A hot galvanized iron strip multi-objective surface roughness prediction method based on a LightGBM algorithm
By using a multi-objective surface roughness prediction method based on the LightGBM algorithm, the problem of traditional models being unable to accurately predict surface roughness under the coordinated changes of parameters in multiple processes is solved. This method achieves high-precision synchronous prediction of surface roughness of hot-dip galvanized steel strip, thereby improving product quality and production efficiency.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- 燕山大学深圳研究院
- Filing Date
- 2026-04-08
- Publication Date
- 2026-07-10
Smart Images
Figure CN122366733A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of metal plastic processing technology, specifically relating to a multi-objective surface roughness prediction method for hot-dip galvanized iron strip based on the LightGBM algorithm. Background Technology
[0002] Hot-dip galvanized steel (GA) strip is an indispensable metal-based composite material in the automotive manufacturing industry. Its surface roughness is one of the core quality attributes that determines paint adhesion, coating uniformity, final appearance, and stamping performance. This indicator not only directly affects the product's visual quality and corrosion resistance but also relates to the stability of subsequent manufacturing processes and the safety and reliability of the entire vehicle structure.
[0003] The production process of hot-dip galvanized steel strip is characterized by multiple processes, high coupling, and nonlinearity, encompassing several key stages such as cold rolling, continuous annealing, hot-dip galvanizing, alloying, and leveling. In each stage, process parameters—such as rolling force, annealing temperature and speed, control of galvanized layer alloying, and leveling elongation—interact in complex ways, collectively influencing the final surface morphology of the strip. This multi-physics, multi-scale coupled process chain makes the formation mechanism of surface roughness extremely complex, especially in the strip width direction, where uneven distribution of process conditions often results in significant transverse non-uniformity.
[0004] However, traditional surface roughness prediction models are mostly limited to local analysis of a single process or statistical modeling only for individual inspection points, making it difficult to fully characterize the roughness distribution across the entire strip width. When faced with coordinated changes in multi-process parameters and dynamic fluctuations in production conditions, the prediction accuracy and robustness of such methods are significantly limited, failing to fully meet the requirements of high-precision, full-width quality control. With the automotive industry's increasing demands for material surface consistency, machinability, and appearance standards, the shortcomings of existing methods have become a bottleneck restricting further optimization of GA strip steel quality. Therefore, there is an urgent need for an advanced method capable of comprehensively and accurately predicting multiple surface roughness objectives. Summary of the Invention
[0005] The purpose of this invention is to provide a method for predicting the surface roughness of hot-dip galvanized iron strip based on the LightGBM algorithm, which enables synchronous and high-precision prediction of surface roughness at multiple measurement points under the coordination of multiple process parameters, in order to solve the above-mentioned problems.
[0006] The present invention achieves the above objectives through the following technical solutions: A method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm includes the following steps: S1: Data collection and feature matrix construction; S2: Data preprocessing; S3: Divide the feature parameter dataset; S4: Construct the LightGBM model; S5: Custom multi-objective loss function; S6: Unified hyperparameter optimization; S7: Surface roughness prediction; S8: Model evaluation and results presentation.
[0007] As a further optimization of the present invention, step S1 specifically includes: Industrial measurement data is continuously collected from the hot-dip galvanized iron strip production line; the data precisely includes: Input characteristics: The hot-dip galvanizing process includes: alloying temperature (°C), alloying speed (m / min), iron content of the upper surface coating (%), Iron content of the lower surface coating (%), the leveling process includes: strip entry thickness (mm), elongation (%), Yield strength (MPa), rolling kilometers of the work rolls (km), where the feature matrix X consists of the following parameters: Output target: 6 measuring points on the upper and lower surfaces of hot-dip galvanized steel strip, specifically the measuring points on the working side of the upper surface. (µm), upper surface operating side measurement point (µm), measuring point at the middle of the upper surface (µm), lower surface working side measuring point (µm), lower surface operating side measurement point (µm), middle measuring point on the lower surface The surface roughness value is (µm); where the characteristic matrix y consists of the following parameters: .
