Enhanced Generative Adversarial Networks for Isa Furnace Matte Grade Prediction under Limited Data

Abstract

Due to the scarcity of modeling samples and the low prediction accuracy of the matte grade prediction model in the copper melting process, a new prediction method is proposed. This method is based on enhanced generative adversarial networks (EGANs) and random forests (RFs). Firstly, the maximum relevance minimum redundancy (MRMR) algorithm is utilized to screen the key influencing factors of matte grade and remove redundant information. Secondly, the GAN data augmentation model containing different activation functions is constructed. And, the generated data fusion criterion based on the root mean squared error (RMSE) and the coefficient of determination (R²) is designed, which can tap into the global character distributions of the copper melting data to improve the quality of the generated data. Finally, a matte grade prediction model based on RF is constructed, and the industrial data collected from the copper smelting process are used to verify the effectiveness of the model. The experimental results show that the proposed method can obtain high-quality generated data, and the prediction accuracy is better than other models. The R² is improved by at least 2.68%, and other indicators such as RMSE, mean absolute error (MAE), and mean absolute percentage error (MAPE) are significantly improved.

1. Introduction

Isa furnace copper smelting is one of the main methods of modern thermal copper refining. In the smelting process, matte grade is a critical process indicator for measuring the quality of the smelting process and can guide the adjustment of operating parameters [1]. A stable matte grade plays a vital role in the subsequent blowing, electrolysis, and sulfuric acid production processes. However, due to the complexity of the Isa furnace melting reaction mechanism, the solid phase, liquid phase, and gas phase coexist in the furnace, resulting in an inability to achieve the real-time detection of matte grade. Existing approaches mostly adopt laboratory testing to detect the value of the matte grade, which results in an inability to adjust the operating parameters in a timely manner [2]. Therefore, constructing an accurate and effective prediction model for matte grade is of great significance in guiding the production operation of copper melting processes.

The available research mainly inputs the production parameters of the copper smelting process into a data-driven model that can achieve the prediction of matte grade. Shouyi Yu et al. [3] proposed a prediction method for matte grade, matte temperature, and iron–silicon ratio in slag in the copper smelting process based on an artificial neural network and achieved the real-time prediction of three main process parameters in the copper smelting process. Weihua Gui et al. [4] constructed a matte grade prediction model based on a fuzzy neural network and improved the gradient descent algorithm to enhance the accuracy of matte grade prediction. Chunhua Yang et al. [5] proposed an intelligent integrated prediction model for matte grade and verified the effectiveness of the model using industrial data. Xiaobo Peng et al. [6] established a dynamic T-S recursive fuzzy neural network prediction model and applied it to the prediction of key parameters in the copper flash melting process. Jianhua Liu et al. [7] proposed a prediction method for key process indicators of the copper flash melting process based on projection seeking regression, utilizing an accelerated genetic algorithm for updating model parameters in real time. Xiaolong Zhang et al. [8] proposed an adaptive Isa furnace copper melting process prediction method based on generalized maximum entropy regression to improve the prediction accuracy. Maoqi He et al. [9] proposed a data-driven modeling method based on the soft measurement of matte grade and predicted matte grade during oxygen-enriched side-blowing melt pool melting. The above matte grade prediction methods mainly integrate traditional machine learning methods to learn the mapping relationship between input variables and matte grade from historical data, so as to predict unknown data. However, in the actual production process of copper smelting, an offline analysis of ore composition and matte grade take a lot of time and money, which leads to the scarcity of modeling samples and limits the development of matte grade prediction technology.

To address the above problem, scholars have proposed data augmentation methods such as geometric transformation, window clipping, adding noise [10], FBG [11], and generative adversarial networks (GANs) [12]. In particular, a generative adversarial network (GAN) can learn high-dimensional and complex data distributions without relying on any prior assumptions, which have a wide range of applications in complex industrial fields. Zherui Ma et al. [13] proposed a virtual sample generation method based on GAN, which improved the accuracy of hydrogen yield prediction during supercritical water gasification hydrogen production. Xiaochen Hao et al. [14] proposed a data augmentation model of calcium oxide in a cement clinker based on the Wasserstein generative adversarial network (WGAN), which solved the problem of low prediction accuracy caused by limited labeled samples. Zhongsheng Chen et al. [15] proposed an industrial data generation method based on the conditional generative adversarial network (CGAN), which improved the performance of soft measurement models. Yajun Wu et al. [16] combined WGAN with the Gradient Penalty and Extreme Gradient Boosting Algorithm, which expanded the experimental data and established the relationship between the process parameters and relative density of aluminum alloy thin-walled parts. And, the influence of process parameters on the relative density was revealed. Zhongsheng Chen et al. [17] proposed a data augmentation method for embedding quantile regression into CGAN and verified the effectiveness of the proposed method through chemical data. Zoujing Yao et al. [18] proposed a generative adversarial data filling model with the addition of a soft measurement module, and verification of the proposed method was accomplished with the collected chemical data.

