Abstract
The present research employs the models based on the relevance vector machine (RVM) approach to predict the unconfined compressive strength (UCS) of the cohesive virgin (fine-grained) soil. For this purpose, the Linear, Polynomial, Gaussian, and Laplacian kernel functions have been implemented in RVM models. Two types of RVM models have been developed: (i) single kernel function based (mentioned by SRVM) and (ii) dual kernel function-based (mentioned by DRVM). Each model has been optimized by each genetic (GA) and particle swarm optimization (PSO) algorithm. Eighty-five data points (75 training + ten testing) have been collected from the literature to train and test the SRVM and DRVM models. The data proportionality method has been used to create six training databases, i.e., 50%, 60%, 70%, 80%, 90%, and 100%, to determine the effect of the quality and quantity of training database on the performance, accuracy, and overfitting of the soft computing models. Ten conventional and three new performance parameters, i.e., a20 index, index of agreement (IOA), and index of scatter (IOS), have measured the performance of models. The present research concludes that (i) a strongly correlated pair of data points affect the performance and accuracy of the model; (ii) GA-optimized SRVM model MD119 has outperformed other SRVM and DRVM models with a20 = 100, IOA = 0.9947, and IOS = 0.0272; (iii) k-fold cross-validation test (k = 10) validates the capabilities of SRVM and DRVM models; (iv) model MD119 has predicted UCS better than GPR model MD11 developed in part 1 of this research; (v) high correlated data points increases the overfitting of the model; (vi) model MD119 has predicted UCS of lab tested soil with a confidence interval of ± 4.0%.
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Data availability
All data, models, and code generated or used during the study appear in the submitted article. The database may be provided on request.
Abbreviations
- a20:
-
A20-index
- a20A :
-
A20-index accuracy
- AD:
-
Anderson–Darling test
- AI:
-
Artificial intelligence
- ANN:
-
Artificial neural networks
- ANOVA:
-
Analysis of variance
- AOC:
-
Area over the curve
- BWOA:
-
Black widow optimization algorithm
- Coeff.:
-
Coefficient
- CSAFR:
-
Ratio of free lime to SAF
- CV:
-
Cross-validation model
- df:
-
Degree of freedom
- DMR:
-
Maximum dry density to the optimum moisture content
- DRVM:
-
Dual kernel function-based RVM
- DRVM-GA:
-
GA-optimized DRVM
- DRVM-PSO:
-
PSO-optimized DRVM
- DS:
-
Degree of saturation
- DST:
-
Direct shear test
- DUW:
-
Dry unit weight
- F :
-
F State value
- F crit:
-
F Critical value
- FC:
-
Fine content
- GA:
-
Genetic algorithm
- GAU:
-
Gaussian kernel
- Gauss-GA:
-
GA-optimized Gaussian
- Gauss-PSO:
-
PSO-optimized Gaussian
- GPR:
-
Gaussian process regression
- G-SRVM:
-
Gaussian kernel function-based SRVM
- ICA:
-
Imperialism competitive algorithm
- IOA:
-
Index of agreement
- IOAA :
-
Index of agreement accuracy
- IOS:
-
Index of scatter
- IOSA :
-
Index of scatter accuracy
- J:
-
Number of models
- LAP:
-
Laplacian kernel
- Lap-GA:
-
GA-optimized Laplacian SRVM
- Lap-PSO:
-
PSO-optimized Laplacian SRVM
- Lap-SRVM:
-
Laplacian kernel function-based SRVM
- lb:
-
Lower boundary
- LB:
-
Lower bound
- LIN:
-
Linear kernel
- Lin-GA:
-
GA-optimized linear SRVM
- Lin-PSO:
-
PSO-optimized linear SRVM
- LMI:
-
Legate McCabe's Index
- L-SRVM:
-
Linear kernel function-based SRVM
- MAE:
-
Mean absolute error
- MAEA :
-
Mean absolute error accuracy
- MAPE:
-
Mean absolute percentage error
- MAPEA :
-
Mean absolute percentage error accuracy
- MD:
-
Model
- ME:
-
Margin of error
- MOE:
-
Mean of error
- MRI:
-
Magnetic resonance imaging
- MS:
-
Mean square
- MVO:
-
Multi-verse optimization algorithm
- NMBE:
-
Normalized mean bias error
- NS:
-
Nash–Sutcliffe efficiency
- o3:
-
Confining pressure
- od:
-
Deviatoric stress
- P:
-
Porosity
- POLY:
-
Polynomial kernel
- Poly-GA:
-
GA-optimized polynomial SRVM
- Poly-PSO:
-
PSO-optimized polynomial SRVM
- PSO:
-
Particle swarm optimization algorithm
- PSO-XGBoost:
-
PSO-extreme gradient boosting
- P-SRVM:
-
Polynomial kernel function-based SRVM
- P-value:
-
Calculated significant value
- r, R :
-
Correlation coefficient
- R 2 :
-
Coefficient of determination
- R A :
-
Correlation coefficient accuracy
- REC:
-
Regression error characteristics curve
- RMSE:
-
Root mean square error
- RMSEA :
-
Root mean square error accuracy
- ROC:
-
Receiver operating characteristic curve
- RSR:
-
Root mean square error to observations' standard deviation ratio
- RSRA :
-
Root mean square error to observations' standard deviation ratio accuracy
- RVM:
-
Relevance vector machine
- SCA:
-
Sine cosine algorithm
- SE:
-
Standard error
- SG:
-
Specific gravity
- SRVM:
-
Single kernel function-based RVM
- SS:
-
Sum of squares
- SSO:
-
Social spider optimization algorithm
- Std Error:
-
Standard error
- StDev:
-
Standard deviation
- t state:
-
T Statistical
- TCS:
-
Triaxial compressive strength
- UA95 :
-
Uncertainly analysis
- uB:
-
Upper boundary
- UB:
-
Upper bound
- UCS:
-
Unconfined compressive strength
- VAF:
-
Variance accounted for
- VAFA :
-
Variance accounted for accuracy
- VIF:
-
Variance inflation factor
- VR:
-
Void ratio
- VST:
-
Vane shear test
- WBC:
-
Width of confidence bound
- WDC:
-
Number of W-D cycles
- WMAPE:
-
Weighted mean absolute percentage error
- WMAPEA :
-
Weighted mean absolute percentage error accuracy
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JK: main author, conceptualization, literature review, manuscript preparation, application of AI models, methodological development, statistical analysis, detailing, and overall analysis; KSG: conceptualization, overall analysis, manuscript finalization, detailed review, and editing.
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Khatti, J., Grover, K.S. Prediction of UCS of fine-grained soil based on machine learning part 2: comparison between hybrid relevance vector machine and Gaussian process regression. Multiscale and Multidiscip. Model. Exp. and Des. 7, 123–163 (2024). https://doi.org/10.1007/s41939-023-00191-8
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DOI: https://doi.org/10.1007/s41939-023-00191-8