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Figure 7. Predicted versus measured Cc for the validation subset.

Figure 7 Predicted versus measured Cc for the validation subset.

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Figure 1. Sample gene expression programming (GEP) model.
Figure 1. Sample gene expression programming (GEP) model.
Figure 4. Distribution of éo.
Figure 4. Distribution of éo.
Table 1. Descriptive statistics for input and output parameters used in the GEP-based developed mode LL: liquid limit; PL: plastic limit.
Table 1. Descriptive statistics for input and output parameters used in the GEP-based developed mode LL: liquid limit; PL: plastic limit.
3.2. Model Structure and Performance The LL and PL represent the two various states of the soil depending on its water content. The €o of soil represents the initial ratio of the volume of voids to the solids. Prediction equations for C, developed by previous studies (see Equation (1)) have clearly indicated that LL, PL, and eg are the three main parameters that influence C; [3-9]. Thus, these parameters were considered in the current study to develop a simplified prediction equation for C,. The main motivation of developing such an equation was that determination of LL, PL, and ep is straightforward compared to performing any consolidation test that directly determines C,. Therefore, the developed model is anticipated to result in considerable savings in terms of testing time, technician costs, and laboratory equipment. It should be noted that LL, PL, and eo are influenced by the natural water content of partially saturated soils, thus making the developed equation applicable to any saturated fine-grained soils [28,39,40]. Mathematically, the developed equation has the following structure:
3.2. Model Structure and Performance The LL and PL represent the two various states of the soil depending on its water content. The €o of soil represents the initial ratio of the volume of voids to the solids. Prediction equations for C, developed by previous studies (see Equation (1)) have clearly indicated that LL, PL, and eg are the three main parameters that influence C; [3-9]. Thus, these parameters were considered in the current study to develop a simplified prediction equation for C,. The main motivation of developing such an equation was that determination of LL, PL, and ep is straightforward compared to performing any consolidation test that directly determines C,. Therefore, the developed model is anticipated to result in considerable savings in terms of testing time, technician costs, and laboratory equipment. It should be noted that LL, PL, and eo are influenced by the natural water content of partially saturated soils, thus making the developed equation applicable to any saturated fine-grained soils [28,39,40]. Mathematically, the developed equation has the following structure:
In these equations, h; and ¢; are measured and predicted output (C,) values, respectively, for the ith data point. Furthermore, h; and t; are averages of the measured and predicted values, respectively, and n is the number of samples [28,29].
In these equations, h; and ¢; are measured and predicted output (C,) values, respectively, for the ith data point. Furthermore, h; and t; are averages of the measured and predicted values, respectively, and n is the number of samples [28,29].
Table 2. Parameters used for implementation of the GEP-based model. In every GEP-based model, the values of the setting parameters have a significant impact on model performance. These parameters include the quantity of genes and chromosomes, in addition to a gene’s head size and the rate of genetic operators. Since minor information was available about GEP parameters in the literature, appropriate settings were selected based on a trial and error scheme (see Table 2). In every GEP-based model, the values of the setting parameters have a significant impact on
Table 2. Parameters used for implementation of the GEP-based model. In every GEP-based model, the values of the setting parameters have a significant impact on model performance. These parameters include the quantity of genes and chromosomes, in addition to a gene’s head size and the rate of genetic operators. Since minor information was available about GEP parameters in the literature, appropriate settings were selected based on a trial and error scheme (see Table 2). In every GEP-based model, the values of the setting parameters have a significant impact on
Figure 6. Predicted versus measured C; for the training subset.
Figure 6. Predicted versus measured C; for the training subset.
Table 3. Model performance. RMSE: root mean squared error; MAE: mean average erro! 3.4. Additional Evaluation of Model Performance
Table 3. Model performance. RMSE: root mean squared error; MAE: mean average erro! 3.4. Additional Evaluation of Model Performance
Table 4. Evaluating the developed GEP-based model using additional statistical parameters.
Table 4. Evaluating the developed GEP-based model using additional statistical parameters.
Figure 8. Predicted versus measured C; for the entire dataset (training + validation).
Figure 8. Predicted versus measured C; for the entire dataset (training + validation).
Table 5. Performance comparison of the current developed GEP-based model to existing models. MEP: multi-expression programming; ANN: artificial neural network. Based on Table 5, the developed GEP-based model outperformed the regression models, since the regression models considered only a small quantity of base functions. Therefore, such models could not be used for the complex interactions of soil parameters (i.e., LL, PL, and eg) and C;. However, the developed GEP-based model considered a variety of base functions and their combination in order to achieve a closed-form equation with high performance. The developed GEP-based model directly considered the experimental data with no prior assumptions. In other words, contrary to traditional regression models, GEP did not assume any predefined shape for the solution equation. The high values of R? presented in Table 5 indicate that the developed GEP-based model was very successful at fitting the measured C, to the input parameters of LL, PL, and eo.
Table 5. Performance comparison of the current developed GEP-based model to existing models. MEP: multi-expression programming; ANN: artificial neural network. Based on Table 5, the developed GEP-based model outperformed the regression models, since the regression models considered only a small quantity of base functions. Therefore, such models could not be used for the complex interactions of soil parameters (i.e., LL, PL, and eg) and C;. However, the developed GEP-based model considered a variety of base functions and their combination in order to achieve a closed-form equation with high performance. The developed GEP-based model directly considered the experimental data with no prior assumptions. In other words, contrary to traditional regression models, GEP did not assume any predefined shape for the solution equation. The high values of R? presented in Table 5 indicate that the developed GEP-based model was very successful at fitting the measured C, to the input parameters of LL, PL, and eo.
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EngineeringCivil EngineeringMaterials ScienceDecision MakingPrediction-based Decisions & PlanningMachine LearningSoil MechanicsModeling and SimulationForecasting and Prediction Tools
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