A novel social network search and LightGBM framework for accurate prediction of blast-induced peak particle velocity

Abstract

The accurate prediction of peak particle velocity (PPV) is essential for effectively managing blast-induced vibrations in mining operations. This study presents a novel PPV prediction method based on the social network search and LightGBM (SNS-LightGBM) deep gradient cooperative learning framework. The SNS algorithm enhances LightGBM’s learning process by optimizing hyperparameters through global search capabilities and balancing model complexity to improve generalization. To assess its performance, five baseline machine learning models and a hybrid model combining SNS-LightGBM were developed for comparison. The predictive performance of these models was evaluated using metrics such as coefficient of determination (R²), mean absolute error (MAE), mean absolute percentage error (MAPE), mean squared error (MSE), and root mean squared error (RMSE). The results indicate that the SNS-LightGBM model substantially improves both the accuracy and stability of PPV predictions. The SNS-LightGBM model outperformed all other models, achieving an R² of 0.975, MAE of 0.086, MAPE of 0.071, MSE of 0.019, and RMSE of 0.138. Additionally, a feature importance analysis revealed that distance and charge weight are the most significant factors influencing PPV, far surpassing other parameters. These findings offer valuable insights for improving the precision of blast vibration prediction and optimizing blasting designs.

Graphical abstract

Keywords

peak particle velocity / social network search / LightGBM / feature importance analysis / predicting performance

1 Introduction

Peak particle velocity (PPV) is a crucial parameter for assessing the impacts of blasting operations, as excessive ground vibrations can result in structural damage, environmental hazards, and safety concerns [1]. In sectors such as mining, construction, and demolition, accurately predicting PPV is essential for optimizing blast designs and minimizing risks to surrounding infrastructure [2,3].

The most common method for predicting PPV is the use of empirical formulas. For example, the formulas developed by the US Bureau of Mines [4] and Langefors and Kihlström [5] are widely used to estimate PPV. These empirical formulas summarize patterns derived from historical data and can provide reasonable predictions in relatively simple geological conditions [6]. However, while these formulas are somewhat effective, they often become inadequate as geological complexity increases. For example, Gui et al. [7] noted that strata heterogeneity and differences in rock types can significantly affect vibration propagation characteristics. Yin et al. [8] also indicated that the propagation speed and attenuation characteristics of blast waves in different media are crucial factors to consider when predicting PPV. Therefore, relying solely on empirical formulas for predicting PPV may not accurately reflect actual vibration conditions in complex geological circumstances.

In recent years, machine learning techniques have emerged as powerful alternatives for PPV prediction, leveraging their capacity to capture nonlinear relationships among various input parameters [9–20]. Many studies have illustrated the effectiveness of models such as artificial neural network (ANN) [21,22], support vector machine (SVM) [23,24], and random forest (RF) [25,26] in predicting PPV with greater accuracy than traditional empirical models. For instance, Iramina et al. [27] demonstrated that ANN significantly outperformed conventional regression models in PPV predictions. Khandelwal [28] also applied SVM for PPV prediction, revealing that machine learning methods could better account for the complex factors affecting blast-induced vibrations. Moreover, researchers have noted that the ability of the RF model to select key features and reduce overfitting enhances the reliability of predictions [29]. Despite these advancements, the performance of traditional machine learning models is heavily reliant on appropriate parameter tuning and optimization, which significantly influences prediction accuracy [30,31]. Additionally, while machine learning models adeptly capture nonlinear trends, their predictions can be sensitive to the quality of training data and the optimization of hyperparameters [32,33].

To overcome these limitations, hybrid models have been developed by combining the strengths of various machine learning algorithms or incorporating optimization techniques to fine-tune model parameters [34–39]. For example, Fathipour-Azar [40] hybrid model based on boosted additive regression combined with random trees was proposed to predict rock strength. Results demonstrated that the hybrid model achieved higher prediction accuracy compared to other models. Liu et al. [41] proposed a hybrid model combining Transformer and UNet for lithology identification, demonstrating that the model achieved a prediction accuracy of 95.57%. Similarly, integrated models are extensively applied in engineering fields [42], such as predicting the load-bearing capacity of composite steel beams in fire scenarios [43,44] and assessing the performance of steel panels under explosive loading conditions [45,46]. These studies demonstrate that hybrid models offer superior adaptability and predictive accuracy in dynamic and complex environments.

Despite the notable advancements of machine learning and hybrid models across various fields, challenges persist in predicting PPV, especially under complex geological conditions. First, although hybrid models excel at managing nonlinear and high-dimensional data, computational efficiency and processing time remain significant challenges when applied to large-scale data sets. Second, existing hybrid models are prone to overfitting in practical applications, particularly when data quality is poor or training data sets are insufficient. Therefore, while hybrid models have demonstrated robust predictive capabilities, further optimization is essential to enhance computational efficiency, reduce overfitting, and improve model generalizability.

