# Extreme Learning Machine Based Prediction of Soil Shear Strength: A Sensitivity Analysis Using Monte Carlo Simulations and Feature Backward Elimination

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Data Collection and Preparation

#### 2.2. Extreme ML-Based Modeling

_{i}is the weights between the input layer and the hidden layer and α

_{j}is the weights between the output layer and the hidden layer, a

_{j}is the critical value of the neurons in the hidden layer, g(.) activation function. Input layer weights (w

_{i,j}) and bias (a

_{j}) are randomly selected. At the beginning of the input layer neuron number (n) and hidden-layer neuron number (m), the activation function (g(.)) is selected. To construct the ELM algorithm, the database was split into a training dataset (70% data) and the remaining data (30%) for building and validation of the ELM model.

#### 2.3. Backward Elimination-Based Sensitivity Analysis

#### 2.4. Monte Carlo Simulations

_{MC}is the number of Monte Carlo runs. This convergence function provides efficient information related to the computational time, reliability results for further statistical analysis.

#### 2.5. Performance Evaluation

_{i}and t are defined as the values and means of the predicted shear strength, respectively, and t

_{i}and t are the values and mean of the actual shear strength, respectively.

## 3. Results

#### 3.1. Validation of ELM with Various Number of Neurons

#### 3.2. Sensitivity Analysis Using Backward Elimination and Monte Carlo Simulations

_{2}, X

_{3}, X

_{4}, X

_{5}, X

_{6}; (ii) X

_{1}, X

_{3}, X

_{4}, X

_{5}, X

_{6}; (iii) X

_{1}, X

_{2}, X

_{4}, X

_{5}, X

_{6}; (iv) X

_{1}, X

_{2}, X

_{3}, X

_{5}, X

_{6}; (v) X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{6}; (vi) X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}. Similarly, the “Scenario 2”, “Scenario 3”, and “Scenario 4” corresponded to the cases with five, four, three input spaces, respectively (Figure 3). The summarized input space and the four scenarios could be illustrated in Figure 3. The following sections are dedicated to each step of the backward elimination process.

#### 3.2.1. Reduction of the Input Space from 6 to 5 Variables (Scenario 1)

_{1}to X

_{6}. The results are plotted in Figure 6 for average values of R, RMSE, and MAE (red squares), standard deviation (blue bars) and min, max values (orange bars). Detailed values with respect to six elimination indicators are summarized in Table 2. For the sake of comparison, the discontinuous black lines represent the corresponding values of the criteria for the case of using all input variables (Scenario 0). On the basis of average values of R, it is observed that the performance of the ELM algorithm in excluding clay content (X

_{1}) slightly decreased from 0.9218 (simulation with six inputs) to 0.9203 (simulation with five inputs except for clay content). For the remaining cases (excluding from X

_{2}to X

_{6}), the performance of ELM decreased more significantly. Similar observations were noticed taking the average values of RMSE and MAE. Indeed, it was found that excluding clay content (X

_{1}) reduced the ELM prediction performance with RMSE decrease from 0.1082 to 0.0925, and MAE decreased from 0.0857 to 0.0722 while comparing the cases of all input variables and without clay content in the input space. Besides, taking the maximum values of R or minimum value of MAE as an indicator, plastic limit (X

_{6}) was the variable to be excluded. However, taking the minimum value of RMSE as an indicator, the specific gravity (X

_{3}) was the variable to be excluded. Finally, the elimination decision was made based on the majority vote between indicators, where clay content (X

_{1}) was selected to be a less important variable compared with other variables for predicting soil shear strength.

#### 3.2.2. Reduction of the Input Space from Five to Four Variables (Scenario 2)

_{2}to X

_{6}). Thus, a number of 5000 simulations (5 input spaces × 1000 simulations) were performed. The results of average and standard deviation values of R, RMSE and MAE are displayed in Figure 7. Detailed values with respect to six indicators are summarized in Table 3. The discontinuous black lines represent the error criteria values for the case without using clay content (X

_{1}) as input variable. On the basis of average values of R, it is observed that the performance of the ELM algorithm in excluding void ratio (X

_{4}) decreased from 0.9203 (simulation with five inputs except clay content) to 0.9188 (simulation with four inputs except clay content and void ratio). For the remaining cases (excluding X

_{2}, X

_{3}, X

_{5}, and X

_{6}), the performance of ELM exhibited lower values (Table 3). Similar remarks were observed for the average values of RMSE and MAE. Indeed, it was found that excluding the void ratio made inconsiderable changes with RMSE (increase from 0.0925 to 0.0957) and MAE (increase from 0.0722 to 0.0751) with respect to the cases of all input variables without clay content as a variable. Interestingly, taking the maximum values of R, or minimum values of RMSE and MAE as indicators, the void ratio was also the variable to be excluded. The elimination at this stage revealed that void ratio (X

_{4}) is a less important variable compared with other variables (X

_{2}, X

_{3}, X

_{5}and X

_{6}) in predicting the soil shear strength.

