Hybrid ELM and MARS-Based Prediction Model for Bearing Capacity of Shallow Foundation
Abstract
:1. Introduction
2. Details of AI-Based Models Used
2.1. MARS
- (a)
- Constructive phase.
- (b)
- Pruning phase.
- (c)
- Selection of optimum MARS.
2.2. ELM
- It avoids a number of issues that are difficult to deal with in traditional methods, such as halting criteria, learning rate, learning epochs, and local minimums.
- In most circumstances, it can provide better generalized performance than backpropagation (BP) since ELM is a one-pass learning technique that does not require re-iteration.
- It may be used to activate practically any nonlinear function.
2.3. PSO
2.4. EO
2.5. Regression Optimization
3. Details of Dataset
4. Research Methodology
5. Results and Discussion
5.1. Configuration of the Models
5.2. Performance Parameters
5.3. Rank Analysis
5.4. Error Matrix
5.5. Sensitivity Analysis
5.6. REC Curves
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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B (m) | D (m) | L/B (-) | γ (KN/m2) | φ (°) | Q (KPa) | |
---|---|---|---|---|---|---|
Mean | 0.11 | 0.08 | 3.92 | 16.45 | 38.95 | 192.84 |
Minimum | 0.06 | 0.03 | 1.00 | 15.70 | 34.00 | 58.50 |
Maximum | 0.15 | 0.15 | 6.00 | 17.10 | 42.50 | 423.60 |
Standard Error | 0.01 | 0.01 | 0.35 | 0.07 | 0.44 | 13.18 |
Standard Deviation | 0.04 | 0.04 | 2.47 | 0.50 | 3.11 | 94.13 |
Sample Variance | 0.00 | 0.00 | 6.09 | 0.25 | 9.67 | 8860.48 |
Kurtosis | −1.55 | −0.82 | −1.94 | −1.28 | −1.21 | −0.38 |
Skewness | −0.03 | 0.61 | −0.37 | −0.22 | −0.45 | 0.65 |
Range | 0.09 | 0.12 | 5.00 | 1.40 | 8.50 | 365.10 |
Parameters | MARS-L |
---|---|
GCV penalty per knot | 0 |
Cubic modelling | 0 (No) |
Self-interactions | 1 (No) |
Maximum interactions | 2 |
Prune | 1 (true) |
No. of in the final model | 15 |
SL.NO | Basic Function | Equation |
---|---|---|
1 | BF1 | max(0, φ − 0.352) |
2 | BF2 | max(0, 0.352 − φ) |
3 | BF3 | BF1 × max(0, D − 0.380) |
4 | BF4 | Bf1 × max(0, 0.380 − D) |
5 | BF5 | max(0, B − 0.379) |
6 | BF6 | max(0, 0.379 − B) |
7 | BF7 | BF5 × max(0, γ − 0.57) |
8 | BF8 | max(0, D − 0.53) |
9 | BF9 | max(0, 0.53 − D) |
Parameters | |||||
---|---|---|---|---|---|
ELM | 25 | ||||
ELM-PSO | 50 | 25 | 100 | −1 | +1 |
ELM-EO | 50 | 25 | 100 | −1 | +1 |
Parameters | Ideal Value | Parameters | Ideal Value |
---|---|---|---|
VAF | 100 | RMSE | 0 |
R2 | 1 | WMAPE | 0 |
PI | 2 | MAE | 0 |
WI | 1 | MBE | 0 |
Adj. R2 | 1 | NMBE | 0 |
NS | 1 | LMI | 0 |
RSR | 0 | Bias | 1 |
Model Statistical Parameters | ELM | ELM-EO | ELM-PSO | MARS | ELM | ELM-EO | ELM-PSO | MARS |
---|---|---|---|---|---|---|---|---|
Testing Performance | Training Performance | |||||||
WMAPE | 0.0797 | 0.0306 | 0.0441 | 0.0498 | 0.0543 | 0.0030 | 0.0127 | 0.0396 |
RMSE | 0.0558 | 0.0170 | 0.0186 | 0.0199 | 0.0248 | 0.0014 | 0.0060 | 0.0180 |
VAF | 93.921 | 99.3963 | 99.3155 | 99.3155 | 99.1566 | 99.9973 | 99.951 | 99.5517 |
R2 | 0.9425 | 0.9945 | 0.9932 | 0.9954 | 0.9915 | 0.9999 | 0.9995 | 0.9955 |
Adj. R2 | 0.9413 | 0.9872 | 0.9840 | 0.9946 | 0.9910 | 0.9999 | 0.9993 | 0.9952 |
NS | 0.9386 | 0.9938 | 0.9926 | 0.9916 | 0.9915 | 0.9999 | 0.9995 | 0.9955 |
PI | 1.8247 | 1.9641 | 1.9586 | 1.9673 | 1.9578 | 1.9985 | 1.9930 | 1.9727 |
RSR | 0.2477 | 0.0785 | 0.0858 | 0.0916 | 0.0919 | 0.0052 | 0.02200 | 0.0669 |