Based on this, the research aims to enhance PPV prediction methodologies by introducing a hybrid model that combines social network search (SNS) with LightGBM. First, SNS was integrated into LightGBM to create the SNS-LightGBM hybrid model. Next, five additional non-optimized models were developed to evaluate the predictive performance of the hybrid model against these non-optimized models in predicting PPV. Finally, the feature importance of each parameter’s influence on PPV was examined. The novelty of this study lies in the integration of SNS with LightGBM for hyperparameter optimization, addressing challenges such as overfitting and computational efficiency, and providing deeper insights into the factors governing PPV through a standardized feature importance framework.

2 Methodology

2.1 Data-driven prediction model

2.1.1 LightGBM

Among various machine learning algorithms, gradient boosting decision trees (GBDT) are widely used for regression and classification tasks due to their excellent nonlinear modeling capabilities [47]. However, traditional GBDT often faces challenges such as low computational efficiency and suboptimal feature selection when dealing with large-scale data. To address these issues, this study selects LightGBM to build a predictive model for blast vibration velocity. LightGBM [48] is an efficient gradient boosting decision tree algorithm developed by Microsoft corporation, specifically designed to overcome the inefficiencies of traditional GBDT in finding the best split points. The principal diagram of LightGBM is shown in Fig.1.

Its main advantage lies in its histogram-based algorithm, which significantly reduces the computation time required to locate split points, thereby enhancing the training and prediction efficiency of the model (Fig.1(a)). Additionally, LightGBM introduces the gradient-based one-side sampling (GOSS) algorithm, which further improves computational efficiency. GOSS prioritizes samples with larger gradients for training while randomly sampling those with smaller gradients, ensuring sample representativeness and increasing attention to outlier data points (Fig.1(b)). This approach enhances the overall predictive accuracy of the model, making it particularly suitable for predicting blast vibration velocity, which exhibits strong nonlinearity and uncertainty.

To optimize computational efficiency, LightGBM also employs exclusive feature bundling (EFB) technology, which reduces feature redundancy and lowers computational complexity by bundling mutually exclusive features together (Fig.1(c)). EFB significantly accelerates the training speed of the model when handling high-dimensional feature data sets. Finally, regarding the tree growth strategy, LightGBM adopts a leaf-wise growth method (Fig.1(d)), which allows control over tree growth by setting a maximum depth to avoid overfitting due to excessive growth. This approach enables LightGBM to excel in regression prediction tasks.

2.1.2 Extreme gradient boosting

Extreme gradient boosting (XGBoost) [49] is a highly efficient ensemble learning algorithm derived from GBDT and is widely utilized for both regression and classification tasks. The fundamental principle of XGBoost involves constructing decision trees through multiple iterations, where each new tree is designed to correct the errors made by the previous model, thereby progressively enhancing prediction accuracy.

In contrast to traditional GBDT, XGBoost integrates several optimizations to improve performance and computational efficiency. First, it incorporates regularization terms (both L1 and L2) to manage tree complexity and mitigate the risk of overfitting. Secondly, XGBoost employs feature subsampling, randomly selecting a subset of features during each training iteration, which reduces feature correlation and bolsters the model’s generalization capability. Additionally, it utilizes a Shrinkage mechanism that reduces the contribution of each tree, preventing rapid convergence and enhancing the model’s robustness.

When applied to the prediction of blast vibration velocity, XGBoost’s robust nonlinear modeling capabilities effectively capture complex relationships within the data. Its efficient parallel computing mechanism enables it to maintain rapid training speeds and strong predictive performance, even with large-scale data sets. Consequently, XGBoost is particularly well-suited for developing predictive models for blast vibration velocity, offering improved accuracy while effectively preventing overfitting.

2.1.3 Random forest

RF [50] is an ensemble learning algorithm that tackles regression and classification tasks by constructing multiple decision trees. The core idea is to average or vote on the predictions from these trees, which enhances the model’s accuracy and robustness. Each tree is trained on different samples and subsets of features to ensure model diversity, thereby reducing the risk of overfitting.

In building a RF model, two randomization mechanisms are employed. First, the Bootstrap sampling method is used to create the training data sets by randomly sampling with replacement from the original data set. Second, when selecting split nodes for each tree, a random subset of features is chosen for partitioning, which prevents certain features from dominating the decision-making process across all trees. This randomness enhances the model’s generalization ability and reduces its reliance on individual features.

In the context of predicting blast vibration velocity, RF effectively captures nonlinear relationships within the data by leveraging the collective strength of multiple decision trees. Additionally, its robustness against noisy data and outliers significantly improves prediction accuracy. By integrating multiple trees, RF also mitigates overfitting issues that can arise from relying on a single decision tree.

