One of the most significant and crucial issues in geotechnical engineering works, such as earth dams, embankments, and landfills to name a few, is slope stability assessment. Better methods are required to anticipate slope collapse because of its fatal effects. The goal of this research is to create a straightforward machine learning (ML) model for examining slope stability under seismic conditions. Four ML algorithms are examined, including Logistic Regression (LR), Quadratic Discriminant Analysis (QDA), Light Gradient Boosting Machine (LGBM), and Linear Discriminant Analysis (LDA). The models are trained and tested on the database containing 700 slopes. 10-fold cross validation is utilized for parameter tuning, model training and performance estimating of machine learning models using training set. The best model is interpreted using the SHapley Additive exPlanations (SHAP) method, which is built on game theories. Among the studied models, the LGBM model is the most accurate model based on ranking technique. Most influential features for slope stability prediction under seismic conditions are detected by the SHAP method as follows: peak ground acceleration, friction angle, and angle of inclination.
Research Article
Straightforward slope stability prediction under seismic conditions using machine learning algorithms
https://doi.org/10.21203/rs.3.rs-3575223/v1
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Slope stability
seismic conditions
machine learning
light gradient boosting machine
An important area of slope engineering investigation is the assessment of slope stability [1]. However, the geological procedure involved in slope displacement and failure is quite complicated [2, 3]. It is difficult to appropriately assess slope stability using the theoretical analysis or finite element method because the components that impact slope stability are imprecise and have uncertainty [4, 5]. The computation of the factor of safety (FOS), which is calculated as the shear strength to the acting shear stress ratio, is a key component of the assessment of slope stability [6]. The main factors that determine the slope's geometry, such as height and slope inclination, as well as the characteristics of the material, such as angle of friction, affect how stable a slope is. A valuable alternative to the physics-based models that train and identify the failure types of slopes under various conditions is now available because of the extraordinary progress of machine learning (ML) algorithms and the vast amount of data accumulated in this area [7]. Some researchers use ML to solve nonlinear complex problems [8, 9]. As an example, based on geotechnical characteristics and past behavior, Lu and Rosenbaum [10] estimated potential ground displacement using Artificial Neural Networks (ANN) and grey systems techniques. Chen et al. [11] employed the Adaptive Neuro-Fuzzy Inference System (ANFIS) and ANN models to predict the slope stability using the data of 53 slope cases and reported ANFIS as the most accurate model. Cheng and Hoang [12] used the support vector classifier (SVC) combined to predict slope collapses. The hyper-parameters of the SVC are tuned by the bee colony optimization algorithm. Fattahi [13] utilized different ANFIS models (grid partitioning, subtractive clustering, and fuzzy c-means clustering) to assess their ability for the estimation of FOS. Zeroual et al. [14] used the ANN to develop a prediction model for the FOS estimation of earth dams. Haghshenas et al. [15] developed a hybrid algorithm using Harmony search and the K-means algorithm to analyze the slope stability based on a limited database made up of 19 case studies. Ahmad et al. [16] developed a new model based on tree-augmented Naive-Bayes to estimate slope stability using a total of 87 case studies. Hoang and Pham [17] introduced a hybrid algorithm for slope stability determination based on the firefly algorithm and the SVC. They employed a dataset that included 168 case studies. According to experimental findings, the new hybrid approach has improved the accuracy of the classification by about 4% when compared to existing benchmark techniques.
