
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71367
10.1038/s41598-024-71367-6
Article
Enhancing shear strength predictions of rocks using a hierarchical ensemble model
Ding Xiaohua 12
Amiri Maryam 3
Hasanipanah Mahdi hasanipanahmahdi@duytan.edu.vn

45
1 https://ror.org/01xt2dr21 grid.411510.0 0000 0000 9030 231X School of Mines, China University of Mining and Technology, Xuzhou, 221116 China
2 grid.411510.0 0000 0000 9030 231X State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou, 221116 China
3 https://ror.org/00ngrq502 grid.411425.7 0000 0004 0417 7516 Department of Computer Engineering, Faculty of Engineering, Arak University, Arak, 38156-8-8349 Iran
4 https://ror.org/05ezss144 grid.444918.4 0000 0004 1794 7022 Institute of Research and Development, Duy Tan University, Da Nang, Vietnam
5 https://ror.org/05ezss144 grid.444918.4 0000 0004 1794 7022 School of Engineering & Technology, Duy Tan University, Da Nang, Vietnam
31 8 2024
31 8 2024
2024
14 2026822 6 2024
22 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Shear strength (SS) parameters are essential for understanding the mechanical behavior of materials, particularly in geotechnical engineering and rock mechanics. This study proposes a novel hierarchical ensemble model (HEM) to predict SS parameters: cohesion (C) and angle of internal friction (φ). The HEM addresses the limitations of traditional machine learning models. Its performance was validated using leave-one-out cross-validation (LOOCV) and out-of-bag (OOB) evaluation methods. The model's accuracy was assessed with R-squared correlation (R2), absolute average relative error percentage (AAREP), Taylor diagrams, and quantile–quantile plots. The computational results demonstrated that the proposed HEM outperforms previous studies using the same database. The model predicted φ and C with R2 values of 0.93 and 0.979, respectively. The AAREP values were 1.96% for φ and 4.7% for C. These results indicate that the HEM significantly improves the prediction quality of φ and C, and exhibits strong generalization capability. Sensitivity analysis revealed that σ_3maxσ3max (maximum principal stress) had the greatest impact on modeling both φ and C. According to uncertainty analysis, the LOOCV and OOB had the widest uncertainty bands for the φ and C parameters, respectively.

Keywords

Shear strength parameters
Hierarchical ensemble model
Prediction models
Machine learning
Subject terms

Civil engineering
Computational science
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The strength of intact rock (SIR) refers to its ability to withstand external forces without undergoing significant deformation or failure. It is an important property in engineering and geotechnical applications, as it influences the stability and load-bearing capacity of rock masses1–3. The SIR is influenced by various factors, including mineral composition, grain size, porosity, and structural features such as bedding planes, joints, and foliation2,3. The presence of these features can significantly affect the strength and anisotropy of the rock3. The SIR is typically characterized by different parameters, including compressive strength, tensile strength, and shear strength (SS)4. Compressive strength is the maximum stress that a rock sample can withstand before it fails under a uniaxial compressive load5. Tensile strength represents the resistance of the rock to failure when subjected to tensile stresses6. SS refers to the ability of the rock to resist deformation and sliding along a plane under shear stresses3. The two primary SS parameters used in geotechnical analysis are cohesion (C) and angle of internal friction (φ). These parameters are typically determined through laboratory tests, such as direct shear tests or triaxial shear tests. The C represents the SS of a material in the absence of any normal stress. It is a measure of the material's ability to resist shear without any external frictional forces. The C is a result of interlocking between particles, chemical bonding, or electrostatic forces within the material. The φ represents the resistance of a material to shear stresses when it is subjected to a normal stress. It is a measure of the frictional resistance between particles within the material. The φ is defined as the angle between the normal stress acting on a plane and the shear stress required to cause failure along that plane7. The SS of a material can be calculated using the Mohr–Coulomb failure criterion, which relates the SS to the normal stress and the SS parameters. The two common tests used to determine SS parameters are direct shear tests and triaxial shear tests3. While laboratory tests are commonly used to determine SS parameters, they do have some disadvantages8. For example, the boundary conditions applied during laboratory tests may not accurately replicate the complex stress and strain conditions that occur in the field. Simplified boundary conditions can influence the SS measurements and may not fully represent the true behavior of the material. On the other hands, during sample preparation and testing, the process of extracting and handling soil or rock samples can cause disturbance and alteration of the natural structure and fabric of the material. Furthermore, conducting laboratory tests to determine SS parameters can be time-consuming and expensive7–9.

