Traditional machine learning-based “electricity-to-carbon conversion” methods typically formulate the accounting problem as a regression model, where electricity data for the target year t, P t is used to estimate the carbon emissions C t for that year. Due to the autocorrelation between historical carbon emissions data C t − s , C t − s + 1 , ⋯ , C t − 1 and the carbon emissions C t of the target year (where s represents the input length of the historical data), this paper incorporates historical emissions as inputs into the “electricity-to-carbon conversion” model. In summary, this relationship can be expressed as shown in Equation 1: C t = f P t , C t − 1 ⋯ , C t − s (1)

Where f ⋅ denotes the “electricity-to-carbon conversion” model. Due to the limited number of observations samples, the model faces a small-sample learning challenge. While some machine learning algorithms can accommodate small-sample learning, the performance of these algorithms is highly sensitive to hyperparameters. Small-sample learning typically lacks sufficient data to effectively optimize hyperparameters, which makes it difficult for traditional machine learning-based “electricity-to-carbon conversion” methods to achieve satisfactory accounting accuracy under small-sample conditions.

Based on the analysis above, optimizing the hyperparameters of machine learning algorithms under small-sample conditions is a critical step to improving accounting accuracy. To address this, we propose an “electricity-to-carbon conversion” method based on electricity data and augmentation pre-training, as outlined in Figure 1. This approach consists of three main steps: feature engineering, model construction, and augmentation pre-training.

First, feature engineering involves analyzing the correlation between collected electricity production and consumption data and carbon emissions data to select highly correlated electricity data as input features for the “electricity-to-carbon conversion” model. Second, an appropriate machine learning algorithm is chosen to build the model, mapping the input features to annual carbon emissions. Finally, to optimize the model under small-sample conditions, we introduce an augmentation pre-training optimization method to obtain the optimal hyperparameters for the “electricity-to-carbon conversion” model.

The “electricity-to-carbon conversion” method relies on the correlation between electricity and carbon emission data. Therefore, it is essential to perform correlation analysis on the collected power generation and consumption data, selecting the electricity data most strongly correlated with carbon emissions as input features for the model.

In this study, the Pearson correlation coefficient and Spearman correlation coefficient are used to quantify the relationship between feature data and target data (Cohen et al., 2009). Their calculations are shown in Equations 2, 3: R p = ∑ i = 1 n x i − x ¯ y i − y ¯ ∑ i = 1 n x i − x ¯ 2 ∑ i = 1 n y i − y ¯ (2) R s = 1 − 6 ∑ i = 1 n d i 2 n n 2 − 1 (3)

Where R p denotes the Pearson correlation coefficient and R s denotes the Spearman correlation coefficient. x i and y i denote the values of the two variables for the i -th sample point, while x ¯ and y ¯ are the means of the two variables. The variable d i denotes the difference in ranks between x i and y i , and n represents the sample size.

The Spearman and Pearson correlation coefficients are used to assess the relationship between feature data and target data, with values ranging from −1 to 1. A value closer to 0 indicates a weaker correlation, while values closer to 1 signify a stronger positive correlation, and values closer to −1 indicate a stronger negative correlation. The Spearman coefficient measures monotonic relationships without requiring linearity or normal distribution of the data, whereas the Pearson coefficient reflects only linear correlations and assumes that the data follows a normal distribution (Hauke and Kossowski, 2011).

Data from the past 17 years (2004–2021) were used in the calculation and Table 1 presents the correlation analysis results between various electricity data and carbon emissions. The data reveals a strong monotonic and linear relationship between total annual carbon emissions and electricity consumption, thermal power generation, and total electricity, while a negative correlation is observed with hydropower generation. Therefore, electricity data, excluding hydropower generation, is selected as input variables for the model.

Figure 2 illustrates the structure of the proposed “electricity-to-carbon conversion” model. The input variables X include this year’s electricity production P G , t (comprising total generation and thermal power generation), historical carbon emissions C t − 1 , C t − 2 , ⋯ C t − s , and total electricity consumption P C , t . The composition of the input variable X is shown in Equation 4: X = X 1 , X 2 ⋯ X n = C t − 1 , C t − 2 ⋯ C t − s , P G , t , P C , t (4)

This study employs the Light Gradient Boosting Machine (LightGBM) as the regression fitting model. LightGBM is a machine learning algorithm based on decision tree methods and gradient boosting, designed to iteratively optimize model residuals and enhance predictive performance (Wang et al., 2017; Ke et al., 2017). Comprising multiple simple decision trees, LightGBM features a simpler structure compared to other machine learning models, making it particularly suitable for small sample learning tasks. Additionally, it utilizes methods such as histogram algorithms and leaf-wise strategies to improve computational efficiency and predictive accuracy. The mathematical model for LightGBM is expressed in Equation 5: C t = ∑ n = 1 N f n d t (5) where C t denotes the estimated annual carbon emissions obtained from the LightGBM model; f n d t denotes the estimate from the n-th regression tree; t represents time; N indicates the total number of regression trees; and n denotes the specific tree number.

