# Novel Approach to Predicting Soil Permeability Coefficient Using Gaussian Process Regression

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## Abstract

**:**

## 1. Introduction

^{3}) as inputs; (2) to divide data into training and testing datasets with due attention to statistical aspects such as the minimum, maximum, mean and standard deviation of the datasets. The splitting of the datasets is performed to find out the predictive ability and generalization performance of developed models and later helps in better evaluating them; (3) to compare the proposed models to the reference models used in the published literature; and (4) to investigate the importance and impact of each input parameter on the soil permeability coefficient.

## 2. Methodology

#### 2.1. Data Catalog

^{3}) [1,28]. It has been widely accepted, among researchers, that the input factors selected by Pham et al. [1,28] constitute a complete and suitable set to estimate “k”. As a result, these input variables were used to create the GPR model in the current study. The same input parameters related to permeability were used to estimate the “k” (×10

^{−9}cm/s) of soil. Researchers have used a different percentage of the available data as the training and testing sets for different problems. For instance, Pham et al. [29] used 60%; Liang et al. [30] used 70%; while Ahmad et al. [31] used 80% of the data for training. In this study, the data set was divided into training (70%) and testing (30%) based on statistically consistency. The statistical consistency of training and testing datasets was based on statistically consistency. The statistical consistency of training and testing datasets has a substantial impact on the results when using soft computing techniques, which improves the performance of the model and helps in evaluating them better. Figure 2 depicts the cumulative percentage and frequency distributions for all of the input and output parameters of the mentioned database utilized in the modeling of soil permeability coefficient. The data points of every input parameter are distributed over its range. The statistical analysis, i.e., minimum (Min), maximum (Max), mean, and standard deviation (Std. Dev) of the training and testing datasets is presented in Table 1.

#### 2.2. Gaussian Process Regression

_{i}, N

_{i}) are distributed independently and identically. For regression, let N⊆ ℜ; then, a GPR on 𝜒 is defined by a mean function μ: 𝜒→ℜ and a covariance function k:𝜒 × 𝜒→ℜ. Kuss [33] is recommended for more information on GPR and other covariance functions.

#### Details of Kernel Functions

- Polynomial (Poly)

- 2.
- Radial basis function (RBF)

- 3.
- Pearson universal kernel (PUK)

#### 2.3. Performance Metrics and Evaluation

^{2}), correlation coefficient (R), mean absolute error (MAE), and root mean square error (RMSE) were utilized. The following formula can be used to compute these parameters:

^{2}and R are used to express the degree of collinearity between estimated and actual data. The correlation coefficient, which ranges from 1 to −1, indicates how closely actual and estimated data are related. If R is equal to 0, there is no linear relationship. If R = 1 or −1, there is a perfect positive or negative linear relationship. R

^{2}indicates how much percentage of variance in estimated data the model can explain. R

^{2}is a number that ranges from 0 to 1, with higher values indicating less error variation and values over 0.5 considered acceptable [37,38]. The MAE indicates the mean of the estimated and actual values. The adjustment has a better effect when the MAE is close to 0, meaning that the prediction model more accurately describes the set of training data [39]. The RMSE is the average magnitudes of the errors in predictions for all observations in a single measure of predictive power. The RMSE is larger than or equal to 0, with 0 signifying that the observed data is statistically perfectly fit. As a result, the lesser the values of MAE and RMSE criteria are, the better the model. Visual representations such as scatter plots were also employed to compare the performance of the established models. The flowchart of the methodology of the present study is shown in Figure 3.

## 3. Results and Discussion

^{2}0.980, 0.929, and 0.912; MAE 0.0023, 0.0028, and 0.0031; and RMSE 0.0038, 0.0047, and 0.0048 for GPR-PUK, GPR-Poly, and GPR-RBF models, respectively, the GPR-PUK outputs were verified to be the most compatible with actual coefficient of soil permeability values. Following that, GPR-Poly confirmed a high level of accuracy. Similarly, the GPR-PUK has the highest value of R (0.9754) and R

^{2}(0.951), then comes the GPR-Poly (R = 0.9624; R

^{2}= 0.926) and the GPR-RBF (R = 0.9387; R

^{2}= 0.881) in the test dataset. The GPR-Poly, on the other hand, has the lowest values of MAE (0.0034), followed by the GPR-PUK (0.0037) and the GPR-RBF (0.0223), and the GPR-RBF has the lowest value of RMSE (0.0047), followed by the GPR-PUK (0.0062) and the GPR-Poly (0.0634).

