Probabilistic Design of Retaining Wall Using Machine Learning Methods
Abstract
:1. Introduction
2. Methods
2.1. Emotional Neural Network
2.2. Symbiotic Organisms Search-Least Square Support Vector Machine
2.2.1. LSSVM
2.2.2. Symbiotic Organisms Search (SOS)
- Generation of the initial population
- Do
- Mutualism
- Commensalism
- Parasitism
- Best solution update
- Until the criteria of stopping are satisfied.
- Data are collected for training the model.
- The LSSVM model is used to analyze the ambiguous nature of input and output. In addition, and are tuned.
- SOS algorithm:This algorithm searches for several combinations of and parameters and makes the best set of these two parameters. In addition, SOS employs mutualism, commensalism, and parasitism phases to improve the fitness value of the solutions reached slowly.
- Evaluation of fitness:For evaluation of the system, a fitness function is developed that measures the accuracy of the learning system. The best combination of and represents the accurate and best fitness value. The dataset is not split randomly, and it is are divided into learning and validation subsets. In addition, to avoid the bias of sampling, 10-fold cross-validation is done. The mean square error (MSE) is utilized by the fitness function for aptness and better representation.
- Criteria of termination:The termination criterion used in this technique is the iteration number inculcated in the SOS algorithm.
- Optimal and parameters:Loop stops and optimal and parameters are reached.
- The optimal set of and parameters are further used for developing the model for testing the data.
- Data testing:The dataset split for testing is tested, and the prediction is used for assessing the performance and accuracy of the model.
2.3. Multivariate Adaptive Regression Splines
2.4. Cross-Validation
3. Case Example
4. Result and Performance Assessment of Models
4.1. Errors and Other Parameters
4.2. Taylor Diagram
4.3. AOC–REC Curve
4.4. R Curve
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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MODELS | MARS | EmNN | SOS–LSSVM |
---|---|---|---|
1. WMAPE | 0.0062 | 0.0726 | 0.0008 |
RANK | 2.0000 | 1.0000 | 3.0000 |
2. NS | 0.9999 | 0.9860 | 1.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
3. RMSE | 0.0017 | 0.0183 | 0.0002 |
RANK | 2.0000 | 1.0000 | 3.0000 |
4. VAF | 99.9914 | 98.7306 | 99.9998 |
RANK | 2.0000 | 1.0000 | 3.0000 |
5. R2 | 0.9999 | 0.9860 | 1.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
6. Adj. R2 | 0.9997 | 0.9674 | 1.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
7. PI | 1.9980 | 1.9364 | 1.9998 |
RANK | 2.0000 | 1.0000 | 3.0000 |
8. RMSD | 0.0017 | 0.0183 | 0.0002 |
RANK | 2.0000 | 1.0000 | 3.0000 |
9. BIAS FACTOR | 0.8729 | 0.8371 | 0.8755 |
RANK | 2.0000 | 1.0000 | 3.0000 |
10. RSR | 0.0106 | 0.1182 | 0.0015 |
RANK | 2.0000 | 1.0000 | 3.0000 |
11. NMBE | −0.3611 | 2.7843 | 0.0196 |
RANK | 2.0000 | 1.0000 | 3.0000 |
12. MAPE | 0.0010 | 0.0062 | 0.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
13. WI | 1.0000 | 0.9966 | 1.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
14. MAE | 0.0013 | 0.0155 | 0.0002 |
RANK | 2.0000 | 1.0000 | 3.0000 |
15. MBE | −0.0008 | 0.0059 | 0.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
16. LMI | 0.9859 | 0.8364 | 0.9984 |
RANK | 2.0000 | 1.0000 | 3.0000 |
17. U95 | 0.0043 | 0.0500 | 0.0005 |
RANK | 2.0000 | 1.0000 | 3.0000 |
18. t-stat | 0.0028 | 0.0142 | 0.0001 |
RANK | 2.0000 | 1.0000 | 3.0000 |
19. GPI | −1.8 × 10−15 | 1.08 × 10−09 | 5.98292 × 10−22 |
RANK | 1.0000 | 2.0000 | 3.0000 |
20. R | 1.0000 | 0.9978 | 1.0000 |
RANK | 2.0000 | 1.0000 | 3.0000 |
21. SI | 0.7733 | 8.5962 | 0.0899 |
RANK | 2.0000 | 1.0000 | 3.0000 |
22. a20-index | 1.0000 | 0.8750 | 1.0000 |
RANK | 2.0000 | 1.0000 | 2.0000 |
23. AOC | 0.0011 | 0.0133 | 0.0001 |
RANK | 2.0000 | 1.0000 | 3.0000 |
TOTAL RANK | 45.0000 | 24.0000 | 68.0000 |
MODEL | Reference β | Model’s β | Model’s Pf |
---|---|---|---|
MARS | 1.6450 | 1.6418 | 0.0500 |
EmNN | 1.6405 | 1.6211 | 0.0525 |
SOS–LSSVM | 1.8661 | 1.8661 | 0.0310 |
BF | Equation |
---|---|
BF1 | max(0, x4 − 0.695789507393035) |
BF2 | max(0, 0.695789507393035 − x4) |
BF3 | max(0, x2 − 0.897423953169214) |
BF4 | max(0, 0.897423953169214 − x2) |
BF5 | max(0, x3 − 0.902027009611503) |
BF6 | max(0, 0.902027009611503 − x3) |
BF7 | BF4 × max(0, x1 − 0.382105313562625) |
BF8 | BF4 × max(0, 0.382105313562625 − x1) |
BF9 | BF2 × max(0, 0.557894812924718 − x1) |
BF10 | BF6 × max(0, x4 − 0.81157903285569) |
BF11 | BF6 × max(0, 0.81157903285569 − x4) |
BF12 | BF4 × max(0, x3 − 0.408783600920691) |
BF13 | BF4 × max(0, 0.408783600920691 − x3) |
BF14 | BF1 × max(0, x1 − 0.760000104402251) |
Main equation | y = 0.597051492431488 + 1.2033558916995 × BF1 − 1.01013033595132 × BF2 + 0.262683442461409 × BF3 − 0.290024870554842 × BF4 − 0.0824750745638103 × BF5 + 0.124577962754319 × BF6 − 0.334859635113939 × BF7 + 0.254991811556967 × BF8 + 0.446434437906744 × BF9 + 0.27641860123112 × BF10 − 0.0700841355527202 × BF11 + 0.0920347477963486 × BF12 − 0.0681746540919311 × BF13 + 0.819393035778277 × BF14 |
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Mishra, P.; Samui, P.; Mahmoudi, E. Probabilistic Design of Retaining Wall Using Machine Learning Methods. Appl. Sci. 2021, 11, 5411. https://doi.org/10.3390/app11125411
Mishra P, Samui P, Mahmoudi E. Probabilistic Design of Retaining Wall Using Machine Learning Methods. Applied Sciences. 2021; 11(12):5411. https://doi.org/10.3390/app11125411
Chicago/Turabian StyleMishra, Pratishtha, Pijush Samui, and Elham Mahmoudi. 2021. "Probabilistic Design of Retaining Wall Using Machine Learning Methods" Applied Sciences 11, no. 12: 5411. https://doi.org/10.3390/app11125411
APA StyleMishra, P., Samui, P., & Mahmoudi, E. (2021). Probabilistic Design of Retaining Wall Using Machine Learning Methods. Applied Sciences, 11(12), 5411. https://doi.org/10.3390/app11125411