# Prediction of Self-Healing of Engineered Cementitious Composite Using Machine Learning Approaches
## 3 Proposed Machine Learning Models
### Support Vector Regression  
Support vector machine (SVM) is a supervised machine learning method first introduced by [PERSON] [52; 53] based on the statistical learning theory [54]. Since then, it has gained popularity due to its attractive features and promising empirical performance [55]. SVM includes two main categories: support vector classification (SVC) and SVR. For classification purposes, SVMs often use a _kernel_ function to map the input data as vectors to a high-dimensional feature space so that an optimal separating hyperplane can be constructed [56]. For regression purposes, the basic idea is to provide a nonlinear function by mapping input data into a high-dimensional feature space, where a special type of hyperplane is constructed. After that, a regression model is established in the hyperplane [57]. Given our dataset, \(T=\{(x_{i},y_{i}),i=1,2,\cdots,n\}\), where \(n=617\) is the size of the sample dataset, \(x_{i}\in R^{n}\) is the input vector representing the selected features of a sample, including the FA, SF, LP, and crack width before self-healing, \(R^{n}\) is the \(n\)-dimensional vector space, and \(y_{i}\in R^{1}\) is the target output indicating the crack width after self-healing that corresponds to \(x_{i}\). The SVR aims to seek an optimum regression function \(f(x)\) with minimal empirical risk, which can be expressed as follows [53]:  
\[f(x)=\langle w,x\rangle+b\quad\text{with}\quad w\in T,b\in R \tag{3}\]  
where \(\langle\cdot,\cdot\rangle\) is denoted as the dot product in \(T\) and \(w\) and \(b\) are the weight vector and bias value, which are estimated by minimizing the empirical risk, that is the distance between the predicted crack width and the target crack width after self-healing. SVR adopts an \(\epsilon\)-insensitive loss function, penalizing predictions that have a distance between the predicted crack width and the target crack width when the self-healing is greater than \(\epsilon\). Therefore, the problem of finding \(w\) and \(b\) to reduce the empirical risk with respect to an \(\epsilon\)-insensitive loss function is equivalent to the convex optimization problem that minimizes the margin (\(w\)) with the full prediction error within the range of \(\epsilon\). Then, this problem can be expressed as [53]:  
\[ \begin{split}\text{minimize}&\frac{1}{2}||w||^{2} \\ \text{subject to}&\left\{\begin{array}{l}y_{i}- \langle w,x_{i}\rangle-b\leq\epsilon\\ \langle w,x_{i}\rangle+b-y_{i}\leq\epsilon\end{array}\right.\end{split}  \tag{4}\]  
By introducing slack variables \(\xi\) and \(\xi_{i}^{*}\) to allow some errors to cope with infeasible solutions of the optimization problem, the formulation can be generated as [53]:  
\[ \begin{split}\text{minimize}&\frac{1}{2}||w||^{2}+C\sum \limits_{i=1}^{n}(\xi+\xi_{i}^{*})\\ \text{subject to}&\left\{\begin{array}{l}y_{i}- \langle w,x_{i}\rangle-b\leq\epsilon+\xi_{i}\\ \langle w,x_{i}\rangle+b-y_{i}\leq\epsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0\end{array}\right.\end{split}  \tag{5}\]  
The constant \(C\) is the penalty value imposed on predictions that lies outside the \(\epsilon\) margin. Lagrange multipliers are included to solve this problem. By constructing the objective function and all constraints, a dual set of variables is introduced as follows [58]:\[L_{P}= \frac{1}{2}||w||^{2}+C\sum_{i=1}^{n}(\xi_{i}+\xi_{i}^{*})-\sum_{i=1}^ {n}(\eta_{i}\xi_{i}+\eta_{i}^{*}\xi_{i}^{*}) \tag{6}\] \[-\sum_{i=1}^{n}\alpha_{i}(e+\xi_{i}-y_{i}+\langle w,x_{i}\rangle+b)\] \[-\sum_{i=1}^{n}\alpha_{i}^{*}(e+\xi_{i}^{*}+y_{i}-\langle w,x_{i} \rangle-b)\] \[s.t. \alpha_{i},\alpha_{i}^{*},\eta_{i},\eta_{i}^{*}\geq 0\]  
where \(L_{P}\) is the Lagrangian and \(\alpha_{i},\alpha_{i}^{*},\eta_{i},\eta_{i}^{*}\) are Lagrange multipliers. The optimality can be achieved by the partial derivatives of \(L_{P}\) with respect to the primal variables following the saddle point condition.