^{2}= 0.980) and testing phase (R

^{2}= 0.951), respectively. As a result, the GPR-PUK model proposed in this study can be utilized to calculate the soil permeability coefficient, as the predicted value agrees well with the actual value, indicating that this approach can accurately and effectively estimate the coefficient of soil permeability.

## 4. Comparison of Performance with Other Methods

^{2}of 0.766, RMSE of 0.0064 and MAE of 0.004, in the case of the training dataset. Whereas, in the testing dataset, there is good agreement between actual and estimated values in the testing dataset, the M5P models’ error values are RMSE = 0.0081 and MAE = 0.0045 and the determination coefficient is high (R

^{2}= 0.766) in the testing dataset. In general, the proposed GPR-PUK (R

^{2}= 0.9754) has better prediction ability and has the highest goodness of fit with the data used in the training and testing datasets when compared to other models in this study.

## 5. Sensitivity Analysis

## 6. Conclusions

^{3}). The available data is divided into two parts: training set (70%) and testing set (30%). The following is a summary of the findings of this study:

- Comparing GPR models’ performance reveals that the GPR-PUK model gives more accurate prediction results with the coefficient of determination being 0.951, achieved from the correlation between experimental and estimated values of k.
- The GPR-PUK model’s estimation of the soil permeability coefficient was found to be more reliable than that of the ANN, SVM, RF, and M5P models reported in the literature.
- The findings of the sensitivity analysis demonstrate that different input factors have varying degrees of significance on the coefficient of soil permeability as w > e > LL > PL > CC > γ.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