In summary, GAN-based augmentation methods provide a new research direction for matte grade prediction under limited data. However, a GAN with a single activation function cannot fully represent the positive and negative characteristics of copper melting data, resulting in a large distribution discrepancy between the generated data and real data, which reduces the prediction model’s performance. For this reason, a novel matte grade prediction method considering limited data is proposed. The main contributions of this paper are summarized as follows:

• To select the most relevant and the least redundant variable subsets, a maximum relevance minimum redundancy (MRMR) algorithm is introduced to screen the variables, which ensures that the selected variables can effectively capture the change pattern of the matte grade and reduce redundant information.
• Three GAN data augmentation models with different activation functions are constructed, and a data fusion criterion generated based on the root mean squared error (RMSE) and the coefficient of determination (R²) is designed, which can improve the global features’ ability to represent a generative network and the quality of the generated data.
• A matte grade prediction model based on GANs and random forest (RF) is proposed. The prediction method is verified by the production data of a copper smelting enterprise in southwest China. The experimental results show that the proposed method has high prediction accuracy.

2. Theory and Methodology

2.1. Maximum Correlation Minimum Redundancy Algorithm

The maximum relevance minimum redundancy (MRMR) algorithm can maximize the correlation between the input variables and the target variables and minimize the correlation between the input variables simultaneously [19].

Assuming that the joint probability density function p(x,y) and the marginal probability density function p(x), p(y) between two variables are known, the expression for the mutual information is shown in Equation (1).

$$I(x,y)={\displaystyle \iint p(x,y)\log \frac{p(x,y)}{p(x)p(y)}dxdy}$$

(1)

Assuming that S is the optimal subset of input variables and |S| is the number of variables in S, the maximum correlation can be defined as Equation (2).

$$\max D(S,y),D={\displaystyle \frac{1}{\left|S\right|}}\sum _{x_i\in S}I\left(x_i;y\right)$$

(2)

However, there is a redundancy issue among the variables based on maximum correlation. Therefore, a minimum redundancy constraint is introduced, as shown in Equation (3).

$$\min R(S),R={\displaystyle \frac{1}{|S{|}^2}}\sum _{x_i,x_j\in S}I\left(x_i;x_j\right)$$

(3)

Combining these two constraints, the maximum correlation minimum redundancy can be obtained. The expression is shown in Equation (4).

$$\max \psi (D,R),\psi =D-R$$

(4)

Assuming that the set of all variables is X, the set of selected variables is S_m−1, where there are m − 1 variables. The incremental search method is used to select the mth variable from the remaining variable set X-S_m−1, which needs to meet the requirements of Equation (4). The expression can be shown in Equation (5).

$$\underset{x_j\in X-S_{m-1}}{\max }\left[I\left(x_j;y\right)-{\displaystyle \frac{1}{m-1}}\sum _{x_i\in S_{m-1}}I\left(x_j;x_i\right)\right]$$

(5)

2.2. The Principle of GAN

The structure of GAN is shown in Figure 1, which mainly consists of a generator and a discriminator [20].

During training, the generator is responsible for learning the distribution mapping of real data, and its output is shown in Equation (6).

$$G(z)={\phi }_G\left(W_G\cdot z+b_G\right)$$

(6)

where φ_G denotes the activation function of the generator; z denotes the random noise sample; and W_G and b_G denote the weights and biases in the generator.

When the input is a real sample, the output of the discriminator is 1; when the input is a pseudo-sample, the output of the discriminator is 0, as shown in Equations (7) and (8).

$$D(x)={\phi }_D\left(W_D\cdot x+b_D\right)=1$$

(7)

$$D(G(z))={\phi }_D\left(W_D\cdot G(z)+b_D\right)=0$$

(8)

where φ_D denotes the activation function of the discriminator; x denotes a real data sample; and W_D and b_D denote the weights and biases in the discriminator.

