xi), i∈ Ij. The objective function can be written as follows: Further the expression is simplified, order Then the objective function can be simplified as follows: Only after the shape of the tree is given can the minimum value of the loss function and the optimal solution of the leaf node be determined. Therefore, it is assumed that the shape of the tree is fixed and the samples of each node are determined by the values of gi, hi, T, and the output ωj of each leaf node should minimize the objective function. From the extreme point of the quadratic function, the ωj expression obtained is: Finally, the final value of the objective function is obtained: The value of the objective function represents the score of the decision tree. The smaller the value is, the better the structure of the decision tree is. Obj represents the sum of the scores of all nodes. 2.2.2Node splittingIn the previous section, only when the shape of the tree is fixed, can we find the minimum objective function ωj. This section discusses the structure of the tree. The difference in the shape of each tree is mainly the difference in the division of nodes. XGBoost relies on the greedy algorithm27 of recursive splitting of nodes to achieve tree generation, in addition to supporting approximation algorithms, that is, approximation of greedy algorithms, to solve the problem of excessive data volume over memory, or parallel computing needs. In each iteration of the greedy algorithm, after sorting the features, it traverses all the feature values of the partition points, takes the best yield (Gain) as the split yield of the feature, and selects the feature with the best split yield as the partition feature of the current node. Node S is divided according to its optimal partition point, and new nodes L and R are split. If Gain>0, split is feasible. The revenue calculation is defined as follows: In the formula, HL, HR, GL, GR and Hj, Gj are consistent with the definition in 1.2.1. The structures of different trees are enumerated, added to the model, and then repeated. To avoid overfitting, the maximum depth of the tree is set. When the revenue brought by the introduced splitting is less than the set threshold, the splitting is stopped and pruning is performed. Then the shape of each XGBoost decision tree is obtained, and combined with the objective function of 1.2.1, the value of each item in the XGBoost model can be obtained. To avoid overfitting, missing values, and sparse features, XGBoost also introduces column sampling and sparse perception strategies. Column sampling means that in the process of node splitting, instead of splitting all the remaining features, a subset is randomly selected from the remaining features as the selected features, which can prevent the joint action of different features and prevent overfitting. Sparse sensing is that XGBoost treats missing values and sparse feature values as missing values together and then treats these as a whole, and the traversal of split nodes will skip this whole, improving the efficiency of operation. 3.RESULTS AND OUTLIER ANALYSIS3.1Analysis of ARIMA carbon emission time series prediction resultsARIMA (p, d, q) model is used to predict the carbon emissions of different terminal energy industries in time series. Firstly, the carbon emission time series data of different terminal energy consumption industries in a central province from 1995 to 2020 are tested for stationarity. If they are not stationary, it is necessary to conduct differential processing to determine the difference order d. Then determine the optimal p and q orders of ARIMA model, and finally test the feasibility of ARIMA (p, d, q) model. 3.1.1Test data for stationarityARIMA model requires time series to have stationarity, which is a property of time series data, that is, the mean and variance of data are consistent in time. Therefore, the ADF test (unit root test, one of the stationarity test methods) is carried out by the division gate. The principle is to test whether the time series contains a unit root, that is, whether the time series starts with a constant term. If there is a unit root, the series is not stable. The ADF test of the original carbon emissions of each Sector is shown in Table 1. (1%, 5%, and 10% in the table are ADF corresponding to 1%, 5%, and 10% levels) Table 1.ADF test of original carbon emissions by Sectors.
As can be seen from the table,p (the probability value corresponding to the ADF test result) of agriculture is less than 0.05, which indicates that the data is stable. In other sectors except agriculture, the corresponding ADF with p greater than 0.05 and ADF (T statistic) greater than 5%, which indicates the ADF test failed and needs to be further differential treatment. The first-difference results are shown in Figure 1. It can be seen from Figure 1 that the values of the six Sectors are different after the difference, and the carbon emission of Sectors 3 was small and fluctuated within 200. The carbon emissions of Sectors 2 and 5 fluctuate within 6000. Sectors 4, 6, and 7 fluctuate within 1000. The carbon emission values of some Sectors were distributed on both sides of the axis, and the periodicity is obvious. ADF test was conducted on the data after the difference. The ADF test for the one-time difference of Sectors 2 to 7 was shown in Table 2. It can be seen from the table that the P-values of Sectors 2 to 7 after the one-time difference are all less than 0.05, indicating that Sectors 2 to 7 were stationary sequences. Table 2.The first difference ADF test results of Sectors 2-7.