2.1.4 Support vector regression

Support vector regression (SVR) [51] is a regression algorithm derived from SVM. Its core principle is to construct a regression model that maximizes the margin while identifying an optimal regression plane that satisfies error constraints. SVR employs an ε-insensitive loss function, which allows prediction errors within a specified threshold to be ignored, thereby enhancing the model’s robustness. Its objective is to strike the best balance between model complexity and prediction accuracy.

SVR utilizes kernel functions to map low-dimensional input data into high-dimensional space, enabling it to effectively tackle complex nonlinear problems. Common kernel functions include linear kernels and radial basis function kernels, which enhance SVR’s performance when dealing with nonlinear data sets. Moreover, by selectively utilizing support vectors, SVR minimizes reliance on the entire data set, thus improving the model’s computational efficiency.

2.1.5 Backpropagation neural network

Backpropagation neural network (BPNN) [52] is a widely used type of feedforward neural network that utilizes the error backpropagation algorithm to optimize model weights. The fundamental principle involves transmitting input data from the input layer to the hidden layer and then to the output layer, calculating the output at each layer. The model then compares these outputs with the true values and adjusts the weights and biases in reverse to minimize the error function.

The core process of BPNN centers on error backpropagation. Initially, the prediction error at the output layer is calculated, and this error is progressively propagated back through the hidden layer to the input layer using the chain rule. This approach updates each connection weight in the network based on its contribution to the total error, typically employing a gradient descent algorithm for optimization, allowing the model to converge gradually.

BPNN is capable of handling complex nonlinear relationships and demonstrates robust learning capabilities due to its multi-layer architecture and nonlinear activation functions. In the context of predicting blast vibration velocity, BPNN can effectively capture the intricate mapping between vibration velocity and relevant factors by training on extensive historical data, thereby enhancing prediction accuracy. Although the training process may be slow, particularly with large data sets, BPNN exhibits strong generalization ability, adeptly addressing the complexities and nonlinearities inherent in predicting blast vibration velocity.

2.2 Model parameter tuning strategy—Social Network Search

The SNS algorithm is a novel optimization method that simulates human emotional behavior in social networks to solve complex optimization problems [53]. Its primary strength lies in its fast convergence and global optimization capabilities. The algorithm draws on four emotional states commonly observed in social networks: imitation, innovation, conversation, and argument, which drive the optimization process by mimicking how users in social networks interact and update their views based on others’ opinions. Basic Principles of SNS are shown in Fig.2. Key emotional states and their optimization mechanisms are as follows.

1) Imitation. Imitation reflects users’ tendency to adopt ideas from others with similar views or connections. The process allows users to adjust their stance by considering the opinions of others. The mathematical formulation is:

(1)O_{n,\mathrm{n}\mathrm{e}\mathrm{w}}=n_1\cdot n_2\cdot \left(O_n-O_m\right)+O_m,

where O_{n,\text{new}}=n_1\cdot n_2\cdot \left(O_n-O_m\right)+O_m is the updated opinion of the nth user. O_n and O_m are the opinions of the nth and mth users, respectively. n₁ and n₂ are random numbers between 0 and 1, which introduce variability in how the user’s opinion is influenced by others.

2) Innovation. Innovation represents users’ ability to generate new ideas by thinking outside the box. When exposed to differing opinions, users may come up with novel solutions. The formula for innovation is:

(2)O_{n,\mathrm{n}\mathrm{e}\mathrm{w}}^{\ast }=r{O}_n^{\ast }+\left(1-n_2\right)\cdot \left[L{O}^{\ast }+n_1\cdot \left(U{O}^{\ast }-L{O}^{\ast }\right)\right],

where O_{n,\mathrm{n}\mathrm{e}\mathrm{w}}^{\ast } and O^{\ast } are the new and old opinions of the nth user. {UO}^{\ast } and {LO}^{\ast } are the upper and lower bounds of the search space. r is a random number between 0 and 1. If the new solution is better, it is adopted; otherwise, the original solution remains.

(3)

3) Conversation. This state models the exchange of ideas in social settings, where users engage in discussions to refine their views. The conversation state helps optimize solutions through comparison and adjustment:

(4)O_{n,\mathrm{n}\mathrm{e}\mathrm{w}}=n_1\cdot \left[sign\left(f_n-f_m\right)\cdot \left(O_n-O_m\right)\right]+O_i,

where O_i determines the direction of adjustment between opinions. [sign (f_n– f_m) × (O_n– O_m)] are the opinions of different users.

4) Argument. Argument occurs when users strongly disagree, leading to polarized views. This state explores more extreme opinions, creating divergence and potentially leading to new solutions:

(5)O_{n,\mathrm{n}\mathrm{e}\mathrm{w}}=O_n+n_1\cdot ⟨\sum _{o=1}^G{O}_o/G-\left\{\left[1+\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\left(n_1\right)\right]\cdot O_n\right\}⟩,

where {\displaystyle \sum _{o=1}^G{O}_o/G} is the average opinion of the group. (1 + round(n₁)) reflects the position of the user.

