# Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model
## 4 Model Development and Verification  
### Least Squares Support Vector Machine  
The LSSVM, first introduced by [PERSON] et al. [26], is a supervised machine learning method based on statistical learning theory. It is well-known that the LSSVM methodology possesses an outstanding feature in learning nonlinear functions, which simply solves a set of linear equations. Therefore, LSSVM, as an extension of SVM equipped with the advantage of fast calculation and generalized capacity, has been widely applied for practical problems such as nonlinearly, high-dimensional input space and small training samples. These problems are generally divided into two categories, i.e., regression problems and classification problems. The focus of the present study is predicting the yield strength and displacement with the LSSVM adopted in this prediction model, which is thus considered as a regression problem. The governing equations and numerical algorithms are described in detail as follows.  
For a set of given nonlinear training samples \(\{x_{i},y_{i}\}\), \(i\) = 1,2,... \(J\), where \(l\) is the number of training samples. \(x_{i}\)\(\in\)\(R^{n}\) is the input vector, and \(y_{i}\)\(\in\)\(R\) is its corresponding output vector. The regression function is obtained by mapping the input space to high-dimensional feature space using the nonlinear mapping function \(\phi(\cdot)\), the equation of the regression function can be written as  
\[f(x)=w^{T}\phi(x)+b \tag{1}\]  
where \(w\) is the weight vector; \(b\) is the bias term (also called the partial vector).  
The objective of LSSVM regression can be transformed into an optimization problem that is expressed as  
\[\begin{array}{c}\min\limits_{w,b,\xi}J(w,\xi)=\frac{1}{2}\|w\|^{2}+\frac{ \gamma}{2}\sum\limits_{i=1}^{l}\xi_{i}^{2}\\ \text{s.t. }y_{i}=w^{T}f(x_{i})+b+\xi_{i}\text{, }i=1,2,\ldots,l\end{array} \tag{2}\]  
where \(\gamma\) is the regularization parameter (also called the penalty parameter); \(\xi\)\(\in\)\(R\) are the error variables.  
The Lagrange function of the optimization problem, i.e., Equation (2) is formulated as follows:  
\[L=J-\sum\limits_{i=1}^{l}\alpha_{i}\Big{[}w^{T}\cdot\phi(x_{i})+b+\xi_{i}-y_{i }\Big{]} \tag{3}\]  
where \(\alpha_{i}\) is the Lagrange multiplier, and sample (\(\alpha_{i}\
eq 0\)) is the support vector.  
The Karush-Kuhn-Tucker (KKT) conditions are employed for the optimal solution of the object function in a nonlinear optimization problem [51; 52]. The resulting regression model of LSSVM ultimately is obtained and can be evaluated by  
\[y(x)=\sum\limits_{i=1}^{l}\alpha_{i}K(x,x_{i})+b \tag{4}\]  
where \(K(x,x_{i})=\phi(x)^{T}\phi(x_{i})\) is the kernel function, which is used for classification by mapping the input data from the featured space into a high-dimensional space. In the present work, the Radial Basis Function (RBF) kernel function is adopted in the implementation of the LSSVM and given by  
\[K(x,x_{i})=e^{-\frac{\|x-x_{i}\|^{2}}{2\alpha^{2}}} \tag{5}\]  
where \(\sigma\) is the RBF kernel function parameter that needs to be determined, which is optimized via an external optimization technique during the training process. The pseudo-code of LSSVM algorithm is presented in Algorithm 1 as follows.  
In the current study, the learning performance of the LSSVM is mainly determined by two tuning parameters, i.e., the RBF kernel function parameter (\(\sigma\)) and the regularization parameter (\(\gamma\)). The RBF kernel function parameter (\(\sigma\)) influences the smoothness of the approximated nonlinear function, and the regularization parameter (\(\gamma\)) controls the penalty imposed on data point that deviates from the regression function. Particle swarm optimization (PSO) is a popular approach that is employed to search for the optimal solution of parameters (\(\sigma\), \(\gamma\)). The minimum computational error can be obtained by selecting the optimal solution (\(\sigma\), \(\gamma\)) during the training process.