[0008] As a further optimization of the present invention, step S2 specifically includes: First, outliers are handled to improve data quality and model performance. Then, data standardization is performed to eliminate differences in the units of measurement between features, thereby improving the model's computational efficiency and convergence speed. Standardization makes the data conform to a normal distribution with a mean of 0 and a standard deviation of 1.
[0009] As a further optimization of the present invention, step S3 specifically includes: The dataset is divided into training and test sets using stratified random sampling to maintain consistent feature distribution. The training set is used to build the model and the test set is used to evaluate the model's generalization performance, with 80% (training) and 20% (test) allocated to each set.
[0010] As a further optimization of the present invention, step S4 specifically includes: For the six roughness prediction targets on the upper and lower surfaces of GA strip (i.e., the roughness values of the working side, operating side, and intermediate measuring point on the upper surface, and the working side, operating side, and intermediate measuring point on the lower surface), a single multi-output LightGBM model is constructed. The output of this model is no longer a single numerical value, but a vector containing the predicted values of all six roughness targets for each sample.
[0011] As a further optimization of the present invention, step S5 specifically includes: This is the key to achieving multiple outputs from a single model; LightGBM training is based on gradient boosting, and each iteration builds a new decision tree by fitting the negative gradient of the residual predicted by the previous model; for this purpose, a loss function needs to be defined so that a single model can optimize all six objectives simultaneously. Loss function definition: Considering the complexity of the hot-dip galvanized steel strip production process and the characteristics of multi-objective prediction, a custom loss function is defined. (For a single sample i), its ultimate goal is to minimize during training; when considering all six roughness objectives ( When all factors are given equal weight (weight wj=1), we use the sum of mean squared errors (MSE) as the loss function; its calculation formula is as follows: in It is the model's predicted value for the j-th roughness target of sample i. These are the true values; to achieve gradient boosting, LightGBM needs to calculate the loss function. For each predicted value (where k is the index of the predicted target value,) The first derivative (gradient) and second derivative (Hessian) of the gradient are as follows: LightGBM internally calculates the gradient values of all samples i and all targets k, and also calculates the Hessian values of all samples and targets. The model uses these calculated gradients and Hessians to iteratively build new decision trees, with the goal of minimizing the overall loss function, thereby ultimately achieving collaborative prediction of the surface roughness of six different measurement points of GA strip steel.
[0012] As a further optimization of the present invention, step S6 specifically includes: The Genetic Algorithm (GA) is used to simulate the biological evolution process within the predefined LightGBM hyperparameter search space. Through operations such as population generation, selection, crossover, and mutation, the hyperparameter combination is iteratively optimized in order to find the optimal set of hyperparameters that can maximize the model's predictive performance. The real-number encoding method of genetic algorithms allows optimization to be performed directly in the continuous search space of parameters. Through selection, crossover, and mutation operations, the search range of each parameter and the iterative parameters of the genetic algorithm (such as population size, crossover rate, and mutation rate) are determined during implementation; and the optimal LightGBM model configuration is found in multiple rounds of evolution. Search Space Definition: Define the search space for the key hyperparameters of the LightGBM model. These parameters include: learning_rate (learning rate, controlling the step size of each iteration), n_estimators (number of base learners, i.e., the number of decision trees), num_leaves (maximum number of leaf nodes per tree, controlling model complexity), max_depth (maximum depth of each tree), min_child_samples (minimum amount of data in a leaf node, preventing overfitting), reg_alpha (L1 regularization coefficient), and reg_lambda (L2 regularization coefficient).
[0013] As a further optimization of the present invention, step S7 specifically includes: Using a single multi-output LightGBM model optimized by a genetic algorithm in the training set, predictions are made for new and unseen GA strip process parameters in the test set. The model will output a prediction vector containing all six roughness targets at once, namely the surface roughness values of multiple measuring points on the upper and lower surfaces of the GA strip. This prediction process utilizes the extreme computational efficiency of LightGBM to achieve millisecond-level prediction response, which fully meets the production line's requirements for real-time performance and low latency.
