# Influence of Sampling Methods on the Accuracy of Machine Learning Predictions Used for Strain-Dependent Slope Stability

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## Abstract

**:**

## 1. Introduction

^{2}= 0.947).

## 2. Enhanced Strain-Dependent Slope Stability Using Machine Learning

#### 2.1. Strain-Dependent Slope Stability (SDSS)

#### 2.2. Implementation of SDSS

#### 2.3. Application of Machine Learning in SDSS

**scikit-learn**package. Note that the settings of the MLP algorithms used within the framework of SDSS are reported in Section 3.3.

## 3. Sampling

#### 3.1. Motivation

#### 3.2. Modified Sampling Approaches

#### 3.3. Influence of Sampling Approach on Accuracy of MLP Prediction

## 4. Influence of Sampling on the FoS for Slopes Subjected to Different Loading

#### 4.1. Methodology

`numgeo`([51,52,53,54] and www.numgeo.de, accessed on 21 December 2023). The simulations were performed in the following steps: first, in the geostatic step, gravitational load (soil weight) was considered and an additional artificial load of 0.1 kPa acting perpendicular to the top surface was applied for the purpose of numerical stability. Secondly, loads q acting at the crest of the slope and on the embedded foundation were increased incrementally up to 200 kPa (example 1) and 600 kPa (example 2). Lastly, the factor of safety of the slopes was quantified based on the SDSS method.

`numgeo`at different loads, stability analyses were conducted using SDSS, quantifying the factor of safety in terms of ${\mathrm{FoS}}_{max}$. SDSS was conducted with shear stresses determined in accordance with direct simple shear (DSS) tests performed at each node. The element test simulations for the DSS tests were carried out using Incremental Driver (ID). Although SDSS could be implemented for different shapes of slip surfaces (e.g., planar, circular, or polyline), only circular slip surfaces with 20 nodes are considered in the current study. As the location of the critical slip surface is unknown a priori, many slip surfaces need to be investigated. In accordance with the discussions by Schmüdderich et al. [33], differential evolution [40] was used as an optimization algorithm to determine the critical slip surface.

#### 4.2. Results and Discussion

_{max}) versus applied vertical loads based on SDSS analyses obtained with ID and MLP using different sampling approaches for both slope examples. As mentioned in Section 2.1, the SDSS analysis can yield two FoS depending on whether the global mobilized shear resistance ratio T reaches a peak value and then decreases. All the results presented and discussed in this section refer to the peak FoS, also denoted as FoS

_{max}. Figure 11-left presents the results of example 1 (Figure 10a), where a uniform load is acting on the crest of the slope. Analyzing the results obtained with ID (continuous line), that the initial level of safety ($\mathrm{FoS}=2.44$) obtained with SDSS for the case of zero surcharge is in good agreement with the rough estimate of $tan{\phi}_{c}/tan\beta =2.43$, with $\beta ={15}^{\circ}$ being the slope inclination. Moreover, FoS decreases with increasing surcharge applied on the slope crest until the stability approaches $\mathrm{FoS}=1.0$ for a surcharge of 200 kPa. Note that convergence of the FE simulations could not be obtained for loads exceeding 200 kPa, which is in good agreement with the FoS falling below the value of 1.0. Based on the evaluation of slope stability using MLP with different sampling methods, it can be observed that there is a strong agreement between MLP and ID for LHS-1 (red points) and LHS-2 (purple points). However, the other four traditional approaches (grid sampling with linear and quadratic spacing, Monte Carlo sampling, and Latin hypercube sampling) exhibit a wider range of variation in the factor of safety (FoS). Overall, grid sampling with linear spacing led to the largest FoS for all loads investigated, while the other traditional sampling approaches showed oscillating deviations. It is also worth noting that the variation of FoS with the increasing vertical load follows an almost linear pattern for the grid sampling with linear spacing, whereas for the LHS-1 and LHS-2 methods, the trend is nonlinear with a quicker reduction of the FoS for external loads between 0 and 120 kPa. At certain loads, accurate predictions of the FoS were achieved, though, for other loads, large deviations were seen. This is seen, for instance, for grid sampling with quadratic spacing at $q=40$ kPa and $q=60$ kPa, resulting in a large overestimation for the former case and an accurate estimate for the latter case.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

DSS | Direct simple shear |

FoS | Factor of safety |

ID | Incremental Driver |

IGS | Intergranular strain |

KNN | K-nearest neighbors |

LHS | Latin hypercube sampling |

ML | Machine learning |

MLP | Multilayer perceptron |

RBF | Radial basis functions |

RF | Random forest |

SDSS | Strain-dependent slope stability |

SVM | Suppor vector machines |

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**Figure 1.**Evaluation of the global mobilized shear resistance ratio T with increasing shear strain for a material (

**left**) without and (

**center**) with strain-hardening and (

**right**) corresponding evolution of FoS.

**Figure 2.**Determination of FoS based on SDSS with element test simulations conducted using Incremental Driver (ID).

**Figure 3.**Flowchart showing replacement of Incremental Driver (ID) simulations by machine learning (ML) algorithms with application to strain-dependent slope stability analysis.