ANN | Artificial neural network |

RF | Random forest |

SVM | Support vector machine |

GPR | Gaussian process regression |

MAE | Mean absolute error |

M5P | M5Prime algorithm |

RMSE | Root mean square error |

PUK | Pearson universal kernel |

RBF | Radial basis function |

XGBoost | Extreme gradient boosting |

R^{2} | Coefficient of determination |

R | Correlation coefficient |

k | Soil permeability coefficient (×10^{−9} cm/s) |

LL | Liquid limit (%) |

PL | Plastic limit (%) |

CC | Clay content (%) |

e | Void ratio |

w | Natural water content (%) |

γ | Specific density (g/cm^{3}) |

## Appendix A

S. No. | CC (%) | w (%) | LL (%) | PL (%) | γ (g/cm^{3}) | e | k (×10^{−9} cm/s) |
---|---|---|---|---|---|---|---|

1 | 44 | 93.73 | 75.62 | 46.8 | 2.59 | 2.453 | 0.029 |

2 | 21.7 | 20.71 | 24.58 | 13.5 | 2.72 | 0.639 | 0.01 |

3 | 51.8 | 20.98 | 38.17 | 20.2 | 2.73 | 0.625 | 0.003 |

4 | 9.7 | 18.02 | 20.51 | 14.2 | 2.68 | 0.605 | 0.007 |

5 | 46.9 | 95.58 | 82.25 | 53 | 2.6 | 2.514 | 0.026 |

6 | 12.7 | 22.71 | 28.5 | 17.8 | 2.69 | 0.671 | 0.01 |

7 | 47.5 | 85.35 | 71.24 | 40.5 | 2.62 | 2.275 | 0.014 |

8 | 59.4 | 24.95 | 41.87 | 22.3 | 2.74 | 0.713 | 0.003 |

9 | 9.2 | 23.97 | 26.52 | 19.8 | 2.67 | 0.723 | 0.008 |

10 | 55.3 | 98.01 | 73.63 | 40.1 | 2.59 | 2.597 | 0.035 |

11 | 44.8 | 79.96 | 75.45 | 43.6 | 2.59 | 2.083 | 0.039 |

12 | 51.1 | 73.75 | 66.96 | 35.8 | 2.61 | 1.966 | 0.061 |

13 | 46.1 | 25.78 | 38.03 | 17.5 | 2.73 | 0.808 | 0.003 |

14 | 56.1 | 83.25 | 78.23 | 41.9 | 2.62 | 2.235 | 0.055 |

15 | 16.1 | 17.52 | 25.85 | 12.2 | 2.69 | 0.546 | 0.01 |

16 | 49 | 25.45 | 48.24 | 24.8 | 2.72 | 0.711 | 0.003 |

17 | 10.7 | 24.53 | 27.22 | 19.6 | 2.69 | 0.713 | 0.007 |

18 | 64 | 78.72 | 75.53 | 39.5 | 2.64 | 2.106 | 0.03 |

19 | 5.7 | 17.35 | 20.34 | 14.25 | 2.66 | 0.494 | 0.006 |

20 | 41.9 | 69.26 | 66.42 | 48.5 | 2.64 | 1.87 | 0.029 |

21 | 9.5 | 18.12 | 21.2 | 14.5 | 2.68 | 0.567 | 0.008 |

22 | 7.6 | 20.23 | 23.62 | 16.8 | 2.69 | 0.64 | 0.007 |

23 | 11 | 20.14 | 22.78 | 16.1 | 2.67 | 0.608 | 0.008 |

24 | 45 | 35.53 | 53.56 | 28.6 | 2.74 | 1.015 | 0.004 |

25 | 8.5 | 20.81 | 25.31 | 18.53 | 2.68 | 0.576 | 0.005 |

26 | 8.6 | 20.12 | 20.82 | 14.8 | 2.67 | 0.599 | 0.007 |

27 | 10.7 | 17.25 | 19.5 | 13.5 | 2.68 | 0.558 | 0.008 |

28 | 8.9 | 21.79 | 24.98 | 19 | 2.68 | 0.654 | 0.007 |

29 | 46.4 | 99.9 | 82.11 | 43.6 | 2.58 | 2.634 | 0.041 |

30 | 9.7 | 17.34 | 20.49 | 14.3 | 2.66 | 0.486 | 0.007 |

31 | 25.9 | 21.23 | 31.18 | 13.2 | 2.72 | 0.609 | 0.005 |

32 | 12.5 | 19.25 | 23.46 | 14.67 | 2.67 | 0.628 | 0.008 |

33 | 8.4 | 19.46 | 22.97 | 17.43 | 2.68 | 0.605 | 0.007 |

34 | 8.1 | 23.28 | 26.8 | 20.36 | 2.68 | 0.707 | 0.011 |

35 | 23.6 | 18.84 | 27.48 | 13.8 | 2.71 | 0.604 | 0.006 |

36 | 63.4 | 73.1 | 68.47 | 35 | 2.61 | 1.933 | 0.028 |

37 | 19 | 18.35 | 23.61 | 13.35 | 2.7 | 0.579 | 0.007 |

38 | 42.5 | 27.28 | 39.99 | 21.74 | 2.72 | 0.789 | 0.003 |

39 | 49.4 | 62.2 | 59.99 | 38.5 | 2.63 | 1.657 | 0.026 |

