# Modeling Undrained Shear Strength of Sensitive Alluvial Soft Clay Using Machine Learning Approach
## 2 Materials and Methods
To eliminate the impact of incompatible data scales on model training time, and therefore make the computations much faster, the data was normalized using the following equation:  
\[x_{n}=\frac{x-x_{min}}{x_{max}-x_{min}} \tag{1}\]  
where \(x\) refers to the original data, \(x_{n}\) refers to the normalized data, \(x_{min}\), and \(x_{max}\) signify the minimum and maximum values detected in the dataset, respectively. The modified output data was simply retrieved in its original form, after model training, utilizing the formula:  
\[x=x_{n}(x_{max}-x_{min})+x_{min} \tag{2}\]  
We used a statistical procedure known as feature selection in order to identify the most important features that have a significant influence on model predictions. The procedure comprises a correlation matrix (CM) alongside a principal component analysis (PCA). Following that, various supervised ML algorithms including linear regression, Gaussian process regression, regression trees, ensembles of regression trees, and support vector machine (SVM) approaches were adopted in the training phase. The SVM approach [22] is widely used for ML modeling of classification or regression problems. With a few minor exceptions, the support vector machine regression (SVR) utilizes the same concepts as the SVM for classification [23; 24; 25; 26]. In the case of regression, a margin of tolerance (\(\epsilon\)) is specified as a rough approximation to the SVM that the issue would have already requested. However, there is a more problematic reason: the algorithm is more intricate and, therefore, it must be considered. The SVM algorithm operates by using the largest margin to identify the most optimum hyperplane that splits data into  
Figure 3: Sensitivity of the soft soil. many classes. We used the regression function _f(x)_ to optimally approximate the supplied training dataset \(\{(x_{i}\,y_{i})\}_{i}^{N}\), where \(x_{n}\) is a multivariate collection of \(N\) observations with observed response values \(y_{n}\). For the simplest case, the function _f(x)_ is expressed as [22]:  
\[f(x)=wx+b \tag{3}\]  
The optimized values of \(w\) and \(b\) can be acquired by minimizing the following expression [22]:  
\[\min\,\frac{1}{2}\|W\|^{2}+C\sum_{i=1}^{N}(\xi_{i}+\xi_{i}^{*}) \tag{4}\]  
Subject to:  
\[ \begin{cases}y_{i}-wx_{i}-b\leq\varepsilon+\xi_{i}\\ wx_{i}+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\ \xi_{i}^{*}\geq 0\end{cases}  \tag{5}\]  
where \(\varepsilon\) is an \(\varepsilon\)-insensitive tube indicating the error tolerance, and \(C\) is a compromise between the empirical error and the general term. The regression function may be stated as follows, using Lagrangian multipliers and optimum constraints [22]:  
\[y=\sum_{i=1}^{N}(\alpha_{i}+\alpha_{i}^{*})K(x_{i},x)+b \tag{6}\]  
where \(K(x_{i},x_{j})\) is the kernel function. Examples of the most famous kernel functions are the linear, polynomial, sigmoidal, Gaussian, and radial basis functions. The latter is expressed as [22]:  
\[K\big{(}x_{i},x_{j}\big{)}=\exp\Biggl{(}-\frac{\|x_{i}-x_{j}\|^{2}}{2\sigma^{2 }}\Biggr{)} \tag{7}\]  
where \(\|x_{i}-x_{j}\|\) denotes the Euclidean distance between the two feature vectors, and \(\sigma\) is the spread of the kernel function's distribution. optimization procedures, and their values cannot be estimated from data. As a result, for correctly configured model hyperparameters, the objective function, known as the loss function, is minimized. In this process, for each training iteration, the mean squared error (MSE) statistics are obtained and computed using:  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2} \tag{11}\]  
After optimizing the model hyperparameters, they are deployed and can be employed to predict soft clay undrained shear strength. The framework of the modeling procedure is further illustrated as shown in Figure 4.