The entire training process of GAN can be regarded as a binary minimax game problem, as shown in Equation (9).

$$\underset{G}{\min }\underset{D}{\max }V(D,G)=E_{x\sim P(x)}[\lg D(x)]+E_{z\sim p_z(z)}[\lg (1-D(G(z)))]$$

(9)

where P_(x) denotes the true data distribution; P_z(z) denotes the distribution of noise; and E denotes the expected value.

2.3. RF Model

RF has been widely applied in nonlinear regression prediction [21]. Assuming that the number of samples in the training set is X and the number of variables is M, the process of prediction using the RF model is shown in Figure 2.

• A training subset is constructed by randomly selecting x samples from the original training set. And, m variables are randomly selected from the training subset, and the corresponding decision tree is constructed.
• Repeating step (1) n times, a total of n training subsets are obtained, forming n decision trees, and all the decision trees are combined together to obtain an RF model.
• The predictions of all the decision trees were averaged as the final prediction of the random forest regression model.

3. A Matte Grade Prediction Model Incorporating EGAN and RF

The overall architecture of the model is shown in Figure 3. The most valuable information for the prediction task is retained through the MRMR algorithm. The generated data closer to the real data distribution is obtained through GANs and data fusion, and the predicted value of the matte grade is obtained via the RF model.

3.1. Variable Selection Based on MRMR Algorithm

The data in this paper come from the production data of a copper smelting enterprise in southwestern China. And, 17 variables were selected as inputs to the prediction model. Based on the chemical reaction mechanism of copper melting, the 17 variables were categorized into positively and negatively correlated variables, as shown in

Table 1. Input variables for matte grade prediction model.

Positive Correlation Variables	Description	Negative Correlation Variables	Description
X1	Oxygen concentration	X7	Fe content of flux silicon material
X2	Blowing air volume	X8	H2O content of flux silicon material
X3	Amount of diesel fuel added	X11	Ca content in copper concentrate
X4	Amount of coal added	X12	Fe content in copper concentrate
X5	Slagging agent silicon addition	X13	Zn content in copper concentrate
X6	Si content of flux silicon material	X14	H2O content in copper concentrate
X9	Copper concentrate addition	X15	S content in copper concentrate
X10	Cu content in copper concentrate	X16	Si content in copper concentrate
Y	Matte grade	X17	As content in copper concentrates

.

The MRMR algorithm was applied to select the key variables of matte grade prediction and reduce the influence of redundant variables on the prediction model. Its specific process can be explained as follows:

• The variable X_j is selected from the input variable set X that has the maximum mutual information with the matte grade and placed into the optimal input variable subset S.
• The incremental search method is used to select the variable X_j that satisfies the condition and place it into the optimal input variable subset S.
• Whether the set threshold reached (ψ(D,R) > 0) is determined. If the threshold condition is met, the selected variable is output; otherwise, the second step is performed.

3.2. Data Augmentation and Data Fusion Based on GANs

In the process of matte grade prediction, we have selected many positive correlation variables and negative correlation variables to improve the prediction accuracy, as shown in Table 1. When constructing the generative model, accurately representing the variables features is a significant way to improve the quality of the generated data.

Activation functions are crucial to represent variables features for GANs. Specifically, compared to Sigmoid, Tanh, and Softmax activation functions, ReLU can speed up model training and effectively solve the problem of gradient vanishing. The ReLU activation function is widely used in GAN [22]. However, ReLU lacks the ability to represent negative features, which makes the generator unable to learn the global features of real data, resulting in a decline in the quality of generated data.

For this reason, relevant scholars have introduced LeakyReLU [23] and PreLU [24] activation functions into the generator and introduced negative slopes on the basis of ReLU, which can assist the generator to characterize negative features.

The diversity of activation functions has a significant impact on the generator in characterizing data features. In this paper, a comprehensive strategy using multiple activation functions is proposed, which can capture the global variable characteristics of the copper melting data. And, three different activation functions are utilized for data generation, as shown in

Table 2. GAN models with different combinations of activation functions.

Model	Generator	Discriminator
RGAN	ReLU × 4 + Sigmoid × 1	LeakyReLU × 4 + Sigmoid × 1
LRGAN	LeakyReLU × 4 + Sigmoid × 1	LeakyReLU × 4 + Sigmoid × 1
PRGAN	PReLU × 4 + Sigmoid × 1	LeakyReLU × 4 + Sigmoid × 1

.