Zhou et al. [18] developed a unique prediction approach that makes use of the gradient boosting machine algorithm to get around the complexities and uncertainties of numerous connected parameters with small unbalanced data samples for the prediction of slope stability. They made use of a thorough analysis of 221 distinct real-world slope failure cases. In order to model relationships between variables that affect the slope stability, Qi and Tang [19] utilized various ML algorithms, including decision trees, random forests, gradient boosting machines, support vector machines, and multilayer perceptron neural networks. The firefly algorithm was then utilized to tune the hyper-parameters. The ideal support vector machine model with the Youden's cutoff was recommended in terms of accuracy. A Naive Bayes Classifier (NBC) was used by Feng et al. [20] to forecast slope stability for those that had circular failures. With only 69 slope examples in the data set, an expectation maximization approach was utilized to do parameter learning for the NBC. According to the model evaluation using 13 new cases, the suggested NBC model performs better in terms of accuracy when compared to the existing technique. A hybrid stacking ensemble strategy was implemented and suggested by Kardani et al. [21] to improve slope stability forecasting. In the hybrid stacking ensemble method, they selected an appropriate meta-classifier from a collection of 11 different individually optimized machine learning algorithms using an artificial bee colony algorithm to discover the optimum combination of base classifiers. Finite element analysis was done in order to create the artificial database for the proposed model's training phase (150 examples). The findings demonstrated that the hybrid stacking ensemble had significantly improved the capacity to forecast slope stability. A hybrid ensemble method, introduced by Qi and Tang [22], uses a genetic algorithm with classifier ensembles to optimize slope stability estimation. The separate algorithms were decision trees, k-nearest neighbors, support vector machines, quadratic discriminant analysis, and gaussian process. The combination approach included weighted majority voting. The 10-fold cross-validation procedure was selected as the validation approach. The hyperparameter tuning and weight tuning processes used grid search and genetic algorithm, respectively.
To analyze and assess slope stability in open-pit mines, Luo et al.[23] created a new framework. A unique hybrid method based on evolutionary optimization of the cubist algorithm (particle swarm optimization and cubist algorithm (PSO-CA)) was created for estimating the FOS in slope stability. 450 scenarios from the Geostudio software for the FOS of a quarry mine were employed. Based on two popular ensemble techniques—parallel learning and sequential learning—Pham et al.[24] constructed ensemble predictors for slope stability assessment using a library of 153 slope instances. In particular, sequential learning ensemble learners outperformed parallel learning learners in terms of accuracy. According to their findings, the ensemble learners using the extreme gradient boosting framework delivered the best performance. Liao[25] investigated the application of multivariate adaptive regression splines (MARS) to grasp the naturally occurring multidimensional and nonlinear interaction between the factors involved in the assessment of slope stability. A comparison of back-propagation neural network and MARS-based machine learning approaches to slope stability estimation was performed. The study used a dataset containing 122 samples that included real slope collapse incidents. Lin et al.[26] developed a variety of ML techniques for FOS prediction. Based on various input parameter patterns, 11 machine learning algorithms are assessed for their capacity to learn the FOS. Support vector and gradient boosting regression are regarded as the top techniques after examining the assessment indicators. Amirkiyaei and Ghasemi[27] developed two tree-based models utilizing a collection of 87 slope cases: a regression tree-based algorithm for prediction of slope safety factor and a classification tree-based algorithm for prediction of slope stability status. Their findings show that the created tree models are highly reliable and efficient instruments for evaluating slope stability. Using 422 groups of slope sample data, Zhang et al.[28] presented the margin distance minimization selective ensemble (MDMSE) model. They claimed that the model can overcome the drawbacks of large risk of judgment error in conventional ML models by objectively evaluating the slope stability using the fundamental geometric and geological variables. According to their findings, the MDMSE prediction model has a clearly superior capacity for generalization than other models, as well as superior recognition precision and a faster recognition rate than other ensemble models.
If a slope is in an earthquake-prone area, the design must account for these challenging circumstances. The impact of shaking depends on how much the soil shear strength is sufficient under dynamic loading or whether shaking causes a considerable reduction in strength. Since shearing sliding causes deformation, slope stability evaluation is required to make sure the factor of safety is sufficient to withstand dynamic loading and reduce the resultant deformation. Despite the importance of slope stability under seismic conditions, little research has been done on the application of artificial intelligence (AI) and its branches to build a reliable model to determine slope coefficients. In summary, Asteris et al. [29] were the only researchers to assess how well modelling algorithms predicted the slope stability. However, due to a lack of more reliable algorithms, AI applications have not yet realized their full potential. This study aims to develop a simple ML model to investigate slope stability under seismic circumstances. Light gradient boosting machine, logistic regression, quadratic discriminant analysis, and linear discriminant analysis are four ML techniques that were investigated in this research. A database comprised of 700 slopes was used to train and test the models. The SHapley Additive exPlanations (SHAP) approach is used to describe the best model.