40 | 23.5 | 21.32 | 32.23 | 16.4 | 2.71 | 0.604 | 0.005 |

41 | 6.1 | 16.97 | 21.01 | 15.87 | 2.66 | 0.556 | 0.007 |

42 | 7.7 | 21.23 | 25.3 | 18.5 | 2.68 | 0.654 | 0.009 |

43 | 9.7 | 18.01 | 20.3 | 14.2 | 2.67 | 0.599 | 0.007 |

44 | 8.5 | 25.49 | 27.49 | 21.32 | 2.67 | 0.723 | 0.008 |

45 | 60.2 | 95.09 | 84.05 | 54.8 | 2.63 | 2.507 | 0.038 |

46 | 40.3 | 20.75 | 40.77 | 18.64 | 2.72 | 0.591 | 0.003 |

47 | 8.4 | 18.25 | 21.08 | 14.5 | 2.69 | 0.592 | 0.008 |

48 | 50.7 | 28.97 | 46.04 | 25.2 | 2.72 | 0.889 | 0.003 |

49 | 8.8 | 17.19 | 19.81 | 14.3 | 2.68 | 0.549 | 0.007 |

50 | 46.6 | 76.77 | 64.83 | 38.17 | 2.63 | 2.023 | 0.025 |

51 | 9.6 | 17.99 | 20.42 | 15 | 2.67 | 0.571 | 0.008 |

52 | 8.6 | 19.9 | 23 | 16.9 | 2.68 | 0.586 | 0.009 |

53 | 9.2 | 17.81 | 21 | 14.3 | 2.68 | 0.506 | 0.01 |

54 | 11.7 | 19.77 | 23.91 | 13.5 | 2.68 | 0.567 | 0.035 |

55 | 9.4 | 17.85 | 20.48 | 14.8 | 2.68 | 0.558 | 0.008 |

56 | 45.1 | 93.19 | 88.93 | 48 | 2.62 | 2.447 | 0.057 |

57 | 46.1 | 70.21 | 65.46 | 33.6 | 2.64 | 1.87 | 0.071 |

58 | 37.4 | 21.13 | 32.44 | 14.2 | 2.71 | 0.642 | 0.003 |

59 | 45.3 | 19.6 | 30.92 | 13.2 | 2.73 | 0.569 | 0.007 |

60 | 19 | 24.55 | 29.08 | 19.6 | 2.68 | 0.707 | 0.017 |

61 | 37.6 | 87.71 | 75.34 | 40.5 | 2.63 | 2.329 | 0.048 |

62 | 8 | 18.05 | 20.99 | 14.3 | 2.68 | 0.595 | 0.01 |

63 | 8.5 | 19.85 | 23.67 | 17.58 | 2.67 | 0.599 | 0.008 |

64 | 9.6 | 18.18 | 22.58 | 16 | 2.68 | 0.567 | 0.006 |

65 | 8.6 | 18.02 | 20.51 | 14.6 | 2.69 | 0.592 | 0.012 |

66 | 8.3 | 18.01 | 21 | 14.2 | 2.67 | 0.599 | 0.007 |

67 | 10.2 | 18.15 | 22.14 | 15.6 | 2.67 | 0.517 | 0.006 |

68 | 8.6 | 24.84 | 29.32 | 22 | 2.68 | 0.752 | 0.012 |

69 | 45.8 | 89.51 | 85.86 | 42.7 | 2.63 | 2.372 | 0.051 |

70 | 38.6 | 22.79 | 35.83 | 15.2 | 2.72 | 0.689 | 0.009 |

71 | 8.2 | 17.12 | 19.7 | 13.8 | 2.67 | 0.571 | 0.01 |

72 | 26.5 | 21.89 | 30.98 | 17.4 | 2.72 | 0.619 | 0.005 |

73 | 24.5 | 18.28 | 28.11 | 12.5 | 2.71 | 0.522 | 0.006 |

74 | 21 | 20.62 | 28.62 | 17.4 | 2.69 | 0.592 | 0.014 |

75 | 9.3 | 21.14 | 23.89 | 18.53 | 2.68 | 0.686 | 0.008 |

76 | 8.4 | 18.02 | 21.1 | 14.5 | 2.67 | 0.552 | 0.009 |

77 | 9.8 | 18.07 | 20.62 | 14.5 | 2.68 | 0.567 | 0.01 |

78 | 30.4 | 22.23 | 39.53 | 18.64 | 2.72 | 0.648 | 0.004 |

79 | 9.8 | 22.03 | 23.92 | 17.8 | 2.68 | 0.644 | 0.008 |

80 | 6.7 | 18.91 | 21.49 | 15 | 2.69 | 0.582 | 0.007 |

81 | 43.4 | 25.6 | 34.5 | 15.6 | 2.73 | 0.717 | 0.005 |

82 | 40.1 | 25.53 | 36.11 | 19.2 | 2.72 | 0.755 | 0.01 |

83 | 8.7 | 15.09 | 18.9 | 12.63 | 2.66 | 0.462 | 0.008 |

84 | 9.4 | 19.64 | 23.8 | 17.2 | 2.67 | 0.648 | 0.009 |

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**Figure 4.**Comparison of the predicted and actual results of various kernel-function-based GPR models in the training dataset: (

**a**) GPR-PUK, (

**b**) GPR-Poly, and (

**c**) GPR-RBF.

**Figure 5.**Comparison of the predicted and actual results of various kernel-function-based GPR models in the testing dataset: (

