## 3 Results and Discussion
### Machine Learning to Predict \(q_{u}\) and \(E_{t}\)
#### 3.4.3 Symbolic Regression Equations  
The EUREQA software is used in this study to uncover latent patterns and relationships in the dataset and generate equations for predicting the unconfined compressive strength (\(q_{u}\)) and elasticity modulus (\(E_{t}\)) [98].  
Proposed Equations of Elasticity Modulus (\(E_{t}\))  
Equations (6)-(8) are formulated via the utilization of the EUREQA software. It is crucial to acknowledge that as the metric is enhanced, the resulting equations tend to become more intricate, necessitating additional time to achieve optimal performance.  
\[E_{t}=\frac{\gamma_{d}*LL*SR^{2}}{\mathbf{e}^{\frac{\gamma_{0}*SR}{74}}}\qquad \qquad\qquad\left(\mathrm{R}^{2}=0.90816;\;\mathrm{RMSE}=615.6;\;\mathrm{MAE}=49 5.03\right) \tag{6}\]  
\[E_{t}=\gamma_{d}*\sqrt{\frac{\mathbf{e}^{\phi_{t}+\gamma_{d}+\frac{LL}{74}}}{ \mathbf{e}^{\frac{\phi_{t}}{74}}}}\qquad\qquad\qquad\left(\mathrm{R}^{2}=0.93394 ;\;\mathrm{RMSE}=522.13;\;\mathrm{MAE}=412.17\right) \tag{7}\]  
\[E_{t}=\gamma_{d}*\frac{LL*SR^{2}+FS*\gamma_{d}^{2}}{SR^{\frac{1}{4}+\frac{5R* \left(\mathrm{R}*\left(\mathrm{R}*\right)\right)}{74}}e^{\frac{\phi_{t}}{74} +\frac  
Figure 19: The cross plots of actual and predicted targets for **(A)** Equation (6), **(B)** Equation (7), and **(C)** Equation (8).  
Figure 20 shows improved alignment of true and predicted data points with the reference line y = x and best-fit trend line (ideal line) for Equations (9)-(11) as metrics (R2, RMSE, and MAE) improve. Equation 11 has the highest metrics, while Equation (9) has the lowest. It should be noted that most data points are located between the 20% error margin lines. This demonstrates that \(q_{u}\) equations are capable of accurately predicting the \(q_{u}\).  
Figure 20: The cross plots of actual and predicted targets for (**A**) Equation (9), (**B**) Equation (10), and (**C**) Equation (11).  
It is crucial to highlight that as metrics are improved, the generated equations become more complex, necessitating more time to achieve optimal performance. EUREQA can generate many equations. Equations (6)-(11) are chosen for their practicality and ability to produce favorable metrics.  
To ensure the validity of the Equations (6)-(11), the following conditions must be met: sand content below 50%, The Afterberg limits ought to be situated slightly above and in proximity to line A on the Casagrande chart, initial degree of saturation exceeding 50%, and a liquid limit within the range of 40-80%.  
When the metrics of ANN and SR are compared (Figure 21), as well as the visual examination of true and predicted targets in Figure 22, artificial neural networks outperform symbolic regression. This observation highlights the superior performance of artificial neural networks in comprehending complex patterns within data, outperforming symbolic regression.  
Prior research using single or multiple regression techniques focused on only one or two inputs, including (\(w_{n}\), \(LL\), and/or \(\gamma_{d}\)) [1; 2]. The highest achieved R-squared values were 0.89 in work by [PERSON] et al. (2015) and 0.606 in work by [PERSON] and [PERSON], (2018), confirming symbolic regression equations' superior performance over single and multiple regression models. Upon applying the equations from [PERSON] and [PERSON] (2018) and [PERSON] et al. (2015)'s works to our dataset, a large number of data points were predicted to have negative values. This happened as a result of the models in [PERSON] and [PERSON] (2018) and [PERSON] et al. (2015)'s works depending only on one or two inputs. Furthermore, their datasets were not sufficiently explored in terms of expansive soil properties. [PERSON] and [PERSON] (2018) concentrated on MH classified soils and specifically avoided expansive soil behavior, whereas [PERSON] et al. (2015) only addressed a small portion of this soil classification. The power of the equations in our work comes from the ability to accurately predict \(q_{u}\) and \(E_{t}\) in expansive soils that are partially or fully saturated.