This publication also makes use of the database that was developed by other researchers [1, 2, 5, 6, 29]. In order to calculate FOS, several models (which are developed in GeoStudio) are used to construct the database. There are 700 slope samples in the database with various input variable values. Slope height (\(H\)), angle of inclination (\(\alpha\)), cohesion (\(C\)), friction angle (\({\varnothing}\)), and peak ground acceleration (PGA)) are the input variables. Table 1 shows the input and output (FOS) variables' characteristics. In order to achieve the goal of analyzing and categorizing all the slope stability instances in the data, the FOS from the dataset is then divided into subgroups of safe or unsafe slope. 80% of the dataset is chosen at random and utilized for training, and the remaining 20% for testing [30, 31]. Five parameters' correlation coefficients are determined in order to better understand the interaction and link between the various input parameters. Figure 1 presents the correlation coefficients of input parameters. The linear correlation between the individual input parameters is discovered to be poor; indicating (1) a complex nonlinear regression relationship between the input variables and output as well as (2) the multicollinearity phenomenon does not exist.
| Input parameters | ||||||
|---|---|---|---|---|---|---|
| Characteristic | \(\varvec{H}\) | \(\varvec{\alpha }\) | \(\varvec{C}\) | \(\mathbf{\varnothing }\) | PGA | FOS |
| Units | (m) | (°) | (kPa) | (°) | (m/s2) | - |
| Range | [15, 30] | [20, 35] | [20, 50] | [20, 40] | [0, 3.9] | [0.78, 2.46] |
| Standard deviation | 5.6 | 5 | 11.18 | 5.88 | 1.07 | 0.35 |
| Mean | 22.33 | 25.18 | 35.3 | 34.07 | 1.18 | 1.20 |
Here, four ML algorithms are used, including Logistic Regression (LR), Quadratic Discriminant Analysis (QDA), Light Gradient Boosting Machine (LGBM), and Linear Discriminant Analysis (LDA). The 10-fold cross validation method is utilized for parameter tuning, model training, and performance estimating of machine learning models in the training phase. The ultimate performance evaluation is performed using the testing set. PyCaret [32], an open-source machine learning library written in Python, is used to build models for this study.
3.1. Logistic Regression (LR)
Despite its name, logistic regression is more of a classification model than a regression model [33]. For situations involving binary and linear classification, logistic regression is a straightforward and more effective approach. It's a classification model that's incredibly simple to implement and performs admirably with linearly distinguishable classes. It is a widely used categorization method in business. Like the perceptron, the logistic regression model is a mathematical technique for binary classification that can also be applied to multiclass classification. The formula for LR is as follows:
$$\ln (P/1 - P)={\beta _0}+{\beta _1}{x_1}+{\beta _2}{x_2}+...+{\beta _n}{x_n}$$
1
where p is the probability and \({x_i}\)is input, and \({\beta _i}\)is regression coefficient.
3.2. Quadratic Discriminant Analysis (QDA)
QDA is developed from straightforward probabilistic models [34]:
$$\begin{gathered} P(y=k|x)=\frac{{P(x|y=k)P(y=x)}}{{P(x)}} \hfill \\ P(x|y=k)=\frac{1}{{{{(2\pi )}^{d/2}}{{\left| {{\sum _k}} \right|}^{1/2}}}}\exp ( - \frac{1}{2}{(x - {\mu _k})^t}\sum _{k}^{{ - 1}}(x - {\mu _k})) \hfill \\ \end{gathered}$$
2
where \(P(y=k|x)\) is the conditional distribution of the class,is a class, is a training sample, is the number of features, \(\sum\) is the covariance matrix, \({\mu _k}\) represents the mean, \(\left| {{\sum _k}} \right|\) is the determinant of \({\sum _k}\), and \(P(x|y=k)\)is the multivariate Gaussian distribution. The predicted class is the one that maximizes\(\log P(y=k|x)\):
$$\begin{gathered} \log P(y=k|x)=\log P(x|y=k)+\log P(y=k)+CT= \hfill \\ - \frac{1}{2}\log \left| {{\sum _k}} \right| - \frac{1}{2}{(x - {\mu _k})^t}\sum _{k}^{{ - 1}}(x - {\mu _k})+\log P(y=k)+CT \hfill \\ \end{gathered}$$
3
Where \(CT\)(a constant term) corresponds to the denominator P(x).