[0014] As a further optimization of the present invention, step S8 specifically includes: The predictive performance of the LightGBM model group is comprehensively evaluated using a series of metrics, including the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE). The specific formulas are shown below: In the formula: N represents the number of samples; This represents the true value of the j-th predicted target in the i-th sample; This represents the model prediction value for the j-th target of the i-th sample; This represents the average true value of the j-th predicted target across all samples.
[0015] The beneficial effects of this invention are as follows: 1. This invention achieves accurate prediction of surface roughness of hot-dip galvanized iron (GA) strip by using a single multi-output LightGBM model for high-precision multi-objective roughness prediction and by using a genetic optimization algorithm to intelligently optimize the model hyperparameters. This significantly improves the stability of product quality and production efficiency, and effectively solves the challenges faced by traditional roughness control technologies. Attached Figure Description
[0016] Figure 1 This is a flowchart of the overall process for a roughness multi-objective prediction method based on the LightGBM algorithm; Figure 2 This is a diagram showing the distribution of GA strip roughness measuring equipment and measuring points; Figure 3 This is the LightGBM model iteration flowchart; Figure 4 It is a scatter plot of the training set and test set relative to the true values. Detailed Implementation
[0017] The present application will now be described in further detail with reference to the accompanying drawings. It should be noted that the following specific embodiments are only used to further illustrate the present application and should not be construed as limiting the scope of protection of the present application. Those skilled in the art can make some non-essential improvements and adjustments to the present application based on the above application content.
[0018] Example 1: As Figure 1 , Figure 2 , Figure 3 As shown, a multi-objective surface roughness prediction method for hot-dip galvanized steel strip based on the LightGBM algorithm includes the following steps: S1: Data collection and feature matrix construction; continuous collection of industrial measurement data from the hot-dip galvanized iron strip production line; the data precisely includes: Input characteristics: The hot-dip galvanizing process includes: alloying temperature (°C), alloying speed (m / min), iron content of the upper surface coating (%), Iron content of the lower surface coating (%), the leveling process includes: strip entry thickness (mm), elongation (%), Yield strength (MPa), rolling kilometers of the work rolls (km), where the feature matrix X consists of the following parameters: Output target: 6 measuring points on the upper and lower surfaces of hot-dip galvanized steel strip, specifically the measuring points on the working side of the upper surface. (µm), upper surface operating side measurement point (µm), measuring point at the middle of the upper surface (µm), lower surface working side measuring point (µm), lower surface operating side measurement point (µm), middle measuring point on the lower surface The surface roughness value is (µm); where the characteristic matrix y consists of the following parameters: .
[0019] According to step S1, industrial measurement data is continuously and automatically collected on the GA strip steel production line using online sensors. The collected data includes: hot-dip galvanizing process data, including alloying temperature. (°C), alloying speed (m / min), iron content of the upper surface coating (%), Iron content of the lower surface coating (%), and leveling process data including: strip entry thickness (mm), elongation (%), Yield strength (MPa), rolling kilometers of the work rolls (km), some data are shown in Table 1. Subsequently, a roughness measuring instrument was used to periodically measure three preset measuring points on each of the upper and lower surfaces of the finished GA strip steel, as shown in Table 1. Figure 2 The figures shown are the measuring points on the working side of the upper surface. (µm), upper surface operating side measurement point (µm), measuring point at the middle of the upper surface (µm), lower surface working side measuring point (µm), lower surface operating side measurement point (µm), middle measuring point on the lower surface Surface roughness values (µm). Some data are shown in Table 2.
[0020] Table 1. On-site data (partial) Table 2 Real-time roughness measurement data (partial) S2: Data preprocessing; First, outliers are processed to improve data quality and model performance; then, data standardization is performed to eliminate dimensional differences between features, thereby improving model computation efficiency and convergence speed; standardization makes the data conform to a normal distribution with a mean of 0 and a standard deviation of 1.