The SNS algorithm’s strength lies in its ability to mimic complex human behaviors within social networks, which allows it to efficiently search high-dimensional spaces for optimal solutions. The four emotional states work in tandem, ensuring a balance between exploration (through innovation and argument) and exploitation (through imitation and conversation). This balance enables the algorithm to find global optima while avoiding premature convergence to suboptimal solutions.

In summary, the SNS algorithm utilizes its global search capability to systematically explore the hyperparameter space, with a focus on key parameters such as learning rate, maximum tree depth, and number of leaf nodes. Through the imitation stage and innovation stage, SNS strikes a balance between the utilization of high-performance configurations and the exploration of novel combinations, ensuring that LightGBM achieves the best trade-off between model complexity and prediction accuracy. This interaction is further enhanced by SNS’s ability to dynamically refine parameter configurations based on LightGBM’s feedback from previous iterations, creating a feedback loop that iteratively improves the model’s performance. Additionally, by leveraging LightGBM’s efficient handling of feature importance, SNS ensures that the optimized parameters are aligned with the most critical data features, thereby enhancing the model’s overall predictive robustness.

2.3 Hybrid social network search and LightGBM model

The performance of LightGBM significantly depends on hyperparameter configuration, necessitating effective optimization methods to identify the best parameter combinations. To address this, a combined prediction model, SNS-LightGBM, has been developed, integrating the SNS optimization algorithm with LightGBM to improve blast vibration velocity predictions. The SNS algorithm’s strong global search capability enables it to quickly discover optimal hyperparameter combinations in complex high-dimensional spaces, enhancing both model accuracy and generalization. The basic working principle of the SNS-LightGBM model is shown in Fig.3. The computational flow of the hybrid model is as follows.

1) Initial Hyperparameter Generation. The hyperparameter space for LightGBM is defined, encompassing parameters like learning rate, maximum tree depth, and number of leaf nodes. Initially, the SNS algorithm randomly generates a set of candidate parameters.

2) SNS Optimization of Hyperparameters. LightGBM’s hyperparameters are progressively updated through four emotional states of the SNS algorithm. In the imitation phase, the algorithm adopts parameters from superior solutions; in the innovation phase, randomness adjusts the parameters; during the dialog phase, comparisons of different parameters facilitate optimization; and in the debate phase, new parameter combinations are created through polarization. To reduce overfitting, the SNS algorithm carefully adjusts parameters such as the maximum depth of the tree, the number of leaf nodes, and the learning rate, ensuring the best balance between model complexity and prediction accuracy through iterative optimization. This measure prevents the model from becoming too complex and reduces the risk of overfitting.

3) Model Training and Evaluation. Each time the SNS algorithm generates a new hyperparameter set, LightGBM is trained with these parameters, and its performance is evaluated using multiple indicators.

4) Iteration and Convergence. The SNS algorithm iteratively optimizes hyperparameter combinations until model performance stabilizes or a predefined stopping criterion is met.

By leveraging the global search capabilities of the SNS algorithm alongside LightGBM’s efficient learning, the SNS-LightGBM model effectively enhances the accuracy and stability of blast vibration velocity predictions. The diversity and dynamic search capabilities of the SNS algorithm ensure the global optimality of parameters, resulting in a regression prediction model with improved generalization ability.

3 Data preparation and processing

The database used in this study comprises 230 sets of blasting vibration data, all collected from the Leye Tunnel in Baise City, China [54]. The Leye Tunnel spans 774 m in length, with geological formations primarily consisting of Quaternary overburden silty clay and underlying Carboniferous limestone. The tunnel’s cross-sectional area is 131.6 m², with a borehole density of 2 holes per square meter, resulting in a total of 280 boreholes. The explosive used was No. 2 rock emulsion explosive (32 mm in diameter). The charging parameters were as follows: the cut hole had a charging length of 3.3 m and a stemming length of 0.9 m; the collapse hole had a charging length of 2.7 m and a stemming length of 0.5 m; and the perimeter hole had a charging length of 0.9 m and a stemming length of 0.5 m. Millisecond-delay initiation was employed using digital electronic detonators, with a delay time ranging from 25 to 50 ms. The cut holes were fired in pairs. To better investigate the blasting vibration patterns in the Leye Tunnel, measuring points were strategically positioned in a straight line extending outward from the tunnel face at specific intervals. This ensured that the vibration data collected from each tunnel blasting section was representative. In total, 230 sets of blasting vibration data were recorded.

