# Predicting Slope Stability Failure through Machine Learning Paradigms

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Machine Learning and Multilinear Regression Algorithms

#### 2.1. Gaussian Processes Regression (GPR)

_{j}as a test sample [29].

_{j}defines mean value representing the most compatible predicted outputs for the test input vector (x

_{j}). Also, K(X, X), kj, ${\sigma}_{n}^{2}$, and y stand for the covariance matrix, the kernel distance between training and testing data, the noise variance, and training observation, respectively. As well as this, the produced variance by Equation (3) (${\sigma}_{j}^{2}$), represents a confidence measure related to the obtained results. Note that, this variance is adversely proportional to the confidence associated with the w

_{j}[30]. The above formulas can be gathered in the form of a linear combination of the s kernel function and the mean estimation for $\overline{f}$(x

_{j}) can be expressed as follows:

#### 2.2. Multiple Linear Regression (MLR)

_{0}, α

_{1}, … α

_{s}are indicative of MLR unknown parameters. Also, the normally distributed random variable is shown by ε in MLR generic formula.

_{0}, α

_{1}, … α

_{s}) of Equation (5). After applying the least-square technique, the practical form of the statistical regression method is given by [31]:

_{0}, a

_{1}, … a

_{s}are the approximated regression coefficients of α

_{0}, α

_{1}, … α

_{s}, respectively. Also, term e describes the approximated error for the sample. Assuming the term e as the difference between the actual and predicted y, the estimate of y is as follows:

#### 2.3. Multi-Layer Perceptron (MLP)

#### 2.4. Simple Linear Regression (SLR)

_{i}) can be calculated at any given x value (x

_{i}).

#### 2.5. Support Vector Regression (SVR)

_{i}) [49]. As explained in previous sentences, the linear regression is implemented in high dimensional feature space using ε-LF. Moreover, the lower value for ${\Vert w\Vert}^{2}$, the less complex model [50]. As for the non-linear problems, the input data are transformed into high-dimensional space by means of a kernel mapping function indicated by γ(x

_{i}). After that, a linear approach is applied to data in the future space by a convex optimization problem [45,51].

_{i}and ϕ

_{i}

^{*}stand for the slack variables measuring the deviation of training data outside ε-LF zone. The precision factor is also shown by ε in the above formula [52]. In other words, only samples with a deviation value more than ε will be considered for error function [51]. Eventually, to calculate the SVR results, a linear combination is established associated with introducing Lagrange multiplier of ρ

_{i}and ρ

_{i}

^{*}:

## 3. Data Collection

_{u}), slope angle (β), setback distance ratio (b/B), and applied surcharge on the footing installed over the slope (w) opted in this work. To create the required dataset, Optum G2 computer software [53,54] was used, which follows a comprehensive finite element method (FEM) (see Figure 2b). Considering the mentioned influential variables, 630 different stages were analyzed, and the safety factor was derived as the output. During the model development, the elastic parameter of Young’s modulus (E) was supposed to be different for each value of Cu. In this regard, E was 1000, 2000, 3500, 5000, 9000, 15,000 and 30,000 kPa for respective C

_{u}values of 25, 50, 75, 100, 200, 300 and 400 kPa. As well as this, the numbers 0.35, 18 KN/m

^{3}, and 0° were allocated to the mechanical parameters Poisson’s ratio, soil unit weight, and internal friction angle, respectively. For more details, the relationship between the computed F safety factor S and its conditioning factors (i.e., C

_{u}, β, b/B, and w) is demonstrated in Figure 3a–d, showing the safety factor featured on the vertical axis versus the respective parameters of Cu, β, b/B, and w, on the horizontal axis. In all diagrams, the slope safety factor varies from 0.8 to 28.55. As is expected, a proportional distribution can be found for the Cu (25, 50, 75, 100, 200, 300, and 400 kPa) and obtained safety factor (see Figure 3a). Adversely, in a general view, when the values of β (15°, 30°, 45°, 60°, and 75°) and w (50, 100, and 150 KN/m

^{2}) are increased, more instability is observed (see Figure 3b,d). In addition, according to Figure 3c, different values of safety factor have been reported as the purposed foundation takes more distance from the edge of slope (b/B is determined by 0, 1, 2, 3, 4, and 5 values).