3.3. Linear Discriminant Analysis (LDA)
The two traditional classifiers LDA and QDA have linear and quadratic judgment surfaces, respectively, as their names imply [34, 35]. These classifiers are appealing since they have easily derived closed-form solutions, are naturally multiclass, and have a good track record in real-world applications. LDA is a subset of QDA in which the Gaussian distributions for each class are considered to maintain the same covariance matrix, that is \({\sum _k}=\sum\)for all k. This converts\(\log P(y=k|x)\) to:
$$\log P(y=k|x)= - \frac{1}{2}{(x - {\mu _k})^t}\sum _{k}^{{ - 1}}(x - {\mu _k})+\log P(y=k)+CT$$
4
3.4. Light Gradient Boosting Machine (LGBM)
Classification and predictive modelling problems can be addressed by using a class of ensemble ML techniques called gradient boosting [36]. One tree is joined to the group at a time, and it is fitted, correcting the prediction errors introduced by the prior models. This form of ensemble machine learning model is known as boosting. The models are fitted using the gradient descent optimization process and any loss function. As the model is trained, it lowers the loss gradient, like a neural network, which is how the term "gradient boosting" comes to be used. LGBM offers a practical and efficient implementation of the gradient boosting technique. By including a sort of autonomous feature selection and concentrating on boosting cases with greater gradients, LGBM expands the gradient boosting technique [37]. This can hasten training significantly and enhance prediction performance. As a result, when using tabular data for classification predictive modelling tasks, LGBM has established itself as the de facto algorithm for machine learning challenges. In this study, LGBM was also used to develop a model for the stability prediction of the slops.
3.5. Performance measuring
The confusion matrix, precision, recall, area under the curve (AUC) of the true positive rate against the false positive rate, accuracy, Matthews correlation coefficient (MCC), F1 score, and Kappa coefficient are used to assess the models' classification results using the test set. Test accuracy is calculated as the ratio of specimens that were properly identified to all samples in the test set. The kappa coefficient is a reliable indicator of the degree of agreement [38, 39]. The accuracy of the minority group is determined by measuring the percentage of positive class forecasts that clearly correspond to the positive group. A metric called recall counts the number of precise positive assumptions that were made from all conceivable positive expectations. Consider the positive forecasts that were incorrect, as opposed to precision, which only considers the real positive forecasts among all other forecasts. The following formula was used to calculate the metrics:
where T, P, F, and N are the abbreviation of the true, positive, false, negative.
The performance of the models developed in this study is compared in two stages: training and testing. For a comparison between different models, accuracy, AUC, recall, precision, F1, kappa, and MCC values (for training set) obtained using the 10-fold cross validation method for the slope stability prediction are shown in Fig. 2. Also, the mean values of the 10-fold method for each metric are shown in a table under each model’s figure. For all the algorithms, it is observed that the AUC is close to 1. However, the other metrics’ values for the various models show variations. For example, the AUC value for the LDA model is 0.993 (train set), but it is associated with a low kappa value of 0.907 compared to other models. This means that enough metrics should be used to determine the predictive power of algorithms. Among the models, it is seen that the LR algorithm outperforms other algorithms with the highest accuracy, AUC, recall, precision, F1, kappa, and MCC values, as shown in Fig. 2. Choosing another model as the second-best model is a difficult task. Because none of the remaining models is superior to the others in terms of all criteria. Table 2 shows the values of different metrics for all models using the same test set. LGBM is superior to other models in terms of all criteria except AUC and recall. In this study, a ranking mechanism is applied to assess the models’ performance more effectively. The models are graded (Table 3) from one to four according to the highest to lowest value for each criterion (values in Table 2). Then the value of each row of Table 3 (related to each model for different criteria) is added together. Using the obtained value in the last column ("sum"), the performance of the models can be compared. The lower this value is, the better the model has performed. From Table 3, it can be observed that LGBM performed best in predicting the slope stability in the testing phase. LR is the second-best model in the testing phase.