Selecting appropriate input parameters is essential for enhancing the accuracy of the predictive model. In this study, the input parameters chosen for the PPV prediction model include measuring point distance, charge quantity, site coefficient, cut hole depth, and number of cut holes. These parameters were selected for the following reasons: the distance of the measuring point significantly influences vibration attenuation; the charge quantity directly affects the intensity of energy release; the site coefficient reflects the influence of varying geological conditions on vibration propagation; and the depth and number of cut holes determine the concentration and distribution characteristics of energy within the rock mass. A comprehensive consideration of these factors provides a strong theoretical basis for constructing an accurate PPV prediction model. To analyze the data distribution of each input parameter, a violin plot analysis was conducted [55,56]. Fig.4 presents the violin plots of the input parameters for the PPV prediction model. The results indicate that the data fall within reasonable distribution ranges, with no obvious extreme outliers or indications of overly concentrated or sparse data. This suggests that the data source is reliable and effectively represents the input parameters for the PPV prediction model under varying conditions. Consequently, this data set provides a robust foundation for model construction, exhibiting good representativeness and reliability.

The analysis was conducted using a correlation matrix to enhance the understanding of relationships between the input parameters [57,58]. Fig.5 illustrates the correlation among the selected parameters, including charge quantity (Q), measuring point distance (R), site factor (K), cut hole depth (D), number of cut holes (N), and their relationship with PPV (V). The correlation analysis reveals a certain degree of rationale behind the chosen input parameters. A strong negative correlation is observed between measuring point distance and PPV (−0.61), which aligns with the established principle that vibration attenuates as distance increases. While the charge quantity exhibits a weak positive correlation with PPV (0.09), its inclusion as a key parameter is justified by physical reasoning, given its direct influence on energy release during blasting. Although the correlations of site coefficient, cut hole depth, and number of cut holes with PPV are not as pronounced, these parameters play an important regulatory role in the propagation of blasting vibrations and energy dissipation. In summary, the selection of these parameters captures the primary factors affecting blasting vibration, adequately reflecting the complex dynamic behavior inherent to real-world blasting processes. This makes the chosen parameters suitable for accurately predicting PPV.

Before training, the data were normalized to ensure consistent scales across all variables, enhancing model stability and performance. Missing values, accounting for 5% of the data set, were addressed using mean imputation. Outlier detection was conducted using the Z-score method, and no extreme values were removed, as they were deemed representative of real-world blasting scenarios.

After constructing the data set, the samples were randomly shuffled and then stratified sampling was applied, allocating 80% of the data to the training set to establish the nonlinear relationship between input parameters and output results. The remaining 20% of the data was used as the test set to evaluate the accuracy of the model’s predictions. To validate the superior performance of the hybrid model, an additional experiment was conducted using 70% of the data for training and 30% for testing. However, the results showed that the 80%–20% split yielded better performance, and thus, it was adopted as the optimal division ratio. Six different models were selected for comparison: SNS-LightGBM, LightGBM, XGB, RF, SVR, and BPNN. Various metrics were employed to evaluate the predictive performance of each model. The prediction process for blasting PPV is illustrated in Fig.6.

To ensure the reliability and robustness of the results, the training and testing process was repeated 10 times using different random splits of the data set. Each iteration involved splitting the data set into 80% for training and 20% for testing. For each run, the performance of all six models was evaluated based on key metrics (e.g., coefficient of determination (R²), mean absolute error (MAE), mean absolute percentage error (MAPE), mean squared error (MSE), root mean squared error (RMSE)). The results presented in this study represent the average values across all iterations to minimize the influence of random variability and ensure a fair comparison.

4 Results and discussion

4.1 Prediction results of the models

The prediction results for the different models are presented in Fig.7. The SNS-LightGBM model displays a high level of fitting accuracy, with most of its training and testing data points closely clustered around the dashed line. This model demonstrates exceptional predictive capability among all evaluated models, exhibiting a more concentrated distribution of data points. In comparison, the LightGBM model also shows a dense distribution of its training and testing data points near the dashed line. However, some points exhibit significant deviations, particularly at higher PPV values (approximately above 3 m/s). Overall, the performance of LightGBM is slightly inferior to that of SNS-LightGBM, especially regarding the accuracy of predictions for elevated PPV values.

The distribution of training and testing data points in the XGB model is more scattered, with the testing data sets displaying noticeable deviations at high PPV values (above 3 m/s). Consequently, the predictive performance of the XGB model is weaker than that of both SNS-LightGBM and LightGBM, suggesting potential shortcomings in accurately predicting high PPV values. The RF model shows a good overall fitting performance, with training and testing data points distributed closely around the dashed line. However, in contrast to SNS-LightGBM and LightGBM, the prediction accuracy of the RF model diminishes at higher PPV values, evidenced by several data points deviating from the ideal diagonal line. The SVR model demonstrates good fitting performance for both the training data sets (orange) and testing data sets (green) at lower PPV values. Nonetheless, substantial deviations are observed in the distribution of testing data sets data points at higher PPV values. While this model performs adequately in low-value regions, it encounters significant errors when predicting elevated PPV values.Similarly, the BPNN model exhibits good fitting results for the training data sets (blue) and testing data sets (purple) at low PPV values. However, at higher PPV values (above approximately 3.5 m/s), several data points diverge significantly from the dashed line. The predictive capability of the BPNN model is diminished in high-value regions, indicating a degree of underfitting.