According to the data fusion method in Reference [25], the three kinds of data are weighted and fused, as shown in Equations (10)–(12).

$$C_i={\displaystyle \frac{R_i^2}{RMS{E}_i}}$$

(10)

$$w_i={\displaystyle \frac{{(C_i)}^m}{{\displaystyle \sum _{i=1}^m}{(C_i)}^m}}$$

(11)

$$D_f={\displaystyle \sum _{i=1}^m{D}_i{w}_i}$$

(12)

where RMSE_i and R²_i represent the root mean square error and the coefficient of determination, respectively. C_i represents the quadratic evaluation index based on the RMSE_i and R²_i, and the larger the value of C_i, the higher the prediction accuracy of the model. D_i and w_i represent the different generated data and their corresponding weights, D_f represents the weighted fusion data, and m represents the number of different GAN models used.

The generated data are used as the extended data of the training set and then input into the RF model to obtain the corresponding RMSE and R² and update the weights of the generated data. The data fusion based on RMSE and R² can obtain new generated data. The steps for data fusion are as follows:

• The key influencing variables of matte grade were obtained by using the MRMR algorithm for variable selection, which was divided into a training set and a test set in the ratio of 8:2.
• The training set D was input into the RGAN model, and the generated data D₁ under RGAN were obtained according to a ratio of real data to generated data of 1:1.
• The generated data D₁ obtained under the RGAN model were used as the extended data of training set D and fed into the RF model, and the evaluation metrics C₁ was computed.
• Steps (2) and (3) were repeated following the same method to obtain the generated data D₂ and D₃ under LRGAN and PRGAN, respectively, as the extended data for training set D. Then, they were input into the RF model and evaluation indexes C₂ and C₃ were computed sequentially.
• Weights w₁, w₂, and w₃, corresponding to each GAN model’s generated data, and the weighted fusion of the three generated data were calculated to obtain the final generated data D_f.

3.3. Matte Grade Prediction Based on Random Forest

The generated data after weighted fusion were used as the extended data of the real training data to obtain new training data. The new training data were then input into the RF prediction model for model training, and finally, the random forest prediction model with the smallest generalization error was obtained and the trained model was used to predict the data in the test set by adding the prediction results of each decision tree and then taking the average value as the final matte grade prediction, as shown in Equation (13).

$$F(x)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{f}_i(x)}$$

(13)

where F(x) is the final matte grade prediction result and f_i(x) is the matte grade prediction result of the ith decision tree.

4. Experimental Results and Analysis

4.1. Data Sources and Experimental Settings

4.1.1. Copper Smelting Mechanism and Data Sources

The acquisition of the copper concentrate is the primary step. Firstly, the copper ore mined from the mine is crushed and ground. Secondly, the broken ore is separated following a flotation process to extract copper-containing minerals and form the copper concentrate. Then, the copper concentrate is concentrated and dried to increase its copper content and remove excess water. Finally, a copper concentrate with a high copper content is obtained for subsequent Isa furnace smelting.

The top-blowing smelting of the Isa process is a key link in the copper smelting process, and the process is shown in Figure 4. The concentrate raw materials, coal, slagging agent, and flux are mixed and added to the molten pool through the furnace top feeding port. Meanwhile, the spray gun is inserted into the molten pool, and the oxygen-enriched air and diesel oil are blown into the molten pool through the spray gun. The solid, liquid, and gas phases are fully mixed and stirred in the molten pool through the entrainment effect, so that the copper concentrate undergoes a chemical reaction to generate heat and maintain the high-temperature environment in the furnace. Finally, matte and slag enter the depleted electric furnace in the form of molten mixing to achieve the separation of copper and slag [26]. The main chemical reaction equations in the Isa furnace are as follows.

1.

Decomposition reaction of sulfides. The main components of copper concentrate are CuFeS₂, CuS, FeS₂, etc. After the copper concentrate enters the Isa furnace, the decomposition reaction of high-valence sulfides occurs rapidly, as shown in Equations (14)–(16).

$$4{\mathrm{CuFeS}}_2=2{\mathrm{Cu}}_2\mathrm{S}+4\mathrm{FeS}+{\mathrm{S}}_2$$

(14)

$$2{\mathrm{FeS}}_2=2\mathrm{FeS}+{\mathrm{S}}_2$$

(15)

$$4\mathrm{CuS}=2{\mathrm{Cu}}_2\mathrm{S}+{\mathrm{S}}_2$$

(16)

The FeS and Cu₂S produced by the decomposition of the above sulfides will continue to be oxidized or form copper matte, and the resulting S₂ will continue to be oxidized into SO₂ into the flue gas, as shown in Equation (17).

$${\mathrm{S}}_2+2{\mathrm{O}}_2=2{\mathrm{SO}}_2$$

(17)

2.