#### 3.2.3. Reduction of the Input Space from Four to Three Variables (Scenario 3)

_{2}, X

_{3}, X

_{5}, and X

_{6}). This induces a total number of 4000 simulations (4 input spaces × 1000 simulations) to be performed. The results of average and standard deviation values of R, RMSE, and MAE are plotted in Figure 8. Detailed simulation results with respect to six indicators are summarized in Table 4. The discontinuous black lines represent the error criteria values for the simulation without using clay content (X

_{1}) and void ratio (X

_{4}) as input variables. With respect to the average values of R, it is observed that the performance of ELM algorithm in excluding plastic limit (X

_{6}) slightly decreased from 0.9188 (simulation with four inputs without clay content and void ratio) to 0.9164 (simulation with three inputs except for clay content, void ratio, and plastic limit). On the contrary, different remarks were observed for the average values of RMSE and MAE. Precisely, it was found that excluding specific gravity made inconsiderable changes with RMSE (increase from 0.0931 to 0.0993) and MAE (increase from 0.0732 to 0.0778). More importantly, taking the maximum values of R, or minimum values of RMSE and MAE as indicators, specific gravity was the variable need to be eliminated. The backward elimination at this stage revealed that the specific gravity (X

_{3}) is less important input variable compared with other variables (X

_{2}, X

_{5}, and X

_{6}) in predicting the shear strength of soil.

#### 3.2.4. Final Input Space with Three Variables (Scenario 4)

_{2}, X

_{5}, and X

_{6}). At this stage, the total number of 3000 simulations (3 input spaces × 1000 simulations) was performed. The results of average and standard deviation values of R, RMSE, and MAE are plotted in Figure 9. Detailed simulation results with respect to six indicators are summarized in Table 5. The discontinuous black lines represent the error criteria values for the simulation without using clay content (X

_{1}), specific gravity (X

_{3}) and void ratio (X

_{4}) as input variables. With respect to the average values of R, it is observed that the performance of ELM algorithm in excluding plastic limit (X

_{6}) slightly decreased from 0.9138 (simulation with four inputs without clay content, specific gravity, and void ratio) to 0.8815 (simulation with two inputs moisture content and liquid limit). On the contrary, different remarks were observed for the average values of RMSE and MAE. Precisely, it was found that excluding the liquid limit made inconsiderable changes with RMSE (increase from 0.0993 to 0.1334) and MAE (increase from 0.0778 to 0.0778). On the other hand, taking the maximum values of R, or minimum values of RMSE and MAE as indicators, the plastic limit was the variable need to be eliminated. Overall, it could be considered that the plastic limit is less important than the liquid limit and moisture content in predicting the shear strength of soil, and thus, it can be concluded that moisture content is the most important factor for the prediction of the shear strength of soil.

## 4. Discussions

#### 4.1. Performance of ELM in Predicting the Shear Strength of Soil

#### 4.2. Reliability of the Predicted Results by Monte Carlo Approach

_{5}), the plastic limit (X

_{6}) or moisture content (X

_{2}) from the input space seemed to highly increase the fluctuations of the statistical analysis. This could be another confirmation to strengthen the conclusion of the backward selection feature in this study, as these three variables were considered as important. The fluctuation analysis of results was in very good agreement with the convergence analysis. Thus, performing backward elimination coupling with Monte Carlo simulation as a support decision indicator, an in-depth point of view on the importance of variables could be revealed.

#### 4.3. Backward Elimination Criteria-Based Sensitivity Analysis

^{2}) to perform further investigation is very common in the literature, for instance, in [17]. The maximum values of R or minimum values of RMSE and MAE only represented the best performance of the ELM algorithm over a certain number of Monte Carlo simulations. Even though the number of 1000 runs in each case was proven to be statistically satisfied, these values could change when increasing the number of simulations. Therefore, the use of maximum values of R or minimum values of RMSE and MAE as backward elimination indicators still needs further investigations.

#### 4.4. Importance of Input Factors for Prediction of Soil Shear Strength

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Location of the study site: Long Phu 1 power plant (https://www.power-technology.com/projects/long-phu-1-thermal-power-plant-soc-trang-province/).

