# Prediction of Swelling Index Using Advanced Machine Learning Techniques for Cohesive Soils
## 2 Materials and Methods
### Statistical Performance Indicators  
The prediction accuracy of the proposed models was evaluated through various statistical performance indicators and using graphical presentation. The statistical performance indicators are Mean absolute error (\(MAE\)), Root mean square error (\(RMSE\)), Index of scattering (\(IOS\)), Nash-Sutcliffe efficiency (\(NSE\)), Pearson correlation coefficient (\(R\)), and Index of agreement (\(IOA\)). They are expressed as follows [50]:  
1. Mean absolute error (\(MAE\)): \[MEA=\frac{1}{N}\sum_{i=1}^{N}\lvert Y_{tar,i}-Y_{out,i}\rvert\ (0<MAE<\infty)\] (12)
2. Root mean square error (\(RMSE\)): \[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}\ (0<RMSE<\infty)\] (13)
3. Index of scattering (\(IOS\)): \[IOS=\frac{\sqrt{\frac{1}{N}\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}}{\overline {Y_{tar}}}\ (0<RMSE<\infty)\] (14)
4. Nash-Sutcliffe efficiency (\(NSE\)): \[NSE=1-\frac{\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}{\sum_{i=1}^{N}\bigl{(}Y_{ tar,i}-\overline{Y_{tar}}\bigr{)}^{2}}\ (-\infty<NSE<1)\] (15)
5. Pearson correlation coefficient (\(R\)): \[R=\frac{\sum_{i=1}^{N}\bigl{(}\bigl{(}Y_{tar,i}-\overline{Y_{tar}}\bigr{)} \bigl{(}Y_{out,i}-\overline{Y_{out}}\bigr{)}\bigr{)}}{\sqrt{\sum_{i=1}^{N} \bigl{(}\bigl{(}Y_{tar,i}-\overline{Y_{tar}}\bigr{)}^{2}\bigl{(}Y_{out,i}- \overline{Y_{out}}\bigr{)}^{2}\bigr{)}}}\ (-1<R<1)\] (16)
6. Index of agreement (\(IOA\)): \[IOA=1-\frac{\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}{\sum_{i=1}^{N}\Bigl{(} \sum_{i=1}^{N}\bigl{|}Y_{out,i}-\overline{Y_{tar}}\bigr{|}+\sum_{i=1}^{N} \bigl{|}Y_{tar,i}-\overline{Y_{tar}}\bigr{|}\Bigr{)}^{2}}\ (0<IOA<1)\] (17)  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Algorithm Parameters** & **Value** \\ \hline \multirow{2}{*}{\(RF\)} & nTrees & nTrees = 100 \\  & mTrees & mTrees = 26 \\ \hline \multirow{2}{*}{\(LASSO\)} & lambda & series of lambda \\ \hline \multirow{2}{*}{\(PLS\)} & PLS components & NumComp = 3 for PSO \\  & & NumComp = 4 for GT and FS \\ \hline \multirow{2}{*}{\(Ridge\)} & regularization parameter lambda & lambda = 1 \\  & regularization parameter lambda & lambda = 1 \\  & kernel function &!linear’ \\  & parameter for kernel & sigma = 2 \(\times\) 10\({}^{-7}\) \\ \hline \multirow{4}{*}{\(PG\)} & Function set & +, \(-\), \(\times\), \(\div\), \(\div\), power, In, sqrt, sin, cos, tan \\  & Population size & 100 up to 500 \\ \cline{1-1}  & Number of generations & 1000 \\ \cline{1-1}  & Genetic operators & Reproduction, crossover, mutation \\ \hline \hline \end{tabular}
\end{table}
Table 3: \(Cont.\)where \(Y_{tar,i}\), \(Y_{out,i}\), \(\overline{Y_{tar}}\), and \(\overline{Y_{out}}\) present the target, output, mean of target, and mean of output swelling index values for N data samples, respectively. Furthermore, the proposed machine learning model having the lowest value of \(RMSE\), \(IOS\), and \(MAE\) and the highest value of \(IOA\), \(NSE\), and \(R\) presents the better one and the closest to the experimental data.  
Subsequently, after selecting the best model using statistical performance indicators, its predictive capacity is assessed using the \(K\)-fold cross-validation approach. This shows more accuracy and robustness for evaluating the capability of performance predictive of the best model by testing the existence of over-fitting and under-fitting problem in data learning [51; 52]. The method consists of separating the data set into \(k\) equal sizes splits. Hence, for each one, \(K-1\)-folds are used for training and the last one for validation. This operation is repeated successively until the use of all split for the validation step [53]. The main advantage of this method is that all collected data are utilized in both the training and validation steps [52]. [PERSON] and [PERSON] (1992) have demonstrated that \(K=10\) or \(K=5\)-fold cross validation is the best choice for model evaluation [51]. In this work, we chose \(K\)-fold cross-validation with \(K=10\) for comparing the predictive capacity of each model.