# Evaluation of Multiple Linear Regression and Machine Learning Approaches to Predict Soil Compaction and Shear Stress Based on Electrical Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O) emissions [28].

## 2. Materials and Methods

#### 2.1. Experimental Data Acquisition

^{−1}. EM38 measures electrical conductivity either vertically (EM38V), with a theoretical depth response curve of 0 to 1.5 m, or horizontally (EM38H), with a theoretical depth response curve of 0 to 1 m [53], and allows for a faster method of determining if conductivities increase or decrease with depth. Soil moisture equaled 30% measured over a short interval after the EM38 survey; temperature was equal to 15 °C.

^{2}and an angle of 60° was used during field measurements with a penetration speed of 0.03 m·s

^{−1}. After measuring electrical parameters and analyzing the results, a total of five management zones were mapped. In a 30 m grid, soil compaction measurements were conducted, and coordinates were assigned to each measured point based on the GPS position of the Eijkelkamp Penetrologger. The nearest point of soil electrical conductivity and magnetic susceptibility measurements were linked to each spot of soil compaction measurement using the least-squares method. Based on previous research [54], soil compaction was estimated at a depth of 0–0.5 m, and between 0.4 and 0.5 m was chosen for further analysis.

#### 2.2. Multiple Linear Regression

_{1}, X

_{2},…, X

_{k}) and a dependent variable (Y) with an explanation and prediction as objectives:

- The explanation objective examines the regression coefficients and their magnitude, sign, and statistical inference for each predictor variable;
- The forecast objective examines the extent to which the explanatory variables can estimate the explicative variable [55].

_{t}= X

_{t}β + ε

_{t}

_{t}is the estimated value at time t, β = (β

_{0}, β

_{1}, …, β

_{k}) is an indication of the relationship between the independent and dependent variables, and X

_{t}= (1, x

_{1}

_{t}, x

_{2}

_{t}, …, x

_{j}

_{t}) is a vector of j-dependent variables at time t. ε

_{t}is a random error term at time t, t = 1, …, N. The error terms should be independent and show a Gaussian distribution behavior. The presumptions of the MLR model could be examined by the Kolmogorov–Smirnov test and the Q* Ljung–Box statistic, as in the timeseries.

#### 2.3. Artificial Neural Networks

#### 2.4. Sensitivity Analysis

#### 2.5. Support Vector Machines

_{1},y

_{1}), (x

_{2},y

_{2}), (x

_{3}, y

_{3}), …, (x

_{n}, y

_{n}) where x∈𝕽

^{m}is the input data and y

_{i}∈𝕽 is the corresponding target. The objective of a support vector machine is to estimate the regression function, defined as:

^{m}is the vector of weights and b is the bias.

#### 2.6. Criteria of Accuracy Assessment of Models

_{pred}is the absolute predicted value, ${\overline{Y}}_{pred}$ is the mean predicted value, Y

_{meas}is the absolute measured value, ${\overline{Y}}_{meas}$ is the mean measured value, and n is the amount of data in a dataset.

- The model is perfect if GA = 1;
- The model is excellent if 0.75 ≤ GA < 1 or 1 < GA ≤ 1.35;
- The model is good if 1.35 < GA ≤ 2 or 0.5 ≤ GA < 0.75;
- The model is poor and unsuitable for prediction if GA > 2 or GA < 0.5.

_{T}is the amount of data in the training dataset and N

_{V}is the amount of data in the validation (testing) dataset. The lower value of OBJ indicates a more accurate model.

## 3. Results

#### 3.1. Multiple Linear Regression

#### 3.2. Artificial Neural Networks

#### 3.3. Sensitivity Analysis (SA)

#### 3.4. Support Vector Machines

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The location of the study area (•) on a map of Europe, Poland, and Lower Silesia province.

**Figure 2.**The relative importance of the input variables of the MLP model on soil compaction and shear stress.

The Parameter | Minimum | Maximum | Mean | Standard Deviation |
---|---|---|---|---|

Soil compaction (depth 0–0.5 m) (MPa) | 0.65 | 2.20 | 1.41 | 0.28 |

Soil compaction (depth 0.4–0.5 m) (MPa) | 0.17 | 3.39 | 1.14 | 0.58 |

Shear stress (kPa) | 96.00 | 248.00 | 163.40 | 32.88 |

Factor | RSC_0.4_0.5 Constant Term = 1.812 | RSC_0_0.5 Constant Term = 1.545 | RSS Constant Term = 124.587 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

b Coefficient | Standard Error b | p-Value | Significance | b Coefficient | Standard Error b | p-Value | Significance | b Coefficient | Standard Error b | p-Value | Significance | |

Apparent soil electrical conductivity 0.5 m (ECa0.5) | 0.390 | 0.094 | 0.017 | + | −0.040 | 0.008 | <0.001 | + | 5.530 | 1.065 | <0.001 | + |