Due to the abundance of data available10,11, the analysis and extraction of valuable information have become crucial12,13. These tasks can be accomplished through the utilization of data mining and machine learning techniques14.

The acceptability of machine learning (ML) methods have been widely highlighted in various geotechnical and civil fields15–27. Hence, some scholars have employed the ML to predict SS in their studies. Khanlari et al.7 employed an artificial neural network (ANN) to estimate SS parameters. In another study, Rezaee et al.28 established some ML methods such as support vector machine (SVM), ANN, and gradient boosting (GB) method to estimate SS. For the same purpose, Shen and Jimenez3 employed a genetic programming (GP) method. Fattahi and Hasanipanah4 investigated the use of two ML models to predict SS. They employed an improved neuro-fuzzy model, and an optimized support vector regression (SVR) in their study. They concluded the effectiveness of proposed models in this field. Zhou et al.8 predicted the SS parameters through two advanced Random Forest (RF) methods. In their study, the whale optimization and sine cosine algorithms were utilized to optimize RF method.

Although common ML methods have been widely employed for prediction purposes and have yielded accurate results, they also come with certain disadvantages. For example, ML methods heavily rely on data for training and making predictions. They require large and representative datasets to effectively capture patterns and generalize well. Insufficient or biased data can lead to inaccurate or biased predictions, limiting the reliability and fairness of the models. The ML models can also be prone to overfitting, which occurs when a model becomes excessively tailored to the training data and performs poorly on unseen data. Striking the right balance between fitting the training data well and generalizing to new data is a challenge in machine learning, and overfitting can lead to poor performance in real-world scenarios. Additionally, they heavily depend on the quality and relevance of the input data. Noisy or biased data can negatively impact the performance and accuracy of the models. Another challenge is limited model capacity. Common ML models might struggle to capture complex patterns and relationships in high-dimensional or nonlinear datasets, leading to suboptimal performance.

In practice, we often encounter training datasets with small sample sizes, high dimensionality, and a high noise-to-signal ratio. Consequently, finding the most suitable hypothesis is often challenging as the hypothesis space contains numerous suboptimal hypotheses that can fit the training data but lack generalization ability.

As shown in Fig. 1, employing multiple predictors on the training data (hi) can result in a narrowed overlap of the hypothesis space H029. This forms the foundation of Ensemble Learning, a type of ML technique that combines multiple learners, known as base-learners, in classification or regression problems30,31. Due to their strong performance, ensembles are widely regarded as one of the top and most influential machine learning methods32.Fig. 1 The hypothesis space of ensembles29.

In the present study, a hierarchical ensemble model (HEM) is proposed to predict two SS parameters (φ and C). As Fig. 2 shows, HEM is a two-level ensemble model that estimates φ in level 1. Then the predicted φ is fed to the ensemble method in level 2 to predict C.Fig. 2 Structure of HEM.

Source of data

In this study, the required datasets were borrowed from the study conducted by Shen and Jimenez3, where a GP method was employed to predict SS parameters (C and φ). The present study aims to develop a new and more accurate model than Shen and Jimenez3 to predict C and φ. In this section, the used datasets are briefly analyzed, and their results are presented. Table 1 provides the descriptive statistics of all input and output parameters. These statistics were crucial for the feature engineering process and significantly impacted model performance. The mean and standard deviation provided insights into each parameter's central tendency and variability, helping us understand the data's distribution. The minimum and maximum values identified potential outliers or extreme values within the dataset. Overall, the descriptive statistics informed key feature engineering steps, enhancing the HEM's accuracy and reliability and resulting in better predictive performance for the shear strength parameters. The input parameters are σci (MPa), σt (MPa), and σ3max, while φ (∘) and C (MPa) are the output parameters. σci represents the compressive strength or the stress acting on a material in compression, σt is the tensile stress, and σ3max is the maximum principal stress. For better understanding, Fig. 3 displays the correlation matrices between the input and output parameters, showing the R-squared correlation (R2) values. In this Figure, the coefficient of determination (R2) between input and output parameters are demonstrated. According to the R2 values, input 3 (σ3max) has the highest relationship with output 1 (φ), while input 1 (σci) has the highest relationship with output 2 (C). Furthermore, Fig. 4 presents the violin plots of input and output parameters. Note that in Table 1, Figs. 2 and 3, inputs 1–3, and outputs 1 and 2 correspond to σci, σt, σ3max, φ and C, respectively.Table 1 Descriptive statistics of all parameters used in this study.