**Figure 2.**Histograms of the parameters used in this study and correlation graphs with the output: (

**a,b**) Clay, (

**c,d**) moisture content, (

**e,f**) specific gravity, (

**g,h**) void ratio, (

**i,j**) liquid limit, (

**k,l**) plastic limit, and (

**m**) shear strength of soil.

**Figure 3.**The process of backward elimination supported by Monte Carlo simulations in this study. (Correlation coefficient (R); Root mean squared error (RMSE); Mean absolute error (MAE)).

**Figure 5.**Validation of ELM algorithm with various numbers of neurons using different validation criteria: (

**a**) R, (

**b**) RMSE, and (

**c**) MAE.

**Figure 6.**ELM performance with a reduction of the input space from six to five variables (Scenario 1) for the case of (

**a**) R; (

**b**) probability density function of R; (

**c**) RMSE; (

**d**) probability density function of RMSE; (

**e**) MAE; and (

**f**) probability density function of MAE.

**Figure 7.**ELM performance with reduction of the input space from five to four variables (Scenario 2) for the case of (

**a**) R; (

**b**) probability density function of R; (

**c**) RMSE; (

**d**) probability density function of RMSE; (

**e**) MAE; and (

**f**) probability density function of MAE.

**Figure 8.**ELM performance with reduction of the input space from four to three variables (Scenario 3) for the case of (

**a**) R; (

**b**) probability density function of R; (

**c**) RMSE; (

**d**) probability density function of RMSE; (

**e**) MAE; and (

**f**) probability density function of MAE.

**Figure 9.**ELM performance with a reduction of the input space with three variables (Scenario 4) for the case of (

**a**) R; (

**b**) probability density function of R; (

**c**) RMSE; (

**d**) probability density function of RMSE; (

**e**) MAE; and (

**f**) probability density function of MAE.

**Figure 10.**Statistical convergence analysis using backward elimination and 1000 Monte Carlo simulations for four scenarios in this study: (

**a**) R for Scenario 1; (

**b**) RMSE for Scenario 1; (

**c**) R for Scenario 2; (

**d**) RMSE for Scenario 2; (

**e**) R for Scenario 3; (

**f**) RMSE for Scenario 3; (

**g**) R for Scenario 4; (

**h**) RMSE for Scenario 4.

Parameter | Clay | Moisture Content | Specific Gravity | Void Ratio | Liquid Limit | Plastic Limit | Soil Shear Strength |
---|---|---|---|---|---|---|---|

Unit | mm | % | - | - | % | % | kG/cm^{2} |

Coding | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | Y |

Min (α) | 0.2000 | 0.7200 | 0.0100 | 0.0210 | 0.7000 | 0.6000 | 0.0368 |

Average | 33.2467 | 31.8336 | 2.6142 | 0.9142 | 42.3649 | 22.1678 | 0.4791 |

Median | 33.2000 | 26.5500 | 2.6900 | 0.7870 | 42.5000 | 21.4000 | 0.4964 |

Max (β) | 77.6000 | 75.1400 | 2.7500 | 2.0890 | 74.9000 | 41.0000 | 0.9307 |

SD* | 16.1388 | 15.2671 | 0.4271 | 0.3935 | 13.2635 | 6.1376 | 0.2036 |

Q_{25} | 20.7000 | 23.6100 | 2.6700 | 0.7090 | 33.5000 | 18.7000 | 0.3978 |

Q_{50} | 33.2000 | 26.5500 | 2.6900 | 0.7870 | 42.5000 | 21.4000 | 0.4964 |

Q_{75} | 47.4000 | 30.9700 | 2.7100 | 0.8850 | 50.4000 | 24.3000 | 0.6287 |

Excluded | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | Full | Decide |
---|---|---|---|---|---|---|---|---|

Mean (R) | 0.9203 | 0.9121 | 0.9136 | 0.9191 | 0.8858 | 0.9167 | 0.9218 | X_{1} |

Max (R) | 0.9581 | 0.9504 | 0.9502 | 0.9604 | 0.9302 | 0.9552 | - | X_{4} |

Mean (RMSE) | 0.0925 | 0.0968 | 0.0944 | 0.0941 | 0.1080 | 0.0961 | 0.1082 | X_{1} |

Min (RMSE) | 0.0675 | 0.0662 | 0.0639 | 0.0641 | 0.0797 | 0.0652 | - | X_{3} |

Mean (MAE) | 0.0722 | 0.0762 | 0.0740 | 0.0743 | 0.0867 | 0.0753 | 0.0857 | X_{1} |