Oxidation reaction of sulfides. In the Isa furnace, in addition to the decomposition reaction, the sulfide will be directly oxidized, the oxidation reaction will occur, and some will be reduced to high-valence oxides, as shown in Equations (18)–(22).

$$4{\mathrm{CuFeS}}_2+5{\mathrm{O}}_2=2{\mathrm{Cu}}_2\mathrm{S}\cdot \mathrm{FeS}+2\mathrm{FeO}+4{\mathrm{SO}}_2$$

(18)

$$2\mathrm{FeS}+3{\mathrm{O}}_2=2\mathrm{FeO}+2{\mathrm{SO}}_2$$

(19)

$$3\mathrm{FeS}+{\mathrm{O}}_2={\mathrm{Fe}}_3{\mathrm{O}}_4+3{\mathrm{SO}}_2$$

(20)

$$4{\mathrm{FeS}}_2+7{\mathrm{O}}_2=2\mathrm{FeO}+2\mathrm{FeS}+6{\mathrm{SO}}_2$$

(21)

$$2{\mathrm{Cu}}_2\mathrm{S}+3{\mathrm{O}}_2=2{\mathrm{Cu}}_2\mathrm{O}+2{\mathrm{SO}}_2$$

(22)

3.

Reaction between sulfides and oxides. The oxidation reaction in the Isa furnace is intense, and the reaction product will contain a small amount of Cu₂O and Fe₃O₄. In the separation process of matte and slag in the electric furnace, the sulfide in the matte will further react with the metal oxide in the slag, as shown in Equations (23) and (24).

$${\mathrm{Cu}}_2\mathrm{O}+\mathrm{FeS}={\mathrm{Cu}}_2\mathrm{S}+\mathrm{FeO}$$

(23)

$$3{\mathrm{Fe}}_3{\mathrm{O}}_4+\mathrm{FeS}=10\mathrm{FeO}+{\mathrm{SO}}_2$$

(24)

4.

Slagging reaction. The impurities in the molten copper are removed by adding a slagging agent to produce a slagging reaction, as shown in Equations (25) and (26).

$$2\mathrm{FeO}+{\mathrm{SiO}}_2=\left(2\mathrm{FeO}\cdot {\mathrm{SiO}}_2\right)$$

(25)

$$\mathrm{FeS}+3{\mathrm{Fe}}_3{\mathrm{O}}_4+5{\mathrm{SiO}}_2=5\left(2\mathrm{FeO}\cdot {\mathrm{SiO}}_2\right)+{\mathrm{SO}}_2$$

(26)

The data in this paper come from the production data of a copper smelting enterprise in southwestern China. The input variables are listed in Table 1. And, 1136 sets of data were obtained. Among them, the first 909 sets of data were selected as the training datasets, and the last 227 sets of data were used as the test datasets.

4.1.2. Model Parameter Settings

The parameter settings of the GAN and RF models are crucial for the performance of the models. The parameters of GAN and RF used in this article are shown in

Table 3. GAN model parameters.

Parameter Name	Parameter Setting
n_epoch	1000
batch_size	64
noise_dim	128
learning rate	0.0005
criterion	BCELoss
optimization algorithm	Adam

and

Table 4. RF model parameters.

Parameter Name	Parameter Setting
n_estimators	150
criterion	MSE
max_depth	45
min_samples_split	2
min_samples_leaf	5
max_features	Auto

. The experiments were implemented in Python 3.9.7 and PyTorch 1.9.1 with the CUDA 11.2 environment running on a computer with Intel (R) Core (TM) i7-10875H CPU @ 2.30 GHz (8CPUs, NVIDIA GeForce RTX 2060, manufacturer: NVIDIA corporation, Santa Clara, CA, USA) processor, 3.4 GHz 64 G RAM. And, the proposed model uses machine learning libraries such as NumPy, Pandas, and Matplotlib.

4.2. Data Preprocessing and Variable Screening

The original data collected from the industrial site have outliers and missing values, and the data need to be preprocessed to improve the data quality. Firstly, in order to avoid the influence of different dimensions of data, the data were normalized, as shown in Equation (27). Secondly, a box plot was used to screen out outliers in the data. The results of the box plot to screen out outliers are shown in Figure 5. Finally, the interpolation method was used to fill the missing values in the data.

$$x^{\prime }={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(27)

where x denotes the data to be normalized; x_min and x_max denote the minimum and maximum values of x, respectively; and $x^{\prime }$ denotes the value after normalization.