Parameters	Mean	Standard deviation (SD)	Minimum	Maximum	
Input 1	84.30774	46.21627	41.47	196.97	
Input 2	5.74528	4.29734	1.75	16.15	
Input 3	28.04005	19.02078	1.47	100	
Output 1	44.3158	4.89075	33.6	56.02	
Output 2	20.15642	9.5979	8.14	49.62	

Fig. 3 Correlation matrixes between input and output parameters.

Fig. 4 Violin plots of input and output parameters.

Methodology

In this section, the background and primary concepts of the proposed HEM are explained, considering ensemble models and their basic concepts in more detail.

Ensemble models

Figure 5 shows the general structure of an ensemble. According to the figure, an ensemble consists of a number of base learners (or weak learners). If hi,1≤i≤n, is a base learner and x∈χ is an input sample, then the outputs hix are combined to form the final result Hx. Therefore, the ensemble is constructed in two phases: (1) generating base learners and (2) combining them33. The base learners can be decision tree, neural networks, or other types of learning algorithms. Ensembles can be divided into two groups: homogeneous and heterogeneous. In homogeneous ensembles, the base learners are of the same type, while heterogeneous ensembles consist of learners of different types. In this paper, our focus is on homogeneous ensembles.Fig. 5 The general architecture of the ensemble33,34.

There are various approaches for the learning and combination steps in ensembles, with two famous strategies being Boosting and Bagging35. In the following section, we will briefly review these strategies. Boosting is a family of algorithms in ensembles that convert weak learners into strong learners33,34. In boosting, different weak learners are applied to differently weighted versions of the training data. At each round of the algorithm, the focus is on samples that have not been learned yet. As a result, the weight of these samples increases in the next round. Essentially, the learners are trained sequentially, with later learners focusing on the mistakes made by earlier ones.

Algorithm 1 (Fig. 6) presents the general boosting procedure33,34. In lines 2–6, the base learning algorithm is applied T times to weighted versions of the distribution data. The Function AdjastDistribution adjusts the weight of samples in the next distribution Dt+1 based on the prediction error ϵt and the current distribution Dt. In line 7, the outputs of base learners htx,1≤t≤T, are combined to generate the final result of the ensemble H(x).Fig. 6 Algorithm 1: the general procedure of Boosting33.

Although boosting is a sequential approach, bagging is a parallel strategy where the base learners are generated and trained simultaneously. In bagging, bootstrap sampling, which involves data resampling with replacement36, is employed. Essentially, the base learners are trained independently on different subsets of the sample size, and their results are then combined. Figure 7 illustrates the procedures of the bagging and boosting strategies. RF37, Random Patches38, and ExtraTrees39 are examples of bagging-based techniques. On the other hand, AdaBoost40, and GB Machine41 are considered as boosting ensembles.Fig. 7 The procedure of the bagging and boosting strategies.

Modeling process

Decision tree-based ensembles are among the most popular and successful techniques in machine learning35. Due to their effectiveness and simplicity, RF37,42 is considered one of the most famous ensembles35. The experimental results reported in Fernández-Delgado et al.32, which examined 179 different models, show that Random Forest-based ensembles perform the best. As shown in Fig. 1, we propose a two-level ensemble model, where the output of the ensemble in level 1 is fed as an input to the ensemble in level 2. Considering the effectiveness and increasing popularity of Random Forest-based ensembles31, we employ RF Regression43,44 as the ensemble in both levels. In the following section, we will provide a more detailed explanation of the approach.