Fig.8 illustrates the distribution of absolute errors for different machine learning models in predicting blasting PPV, allowing for an assessment of the prediction accuracy of each model on both the training and testing data sets. The error distribution of the SNS-LightGBM model is relatively concentrated, with most errors for both the training and testing data sets falling within the range of −0.2 to 0.2. Notably, the majority of errors are close to zero, and the error curve is sharp, indicating that the model exhibits small prediction errors and high accuracy. In contrast, the error distribution for the LightGBM model is broader, with errors concentrated between −0.5 and 0.5. Although the errors are slightly larger than those observed for the SNS-LightGBM model, the curve remains relatively symmetric. This symmetry suggests that the overall predictive performance of the LightGBM model on both the training and testing data sets is still quite good.

The error distribution of the XGB model primarily concentrates between −0.5 and 0.5; however, a higher number of larger error data points are observed in the testing data sets, with some errors reaching −2. The error distribution curve exhibits a slight skew toward the negative side, indicating that this model experiences underfitting in certain cases, particularly with significant prediction errors for high PPV values. In comparison, the RF model has a broader error distribution, with error ranges for the training and testing data sets spanning from −2.5 to 1. A notable shift in the negative error portion suggests that this model produces larger errors in some predictions and does not perform as effectively as the SNS-LightGBM and LightGBM models.

The SVR model presents a more dispersed error distribution, with most errors in the training and testing data sets ranging between −1 and 1. The testing data sets reveals a significant spread of errors in both positive and negative directions, resulting in a poor concentration of errors. This indicates that the model has larger errors when managing complex data, reflecting lower predictive stability. The error distribution for the BPNN model is more relaxed, with most errors for the training and testing data sets concentrated between −0.6 and 0.6. However, there is a significant increase in errors between 0.6 and 0.8 in the testing data sets, indicating strong dispersion, particularly when predicting high PPV values, which results in substantial errors.

Fig.9 displays the relative error distribution for each model on the training and testing data sets. For the training data sets, the SNS-LightGBM model exhibits the lowest relative error, with the interquartile range (25%–75%) concentrated between 0% and 20%, and a few outliers present. The median is close to 0%, with an extremely low average value, reflecting excellent training accuracy. LightGBM demonstrates slightly higher errors but maintains stable performance with fewer outliers. In contrast, the XGB model shows increased errors, with the 25%–75% range extending from 0% to 30% and a greater number of outliers, indicating subpar performance for certain samples. The RF model displays a significant increase in error, with a range of 0% to 40% and a notable rise in outliers, reflecting lower model accuracy. The SVR model exhibits a wide error distribution, ranging from 0% to 50%, with a higher median and numerous outliers, indicating poor fitting performance. The BPNN model has the largest errors, with a range of 0% to 75% and the highest number of outliers, suggesting severe underfitting.

On the testing data sets, the SNS-LightGBM model continues to perform the best, with an error range of 0% to 10%, and both the median and average values remain extremely low, indicating excellent generalization ability and minimal outliers. LightGBM, while slightly less effective, maintains a strong performance with an error range of 0% to 15%. The errors for the XGB model have risen to a range of 0% to 25%, accompanied by an increase in outliers, indicating diminished generalization ability. The RF model’s errors increased to a range of 0% to 40%, reflecting poor performance. The SVR model continues to exhibit a wide error distribution, with a range of 0% to 40% and an increase in outliers, further indicating poor generalization ability. Finally, the BPNN model performs the worst, exhibiting an error range of 0% to 50% and widespread outliers, demonstrating severe underfitting and very weak generalization capability.

4.2 Comparison of predictive performance

Fig.10 illustrates the comparison of prediction performance among six models on the training data sets and the testing data sets. The evaluation metrics include MAE, MAPE, MSE, RMSE, and the R². A radar chart visually represents the performance of each model across these metrics, with axes ranging from 0 to 1. For MAE, MAPE, MSE, and RMSE, values closer to 1 indicate worse performance, while values closer to 0 signify better performance [59]. Conversely, the interpretation of R² is reversed, where values approaching 1 indicate better performance.

In the training data sets, both the SNS-LightGBM and LightGBM models demonstrate excellent performance across all metrics. Their five error indicators (MAE, MAPE, MSE, RMSE) are close to 0, while their R² values approach 1, indicating high predictive accuracy and strong fitting capability. The XGB model also performs relatively well, although it is slightly inferior to SNS-LightGBM and LightGBM. Despite higher error metrics, XGB still surpasses other models, reflecting a robust fitting capability on the training data. The RF model shows inferior performance compared to XGB, with increased error metrics, particularly higher MAPE and RMSE values. This suggests that while the RF model has some predictive ability on the training data sets, it exhibits larger errors and average fitting performance. In contrast, the SVR and BPNN models demonstrate the poorest performance on the training data sets, particularly the BPNN model, which exhibits significantly higher error metrics than the other models, with its R² value approaching 0. This indicates very poor fitting performance and substantial prediction errors on the training data.