## 4. Results and Discussion

^{2}), root mean square error (RMSE), mean absolute error (MAE), and root relative squared error (RRSE in%). A colour intensity rating is also generated to display a hued ranking of the implemented models. The quality of results is adversely proportional to the intensity of the colour (green) in the last column of each table. In contrast, more intense colour (red) indicates a more proper performance for all other columns. This is noteworthy that these criteria have been widely used in earlier studies (e.g., Moayedi and Hayati [37], Vakili, et al. [55] and Moayedi and Armaghani [56]). Equations (14)–(18) presented the formulation of R

^{2}, MAE, RMSE, RAE, RRSE, respectively.

_{i observed}and Y

_{i predicted}indicates the actual measurement and estimated slope safety factor, respectively. The term S is the defined of the number of data; ${\overline{Y}}_{\mathrm{observed}}$ is the mean of the real values of safety factor. The obtained values of R

^{2}, MAE, RMSE, RAE, RRSE for safety factor estimation are tabulated in Table 1 and Table 2, respectively, for the training and testing phases. In a glance, MLP is qualified as the first ranking model for both tables. For the training results, based on the R

^{2}(0.9467, 0.9586, 0.9937, 0.9019, and 0.9529), RMSE (1.9957, 1.7366, 0.7131, 2.6334, and 1.9183), and RRSE (32.7404%, 28.4887%, 11.6985%, 43.2016%, and 31.4703%), respectively for GPR, MLR, MLP, SLR, and SVR models, the MLP outputs have shown the best accommodation with the actual values of safety factor. After that, SVR and MLR have also presented a high level of accuracy. In addition, GP (4th ranking) and SLR (5th ranking) have shown an acceptable rate of accuracy concerning the calculated MAE (1.5598 and 1.7013, respectively) values. The sole distinction for the final results (i.e., last column) in Table 1 and Table 2 refers to the ranking gained by MLR and SVR predictive models. As is clear, SVR as the second-accurate model has shown more sensitivity for the testing stage compared to the MLR. In addition, the obtained results for the testing dataset confirms the higher capability of MLP. Considering the respective values of R

^{2}(0.9509, 0.9649, 0.9939, 0.9265, and 0.9653), MAE (1.5291, 1.1949, 0.5155, 1.5387, and 1.0364), and RAE (30.9081%, 24.1272%, 10.4047%, 31.0892%, and 20.9366%), it can be concluded that SVR is the second-precise model, and MLR has outperformed GP and SLR.

^{2}, MAE, RMSE, RAE, RRSE single ranking for training and testing dataset). Based on the obtained total scores of 20, 35, 50, 10, and 35 (respectively for GPR, MLR, MLP, SLR, and SVR), the superiority of MLP can be deduced (i.e., the most significant total rank). The multiple linear regression and support vector regression methods have commonly been labelled as the second-accurate models. After these, GPR and SLR have shown a good quality of safety factor estimation with respective total scores of 20 and 10. The remarkable point about Table 3 is that an equal individual score for all indices has featured for MLP, GPR, and SLR predictive model. The MLP and SLR, for instance, had the highest (5) and lowest (1) level of accuracy, based on the R

^{2}, MAE, RMSE, RAE, and RRSE indices simultaneously.

^{2}= 1).

^{2}= 0.9937 and 0.9939, and RMSE = 0.7131 and 0.7039, respectively for training and testing phases) and MLR (R

^{2}= 0.9586 and 0.9649, and RMSE = 1.7366 and 1.5891, respectively for training and testing phases) predictive models are eligible enough to provide an appropriate estimation for safety factor. In addition, the obtained values of RRSE for training (11.6985% and 28.4887%, respectively for MLP and MLR) and testing (11.8116% and 26.4613%, respectively for MLP and MLR) datasets prove a high level of accuracy for these techniques. With this in mind, in this part of the current study, it was aimed to extract the equation of developed MLR and MLP models to be used in stability assessment of similar slopes. The MLR attempts to fit a linear relationship to data. The formula derived from the MLR and MLP models is presented in Equations (19) and (20), respectively.

_{MLP}= (−1.12353500504828 × Y

_{1}) − (2.38866337313669 × Y

_{2}) + 1.77734928298793

_{1}and Y

_{2}are calculated from the below equations:

## 5. Design Charts

_{u}) (on the horizontal axis). Note that these figures are provided as an appendix. Each single chart illustrates the comparison between the real (i.e., linear) and predicted (i.e., points) values of safety factors, for different values of b/B ratio ((a), (b), (c), (d), (e), and (f), respectively for b/B = 0, b/B = 1, b/B = 2, b/B = 3, b/B = 4, and b/B = 5). Note that, in all charts, the data have been divided into three parts concerning the w (applied surcharge on the rigid foundation) values, which were 50, 100, and 150 kPa. Based on the coefficient of determination (i.e., R

^{2}) computed for each case, the effectiveness of estimation carried out in this study can be deduced.