Regression Tree: Let's consider a scalar output Y where Y∈R, and a p-vector of variables X. A regression tree partitions the X-space into J disjoint regions, denoted as R1,R2,⋯,RJ. Within each region Rj, the regression tree provides a fitted value E(Y|X∈Rj). In other words, it calculates the mean of the output values for the training observations within Rj45. Figure 8 illustrates a sample regression tree and its corresponding prediction surface in a two-dimensional space, where ti,1≤i≤5, represents thresholds.Fig. 8 A sample regression tree (a) and the corresponding prediction surface (b)45.

The creation of regression trees involves the application of a recursive partitioning (RP) algorithm. This algorithm constructs a tree by iteratively dividing the training sample into increasingly smaller subsets. However, since the resulting trees may become too complex, they have a tendency to overfit the training data. To address this issue, a common approach is to first grow a large tree TL and then prune it to obtain a subtree that minimizes the test error rate. Cost complexity pruning is a pruning strategy based on a nonnegative tuning parameter α. For each value of α, it aims to find a subtree T⊂TL in a way that Eq. (1) is minimized. In Eq. (1), |T| represents the number of terminal nodes of T, Rm corresponds to the region associated with the mth terminal node, and y^Rm is the mean of the training observations in Rm. The parameter α controls the trade-off between the complexity of the subtree and overfitting of the data. The value of α can be selected using a validation set or through cross-validation. Algorithm 2 (Fig. 9) outlines the process of building a regression tree45.1 ∑m=1T∑i:xi∈Rmyi-y^Rm2+αT

Fig. 9 Algorithm 2: The algorithm of building a regression tree45.

RF Regression: RF regression is an ensemble of C regression trees {T1X,⋯,TCX}, where X=x1,x2,⋯,xp is a p-dimensional vector of input features. The RF regression generates the C outputs {y^1=T1(X),⋯,y^C=TC(X)} corresponding to the C trees44. As shown in Eq. (2), the RF regression ψ employs an aggregation function G that combines the C trees {T1,T2,...,TC} to generate a single output Y^ as follows31:2 Y^=ψX=G(T1(X),⋯,TC(X))

The aggregation is performed by taking a linear combination of the predictions, as described in Eq. (3), where hi(x) is the weighting function that can be constant or non-constant30.3 GT1(X),⋯,TC(X)=∑i=1C[hiX∗TiX]

Algorithm 3 (Fig. 10) presents the procedure for random forest regression. The inputs of the algorithm are the training dataset D, the number of regression trees C, and the number of features considered at each split, m.Fig. 10 Algorithm 3: The algorithm of RF regression35.

It's important to note that when building a RF, the algorithm is not allowed to consider all features at each split. Instead, a random sample of m features is chosen as split candidates from the full set of p features at each split. This helps to decorrelate the regression trees45. In lines 1–4, each tree is built and trained on different training datasets resulting from bootstrap sampler with only a random subset of m features. In line 5, the final prediction of a sample is computed by averaging the outputs of the trees. The value of m is typically chosen as p.

In our problem, we have three inputs σci, σt, and σ3max, and two outputs φ and C. A traditional approach would be to predict each output separately based on the three inputs, ignoring any potential correlations between the outputs.

However, in the proposed approach (HEM), two hierarchical ensembles are utilized, both based on RF regression. The first RF receives σci, σt, and σ3max, as inputs and predicts φ. To predict C, HEM incorporates the three previous inputs along with the predicted φ as inputs to the ensemble in level 2. The hierarchical structure of the ensemble model significantly enhances its ability to capture complex interactions between input parameters and shear strength predictions. By decomposing the prediction task into two levels, the model first predicts the parameter φ, effectively capturing the initial relationships and non-linear interactions among input features. This prediction of φ is then utilized in the second level to predict the parameter C, allowing the model to integrate the estimated φ with other input features and further refine the predictions. This sequential approach enables the model to capture intricate dependencies and interactions that might not be apparent in a single, flat model. Additionally, the hierarchical design allows for iterative refinement, where the second level’s prediction can be adjusted based on the initial φ estimates, improving the accuracy and reliability of the final predictions. The flexibility of the Random Forest ensemble method at each level also contributes to better handling of complex patterns and interactions, reducing overfitting and enhancing generalization across varying complexities in the data.