The objective function of the LightGBM model comprises a loss function and a regularization term. The expression for the objective function of the t-th tree is provided in where Equation 6. The expressions for the loss function and the regularization penalty are given in where Equation 7 and where Equation 8, respectively. Y n = L n + Ω n (6) L n = ∑ t = 1 T C t − C t n − 1 + f n d t 2 (7) Ω n = γ J + 1 2 λ ∑ j = 1 J w j 2 (8) where Y n denotes the objective function of the n-th tree, and L n and Ω n represent the loss function and regularization penalty of the n-th tree, respectively. J denotes the number of leaf nodes, while w signifies the weight values of these nodes. j refers to the j-th leaf node, and λ represents the penalty coefficient for the leaf nodes. T denotes the total number of samples, and C t n − 1 indicates the estimate of the i-th sample from the n−1-th tree.

The input variable X and annual carbon emissions Y are designated as the feature set and target set, respectively, for training the LightGBM model. The final LightGBM model is obtained after multiple iterations, converging to the minimum loss.

The hyperparameters required for LightGBM are shown in Table 2. The selected hyperparameters have a direct and significant impact on the model’s fitting performance. To ensure that the “electricity-to-carbon conversion” model has good generalization performance and regression fitting ability, effective optimization of the hyperparameters is necessary. However, in cases with insufficient sample sizes, model optimization can easily fall into local optima. To enable more thorough and effective optimization under small sample conditions, this paper proposes a model optimization method based on augmentation pre-training. Figure 3 illustrates the processing flow of the augmentation pre-training method, which generates a large amount of augmented data through an interpolation augmentation module, allowing the LightGBM model to undergo pre-training and optimization on this augmented dataset to obtain a set of optimal hyperparameters for formal training.

Interpolation augmentation is the primary step of augmentation pre-training. As shown in Figure 4, this method first uses numerical interpolation to fit discrete data points into a continuous numerical function y t and then performs high-frequency sampling on this function to obtain a large amount of augmented data. In the interpolation method, the commonly used piecewise linear interpolation approximates y t by connecting adjacent interpolation points with line segments. Its mathematical modeling is represented by Equations 9, 10: y h t = ∑ j = 0 n y j l j t (9) l j t = t − t j − 1 t j − t j − 1 , t j − 1 ≤ t ≤ t j , j ≠ 0 t − t j + 1 t j − t j + 1 , t j ≤ t ≤ t j + 1 , j ≠ n 0 , t ∉ t j − 1 , t j + 1 (10)

Where, t represents time, serving as the independent variable for the fitted function y h t , while l j t represents the interpolation basis function. t j and y j denote the j-th group of sampling points in the dataset.

Assuming that when y t ∈ C 2 t 1 , t 2 is given, there exists an upper bound on the error between the piecewise linear interpolation function y h t and the actual function y t . The upper bound of the error, max t 1 ≤ t ≤ t n y t − y h t , can be estimated using Equation 11. max t 1 ≤ t ≤ t n y t − y h t = max t 1 ≤ t ≤ t n y ″ t 8 h 2 (11)

Where, h denotes the sampling interval, i.e., h = t j − t j − 1 , j ≠ 0 . As indicated by Equation 11, there exists an upper limit on the error between the augmented data and the real data, which ensures the reliability of the numerical augmentation method.

Model optimization is the core step of augmentation pre-training, achieved through an iterative process of “training-evaluation-feedback” adjustment for hyperparameter optimization. In each iteration, the LightGBM is first trained on the augmented training set, then the fitting performance is evaluated on the augmented validation set, and finally, hyperparameters are optimized based on the evaluation results.

To efficiently optimize hyperparameters in a complex parameter space, this study proposes an improved particle swarm optimization algorithm based on adaptive inertia weight and local neighborhood strategy (Adaptive Inertia and Local Neighborhood Particle Swarm Optimization, AILN-PSO). The PSO algorithm is a classic swarm intelligence optimization method that effectively addresses continuous optimization problems but suffers from low computational efficiency and a tendency to fall into local optima (Marini and Walczak, 2015). Therefore, we employ adaptive inertia weights and local neighborhood strategies to enhance the algorithm’s computational efficiency and optimization accuracy.

In standard PSO, the inertia weight controls the search step of the particles. When the weight is large, particles tend to explore a larger search space; when the weight is small, particles focus more on local searches. The adaptive inertia weight dynamically balances exploration and exploitation by adjusting the weight value at different stages of the algorithm. To improve efficiency, this study adjusts the inertia weight to a function that decreases as the number of iterations increases, allowing the search to transition from global to local. The expression for the adaptive inertia weight is shown in Equation 12. ω k = ω max − ω max − ω min K max × k (12)

Where, ω max and ω min denote the maximum and minimum values of the inertia weight, respectively, while K max denotes the maximum number of iterations, and k is the current iteration count.