**Figure 4.**Influence of constraints on the distribution of samples using Latin hypercube sampling: (

**top row**) initial distribution of 60,000 samples, (

**bottom row**) distribution of 10,000 valid samples accounting for constraints.

**Figure 5.**Distribution of 60,000 valid samples generated using (

**1st row**) grid sampling with linear spacing, (

**2nd row**) grid sampling with quadratic spacing, (

**3rd row**) Monte Carlo sampling, and (

**4th row**) Latin hypercube sampling.

**Figure 7.**Sample distribution in the input parameter spaces using the modified sampling approaches: (

**top row**) modified Latin hypercube sampling 1, (

**bottom row**) modified Latin hypercube sampling 2.

**Figure 8.**Histogram of the input parameters using (

**1st row**) grid sampling with linear spacing, (

**2nd row**) grid sampling with quadratic spacing, (

**3rd row**) Monte Carlo sampling, (

**4th row**) Latin hypercube sampling, (

**5th row**) modified Latin hypercube sampling 1, and (

**6th row**) modified Latin hypercube sampling 2.

**Figure 9.**Comparison of shear stress vs. shear strain curves (DSS test) at different mean stresses obtained from Incremental Driver and multi-layer perceptron (MLP) using different sampling approaches.

**Figure 10.**Schematic drawing of the slopes under consideration: (

**a**) uniform distributed load and (

**b**) embedded foundation.

**Figure 11.**Comparison of ${\mathrm{FoS}}_{max}$ based on SDSS using ID and MLP trained with different sampling approaches for (

**left**) example 1 and (

**right**) example 2.

**Figure 12.**Influence of surcharge on location of critical slip surface obtained using LHS-2 for (

**top row**) example 1 and (

**middle**and

**bottom row**) example 2 as well as comparison of LHS and LHS-2 for example 2, $q=300$ kPa.

**Figure 13.**Influence of the sampling method on the type of critical slip surface as function of surcharge for the example 2.

**Figure 14.**Relative deviations for nodes along the critical slip surface for (

**a**) example 1 without surcharge ($q=0$) and (

**b**) example 2 with $q=500$ kPa.

$\mathit{p}\le 20\phantom{\rule{0.166667em}{0ex}}\mathbf{kPa}$ | $20\phantom{\rule{0.166667em}{0ex}}\mathbf{kPa}\le \mathit{p}\le 100\phantom{\rule{0.166667em}{0ex}}\mathbf{kPa}$ | $\mathit{p}\ge 100\phantom{\rule{0.166667em}{0ex}}\mathbf{kPa}$ | Overall | |
---|---|---|---|---|

Grid sampling with linear spacing | 0.000 | 0.313 | 0.858 | 0.773 |

Grid sampling with quadratic spacing | 0.000 | 0.750 | 0.934 | 0.887 |

Monte Carlo sampling | 0.011 | 0.642 | 0.938 | 0.879 |

Latin hypercube sampling | 0.003 | 0.603 | 0.878 | 0.823 |

Modified Latin hypercube sampling-1 | 0.328 | 0.856 | 0.936 | 0.913 |

Modified Latin hypercube sampling-2 | 0.769 | 0.823 | 0.917 | 0.907 |

${\mathit{\phi}}_{\mathit{c}}$ in ° | h_{s} in MPa | n | ${\mathit{e}}_{\mathit{d}0}$ | ${\mathit{e}}_{\mathit{c}0}$ | ${\mathit{e}}_{\mathit{i}0}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|---|

33.1 | 4000 | 0.27 | 0.677 | 1.054 | 1.212 | 0.14 | 2.5 |

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**MDPI and ACS Style**

Shakya, S.; Schmüdderich, C.; Machaček, J.; Prada-Sarmiento, L.F.; Wichtmann, T.
Influence of Sampling Methods on the Accuracy of Machine Learning Predictions Used for Strain-Dependent Slope Stability. *Geosciences* **2024**, *14*, 44.
https://doi.org/10.3390/geosciences14020044

**AMA Style**

Shakya S, Schmüdderich C, Machaček J, Prada-Sarmiento LF, Wichtmann T.
Influence of Sampling Methods on the Accuracy of Machine Learning Predictions Used for Strain-Dependent Slope Stability. *Geosciences*. 2024; 14(2):44.
https://doi.org/10.3390/geosciences14020044

**Chicago/Turabian Style**

Shakya, Sudan, Christoph Schmüdderich, Jan Machaček, Luis Felipe Prada-Sarmiento, and Torsten Wichtmann.
2024. "Influence of Sampling Methods on the Accuracy of Machine Learning Predictions Used for Strain-Dependent Slope Stability" *Geosciences* 14, no. 2: 44.
https://doi.org/10.3390/geosciences14020044