Results and discussion

Analysis of the results

To optimize the performance of HEM, we conducted meticulous hyperparameter tuning, focusing specifically on the number of regression trees used in the Random Forest algorithm. Given the dataset of 212 instances, we employed a grid search approach to evaluate a practical range of tree counts: 50, 100, 150, and 200. This range was selected to balance computational efficiency with robust performance. Each configuration was assessed using both leave-one-out cross-validation (LOOCV) and out-of-bag (OOB) methods. With only three input features, feature selection at each split was not a critical factor, and the default settings of the Random Forest algorithm were sufficient.

LOOCV involves using each data point as a test set while the remaining data points are used for training, repeating this process 212 times. This method ensures that every data point is used exactly once for validation, providing a thorough assessment of model performance. OOB estimation averages the outputs of trees not trained on a given sample, offering an unbiased estimate of the generalization error and leveraging the inherent structure of Random Forest models. These methods enhance the robustness of the model by ensuring efficient use of limited data and reducing overfitting risks. LOOCV provides a detailed evaluation, while OOB offers an unbiased estimate of generalization error. Integrating both methods ensures a rigorous validation framework, reinforcing the reliability of the predictions for the SS parameters (φ and C). Various statistical functions46–51 were used to evaluate HEM in the OOB and LOOCV phases, including Mean Absolute Percentage Error (MAPE), Kling-Gupta Efficiency (KGE), R2, Root Mean Square Error (RMSE), and Absolute Average Relative Error Percentage (AAREP). Table 2 presents the MAPE, KGE, R2, RMSE, and AAREP values for HEM-train (HEM-t), OOB, and LOOCV phases. Furthermore, for each phase, the comprehensive rank was indicated, and finally the total rank of each phase was demonstrated. From Table 2, the HEM method provided results with the highest accuracy in both the validations OOB and LOOCV for both φ and C prediction, giving the lowest MAPE, RMSE and AAREP, as well as the highest KGE and R2.Table 2 Values of metrics used in this study obtained from HEM in the different evaluation phases.

Output	Phase	KGE	Rank	MAPE (%)	Rank	R2	Rank	RMSE	Rank	AAREP (%)	Rank	Total	
Output 1 (φ)	HEM-t	0.883	3	2	3	0.93	3	1.344	3	1.96	3	15	
OOB	0.814	1	3.1	1	0.84	2	1.993	1	3.13	1	6	
LOOCV	0.848	2	2.9	2	0.84	2	1.962	2	2.89	2	10	
Output 2 (C)	HEM-t	0.964	2	4.8	3	0.979	3	1.416	3	4.7	3	14	
OOB	0.951	1	7.9	1	0.95	1	2.153	1	7.8	1	5	
LOOCV	0.974	3	5.3	2	0.973	2	1.58	2	5.3	2	11	

The impact of hyperparameter tuning was substantial, leading to significant improvements in model performance. The optimal hyperparameters resulted in lower MAPE, RMSE, and AAREP values, as well as higher KGE and R2 values. This careful tuning reduced overfitting by ensuring the model did not become excessively complex. As shown in Table 2, the consistent performance across the HEM-t, LOOCV, and OOB phases suggests an effective balance between bias and variance, demonstrating that the model achieved optimal generalization without high bias or high variance. This close alignment of performance metrics across the different phases confirms the robustness and reliability of the model in making accurate predictions.

Additionally, Fig. 11 shows the performance of HEM-t, and two OOB and LOOCV evaluation methods based on their comprehensive rankings. From Table 2 and Fig. 11, for both φ and C parameters, the HEM-t yielded the best performance with the highest total rank, followed by LOOCV and OOB evaluation methods.Fig. 11 Performance of HEM-t, OOB and LOOCV phases based on their comprehensive rankings.

To further evaluate the performance of the predictive methods for both φ and C parameters, quantile–quantile (QQ) plots are generated, as shown in Fig. 12. To construct QQ plots, the residuals of each model are computed by subtracting the observed values from the predicted values. Based on the obtained residual values, the QQ plots can be drawn accordingly. For both φ (on the left side of the figure in blue) and C (on the right side of the figure in green), the proposed HEM-t model exhibited better performance in evaluation methods. Concerning the evaluation methods, the performance of LOOCV was superior to the OOB method.Fig. 12 QQ plots obtained from HEM-t, OOB and LOOCV steps for φ (left), and C (right).