Bias | 1.0237 | 1.1431 | 0.9178 | 1.0876 | 0.9723 | 0.9731 | 0.9799 | 0.9383 |
NMBE | −1.0848 | 0.6977 | 1.205 | 1.8913 | 0.1616 | 0.0087 | 0.04750 | 0.1163 |
WI | 0.9830 | 0.9984 | 0.9982 | 0.9978 | 0.9979 | 0.9999 | 0.9998 | 0.9988 |
MAE | 0.0398 | 0.01088 | 0.0157 | 0.0177 | 0.0202 | 0.0012 | 0.0048 | 0.0150 |
MBE | −0.0054 | 0.00248 | −0.0049 | 0.0067 | 0.0006 | 3.24 × 10−5 | 0.00018 | 0.00044 |
LMI | 0.7823 | 0.93410 | 0.9052 | 0.8929 | 0.9114 | 0.9950 | 0.9791 | 0.9343 |
Model Statistical Parameters | ELM | ELM-EO | ELM-PSO | MARS | ELM | ELM-EO | ELM-PSO | MARS | |
---|---|---|---|---|---|---|---|---|---|
Testing Performance | Training Performance | ||||||||
WMAPE | Value | 0.0797 | 0.0306 | 0.0441 | 0.0498 | 0.0543 | 0.0030 | 0.0127 | 0.0396 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
RMSE | Value | 0.0558 | 0.0170 | 0.0186 | 0.0199 | 0.0248 | 0.0014 | 0.0060 | 0.0180 |
Score | 1 | 4 | 3 | 3 | 1 | 4 | 3 | 3 | |
VAF | Value | 93.921 | 99.3963 | 99.3155 | 99.3155 | 99.1566 | 99.9973 | 99.951 | 99.5517 |
Score | 1 | 4 | 3 | 3 | 1 | 4 | 3 | 3 | |
R2 | Value | 0.9425 | 0.9945 | 0.9932 | 0.9954 | 0.9915 | 0.9999 | 0.9995 | 0.9955 |
Score | 1 | 3 | 2 | 4 | 1 | 4 | 3 | 2 | |
Adj. R2 | Value | 0.9413 | 0.9872 | 0.9840 | 0.9946 | 0.9910 | 0.9999 | 0.9993 | 0.9952 |
Score | 1 | 3 | 2 | 4 | 1 | 4 | 3 | 2 | |
NS | Value | 0.9386 | 0.9938 | 0.9926 | 0.9916 | 0.9915 | 0.9999 | 0.9995 | 0.9955 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
PI | Value | 1.8247 | 1.9641 | 1.9586 | 1.9673 | 1.9578 | 1.9985 | 1.9930 | 1.9727 |
Score | 1 | 3 | 2 | 4 | 1 | 4 | 3 | 2 | |
RSR | Value | 0.2477 | 0.0785 | 0.0858 | 0.0916 | 0.0919 | 0.0052 | 0.02200 | 0.0669 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
Bias | Value | 1.0237 | 1.1431 | 0.9178 | 1.0876 | 0.9723 | 0.9731 | 0.9799 | 0.9383 |
Score | 2 | 3 | 1 | 3 | 2 | 3 | 4 | 1 | |
NMBE | Value | −1.0848 | 0.6977 | 12.0509 | 1.8913 | 0.1616 | 0.0087 | 0.04750 | 0.1163 |
Score | 1 | 2 | 4 | 3 | 4 | 1 | 2 | 3 | |
WI | Value | 0.9830 | 0.9984 | 0.9982 | 0.9978 | 0.9979 | 0.9999 | 0.9998 | 0.9988 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
MAE | Value | 0.0398 | 0.01088 | 0.0157 | 0.0177 | 0.0202 | 0.0012 | 0.0048 | 0.0150 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
MBE | Value | −0.0054 | 0.00248 | −0.0049 | 0.0067 | 0.0006 | 3.24 × 10−5 | 0.00018 | 0.00044 |
Score | 4 | 2 | 3 | 1 | 1 | 4 | 3 | 2 | |
LMI | Value | 0.7823 | 0.93410 | 0.9052 | 0.8929 | 0.9114 | 0.9950 | 0.9791 | 0.9343 |
Score | 1 | 4 | 3 | 2 | 1 | 4 | 3 | 2 | |
Total | 18 | 49 | 38 | 37 | 18 | 52 | 42 | 30 |
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Kumar, M.; Kumar, V.; Biswas, R.; Samui, P.; Kaloop, M.R.; Alzara, M.; Yosri, A.M. Hybrid ELM and MARS-Based Prediction Model for Bearing Capacity of Shallow Foundation. Processes 2022, 10, 1013. https://doi.org/10.3390/pr10051013
Kumar M, Kumar V, Biswas R, Samui P, Kaloop MR, Alzara M, Yosri AM. Hybrid ELM and MARS-Based Prediction Model for Bearing Capacity of Shallow Foundation. Processes. 2022; 10(5):1013. https://doi.org/10.3390/pr10051013
Chicago/Turabian StyleKumar, Manish, Vinay Kumar, Rahul Biswas, Pijush Samui, Mosbeh R. Kaloop, Majed Alzara, and Ahmed M. Yosri. 2022. "Hybrid ELM and MARS-Based Prediction Model for Bearing Capacity of Shallow Foundation" Processes 10, no. 5: 1013. https://doi.org/10.3390/pr10051013