3.1.2The optimal p and q orders of ARIMA were determinedARMA is a linear combination of AR model (autoregressive model) and MA model (moving average model), and ARIMA is a differential data stabilization based on ARMA. AIC criteria28 (minimum information content criteria, to balance the relationship between the goodness of fit and the number of parameters of the model) were used for each Sector to determine the p and q parameters of the ARIMA model. In order to avoid over-fitting, AIC values corresponding to each pair of p and q values were calculated, and p and q values meeting the minimum AIC values were iteratively found. The optimal ARIMA (p,d,q) model was determined. The order of ARIMA model iteratively screened by various Sectors based on AIC criterion is shown in Table 3. Table 3.Order of ARIMA model screened by AIC criteria of each Sector.
It can be seen from Table 3 that under the AIC criterion, the model corresponding to Sector 1 with the lowest AIC value was ARIMA (1, 0, 1). The minimum AIC model corresponding to Sector 2 was ARIMA (3,1,2). In the same way, the corresponding AIC minimum model and the applicable ARIMA(p,d,q) can be obtained for Sectors 3 -7. 3.1.3Feasibility test of ARIMA modelThe data from 1995 to 2017 were used as the training set, and the data from 2018 to 2020 were used as the test set. The MAPE values were used to characterize the accuracy of model predictions. The MAPE value predicted by the model from 2018 to 2020 is shown in Figure 2. Figure 2.The MAPE values between the real and predicted carbon emissions of each Sector in the ARIMA model for 2018-2020.  As can be seen from Figure 2, except for Sectors 5 and 7 in 2018 and 2019, the MAPE value of ARIMA prediction results of other Sectors was all within 20%, indicating that ARIMA was suitable for the carbon emission timing prediction of most end energy consumption industries. The predicted outliers of Sectors 5 and Sectors 7 require further analysis. 3.2ARIMA carbon emission prediction outlier analysisThe possible causes of outliers were analyzed from the distribution curve of carbon emissions of the outlier Sectors over time. The real carbon emissions of Sectors 5 from 1995 to 2020 and the predicted values of the test set are shown in Figure 3. Figure 3.Distribution of the real carbon emissions of Sectors 5 and the predicted values of the test set from 1995 to 2020.  As shown in Figure 3, the carbon emission of Sector 5 shows an overall downward trend over time, with certain cyclical characteristics. Among them, the predicted values of carbon emissions in 2018 and 2019 are 27.227 million tons and 23.354 million tons respectively, which is larger than the real value. This phenomenon may be caused by the inaccurate p and q values obtained by these two Sectors according to AIC standards, while BIC standards28 (Bayes criteria) add penalty items to the basis of AIC standards, and the more parameters, the more significant the penalty term, the BIC criterion was first used to correct the outlier, and the parameters p and q with the smallest BIC values were also selected to avoid overfitting. The p and q values determined by AIC criteria were iteratively searched for the p and q values that met the minimum BIC values. When Sectors 5 used the ARIMA (4,1,4) model, the MAPE values from 2018 to 2020 were 6.76%, 46.17%, and 6.76%, respectively. When Sectors 5 uses the ARIMA (4,1,3) model, the MAPE values from 2018 to 2020 are 31.34%, 17.1%, and 0.47% respectively, which is a certain improvement compared with the predicted values under AIC criteria, but the accuracy still needs to be further improved. Therefore, it can be concluded that ARIMA cannot meet the carbon emission forecast of Sector 5, and other forecasting methods need to be found. Similarly, the outliers of Sector 7 were analyzed, and the changes in the real and predicted carbon emission values with time were shown in Figure 4. In Figure 4, the carbon emissions of Sector 7 showed an overall upward trend with time, and a plateau occurred in 2018, 2019, and 2020, and the cyclical performance was not obvious. The forecast value of Sector 7 in 2018 was 414.6 million tons, which was too large. The phenomenon may be caused by ARIMA (5,1,5) model, judging exponential growth in 2018. The BIC criterion was also selected to determine the order, and the ARIMA (5,1,5) model was still the best under the BIC criterion, which was consistent with the result obtained by AIC criterion, indicating that AIC criterion and BIC criterion have weak guiding value for the selection of optimal p and q of ARIMA (p, d, q) model of Sectors 5. p and q values with good fitting effect were selected around the determined parameters p and q, and the ARIMA (3,1,5) model was selected. The predicted carbon emissions of Sectors 7 from 2018 to 2020 were 30.84 million tons, 4.0 million tons and 34.30 million tons, respectively, and MAPE values were 13.93%, 13.82% and 4.78%, respectively. MAPE is all below 15%, indicating that ARIMA (3,1,5) model has a good fitting effect for Sectors 7 and can be used for further prediction. To sum up, ARIMA is obsessed with exploring the internal relationships of time series with linear relationships and fails to consider the nonlinear relationships, resulting in the prediction of outliers of Sectors 5. Machine learning has a strong ability to deal with nonlinear relationships, so the XGBoost model, which has been popular in machine learning in recent years, was introduced to predict the carbon emissions of different end-use energy consumption industries. 