**a**) GPR-PUK, (

**b**) GPR-Poly, and (

**c**) GPR-RBF.

Dataset | Parameters | Clay Content, CC (%) | Water Content, w (%) | Liquid Limit, LL | Plastic Limit, PL | Specific Density, γ (g/cm^{3}) | Void Ratio, e | Permeability Coefficient, k (×10^{−9} cm/s) |
---|---|---|---|---|---|---|---|---|

Training | Min | 5.7 | 16.97 | 19.5 | 12.2 | 2.58 | 0.486 | 0.003 |

Mean | 28.056 | 37.82 | 40.219 | 23.882 | 2.6715 | 1.0576 | 0.016 | |

Max | 64 | 99.9 | 88.93 | 54.8 | 2.74 | 2.634 | 0.071 | |

Std. Dev | 19.761 | 28.62 | 22.228 | 12.347 | 0.0413 | 0.7234 | 0.016 | |

Testing | Min | 6.7 | 15.09 | 18.9 | 12.5 | 2.63 | 0.462 | 0.004 |

Mean | 18.36 | 25.75 | 30.304 | 18.279 | 2.6836 | 0.7553 | 0.012 | |

Max | 45.8 | 89.51 | 85.86 | 42.7 | 2.73 | 2.372 | 0.051 | |

Std. Dev | 13.337 | 19.13 | 16.272 | 7.3879 | 0.0256 | 0.4856 | 0.012 |

Model | Optimal Tuning Parameters |
---|---|

PUK kernel | {noise = 0.6, ω = 0.1, σ = 0.1} |

Poly kernel | {noise = 0.02} |

RBF kernel | $\{\mathrm{noise}=0.04,\lambda $ = 0.6} |

Model | Training | Testing | Reference | ||||||
---|---|---|---|---|---|---|---|---|---|

R | R^{2} | MAE | RMSE | R | R^{2} | MAE | RMSE | ||

RF | 0.972 | - | 0.0023 | 0.0035 | 0.851 | - | 0.0049 | 0.0084 | [1] |

ANN | 0.948 | - | 0.0027 | 0.0047 | 0.845 | - | 0.005 | 0.001 | |

SVM | 0.861 | - | 0.0056 | 0.0078 | 0.844 | - | 0.0064 | 0.0098 | |

M5P | - | 0.792 | 0.004 | 0.0064 | - | 0.766 | 0.0045 | 0.0081 | [28] |

GPR (PUK) | 0.9901 | 0.980 | 0.0023 | 0.0038 | 0.9754 | 0.951 | 0.0037 | 0.0062 | Present study |

GPR (Poly kernel) | 0.964 | 0.929 | 0.0028 | 0.0047 | 0.9624 | 0.926 | 0.0223 | 0.0634 | |

GPR (RBF) | 0.9548 | 0.912 | 0.0031 | 0.0048 | 0.9387 | 0.881 | 0.0034 | 0.0047 | |

CatBoost | 0.960 | 0.922 | 0.0031 | 0.0052 | 0.958 | 0.9178 | 0.0013 | 0.0031 |

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## Share and Cite

**MDPI and ACS Style**

Ahmad, M.; Keawsawasvong, S.; Bin Ibrahim, M.R.; Waseem, M.; Kashyzadeh, K.R.; Sabri, M.M.S.
Novel Approach to Predicting Soil Permeability Coefficient Using Gaussian Process Regression. *Sustainability* **2022**, *14*, 8781.
https://doi.org/10.3390/su14148781

**AMA Style**

Ahmad M, Keawsawasvong S, Bin Ibrahim MR, Waseem M, Kashyzadeh KR, Sabri MMS.
Novel Approach to Predicting Soil Permeability Coefficient Using Gaussian Process Regression. *Sustainability*. 2022; 14(14):8781.
https://doi.org/10.3390/su14148781

**Chicago/Turabian Style**

Ahmad, Mahmood, Suraparb Keawsawasvong, Mohd Rasdan Bin Ibrahim, Muhammad Waseem, Kazem Reza Kashyzadeh, and Mohanad Muayad Sabri Sabri.
2022. "Novel Approach to Predicting Soil Permeability Coefficient Using Gaussian Process Regression" *Sustainability* 14, no. 14: 8781.
https://doi.org/10.3390/su14148781