## 6. Conclusions

_{u}, β, b/B, and w were considered in this study. To create the eligible dataset, referring to possible values that can be received by mentioned effective parameters, 630 different stages were defined and analyzed in Optum G2 software. The factor of safety was picked as the output of this operation. In the next step, the acquired dataset was divided into training (80% of the entire dataset) and validation (20% of the entire dataset) phases to train and validate the efficiency of GPR, MLR, MLP, SLR, and SVR approaches. The implementation of models was carried out in WEKA software, which is a prominent tool for machine learning and classification applications. For the training phase, the R

^{2}(0.9467, 0.9586, 0.9937, 0.9019, and 0.9529), MAE (1.5598, 1.2527, 0.4940, 1.7013, and 1.161), RMSE (1.9957, 1.7366, 0.7131, 2.6334, and 1.9183), RAE (31.1929%, 25.0515%, 9.8796%, 34.0224%, and 23.2182%), and RRSE (32.7404%, 28.4887%, 11.6985%, 43.2016%, and 31.4703%), were obtained, respectively for GPR, MLR, MLP, SLR, and SVR. Similarly, for the testing phase, we acquired R

^{2}(0.9509, 0.9649, 0.9939, 0.9265, and 0.9653), MAE (1.5291, 1.1949, 0.5155, 1.5387, and 1.0364), RMSE (1.9447, 1.5891, 0.7039, 2.2618, and 1.6362), RAE (30.9081%, 24.1272%, 10.4047%, 31.0892%, and 20.9366%), and RRSE (32.3841%, 26.4613%, 11.8116%, 37.6639%, and 27.2470%), respectively for GPR, MLR, MLP, SLR, and SVR. Referring to the indices mentioned above, the advantage of MLP is deduced, compared to the other applied machine learning methods. In addition, it can be seen that there is a slight difference between the performance of MLR and SVR predictive models, and GPR outperforms SLR. In the next section, the equation of implemented MLP and MLR (i.e., for their optimal condition) was derived to be used in similar slope stability problems. In addition, due to the highest rate of success for MLP prediction, in the last part, the outputs produced by this model were more particularly compared to the actual values of safety factor within design charts generated for different values of Cu, β, b/B, and w.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**A schematic view of the (

**a**) slope geometry and (

**b**) the Optum G2 ε

_{xx}result, horizontal strain diagram for finite element method (FEM) model with Cu = 75, β = 15, b/B = 1, and w = 50 kPa.

**Figure 3.**The relationship between the input parameters versus an obtained factor of safety (vertical axis). (

**a**) Cu versus safety factor, (

**b**) β versus safety factor, (

**c**) b/B versus safety factor, (

**d**) surcharge (w) versus safety factor.

**Figure 5.**The results of correlation obtained for the training datasets after conducting various models in predicting safety factor (

**a**) Gaussian processes regression, (

**b**) multiple linear regression, (

**c**) multi-layer perceptron, (

**d**) simple linear regression, (

**e**) support vector regression.

**Figure 6.**The results of correlation obtained for the testing datasets after conducting various models in predicting safety factor: (