Using the MRMR algorithm, 12 key variables required for predicting matte grade were chosen: X₁, X₂, X₃, X₄, X₅, X₆, X₉, X₁₀, X₁₁, X₁₂, X₁₅, X₁₆. X₇, and X₈ were not selected, probably because X₆ is sufficient to characterize the main components of the flux silicon, resulting in high redundancy among X₇, X₈, and X₆. Due to the high redundancy between the information contained in X₁₃ and X₁₇ and the information represented by other elements, X₁₃ and X₁₇ were not selected.

4.3. Data Enhancement and Data Fusion Results and Analysis

4.3.1. Quality Evaluation of Generated Data

In this paper, the Pearson correlation coefficient (PCC) is used to measure the similarity between the real data and the generated data, and a higher similarity indicates a higher quality of the generated data [13]. The results of the PCC between the input variables and matte grade in D₁, D₂, D₃, and D (D₁ is the data generated by RGAN, D₂ is the data generated by LRGAN, D₃ is the data generated by PRGAN, and D is the real data) are shown in Figure 6a.

Three different combinations of activation functions are used in the GAN, resulting in different degrees of similarity between the three generated data and the real data. In Figure 6a, the PPCs between variables X₃, X₁₀, X₁₁, and X₁₆ and matte grade in D₁ are closer to those calculated in D. And, the PPCs between variables X₄, X₆, X₉, and X₁₅ and matte grade in D₂ are closer to those calculated in D. Similarly, the PPCs between variables X₁, X₂, X₅, and X₁₂ and matte grade in D₃ are closer to those calculated in D.

In summary, the data generated by the GANs with different activation functions can represent some variables similarly to real data. But, none of the GANs with a single activation function fail to globally characterize the positive features and negative features. The generated data and real data have local similarity. Therefore, a data fusion criterion based on RMSE and R² was further designed to fuse the data generated by the three GANs.

In order to determine the optimum data fusion strategy, four different data fusion experiments were designed, as shown in

Table 5. Different data fusion experiments.

Data to Be Fused	The Data Obtained after Fusion
D1D2	Df1
D1D3	Df2
D2D3	Df3
D1D2D3	Df4

.

Firstly, the generated data D₁ and D₂ under Nash equilibrium were obtained using RGAN and LRGAN, respectively. And then, the weights of D₁ and D₂ were calculated using Equations (10) and (11), respectively. The results are shown in

Table 6. Generated data weight calculation results 1.

Model	RMSE	R2	Ci	Wi
MRMR_RGAN_RF (D1)	0.7241	0.8237	1.1376	0.4652
MRMR_LRGAN_RF (D2)	0.7038	0.8585	1.2198	0.5348

. Finally, the weighted fusion of D₁ and D₂ based on RMS and R² was performed using Equation (12) to obtain the fused data D_f1.

Similarly, we calculated the correlation weights of the other fusion experiments. And, D_f2, D_f3, and D_f4 were obtained, as shown in

Table 7. Generated data weight calculation results 2.

Model	RMSE	R2	Ci	Wi
MRMR_RGAN_RF (D1)	0.7241	0.8237	1.1376	0.5243
MRMR_PRGAN_RF (D3)	0.7414	0.8033	1.0835	0.4757

,

Table 8. Generated data weight calculation results 3.

Model	RMSE	R2	Ci	Wi
MRMR_LRGAN_RF (D2)	0.7038	0.8585	1.2198	0.5590
MRMR_PRGAN_RF (D3)	0.7414	0.8033	1.0835	0.4410

and

Table 9. Generated data weight calculation results 4.

Model	RMSE	R2	Ci	Wi
MRMR_RGAN_RF (D1)	0.7241	0.8237	1.1376	0.3229
MRMR_LRGAN_RF (D2)	0.7038	0.8585	1.2198	0.3981
MRMR_PRGAN_RF (D3)	0.7414	0.8033	1.0835	0.2790

.

The PCC between the input variables and matte grade in D_f1, D_f2, D_f3, and D_f4 were calculated. And, the variables closest to real data were found in the fused data, as shown in Figure 6b.

From Figure 6b, it can be seen that there are eight process variables X₁, X₂, X₄, X₅, X₁₀, X₁₁, X₁₅, and X₁₆ in D_f4 with a matte grade. The PPCs are closer to those calculated in D. D_f4 is the most similar to the real data. It is indirectly illustrated that the D_f4 obtained from the data fusion criterion based on RMSE and R² can adequately characterize the positive and negative features of copper melting data, thus improving the quality of the generated data.