On the testing data sets, both the SNS-LightGBM and LightGBM models maintain their superior performance. Notably, SNS-LightGBM exhibits low error values across all five metrics, demonstrating its strong generalization ability on the test data. LightGBM closely follows, also displaying stable performance. Although the performance of the XGB model declines slightly compared to its training data sets results, it continues to outperform the RF, SVR, and BPNN overall. Despite relatively high values for MAE, MAPE, and RMSE, XGB’s performance remains within an acceptable range, indicating a decent generalization ability on the testing data sets. In contrast, the error metrics for RF and SVR on the testing data sets are notably high, with RF’s MAPE and RMSE values significantly exceeding those of the other models. This suggests poor generalization ability, indicating that this model does not perform adequately when handling new data. The BPNN model exhibits the weakest performance on the testing data sets, with all five error metrics remaining very high, particularly for MAE, MAPE, and MSE. This highlights its extremely poor generalization ability and challenges in accurately predicting new data.

Fig.11 presents the comprehensive evaluation results of the six models on both the training and testing data sets. Each model’s performance is quantified with a score, where higher scores indicate better performance across various statistical metrics. In each metric, the model ranked first is awarded 6 points, while the model ranked last receives only 1 point. This scoring system effectively captures the relative performance of the models across various indicators, distinguishing those that excel from those that underperform. This method is not only intuitive and easy to comprehend but also facilitates comprehensive comparisons for the final selection of models [60].

In the training data sets, SNS-LightGBM receives the highest score, reflecting excellent performance across all evaluation metrics, strong fitting capability, and good predictive accuracy, as evidenced by its high R² and low RMSE, MSE, MAPE, and MAE values. LightGBM follows closely with a score of 25, slightly below SNS-LightGBM, but performs well on the RMSE and MAE metrics. The XGB model scores 20, which is significant compared to the first two models, despite some gaps in R² and MAE. RF and SVR receive lower scores, indicating their limitations on the training data sets, especially RF, which performs poorly on the MAE and MAPE metrics. Lastly, BPNN finishes with the lowest score of 5, reflecting its extremely poor predictive ability on the training data sets.

In the testing data sets, the SNS-LightGBM model maintains the highest score, showcasing its strong generalization ability. The LightGBM model scores 25, performing excellently but exhibiting a slight decrease compared to the training data sets, potentially due to limited generalization capability. The XGB model scores 17, indicating some overfitting issues. The RF model scores 14, suggesting acceptable performance on the testing data sets, though it remains insufficient, particularly concerning the MAE and MAPE metrics. The SVR model scores 15, while the BPNN continues to receive the lowest score, highlighting its poor predictive ability.

In summary, the analysis clearly indicates that SNS-LightGBM and LightGBM are the top choices, both demonstrating good fitting and generalization capabilities. In contrast, BPNN and SVR do not meet performance expectations in either the training or testing data sets, making them unsuitable for use.

4.3 Feature importance analysis

Sensitivity analysis and feature importance analysis are commonly used methods to examine the relationship between input parameters and prediction outcomes [61,62]. In comparison, feature importance analysis is more directly tied to model performance, offering a straightforward and intuitive approach that aids in feature selection, particularly for complex machine learning models. To comprehensively assess the contribution of various features to predicting blasting vibration velocity in the model, it is necessary to employ multiple methods for measuring feature importance [63]. However, these methods typically rely on different metrics, such as the number of splits, information gain, and coverage [64]. Therefore, it is essential to standardize and effectively combine these metrics. Based on this premise, the study proposes a comprehensive feature importance method, which calculates a combined importance score for each feature through a weighted average after standardizing the importance based on split count, gain, and coverage.

1) Feature importance based on split count

Feature importance based on split count quantifies how frequently a feature is used for splitting nodes in a decision tree. The principle is as follows: the more frequently a feature is employed for splitting nodes, the more critical it is to the model’s decision-making process:

(6)I_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t},j}=\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}j,

where I_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t},j} represents the split count importance of feature j.

2) Feature importance based on gain

Gain-based feature importance assesses a feature’s importance by the information gain it provides when making a split. The larger the gain, the greater the feature’s contribution to optimizing the model’s objective function:

(7)I_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n},j}=\sum _{t\in T_J}\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}(t),

where I_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n},j} represents the gain importance of feature j, T_j is the set of all split nodes containing feature j, and gain(t) is the gain value at node t.

3) Feature importance based on coverage

Coverage-based feature importance measures the number of data samples affected after a feature is used to split nodes. The more samples covered, the greater the feature’s impact on the overall data set:

(8)I_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r},j}=\sum _{t\in T_j}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}(t) ,

where I_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r},j} represents the coverage importance of feature j, and cover(t) indicates the number of samples affected after node t is split.