**a**) Gaussian processes regression, (

**b**) multiple linear regression, (

**c**) multi-layer perceptron, (

**d**) simple linear regression, (

**e**) support vector regression.

**Figure 8.**Comparison of the MLP estimation and real values of FS for the β = 15°, (

**a**) b/B = 0, (

**b**) b/B = 1, (

**c**) b/B = 2, (

**d**) b/B = 3, (

**e**) b/B = 4, (

**f**) b/B = 5.

**Figure 9.**Comparison of the MLP estimation and real values of FS for the β = 30°, (

**a**) b/B = 0, (

**b**) b/B = 1, (

**c**) b/B = 2, (

**d**) b/B = 3, (

**e**) b/B = 4, (

**f**) b/B = 5.

**Figure 10.**Comparison of the MLP estimation and real values of FS for the β = 45°, (

**a**) b/B = 0, (

**b**) b/B = 1, (

**c**) b/B = 2, (

**d**) b/B = 3, (

**e**) b/B = 4, (

**f**) b/B = 5.

**Figure 11.**Comparison of the MLP estimation and real values of FS for the β = 60°, (

**a**) b/B = 0, (

**b**) b/B = 1, (

**c**) b/B = 2, (

**d**) b/B = 3, (

**e**) b/B = 4, (

**f**) b/B = 5.

**Figure 12.**Comparison of the MLP estimation and real values of FS for the β = 75°, (

**a**) b/B = 0, (

**b**) b/B = 1, (

**c**) b/B = 2, (

**d**) b/B = 3, (

**e**) b/B = 4, (

**f**) b/B = 5.

Proposed Models | Network Results | Ranking the Predicted Models | Total Ranking Score | Rank | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | RAE (%) | RRSE (%) | R^{2} | MAE | RMSE | RAE (%) | RRSE (%) | |||

Gaussian Processes | 0.9467 | 1.5598 | 1.9957 | 31.1929 | 32.7404 | 2 | 2 | 2 | 2 | 2 | 10 | 4 |

Multiple Linear Regression | 0.9586 | 1.2527 | 1.7366 | 25.0515 | 28.4887 | 4 | 3 | 4 | 3 | 4 | 18 | 2 |

Multi-layer Perceptron | 0.9937 | 0.494 | 0.7131 | 9.8796 | 11.6985 | 5 | 5 | 5 | 5 | 5 | 25 | 1 |

Simple Linear Regression | 0.9019 | 1.7013 | 2.6334 | 34.0224 | 43.2016 | 1 | 1 | 1 | 1 | 1 | 5 | 5 |

Support Vector Regression | 0.9529 | 1.161 | 1.9183 | 23.2182 | 31.4703 | 3 | 4 | 3 | 4 | 3 | 17 | 3 |

Proposed Models | Network Results | Ranking the Predicted Models | Total Ranking Score | Rank | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | RAE (%) | RRSE (%) | R^{2} | MAE | RMSE | RAE (%) | RRSE (%) | |||

Gaussian Processes | 0.9509 | 1.5291 | 1.9447 | 30.9081 | 32.3841 | 2 | 2 | 2 | 2 | 2 | 10 | 4 |

Multiple Linear Regression | 0.9649 | 1.1949 | 1.5891 | 24.1272 | 26.4613 | 3 | 3 | 4 | 3 | 4 | 17 | 3 |

Multi-layer Perceptron | 0.9939 | 0.5155 | 0.7039 | 10.4047 | 11.8116 | 5 | 5 | 5 | 5 | 5 | 25 | 1 |

Simple Linear Regression | 0.9265 | 1.5387 | 2.2618 | 31.0892 | 37.6639 | 1 | 1 | 1 | 1 | 1 | 5 | 5 |

Support Vector Regression | 0.9653 | 1.0364 | 1.6362 | 20.9366 | 27.247 | 4 | 4 | 3 | 4 | 3 | 18 | 2 |

Proposed Models | Network Result | Total Rank | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Training Dataset | Testing Dataset | ||||||||||

R^{2} | MAE | RMSE | RAE (%) | RRSE (%) | R^{2} | MAE | RMSE | RAE | RRSE | ||

Gaussian Processes | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 20 |

Multiple Linear Regression | 4 | 3 | 4 | 3 | 4 | 3 | 3 | 4 | 3 | 4 | 35 |

Multi-layer Perceptron | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 50 |

Simple Linear Regression | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 |

Support Vector Regression | 3 | 4 | 3 | 4 | 3 | 4 | 4 | 3 | 4 | 3 | 35 |

^{2}: Correlation coefficient; MAE: Mean absolute error; RMSE: Root mean squared error; RAE: “Relative absolute error”. RRSE: Root relative squared error.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tien Bui, D.; Moayedi, H.; Gör, M.; Jaafari, A.; Foong, L.K. Predicting Slope Stability Failure through Machine Learning Paradigms. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 395.
https://doi.org/10.3390/ijgi8090395

**AMA Style**

Tien Bui D, Moayedi H, Gör M, Jaafari A, Foong LK. Predicting Slope Stability Failure through Machine Learning Paradigms. *ISPRS International Journal of Geo-Information*. 2019; 8(9):395.
https://doi.org/10.3390/ijgi8090395

**Chicago/Turabian Style**

Tien Bui, Dieu, Hossein Moayedi, Mesut Gör, Abolfazl Jaafari, and Loke Kok Foong. 2019. "Predicting Slope Stability Failure through Machine Learning Paradigms" *ISPRS International Journal of Geo-Information* 8, no. 9: 395.
https://doi.org/10.3390/ijgi8090395