In standard PSO, all particles update their positions based on the global best particle. However, the global best particle may sometimes limit the particles’ ability to escape local optima. The local neighborhood strategy enhances the diversity of the population by allowing each particle to update its position based only on the best particle in its neighborhood, which helps to avoid getting trapped in local optima. Under this strategy, the update velocity and position of the particles are represented by Equations 13, 14, respectively. v a k + 1 = ω k ⋅ v a k + c 1 ⋅ r 1 ⋅ p a k − x a k + c 2 ⋅ r 2 ⋅ l a k − x a k (13) x a k + 1 = x a k + v a k + 1 (14)

Where, x a k and v a k represent the position and velocity of particle A at the k-th iteration, while c 1 and c 2 are the acceleration coefficients. r 1 and r 2 are random numbers. p a k denotes the historical best position of particle a and l a k indicates the best position of particle A within its local neighborhood. In this study, the Mean Squared Error (MSE) of the pre-trained model on the augmented validation set is used as the evaluation metric. The maximum number of iterations for AILN-PSO is set to 300, and an early stopping mechanism is implemented. Specifically, if the evaluation value does not improve for 10 consecutive iterations, it is considered that the optimization process has converged, leading to the termination of training and the output of the optimized hyperparameters.

The carbon emission dataset used in this study is sourced from the China Carbon Accounting Database (CEADs) (Shan et al., 2018; Shan et al., 2020; Xu et al., 2024), while the electricity data comes from the National Bureau of Statistics (National Bureau of Statistics of China, 2024). The electricity data includes electricity consumption, total power generation, thermal power generation, and hydropower generation. The dataset spans from 1997 to 2021, with an annual granularity.

To comprehensively evaluate the performance of the proposed method, various machine learning models suitable for small sample learning were selected for comparison. The specific models are as follows:

1) ARMA: The Auto-Regressive Moving Average (ARMA) model is a commonly used time series model that extrapolates future data points based on past time series. ARMA uses only historical carbon emissions as input data and does not include electricity data.

The length of historical data has a critical impact on model performance: an appropriate historical data length allows the model to capture and learn the changing trends in carbon emissions, while a length that is too long increases feature complexity and reduces the amount of training samples. Therefore, this study first discusses the selection of historical data length by comparing the performance of models with different historical data lengths to determine the optimal length.

Table 3 presents the hyperparameters to be optimized of MLP and SVM model. To validate the effectiveness of the proposed augmentation pre-training method in small sample learning, the hyperparameter optimization method based on augmentation pre-training is compared with the hyperparameter optimization method using the validation set. In augmentation pre-training optimization, the model’s hyperparameter optimization is conducted on the augmented dataset; whereas in validation set optimization, the hyperparameters are optimized on the validation set. In both methods, the dataset is divided into training, validation, and test sets, with proportions of 6:2:2.

This study employs two statistical metrics to assess the goodness of fit of the “electricity-to-carbon conversion” model: Mean Absolute Percentage Error (MAPE) and Root-Mean-Square Error (RMSE). These two metrics are commonly used in regression analysis, with smaller values indicating less deviation between the fitted values and the actual values. RMSE is an error evaluation metric that measures the deviation between predicted values and target values, with smaller numbers indicating closer alignment between model predictions and target values. MAPE is a relative error evaluation metric suitable for comparing the error rates between different model predictions and target values, where smaller values signify lower errors. Their expressions are given in Equations 15, 16: R M S E = 1 n ∑ i = 1 n y ^ i − y i 2 (15) M A P E = 100 % n ∑ i = 1 n y ^ i − y i y i (16)

Where, y i denotes the i-th output value of the model, y i denotes the actual value at the i-th point, and n indicates the number of samples.

The accounting results of carbon emissions are shown in Tables 4, 5.

Tables 4, 5 present the performance evaluation data of each model under different historical data lengths, revealing significant differences in performance across methods. Shorter data lengths provide better accounting accuracy due to the availability of more training samples and the substantial variations in carbon emission trends over different periods. Additionally, each machine learning model achieved better results using the augmentation pre-training optimization method.

A comparison of the models under the optimal method is shown in Table 6, detailing their fitting performance. It is evident from the table that all models outperform ARMA, which relies solely on the autocorrelation characteristics of historical carbon emissions without incorporating power features. This indicates that power features significantly enhance accounting accuracy. For MLP, the training data volume is still too limited, hindering its learning effectiveness. SVM and LightGBM show good accounting results, demonstrating their superiority in small sample learning tasks. LightGBM, with its simpler structure, more easily identifies suitable hyperparameters, yielding good fitting performance under both optimization methods. In contrast, SVM has numerous and complex hyperparameters that require richer samples for optimization, resulting in suboptimal performance on the small sample validation set, although significant accuracy improvements were observed after augmentation pre-training optimization.

To provide a more intuitive comparison of the differences between the various models’ “electricity-to-carbon” performance, the accounting results of all models utilizing the best method on the test set are compared with the actual values, as shown in Figure 5 and Table 7.

From the analysis of Figure 5, it is evident that ARMA and MLP exhibit significant deviations between their measured values and target values, with some accountings being either too high or too low. Although the SVM regression model shows better evaluation results, there remains a noticeable deviation between many fitted values and the target values. In contrast, LightGBM’s fitted values align more closely with the target values, demonstrating a tighter distribution of accounting values relative to the actual target values.