Table 2. Performance evaluation of various models– testing phase
|
Model |
Accuracy |
AUC |
Recall |
Precision |
F1 |
Kappa |
MCC |
|
LR |
0.978 |
0.997 |
0.988 |
0.978 |
0.983 |
0.953 |
0.953 |
|
QDA |
0.950 |
0.992 |
0.977 |
0.946 |
0.961 |
0.889 |
0.890 |
|
LGBM |
0.985 |
0.995 |
0.977 |
1.000 |
0.988 |
0.969 |
0.969 |
|
LDA |
0.935 |
0.981 |
0.911 |
0.988 |
0.948 |
0.864 |
0.869 |
Table 3. Model ranking - testing phase
|
Model |
Accuracy |
AUC |
Recall |
Precision |
F1 |
Kappa |
MCC |
Sum |
|
LR |
2 |
1 |
1 |
3 |
2 |
2 |
2 |
13 |
|
QDA |
3 |
3 |
2 |
4 |
3 |
3 |
3 |
21 |
|
LGBM |
1 |
2 |
2 |
1 |
1 |
1 |
1 |
9 |
|
LDA |
4 |
4 |
3 |
2 |
4 |
4 |
4 |
25 |
4.1. Comparison with previous studies
The use of artificial intelligence in slope stability approaches is examined in previous studies. This section compares the findings of the most current study that made use of the same database. Asteris et al. [29] used Decision Tree Algorithm (DTA), Random Forest Algorithm (RFA), and AdaBoost algorithm. DTA is an AI method that divides predictor parameters into homogeneous groups using conditional evaluation criteria. Finding a set of decision guidelines for extrapolating an output from a set of input constraints is the goal of DTA specifications. RFA ensemble modelling technique is employed for categorizing and regression. It operates by training a large number of DTAs and then producing the group that is the mean estimate of the individual trees. To create each DTA, data from a random variable were employed. Classification models calculate the value produced by distinct trees based on voting. The fundamental RFA makes use of the random subspace technique. The AdaBoost makes an effort to utilize scaled derivatives of the original testing set rather than subsamples. The AdaBoost system learns by employing a number of base learners or classifiers. The algorithm's weights for the training set are assigned in each round based on how prior classifiers performed. The samples or data sets that had previously been incorrectly categorized in this situation are then used by the algorithm. The ranking of reported indicators for classification was investigated in this section. Table 4 provides results obtained using DTA, RFA, and AdaBoost. According to Table 4, AdaBoost performed well in terms of accuracy, F1, precision, and recall. DTA and RFA performed worse than AdaBoost in terms of other indices, while having strong AUC scores. Overall, AdaBoost had performed best in all indices except AUC. AdaBoost has fared poorly as compared to the model proposed in this study. LGBM outperformed AdaBoost with the statical values of AUC = 0.995, accuracy = 0.985, recall = 0.977, precision = 1.0, and F1 = 0.988 (Table 2).
|
Model |
Indicators |
Rank |
|||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
|
AUC |
Accuracy |
F1 |
Precision |
Recall |
AUC |
Accuracy |
F1 |
Precision |
Recall |
Total |
|
|
DTA (Asteris et al., 2022) |
0.968 |
0.891 |
0.895 |
0.908 |
0.891 |
1 |
3 |
3 |
3 |
3 |
13 |
|
RFA (Asteris et al., 2022) |
0.961 |
0.914 |
0.915 |
0.916 |
0.914 |
2 |
2 |
2 |
2 |
2 |
10 |
|
AdaBoost (Asteris et al., 2022) |
0.910 |
0.931 |
0.931 |
0.931 |
0.931 |
3 |
1 |
1 |
1 |
1 |
7 |
4.2. Interpretability of the LGBM model
Because of their extremely intricate and non-linear structure, ML models like LGBM frequently act as "black boxes” [40, 41]. Due to their hierarchical nature, tree models are explicable, although their visuals could be challenging to understand. In this regard, the SHapley Additive exPlanations (SHAP)[42] method is employed in this study. SHAP can be a very helpful model-neutral tool for deciphering intricate ML models with several variables. The LGBM model is interpreted in this section using SHAP because LGBM performs the best in terms of slope stability forecasting. Figure 7 displays the plot of the SHAP results for each feature used to predict the slope stability. Each feature's color denotes its value, and the x-axis value that corresponds to it denotes how much it contributed to the output property. For example, for the PGA, red dots on the left and blue dots on the right, respectively, denote high and low values of the PGA. Similarly, the negative SHAP values of the PGA suggest that a high PGA can decrease the slope stability. The SHAP values for and attributes are often centered near zero. These low values indicate that they are less significant to the model's outcome. In addition, Fig. 7 shows input features based on the most dominant features from top to down. In other words, the y-axis shows the input features in their order of importance. As can be observed in Fig. 7, the three most influential inputs are PGA, \(\phi\), and \(\alpha\), respectively. The distribution of SHAP values for cohesion () and friction angle (\(\phi\)) point out that high cohesion and friction angle can increase the slope stability.