Min (MAE) | 0.0506 | 0.0500 | 0.0489 | 0.0502 | 0.0601 | 0.0482 | - | X_{6} |

Excluded | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | Full | Decide |
---|---|---|---|---|---|---|---|

Mean (R) | 0.9089 | 0.9149 | 0.9188 | 0.8868 | 0.9113 | 0.9203 | X_{4} |

Max (R) | 0.9528 | 0.9503 | 0.9533 | 0.9333 | 0.9503 | - | X_{4} |

Mean (RMSE) | 0.0957 | 0.0969 | 0.0931 | 0.1083 | 0.1002 | 0.0925 | X_{4} |

Min (RMSE) | 0.0694 | 0.0683 | 0.0644 | 0.0772 | 0.0650 | - | X_{4} |

Mean (MAE) | 0.0751 | 0.0760 | 0.0732 | 0.0861 | 0.0793 | 0.0722 | X_{4} |

Min (MAE) | 0.0508 | 0.0507 | 0.0495 | 0.0593 | 0.0511 | - | X_{4} |

Excluded | X_{2} | X_{3} | X_{5} | X_{6} | Full | Decide |
---|---|---|---|---|---|---|

Mean (R) | 0.7985 | 0.9138 | 0.8897 | 0.9164 | 0.9188 | X_{6} |

Max (R) | 0.8864 | 0.9574 | 0.9360 | 0.9535 | - | X_{3} |

Mean (RMSE) | 0.1722 | 0.0993 | 0.1219 | 0.1031 | 0.0931 | X_{3} |

Min (RMSE) | 0.0990 | 0.0633 | 0.0792 | 0.0670 | - | X_{3} |

Mean (MAE) | 0.1429 | 0.0778 | 0.1002 | 0.0827 | 0.0732 | X_{3} |

Min (MAE) | 0.0758 | 0.0480 | 0.0591 | 0.0516 | - | X_{3} |

Excluded | X_{2} | X_{5} | X_{6} | Full | Decide |
---|---|---|---|---|---|

Mean(R) | 0.5499 | 0.8571 | 0.8815 | 0.9138 | X_{6} |

Max(R) | 0.7851 | 0.9322 | 0.9481 | - | X_{6} |

Mean(RMSE) | 0.2673 | 0.1334 | 0.1397 | 0.0993 | X_{5} |

Min(RMSE) | 0.1296 | 0.0782 | 0.0684 | - | X_{6} |

Mean(MAE) | 0.2291 | 0.1093 | 0.1162 | 0.0778 | X_{5} |

Min(MAE) | 0.0979 | 0.0627 | 0.0509 | - | X_{6} |

Order of Importance | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Scenario 1 | X_{1} | X_{4} | X_{3} | X_{6} | X_{2} | X_{5} |

Scenario 2 | X_{4} | X_{2} | X_{3} | X_{6} | X_{5} | - |

Scenario 3 | X_{3} | X_{6} | X_{5} | X_{2} | - | - |

Scenario 4 | X_{5} | X_{6} | X_{2} | - | - | - |

© 2020 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**

Pham, B.T.; Nguyen-Thoi, T.; Ly, H.-B.; Nguyen, M.D.; Al-Ansari, N.; Tran, V.-Q.; Le, T.-T.
Extreme Learning Machine Based Prediction of Soil Shear Strength: A Sensitivity Analysis Using Monte Carlo Simulations and Feature Backward Elimination. *Sustainability* **2020**, *12*, 2339.
https://doi.org/10.3390/su12062339

**AMA Style**

Pham BT, Nguyen-Thoi T, Ly H-B, Nguyen MD, Al-Ansari N, Tran V-Q, Le T-T.
Extreme Learning Machine Based Prediction of Soil Shear Strength: A Sensitivity Analysis Using Monte Carlo Simulations and Feature Backward Elimination. *Sustainability*. 2020; 12(6):2339.
https://doi.org/10.3390/su12062339

**Chicago/Turabian Style**

Pham, Binh Thai, Trung Nguyen-Thoi, Hai-Bang Ly, Manh Duc Nguyen, Nadhir Al-Ansari, Van-Quan Tran, and Tien-Thinh Le.
2020. "Extreme Learning Machine Based Prediction of Soil Shear Strength: A Sensitivity Analysis Using Monte Carlo Simulations and Feature Backward Elimination" *Sustainability* 12, no. 6: 2339.
https://doi.org/10.3390/su12062339