[0021] According to step S2, a rigorous preprocessing procedure was performed on the dataset to systematically identify and remove outliers and duplicates, ensuring data quality and the reliability of model performance. Subsequently, the StandardScaler library in Python was used to standardize all numerical features, making their mean 0 and standard deviation 1, thereby eliminating the dimensional differences between features and optimizing the data distribution for model training.
[0022] S3: Divide the feature parameter dataset; use stratified random sampling to divide the dataset into training and test sets to maintain consistent feature distribution; divide it according to an 80% (training) and 20% (test) ratio, with the training set used to build the model and the test set used to evaluate the model's generalization performance.
[0023] According to step S3, a stratified random sampling strategy was used to divide the preprocessed complete dataset. Specifically, 80% of the dataset was divided into a training set (for model learning and hyperparameter optimization), and the remaining 20% was divided into an independent test set (for final model performance evaluation). This ensures that the distribution ratio of each target variable (i.e., the roughness value of each measurement point) in the training and test sets is consistent with that of the original dataset, thereby avoiding evaluation bias caused by uneven data distribution.
[0024] S4: Construct a LightGBM model; For the six roughness prediction targets on the upper and lower surfaces of the GA strip (i.e., the roughness values of the working side, operating side, and intermediate measuring point on the upper surface, and the working side, operating side, and intermediate measuring point on the lower surface), construct a single multi-output LightGBM model. The output of this model is no longer a single numerical value, but a vector containing the predicted values of all six roughness targets for each sample.
[0025] S5: Custom multi-objective loss function; this is the key to achieving multiple outputs from a single model; LightGBM training is based on gradient boosting, and each iteration builds a new decision tree by fitting the negative gradient of the residual predicted by the previous model; for this purpose, a loss function needs to be defined so that a single model can optimize all six objectives simultaneously; Loss function definition: Considering the complexity of the hot-dip galvanized steel strip production process and the characteristics of multi-objective prediction, a custom loss function is defined. (For a single sample i), its ultimate goal is to minimize during training; when considering all six roughness objectives ( When all factors are given equal weight (weight wj=1), we use the sum of mean squared errors (MSE) as the loss function; its calculation formula is as follows: in It is the model's predicted value for the j-th roughness target of sample i. These are the true values; to achieve gradient boosting, LightGBM needs to calculate the loss function. For each predicted value (where k is the index of the predicted target value,) The first derivative (gradient) and second derivative (Hessian) of the gradient are as follows: LightGBM internally calculates the gradient values of all samples i and all targets k, and also calculates the Hessian values of all samples and targets. The model uses these calculated gradients and Hessians to iteratively build new decision trees, with the goal of minimizing the overall loss function, thereby ultimately achieving collaborative prediction of the surface roughness of six different measurement points of GA strip steel.
[0026] S6: Unified hyperparameter optimization; adopts a genetic algorithm (GA) within the pre-defined LightGBM hyperparameter search space, simulates the biological evolution process, and iteratively optimizes the hyperparameter combination through operations such as population generation, selection, crossover and mutation, in order to find the optimal set of hyperparameters that can maximize the model's predictive performance. The real-number encoding method of genetic algorithms allows optimization to be performed directly in the continuous search space of parameters. Through selection, crossover, and mutation operations, the search range of each parameter and the iterative parameters of the genetic algorithm (such as population size, crossover rate, and mutation rate) are determined during implementation; and the optimal LightGBM model configuration is found in multiple rounds of evolution. Search Space Definition: Define the search space for the key hyperparameters of the LightGBM model. These parameters include: learning_rate (learning rate, controlling the step size of each iteration), n_estimators (number of base learners, i.e., the number of decision trees), num_leaves (maximum number of leaf nodes per tree, controlling model complexity), max_depth (maximum depth of each tree), min_child_samples (minimum amount of data in a leaf node, preventing overfitting), reg_alpha (L1 regularization coefficient), and reg_lambda (L2 regularization coefficient).