The usage of ML models in engineering practice is encouraged by creating an interface that is user-friendly. The Streamlit library was utilized in this study to create a GUI. In Fig. 4, it is depicted. By just pressing a button, the slope stability is predicted. It will unquestionably save a lot of time and work. Anyone can access the code for the GUI by going to the specified URL (https://msbearthquake-slopestability--streamslope-tk9bb3.streamlit.app) without having to download any additional programs.
Based on design criteria and engineering qualities, geotechnical engineers frequently use analytical and empirical approaches to assess the safety factor of soil or rock material. The job of creating a sufficient model to effectively mimic site-specific engineering geological characteristics and adhere to the proper design methodology in order to eradicate failure potential and suggest the most cost-effective solution is hard. This study employed ML algorithms for the prediction of the slope stability under seismic conditions, including Logistic Regression (LR), Quadratic Discriminant Analysis (QDA), Light Gradient Boosting Machine (LGBM), and Linear Discriminant Analysis (LDA). Inputs vector constituted slope height (\(H\)), angle of inclination (\(\alpha\)), cohesion (\(C\)), friction angle (\({\varnothing}\)), and peak ground acceleration (PGA)). The models are tuned through 10-fold cross validation. Based on this study, the following results can be presented:
It has been noted that the AUC for all methods is quite near to 1. The values of the other metrics for the various models, however, differ. For instance, the LDA model's AUC value is 0.993 (train set), yet in comparison to other models, it has a low kappa value of 0.907. This means that sufficient criteria must be employed to assess the predictive capability of algorithms.
It can be observed that the LR method surpasses other algorithms in the training phase in terms of the accuracy, AUC, recall, precision, F1, kappa, and MCC values among the models.
Since no model is better than the others in terms of all criteria in testing phase, a ranking system is used to evaluate the models’ performance more accurately. According to the ranking technique, LGBM did the best job in predicting the stability of the slope during testing.
The well-tuned model (LGBM) can be effectively interpreted by its SHAP values. The SHAP method found that the peak ground acceleration, the friction angle, and the slope angle are the three most important factors for slope stability.
This current work can advance the state-of-the-art for slope design and stability testing. The results of this study are still limited to the range of input parameters that were looked at.
Author Contributions: Conceptualization, M.S.B., M.M.B, D.J.A., B.G., E.T.M., and M.M.S.S.; methodology, M.S.B., M.M.B. and D.J.A.; software, M.S.B., M.M.B and D.J.A.; validation, M.S.B., M.M.B; formal analysis, M.S.B., M.M.B; resources, B.G., writing—original draft preparation, M.S.B., M.M.S.S., B.G., E.T.M. and D.J.A.; writing—review and editing, M.S.B., M.M.S.S. B.G., E.T.M., and D.J.A.; supervision, D.J.A.; funding acquisition, M.M.S.S. and. All authors have read and agreed to the published version of the manuscript.
Funding: Funding: The research is partially funded by the Ministry of Science and Higher Education of the Russian Federation under the strategic academic leadership program ‘Priority 2030’ (Agreement 075-15-2021-1333 dated 09/30/2021).
Data Availability Statement: The data that support the findings of this study are available from the corresponding author, upon reasonable request.
Acknowledgments: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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No competing interests reported.