4) Comprehensive feature importance calculation

Due to the different dimensionalities of the three types of feature importance, direct comparison is not feasible. Thus, the Min-Max normalization method is employed to standardize the importance of each feature, mapping it to the range [0,1]:

(9)I_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d},j}=\frac{I_j-min(I)}{max(I)-min(I)},

where I_j is the original importance of feature j, and I_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d},j} is the standardized feature importance.

To further combine these different sources of feature importance, the standardized metrics are processed through a weighted average. Let {\omega }_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}, {\omega }_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}, and {\omega }_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}} represent the importance weights for split count, gain, and coverage, respectively. The combined comprehensive feature importance is then calculated as follows:

(10)\begin{array}{rl}{I}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d},j}=& {\omega }_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}\cdot I_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{r},j}+{\omega }_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}\cdot I_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{r},j}\\ & +{\omega }_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}\cdot I_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{r},j}.\end{array}

This formula quantifies feature importance from three distinct sources into a unified comprehensive score, allowing for a more accurate assessment of each feature’s contribution to the model. It provides a more reliable basis for feature selection and model interpretation.

The feature importance analysis, as depicted in Fig.12, reveals that distance is the most influential factor, with an importance score of 0.41. This aligns with the physical principles of wave propagation, where blasting-induced vibrations dissipate energy as they travel through the medium, leading to an exponential attenuation of vibration intensity with increasing distance. This phenomenon is well-documented in empirical studies, as the reduction in vibration intensity is governed by geometric spreading and material damping effects [65–70]. The second most critical factor is the charge quantity, with an importance score of 0.25. This reflects its direct relationship to the energy released during blasting, as a larger charge results in greater energy input into the rock mass, thereby generating stronger vibrations. From an operational perspective, optimizing charge quantity is crucial not only for controlling vibration but also for achieving the desired fragmentation [71]. The site coefficient, with an importance score of 0.16, represents the influence of geological and geotechnical conditions, such as rock type, joint orientation, and layer stiffness, on vibration propagation. Although less impactful than distance and charge quantity, it plays a pivotal role in site-specific vibration behavior. Rocks with higher density or lower damping properties tend to transmit vibrations more efficiently, thus increasing the intensity at given distances [72]. The cut hole depth (0.11) and the number of cut holes (0.07) are of lesser importance. These factors influence vibration indirectly by affecting the distribution of blasting energy and the rock fragmentation pattern. Deeper cut holes and a higher number of cut holes typically lead to more focused energy release, which can locally amplify vibrations but have a more limited overall effect compared to distance or charge quantity.

In summary, distance and charge quantity emerge as dominant factors due to their fundamental role in vibration generation and attenuation. Geological conditions (captured by the site coefficient) and blasting configurations (cut hole depth and number of cut holes) contribute to secondary effects by modulating how energy is transmitted and distributed during blasting. This analysis underscores the importance of considering both physical laws and operational parameters when predicting and managing blasting-induced vibrations.

5 Conclusions

This study proposes a PPV prediction method based on the SNS-LightGBM deep gradient collaborative learning framework. Five unoptimized machine learning models and one hybrid model (SNS-LightGBM) were developed for comparative analysis. The predictive performance of these models is comprehensively evaluated using metrics such as R², MAE, MAPE, MSE, and RMSE. Additionally, a comprehensive feature importance analysis method is developed to quantitatively assess the significance of each input feature. The main conclusions of the study are as follows.

1) The global search capability of the SNS algorithm, combined with the efficient learning ability of LightGBM, results in the SNS-LightGBM model significantly improving the accuracy and stability of PPV predictions. The diversity and dynamic search mechanism of the SNS algorithm ensure the global optimality of parameters, leading to a regression prediction model with enhanced generalization capability.

2) The proposed SNS-LightGBM model exhibits excellent performance in PPV prediction. In the performance evaluation, the ranking of model performance on the test set is as follows: SNS-LightGBM > LightGBM > XGB > RF > SVR > BPNN. Notably, the evaluation metrics for the SNS-LightGBM model on the test set are R² = 0.975, MAE = 0.086, MAPE = 0.071, MSE = 0.019, and RMSE = 0.138, demonstrating its high accuracy in PPV prediction.

3) The importance ranking of input parameters for PPV prediction is as follows: distance > charge quantity > site factor > hole depth > number of holes > saturation moisture content. It is noteworthy that the impact of distance and charge quantity on PPV is significantly greater than that of other parameters, indicating their critical role in the predictions.

4) Future research could explore hybrid approaches that combine physical equations with data-driven models, such as physics-informed neural networks [73,74]. By incorporating physical constraints into machine learning models, this approach could significantly enhance prediction accuracy and physical consistency under complex geological conditions. Additionally, results from physical solvers can serve as data sources, complementing the efficiency of machine learning models to address high-dimensional and complex problems effectively. This integration of physics and data-driven strategies presents a promising direction for future predictions in mining engineering.

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