[0027] Then, according to steps S4, S5, and S6, as follows: Figure 3 As shown, a LightGBM model is constructed and initialized. During model training, based on a custom multi-objective loss function, LightGBM constructs a decision tree by aggregating the total gradient per sample and Hessian of all objectives calculated by this function. Simultaneously, a genetic optimization algorithm is used to search within a pre-defined hyperparameter search space. Through continuous trials, considering both prediction accuracy and computational time, the optimal parameter combination for the LightGBM algorithm under this multi-objective prediction task is finally determined. The parameter settings and hyperparameter optimization ranges are shown in Tables 3 and 4.
[0028] Table 3 Genetic Algorithm Parameter Settings Table 4 LightGBM Hyperparameter Search Range S7: Surface roughness prediction; Using a single multi-output LightGBM model optimized by a genetic algorithm in the training set, predictions are made for new and unseen GA strip process parameters in the test set; The model will output a prediction vector containing all six roughness targets at once, i.e., the surface roughness values of multiple measuring points on the upper and lower surfaces of the GA strip; This prediction process utilizes the extreme computational efficiency of LightGBM to achieve millisecond-level prediction response, fully meeting the production line's requirements for real-time performance and low latency.
[0029] Subsequently, according to step S7, the already trained LightGBM model is used to predict the new and unseen GA strip process parameters input in the test set. The model will simultaneously output a surface roughness value containing six measuring points on the upper and lower surfaces of the GA strip.
[0030] S8: Model Evaluation and Results Presentation; The predictive performance of the LightGBM model group is comprehensively evaluated using a series of indicators such as the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE). The specific formulas are shown below: In the formula: N represents the number of samples; This represents the true value of the j-th predicted target in the i-th sample; This represents the model prediction value for the j-th target of the i-th sample; This represents the average true value of the j-th predicted target across all samples.
[0031] According to step S8, the results are shown in Table 5. On the training set, the model achieved an MAE of 0.020, an RMSE of 0.025, and an R² of 0.988; on the test set, the model achieved an MAE of 0.035, an RMSE of 0.042, and an R² of 0.965. Simultaneously, scatter plots of the training and test sets relative to the true values were generated, as shown below. Figure 4 As shown in the figure, the results fully demonstrate that this model can not only accurately capture the inherent patterns in the training data, but also has excellent generalization ability, enabling it to make high-precision predictions on unseen new data, thus ensuring its reliability and effectiveness in practical production applications.
[0032] Table 5. Coefficients of determination and normalized mean square errors for training and test sets. The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.
Claims
1. A method for predicting the multi-objective surface roughness of hot-dip galvanized iron strip based on the LightGBM algorithm, characterized in that: Includes the following steps: S1: Data collection and feature matrix construction; S2: Data preprocessing; S3: Divide the feature parameter dataset; S4: Construct the LightGBM model; S5: Custom multi-objective loss function; S6: Unified hyperparameter optimization; S7: Surface roughness prediction; S8: Model evaluation and results presentation.
2. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 1, characterized in that: Step S1 specifically includes: Industrial measurement data is continuously collected from the hot-dip galvanized iron strip production line; the data precisely includes: Input characteristics: The hot-dip galvanizing process includes: alloying temperature (°C), alloying speed (m / min), iron content of the upper surface coating (%), Iron content of the lower surface coating (%), the leveling process includes: strip entry thickness (mm), elongation (%), Yield strength (MPa), rolling kilometers of the work rolls (km), where the feature matrix X consists of the following parameters: Output target: 6 measuring points on the upper and lower surfaces of hot-dip galvanized steel strip, specifically the measuring points on the working side of the upper surface. (µm), upper surface operating side measurement point (µm), measuring point at the middle of the upper surface (µm), lower surface working side measuring point (µm), lower surface operating side measurement point (µm), middle measuring point on the lower surface The surface roughness value is (µm); where the characteristic matrix y consists of the following parameters: 。 3. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 2, characterized in that: Step S2 specifically includes: First, outliers are handled using the IQR (interquartile range) method to improve data quality and model performance. Then, data standardization is performed to eliminate dimensional differences between features, thereby improving model computation efficiency and convergence speed, and ensuring that the data has zero mean and unit variance.
4. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 3, characterized in that: Step S3 specifically includes: The dataset is divided into training and test sets by stratified sampling based on production batches to maintain consistent feature distribution. The training set is used to build the model and the test set is used to evaluate the generalization performance of the model, with 80% (training) and 20% (test) allocated to each set.
5. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 4, characterized in that: Step S4 specifically includes: For the six roughness prediction targets on the upper and lower surfaces of GA strip (i.e., the roughness values of the working side, operating side, and intermediate measuring point on the upper surface, and the working side, operating side, and intermediate measuring point on the lower surface), a single multi-output LightGBM model is constructed. The output of this model is no longer a single numerical value, but a vector containing the predicted values of all six roughness targets for each sample.
6. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 5, characterized in that: Step S5 specifically includes: This is the key to achieving multiple outputs from a single model; LightGBM training is based on gradient boosting, and each iteration builds a new decision tree by fitting the negative gradient of the residual predicted by the previous model; for this purpose, a loss function needs to be defined so that a single model can optimize all six objectives simultaneously. Loss function definition: Considering the complexity of the hot-dip galvanized steel strip production process and the characteristics of multi-objective prediction, a custom loss function is defined. (For a single sample i), its ultimate goal is to minimize during training; when considering all six roughness objectives ( When all factors are given equal weight (weight wj=1), we use the sum of mean squared errors (MSE) as the loss function; its calculation formula is as follows: in It is the model's predicted value for the j-th roughness target of sample i. These are the true values; to achieve gradient boosting, LightGBM needs to calculate the loss function. For each predicted value (where k is the index of the predicted target value,) The first derivative (gradient) and second derivative (Hessian) of the gradient are as follows: LightGBM internally calculates the gradient values of all samples i and all targets k, and also calculates the Hessian values of all samples and targets. The model uses these calculated gradients and Hessians to iteratively build new decision trees, with the goal of minimizing the overall loss function, thereby ultimately achieving collaborative prediction of the surface roughness of six different measurement points of GA strip steel.
7. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 6, characterized in that: Step S6 specifically includes: The Genetic Algorithm (GA) is used to simulate the biological evolution process within the predefined LightGBM hyperparameter search space. Through operations such as population generation, selection, crossover, and mutation, the hyperparameter combination is iteratively optimized in order to find the optimal set of hyperparameters that can maximize the model's predictive performance. The real-number encoding method of genetic algorithms allows optimization to be performed directly in the continuous search space of parameters. Through selection, crossover, and mutation operations, the search range of each parameter and the iterative parameters of the genetic algorithm (such as population size, crossover rate, and mutation rate) are determined during implementation; and the optimal LightGBM model configuration is found in multiple rounds of evolution. Search Space Definition: Define the search space for the key hyperparameters of the LightGBM model. These parameters include: learning_rate (learning rate, controlling the step size of each iteration), n_estimators (number of base learners, i.e., the number of decision trees), num_leaves (maximum number of leaf nodes per tree, controlling model complexity), max_depth (maximum depth of each tree), min_child_samples (minimum amount of data in a leaf node, preventing overfitting), reg_alpha (L1 regularization coefficient), and reg_lambda (L2 regularization coefficient).
8. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 7, characterized in that: Step S7 specifically includes: Using a single multi-output LightGBM model optimized by a genetic algorithm in the training set, predictions are made for new and unseen GA strip process parameters in the test set. The model will output a prediction vector containing all six roughness targets at once, namely the surface roughness values of multiple measuring points on the upper and lower surfaces of the GA strip. This prediction process utilizes the extreme computational efficiency of LightGBM to achieve a fast prediction response, fully meeting the production line's requirements for real-time performance and low latency.
9. The method for predicting the multi-objective surface roughness of hot-dip galvanized steel strip based on the LightGBM algorithm according to claim 8, characterized in that: Step S8 specifically includes: The predictive performance of the LightGBM model group is comprehensively evaluated using a series of metrics, including the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE). The specific formulas are shown below: In the formula: N represents the number of samples; This represents the true value of the j-th predicted target in the i-th sample; This represents the model prediction value for the j-th target of the i-th sample; This represents the average true value of the j-th predicted target across all samples.