Magnetic susceptibility 0.5 m (MS0.5) | −0.837 | 0.089 | 0.749 | − | −1.868 | 1.238 | 0.133 | − | 146.434 | 147.712 | 0.323 | - |

Apparent soil electrical conductivity 1 m (ECa1) | 0.390 | 0.132 | 0.038 | + | 0.009 | 0.005 | 0.083 | − | −1.192 | 0.644 | 0.066 | - |

Magnetic susceptibility 1 m (MS1) | −0.077 | 0.139 | 0.502 | − | −0.025 | 0.054 | 0.642 | − | −18.247 | 6.549 | 0.006 | + |

Model | RMSE | MAE | MAPE | NSC | R |
---|---|---|---|---|---|

RSC_0.4_0.5 | 0.535 | 0.401 | 28.468 | 0.152 | 0.408 |

RSC_0_0.5 | 0.335 | 0.261 | 18.187 | 0.072 | 0.469 |

RSS | 37.794 | 30.433 | 21.299 | 0.073 | 0.423 |

Model | Model Structure | Train | Validation | GA | OBJ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | MAPE | NSC | R | RMSE | MAE | MAPE | NSC | R | ||||

MLPSC_0_0.5 | 4-12-1 | 0.246 | 0.191 | 15.416 | 0.297 | 0.545 | 0.153 | 0.134 | 9.550 | 0.555 | 0.790 | 0.621 | 0.281 |

MLPSC_0.4_0.5 | 4-10-1 | 0.471 | 0.355 | 21.302 | 0.319 | 0.567 | 0.387 | 0.323 | 20.246 | 0.546 | 0.772 | 0.821 | 0.562 |

MLP_SS | 4-19-1 | 29.363 | 23.289 | 14.699 | 0.236 | 0.486 | 24.210 | 20.120 | 12.912 | 0.408 | 0.680 | 0.824 | 38.280 |

Model | Model Structure | Train | Validation | GA | OBJ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | MAPE | NSC | R | RMSE | MAE | MAPE | NSC | R | ||||

RBFSC_0_0.5 | 4-17-1 | 0.264 | 0.210 | 16.779 | 0.187 | 0.432 | 0.160 | 0.138 | 9.398 | 0.603 | 0.812 | 0.606 | 0.322 |

RBFSC_0.4_0.5 | 4-16-1 | 0.526 | 0.387 | 31.574 | 0.149 | 0.386 | 0.405 | 0.316 | 17.085 | 0.511 | 0.846 | 0.769 | 0.663 |

RBF_SS | 4-25-1 | 29.637 | 22.765 | 14.511 | 0.221 | 0.470 | 26.484 | 22.313 | 15.314 | 0.311 | 0.648 | 0.893 | 39.917 |

Model | Train | Validation | GA | OBJ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | MAPE | NSC | R | RMSE | MAE | MAPE | NSC | R | |||

SVMSC_0_0.5 | 0.251 | 0.198 | 15.605 | 0.208 | 0.457 | 0.216 | 0.074 | 6.415 | 0.242 | 0.709 | 0.860 | 0.281 |

SVMSC_0.4_0.5 | 0.539 | 0.393 | 29.897 | 0.187 | 0.437 | 0.437 | 0.207 | 14.061 | 0.086 | 0.555 | 0.810 | 0.636 |

SVM_SS | 29.690 | 23.624 | 15.466 | 0.228 | 0.478 | 31.125 | 15.345 | 9.044 | 0.016 | 0.243 | 1.048 | 43.642 |

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**MDPI and ACS Style**

Pentoś, K.; Mbah, J.T.; Pieczarka, K.; Niedbała, G.; Wojciechowski, T.
Evaluation of Multiple Linear Regression and Machine Learning Approaches to Predict Soil Compaction and Shear Stress Based on Electrical Parameters. *Appl. Sci.* **2022**, *12*, 8791.
https://doi.org/10.3390/app12178791

**AMA Style**

Pentoś K, Mbah JT, Pieczarka K, Niedbała G, Wojciechowski T.
Evaluation of Multiple Linear Regression and Machine Learning Approaches to Predict Soil Compaction and Shear Stress Based on Electrical Parameters. *Applied Sciences*. 2022; 12(17):8791.
https://doi.org/10.3390/app12178791

**Chicago/Turabian Style**

Pentoś, Katarzyna, Jasper Tembeck Mbah, Krzysztof Pieczarka, Gniewko Niedbała, and Tomasz Wojciechowski.
2022. "Evaluation of Multiple Linear Regression and Machine Learning Approaches to Predict Soil Compaction and Shear Stress Based on Electrical Parameters" *Applied Sciences* 12, no. 17: 8791.
https://doi.org/10.3390/app12178791