## 3 The Sequential Sampling Method
### Infill Criterion II: Expected Improvement  
The expected improvement (EI) is an infill criterion to compute how much improvement of the current kriging model can be achieved if an infill sample is added. Let \(Y(\mathbf{x})\) be a random variable representing the uncertainty in the prediction of sample \(\mathbf{x}\). The uncertainty in the prediction is represented by the Gaussian distribution with a mean (\(\hat{y}(\mathbf{x})\)) and prediction variance (\(\hat{s}^{2}(\mathbf{x})\)). If the current best value is the minimum value (\(y_{min}\)) thus far, then the improvement on the minimum (\(I(\mathbf{x})\)) can be defined as  
\[\mathbf{I}(\mathbf{x})=y_{min}-Y(\mathbf{x}). \tag{13}\]  
Since \(Y(\mathbf{x})\) is the Gaussian distribution, Equation (13) can be integrated to compute the expectation of \(\mathbf{I}(\mathbf{x})\). Then, EI on minimum (\(EI_{min}(\mathbf{x})\)) can be defined as  
\[EI_{min}(\mathbf{x})=(y_{min}-\hat{y}(\mathbf{x}))\Bigg{[}\frac{1}{2}+\frac{1}{2}\text {erf}\Bigg{(}\frac{y_{min}-\hat{y}(\mathbf{x})}{\hat{s}(\mathbf{x})\sqrt{2}}\Bigg{)} \Bigg{]}+\hat{s}(\mathbf{x})\frac{1}{\sqrt{2\pi}}\exp\Bigg{[}-\frac{(y_{min}-\hat {y}(\mathbf{x}))^{2}}{2\hat{s}^{2}(\mathbf{x})}\Bigg{]} \tag{14}\]  
where \(\text{erf}(\cdot)\) denotes the error function for the cumulative Gaussian distribution of the Kriging model at sample \(\mathbf{x}\). The first term in Equation (14) tends to be bigger when the mean (\(\hat{y}(\mathbf{x})\)) is likely to be better than the current minimum (that is, exploitation). On the other hand, the second term in Equation (14) plays a role in exploration by getting bigger with higher prediction variance values (\(\hat{s}^{2}(\mathbf{x})\)). In this context, \(EI_{min}(\mathbf{x})\) searches an infill sample which improves both exploitation and exploration in the response surface. The infill sample of the \(EI_{min}(\mathbf{x})\) is chosen as  
\[\mathbf{x}_{EI_{min}}^{n+1}=\underset{\mathbf{x}}{\text{argmax}}EI_{min}(\mathbf{x}). \tag{15}\]  
EI on maximum (\(EI_{max}(\mathbf{x})\)) can be defined by replacing \(y_{min}-Y(\mathbf{x})\) with \(Y(\mathbf{x})-y_{max}\). Then, \(EI_{max}(\mathbf{x})\) can be defined as the following equation:  
\[EI_{max}(\mathbf{x})=(\hat{y}(\mathbf{x})-y_{max})\Bigg{[}\frac{1}{2}+\frac{1}{2}\text {erf}\Bigg{(}\frac{\hat{y}(\mathbf{x})-y_{max}}{\hat{s}(\mathbf{x})\sqrt{2}}\Bigg{)} \Bigg{]}+\hat{s}(\mathbf{x})\frac{1}{\sqrt{2\pi}}\exp\Bigg{[}-\frac{(\hat{y}(\mathbf{ x})-y_{max})^{2}}{2\hat{s}^{2}(\mathbf{x})}\Bigg{]}. \tag{16}\]  
The infill sample of the \(EI_{max}(\mathbf{x})\) is chosen as  
\[\mathbf{x}_{EI_{min}}^{n+1}=\underset{\mathbf{x}}{\text{argmax}}EI_{max}(\mathbf{x}). \tag{17}\]  
Figure 4 shows the \(EI_{min}(\mathbf{x})\) and \(EI_{max}(\mathbf{x})\) from the kriging model constructed by the three training samples. The values of \(EI_{min}(\mathbf{x})\) are becoming larger near \(x=5\) since this region has the greatest  
Figure 3: The mean squared error (_MSE_).  
prediction variance (\(\hat{s}^{2}(\mathbf{x})\)) and is near the best observed value (y(6)) thus far. In other words, the region near \(x=5\) maximizes both terms in Equation (14) (Figure 4a). A similar observation can be found in \(EI_{max}(x)\) (Figure 4b).