4.3.2. Visualization of Generated Data

To intuitively evaluate the quality of the data D_f4, the frequency distribution histograms of the 12 input variables and matte grade in the real data D and the generated data D_f4 are completed, respectively, as shown in Figure 7.

Secondly, the normal distribution curves of the 12 input variables and matte grade in the real data D and the generated data D_f4 are shown in Figure 8 and Figure 9b; it can be seen that there is a large overlap between the generated data and real data, which further illustrates that the generated data are reliable.

Finally, from the perspective of inter-variable relationships, the 12 input variables of the generated data D_f4 and the real data D were reduced to two dimensions using principal component analysis (PCA) dimensionality reduction, as shown in Figure 9c. We can see that the generated data and real data have a large overlap in the middle portion of the graphs, which indicates that the generated data are basically in agreement with real data and further illustrates the accuracy and reliability of the generated data after fusion.

4.4. Prediction Results and Analysis of Matte Grade

4.4.1. Evaluation Index of Matte Grade Prediction Model

There are four model evaluation metrics used in this paper, which are mean absolute error (MAE), mean absolute percentage error (MAPE), RMSE, and R². MAE is the average value of the prediction error, which reflects how well the problem of offsetting errors can be avoided. RMSE measures the deviation of the predicted value from the true value. MAPE is a relative measure, which reflects the deviation of the predicted value relative to the true value. And, R² represents the extent to which the independent variable explains the dependent variable. The smaller MAE, RMSE, and MAPE, the smaller the deviation of the predicted value from the true value. The smaller deviation, the better the model prediction effect. The closer the value of R² is to 1, the stronger the model fitting ability and the better the model prediction effect [27]. The formulas for the four evaluation indicators are shown in Equations (28)–(31).

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}\sum _{i=1}^n}{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(28)

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(29)

$$MAPE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|\times 100$$

(30)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{\left(y_i-{\widehat{y}}_i\right)}^2}{{\displaystyle \sum _{i=1}^n}{\left(y_i-\overline{y}\right)}^2}}$$

(31)

where $\widehat{y}$ denotes the predicted value; y denotes the true value; $\overline{y}$ is the mean of the real data; and n denotes the number of samples in the test set.

4.4.2. Visualization of Matte Grade Prediction Results

In order to reflect the advantage of RF in matte grade prediction, the RF, Linear Regression (LR), Support Vector Regression (SVR), Long and Short-Term Memory Network (LSTM), and Backpropagation Neural Network (BP) models were used for matte grade prediction. In order to more accurately evaluate the generalization and stability of the model, we used a five-fold cross validation method. The performances of each model on the test set are shown in

Table 10. Comparison of matte grade prediction experimental results from different models.

Model	RMSE	MAE	MAPE	R2
LR	1.0826	0.4568	0.7893	0.6753
SVR	0.9579	0.4288	0.7341	0.7061
LSTM	0.8938	0.4093	0.6873	0.7269
BP	0.9361	0.4206	0.7198	0.7195
RF	0.8498	0.3691	0.6251	0.7628

, and the results indicate that the RF model has the best prediction performance, with an RMSE of 0.8498, MAE of 0.3691, MAPE of 0.6251, and R² of 0.7628.

In order to verify the effectiveness of the method proposed in this paper, the ablation experiments conducted in this paper and their results are shown in

Table 11. Comparison of results of ablation experiments.

Model	RMSE	MAE	MAPE	R2
MRMR_RF	0.7931	0.3447	0.5803	0.7813
MRMR_RGAN_RF	0.7241	0.3152	0.5398	0.8237
MRMR_LRGAN_RF	0.7038	0.2985	0.5097	0.8585
MRMR_PGAN_RF	0.7414	0.3253	0.5513	0.8033
MRMR_RGAN + LRGAN_RF	0.6876	0.2911	0.4921	0.8727
MRMR_RGAN + PRGAN_RF	0.7164	0.3114	0.5327	0.8476
MRMR_LRGAN + PRGAN_RF	0.7011	0.3081	0.5193	0.8623
MRMR_EGAN(RGAN + LRGAN + PRGAN)_RF	0.6653	0.2805	0.4837	0.8815

.

Firstly, the RF and MRMR_RF methods were used to predict the matte grade, and the experimental results are shown in Figure 10a. In MRMR_RF model, RMSE is reduced by 6.67%, MAE is reduced by 6.61%, MAPE is reduced by 7.17%, and R² is increased by 2.43%. This shows that the MRMR-based variable screening method can find important input variables and improve the prediction accuracy of the matte grade.