The Taylor diagram (TD) is a graphical representation that allows for the comparison of one or more methods based on their root mean square difference (RMSD), correlation, and SD with respect to a target (observed) method. It provides a comprehensive view of the performance of different methods relative to the target. In this study, the prepared database was firstly trained by HEM method, called HEM-t, and then tested by OOB, and LOOCV evaluation methods. Figure 13 presents the TDs for predicting both φ and C parameters. From the Taylor diagrams, it is evident that for the prediction of φ (left diagram), the HEM-t phase demonstrates superior performance compared to the OOB and LOOCV phases, with the latter two methods showing relatively similar performance in the testing phase. For the prediction of C (right diagram), the HEM-t method again exhibits the best performance, followed by the LOOCV method, and then the OOB method. This indicates that the developed model performed better during the training phase, and in the testing phase, the LOOCV evaluation method outperformed the OOB evaluation method.Fig. 13 Taylor diagram obtained from HEM-t, OOB and LOOCV phases for φ (left), and C (right).

Discussion

It is worth mentioning that the datasets used in this study were obtained from Shen and Jimenez3, who utilized GP and regression models to predict φ and C. According to their findings, the regression model achieved higher accuracy (with an AAREP of 9.5%) compared to GP (with an AAREP of 11.3%). However, based on the results presented in Table 2, the proposed HEM in this study demonstrated improved prediction quality for φ and C parameters, with AAREPs of 2.89% and 5.3% respectively. Note that, the mentioned values are related to best method in the testing phase (LOOCV evaluation method), as shown in Table 2. These findings indicate a significant enhancement in the performance prediction of φ and C compared to the results presented by Shen and Jimenez3.

In this study, a sensitivity analysis was performed to indicate the effect of each input parameter on the modeling of output parameters. For Output 1 (φ), the σci parameter was initially excluded from the HEM model, and considering only two other input parameters, the modeling was implemented. In the next step, the σt parameter was removed from the HEM model, and using σci and σ3max, the modeling was performed. In the final step, the σ3max parameter was excluded, and the modeling was carried out based on σci and σt parameters as inputs. Based on the obtained results, when the σci was removed from the HEM model, the proposed method herein predicted φ with a mean square error (MSE) of 1.8. However, when the σt and σ3max parameters were removed from the HEM model, the MSE values were 1.807 and 15.71, respectively. This finding demonstrates that removing the σ3max parameter significantly reduced the accuracy of the HEM model. Therefore, this parameter can be identified as the most crucial factor in HEM model. Similar analyses were performed for Output 2 (C). According to the findings, when each of the σci, σt, and σ3max parameters were excluded from the modeling, the proposed method herein predicted the C parameters with MSE values of 1.967, 1.961, and 2.389, respectively. These results indicate that when C is considered as the output parameter, the σ3max parameter has the greatest impact on HEM performance and can be deemed the most influential parameter in C modeling.

To quantitatively assess the uncertainty of the models in predicting φ and C, an uncertainty analysis was conducted. Initially, the error for each sample (observed-predicted) was calculated. Subsequently, the mean error (e¯) and the standard deviation of the prediction error (Se) were computed using the following formulas;4 e¯=∑i=1nei

5 Se=∑i=1n(ei-e¯)2n-1

A positive e¯ signifies that the model overestimates, whereas a negative e¯ indicates underestimation. High Se values suggest a significant variation in error values, thus increasing the model's prediction uncertainty. A 95% confidence interval can be derived using ± 1.96 Se. Tables 3 and 4 present the mean error (e¯), the standard deviation of error (Se), and the 95% error confidence band for all models concerning φ and C, respectively. As shown in Table 3, HEM-t and OOB have negative e¯ values, indicating underestimation, while LOOCV has a positive value, indicating overestimation. Additionally, Table 4 shows that all three phases (HEM-t, OOB, and LOOCV) have negative e¯ values, indicating underestimation.Table 3 Uncertainty predicts of output 1 (φ) based on various phases.

Phase	e¯	Se	Width of uncertainty band	
HEM-t	− 0.871	1.603	 ± 3.141	
LOOCV	7.794	8.021	 ± 15.721	
OOB	− 2.938	3.547	 ± 6.953	

Table 4 Uncertainty predicts of output 2 (C) based on various phases.

