## 2 FEA and Optimal Design
### Optimal Design of Bicycle Handle Frame with Monitor
#### 2.3.2 Optimal Design Using SVR  
SVM is a machine learning method that is actively used in pattern recognition and data analysis. It is widely used as a classification or prediction method due to its advantage of low probability of overfitting, and high prediction accuracy [10]. It is largely classified into support vector classification (SVC), and continuous variable support vector regression (SVR) [11]. SVC calculates the cost of the model based on errors included within the margin range. Meanwhile, SVR calculates the cost of the model based on errors for out-of-margin values, and is a generalized method used to predict real values by introducing an \(\varepsilon\)-insensitivity loss function to SVM [12]. As shown in Figure 9, the difference between the actual value and the predicted value does not deviate from the error allowance rate, \(\varepsilon\), and aims to maximize the margin. The equation of SVR is the same as Equation (1), where \(x\) is the input variable, \(y\) is the output variable, \(\xi_{i}\) is the distance outside the range \(\varepsilon\), and \(y_{i}\) is the penalty value given when outside the range \(\varepsilon\). Here, the constant \(C\) and \(\varepsilon\) values must be appropriately assigned to secure good performance of the prediction model.  
Figure 8: Design variables of the correlation analysis.  
Figure 7: Sensitivity analysis results.  
\[\begin{array}{l}\frac{1}{2}\|w\|^{2}+C\sum\limits_{i=0}^{l}\left(\xi_{i}+\xi_{i}^ {*}\right)\\ y_{i}-wx_{i}-b\leq\varepsilon+\xi_{i}\\ wx_{i}+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i}\,,\,\xi_{i}^{*}\geq 0\end{array} \tag{1}\]  
In this study, we derived a predictive model using Python 3.7, and its machine learning library, Scikit-learn [13]. Here, SVR uses a kernel function to classify the low-dimensional nonlinear data by moving it to a high-dimensional level, and the kernel types are generally either linear polynomial or radial basis function methods. To determine the kernel function suitable for the data used in this study, we compared these two types of functions: polynomial, and radial basis function kernel. Herein, \(C\), which represents the penalty value for the error interval, was fixed at 1, \(\varepsilon\) at 0.001, and \(\gamma\) at 0.01, and the corresponding predicted performance was evaluated. Typically, mean square error (MSE) is widely used as an indicator for regression evaluation. The two kernels were compared in terms of their MSE, as expressed in Equation (2). The symbol \(\hat{y}_{i}\) denotes a predicted value, whereas \(y_{i}\) denotes a reference value derived by analysis. This shows that the closer the value is to 0, the better the predictive performance. Subsequently, the radial basis function kernels showed better performance. Therefore, radial basis function kernel was used to derive an optimal prediction model in the current study. The radial basis function kernel sequentially performs learning using the GridSearchCV method by setting three parameter range values, that is, \(C\), \(\varepsilon\), and \(\gamma\). Each parameter variable value was designated as shown in Table 3, and a total of 27 result values were compared in terms of their MSE results. The results of SVR could vary according to the data scale, and the five design variables used as SVR input values could affect the prediction results because they were all different in scale. Accordingly, the range of data characteristic values was adjusted equally by the standard scaler method, using their mean and standard deviation.  
\[\text{MSE}=\frac{1}{n}\sum\limits_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{2} \tag{2}\]  
Figure 9: Concept of support vector regression.