Secondly, MRMR_RGAN_ RF, MRMR_LRGAN_RF, and MRMR_PRGAN_RF were used to predict the grade of matte. The experimental results are shown in Figure 10b. Compared to the MRMR_RF model, the RMSE is reduced by at least 6.52%, the MAE is reduced by at least 5.63%, the MAPE is reduced by at least 5.00%, and the R² is increased by at least 2.82%, indicating that the GAN-based data enhancement method can effectively improve the prediction accuracy of the matte grade.

Finally, MRMR_RGAN + LRGAN_RF, MRMR_RGAN + PRGAN_RF, MRMR_ LRGAN + PRGAN_RF, and MRMR_EGAN_RF were used to predict the matte grade. The experimental results are shown in Figure 10c. The proposed method has the highest prediction accuracy. Compared to single GAN data enhancement models, RMSE is reduced by at least 5.47%, MAE is reduced by at least 6.03%, MAPE is reduced by at least 5.10%, and R² is increased by at least 2.68%, which proves that data fusion can further improve the prediction accuracy of the matte grade.

Furthermore, to visualize the distribution of the prediction errors of different models for matte grade, the difference between the predicted value and the real value of the matte grade was calculated, and the error distribution curve is presented in Figure 11a. Compared with other models, the prediction error of MRMR_EGAN_RF is the smallest. The real value and predicted value of the matte grade obtained by different models are taken as horizontal and vertical coordinates, respectively, and the scatter plot is drawn, as shown in Figure 11b. Among them, the blue regression line represents the prediction results of the MRMR_EGAN_RF model. Compared with the prediction results of matte grade obtained by other methods, the scatter plot of the proposed method has a large number of points distributed near y = x, and the predicted matte grade is between 53% and 62%. The prediction accuracy of the proposed method is the highest, which proves the effectiveness of the proposed method.

4.5. Interpretability Analysis

To improve the interpretability of the prediction results, the SHAP theory is used to explain and analyze the prediction results of the MRMR-EGAN-RF model [28].

By using the SHAP method, the contribution of different parameters to predicting the matte grade can be explored. Figure 12a shows the SHAP values for each parameter. The positive/negative SHAP value of each parameter indicates that the parameter promotes/suppresses the predicted matte value, and the color of the dots represents the size of the specific value. For a certain parameter, the wider the horizontal coverage range, the greater the impact of the parameter on the prediction results, so the parameters are more important. The average of the absolute SHAP values of all data were calculated for each parameter for the parameter importance ranking, as shown in Figure 12b.

The interpretation of matte grade prediction for two different test groups of samples is shown in Figure 13. The values in the graph intuitively represent the specific impact of each variable on the predicted matte grade. The results indicate that the oxygen concentration and the Cu content in the copper concentrate have a significant positive impact on the predicted matte grade. When the oxygen enrichment concentration and copper content in the copper concentrate are high, it will push the predicted value of the model towards the direction of increasing the base value because a higher oxygen enrichment concentration can improve the oxidation reaction rate. The Isa furnace can often maintain a higher temperature, thereby improving the fluidity of the slag and more effectively removing impurities from the melt, thereby improving the matte grade. The above analysis is consistent with the experimental results, indicating that SHAP theory can improve the interpretability of the prediction models, thereby guiding users to optimize the copper smelting process.

5. Conclusions

Aiming to address the problem of the low prediction accuracy of a matte grade prediction model due to the scarcity of samples in the copper melting process, this paper proposes the MRMR_EGAN_RF model for matte grade prediction. The MRMR algorithm is introduced to select variables that have a great influence on the matte grade. And, we use GANs with different activation functions to enhance the data and fuse different generated data based on RMSE and R² to improve the quality of the generated data. A matte grade prediction model based on RF is proposed. The actual data collected from the industrial site are utilized to carry out the application validation and comparative analysis. The experimental results show that the proposed method can obtain high-quality generated data, and the prediction accuracy is better than other models. The R² is improved by at least 2.68%, and other indicators such as RMSE, MAE and MAPE are significantly improved.

In future studies, we will use sensitivity analysis techniques to determine the parameters that have a significant impact on matte grade. Through a weighted analysis of the parameters, a more effective decision-making in the copper smelting process will be completed. At the same time, in order to effectively capture the key information of input variables on matte grade at different time points, we will further explore the time correlation between input variables and matte grade in the copper smelting process, construct a time attention mechanism, and improve the accuracy of matte grade prediction.