Phase	e¯	Se	Width of uncertainty band	
HEM-t	− 11.637	11.696	 ± 22.924	
LOOCV	− 2.164	2.677	 ± 5.248	
OOB	− 29.514	29.523	 ± 57.865	

For the φ parameter (Table 3), HEM-t has the narrowest uncertainty band (± 3.141), followed by OOB (± 6.953). LOOCV has the widest uncertainty band (± 15.721), indicating the highest uncertainty in its results. For the C parameter (Table 4), LOOCV has the narrowest uncertainty band (± 5.248), followed by HEM-t (± 22.924). OOB has the widest uncertainty band (± 57.865), indicating the highest uncertainty in its results.

Conclusions

The parameters φ and C, known as the SS parameters, are crucial inputs in geotechnical analyses, including slope stability calculations, bearing capacity analysis, and the design of foundations and retaining structures. These parameters provide insights into the behavior of soil and rock materials under various loading conditions, ensuring the stability and safety of geotechnical structures. In this study, a novel prediction model called the HEM method was proposed to predict φ and C parameters. Additionally, two validation tests, namely OOB and LOOCV, were employed for comparative purposes. The performance of the HEM in three evaluation phases was evaluated using statistical functions such as MAPE, KGE, R2, RMSE, and AAREP. Furthermore, QQ plots and Taylor diagrams were generated to compare the results obtained from the predictive methods. The results indicate that the proposed methods in this study performed well in predicting both φ and C parameters. However, the proposed HEM method exhibited excellent performance in both training and testing phases. Between the two evaluation methods, the performance of LOOCV was better than OOB. As mentioned earlier, the datasets used in this study were obtained from Shen and Jimenez3. Our findings demonstrate that the performance of the proposed HEM method was significantly better than the models employed by Shen and Jimenez3. This clearly highlights the robustness of the proposed HEM method in predicting φ and C parameters. Derived from our findings, one can infer that the HEM method is a powerful tool and is recommended for prediction purposes in other geotechnical tasks. Furthermore, based on sensitivity analysis, σ3max had the greatest impact on HEM performance for both φ and C predictions. Additionally, the uncertainty of the prediction models in both φ and C parameters were calculated, and according to the results, for the φ and C parameters, the LOOCV and OOB had the widest uncertainty band, respectively. While the proposed HEM has demonstrated excellent performance in predicting shear strength parameters and shows promise for application in other fields of geotechnical engineering, there remain areas that warrant further exploration. Firstly, although the HEM addresses some limitations of machine learning methods used in the literature, continuous improvements in data collection and preprocessing techniques could enhance the model's robustness and generalizability. Future research should focus on expanding the database with more diverse and representative samples to further validate the model across different geotechnical scenarios. Secondly, while the current study highlights the accuracy of the HEM, exploring other ensemble methods and hybrid models might yield additional insights and improvements. Comparative studies involving different algorithms could help identify the best approaches for specific applications within geotechnical engineering.

In future work, we plan to enhance our model by incorporating more comprehensive datasets, experimenting with alternative ensemble techniques, and continuously refining the HEM based on feedback and emerging research in the field. This iterative approach will ensure that our model remains at the forefront of predictive accuracy and applicability in geotechnical engineering.

Additionally, we will focus on integrating HEM with existing predictive tools and software in geotechnical engineering. This will involve ensuring compatibility with common geotechnical data formats such as CSV and Excel files, developing APIs and scripting support for real-time data integration and batch processing, and creating plugins or extensions for popular geotechnical software platforms. Establishing interoperability protocols will be a key aspect of this integration process. By addressing these steps, we aim to facilitate the seamless incorporation of HEM into existing workflows, enhancing its practical applicability and usability in the field of geotechnical engineering.

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (Grant No. 52174131; 52474153), and Nation Key R&D Program of China (2023YFC2907305).

Author contributions

Xiaohua Ding: investigation, methodology, validation, writing—review and editing. Maryam Amiri: methodology, validation, writing—review and editing. Mahdi Hasanipanah: conceptualization, data curation, investigation, supervision, writing—review and editing.

Data availability

The data of this research can be available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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