3.3Analysis of XGBoost carbon emission time series prediction resultsXGBoost is the current master of machine learning. For the prediction of carbon emissions in different end-energy consumption industries, the time series data from 1995 to 2020 were processed first, and then the periodicity and trend of the time series were enhanced, and then the predicted value of the XGBoost model was tested. Finally, the carbon emissions of different end energy consumption industries from 2018 to 2020 are predicted and compared with the real value to measure the feasibility of XGBoost model. 3.3.1Data processing and preprocessingThe data formats need to be uniform, all two decimal places were reserved, and the code was used to check whether the data was successfully read. 3.3.2Periodicity and trend enhancement of time seriesUsing XGBoost to make a time series forecast requires strong trends and periodicity of data. If the periodicity and trend of the data itself were not obvious, the past data can be used as the supplement of the trend item, and the past average value can be used as the supplement of the periodic item so that the periodicity and trend of the time series can be better highlighted. The characteristics of the original data, the year before the original data, the past two years, and the average of the previous two years are collected, in which the characteristics of the year before 1996, the past two years, and the average of the previous two years were represented by the characteristics of 1995. A data set characterized by carbon emissions and all independent variables, the independent variables of the previous year, the independent variables of the past two years, and the average of all independent variables of the previous two years was obtained. 3.3.3Testing of XGBoost predicted valuesThis paper belongs to small sample prediction, and the training set and test set are obtained by randomly dividing the data according to the ratio of 8:2. In the training set, the characteristics of the current year, the characteristics of the past year, the characteristics of the past two years, and the average characteristics of the past two years were used to make predictions, and the MAPE value was compared with the real value. The real and predicted MAPE values of the test set randomly divided by various sectors of XGBoost were shown in Figure 5. In Figure 5, the MAPE value of the real value and predicted value of a test set in Sector 4 was 21.38%, while the MPAE value of other Sectors was lower than 20%, indicating that the test set data was well predicted. 3.3.4XGBoost predicts test set carbon emissionsSimilarly, taking the data before 2018 as the training set and 2018, 2019, and 2020 as the test set, the MAPE values of the predicted and real XGBoost values of each Sector from 2018 to 2020 are shown in Figure 6. In Figure 6, the MAPE values of Sectors 1 to 6 in the test set are all lower than 20%, and only the accuracy of Sector 7 in 2019 does not meet the requirements, so the outliers of Sectors 7 need to be analyzed. 3.4XGBoost carbon emission prediction outlier analysisTwo approaches were taken for Sector 7 in XGBoost, one was to adjust the number of decision trees from 15 to 1000 tests, and the other was to replace each feature of the previous two-year average with the previous three-year average and the previous five-year average (enhancing cyclicality) without considering the past one year or the past two years (weakening trend). When the first method was used, the resulting MAPE values for 2018 were stable at around 63%, and the MAPE values for 2019 and 2020 were also stable at around 11% and 0.3%. When the second method was used, the same result was obtained as the first method. The MAPE of the five groups of test sets randomly divided in step 2 has been stable below 20%. The failure of method 1 was most likely because the XGBoost trained by sector 7 before 2018 determined that the error between prediction and the true value was less than 0.1% after 15 decision trees, the predicted value is modified to less than 0.1% by the generated decision tree, which does not modify the predicted value. The failure of method 2 may be that the addition of past features and past average features does not affect the selection of features in Sectors 7. Therefore, XGBoost cannot meet the carbon emission prediction of Sector 7, but the prediction of Sector 5 was more accurate. To sum up, XGBoost also has limitations and cannot meet the carbon emission forecast of Sectors 7. Carbon emissions of different energy industries need to be considered comprehensively by ARIMA to complement each other. 4.CONCLUSIONAccording to the existing research results, the outliers obtained by the ARIMA model were in Sectors 5 and 7, the outliers in Sector 5 were not adjustable, and the prediction accuracy of the outliers in Sector 7 was significantly improved after other parameters p and q were changed. The outlier obtained by XGBoost model was in Sector 7, and the outlier was not adjustable. Based on the above results, the main conclusions were as follows:
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