# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 3 Response-Adaptive Experiment Design
### Experimental Design for Reduction of Type II Error  
As explained in Section 3.1, RAED selects the next experimental setting as the condition where the calculated area metric (\(U\)) is smaller than other conditions. The lower area metric indicates that the predicted PDF is not closer to the mother distribution of the experimental data. In this section, with deductive reasoning, the relation between the low area metric and the reduction of type II error is studied. The type II errors of two hypothetical experimental conditions are compared. It is worth noting that any variation of the experimental conditions can be made. The use of a particular  
Figure 4: Example of RAED: Two-level factorial design with two design variables. set of experimental conditions does not limit the applicability of the proposed RAED method to extended problems. Experimental condition I: The mother distribution of the experimental results (i.e., performance of interest (PoI)) is a standard normal distribution (\(f_{0}\sim\) Normal (0, 1)) and the PDF of the predicted results is a normal distribution (e.g., \(f_{1}\sim\) Normal (0.56, 1)), as shown in Figure 5. A predefined number of experimental data (e.g., eighteen) are randomly obtained from the \(f_{0}\) to calculate the area metric. The number of experimental data is determined by considering the additional experimental resources available. Experimental condition II: The mother distribution of the experimental results is also a standard normal distribution (\(f_{0}\)). However, the PDF of the predicted results is a normal distribution (e.g.,\(f_{2}\sim\) Normal (1.05, 1)), as shown in Figure 5. A predefined number of experimental data (e.g., eighteen) are randomly obtained from the \(f_{0}\) to calculate the area metric. It is certain that the predicted model, \(f_{2}\), is more invalid and has a lower area metric than the predicted model, \(f_{1}\), when they are compared with the mother distribution of the experimental results, \(f_{0}\). The type II error of each experimental condition can be calculated using the PDF of the area metric (\(f_{u,u}(x)\)) and the following equation. The null hypothesis is the claim that the computational model is valid, while the alternative hypothesis is the claim that the computational model is not valid. \[\text{Type II error}=\int_{0}^{D_{n}(\alpha)}f_{u,u}(x)dx \tag{8}\]  
For the quantification, it is assumed that type I error is 0.1 (the significant level in Equation (3)). Figure 6 shows the PDF of the area metric of experimental conditions I and II. The procedure for constructing the PDFs of the area metric is described as follows. Figure 5: PDFs of the experimental and predicted results. Figure 6: PDFs of the area metric (\(U\)) for experimental conditions I and II. Step 1: Determine a virtual sampling size (\(k\)) and the experimental data size (\(n\)). Here, \(k\) and \(n\) are 5000 and 18, respectively. Step 2: Generate samples (\(x_{1}\), \(x_{2}\),..., \(x_{n}\)) randomly from the mother distribution of the experimental data. Step 3: Calculate the CDF values (\(u_{1}\), \(u_{2}\),..., \(u_{n}\)) corresponding to \(y_{1}\) using Equation (1). Step 4: Calculate the area metric (\(U\)) using Equation (2). Step 5: Repeat Steps 2 to 4, \(k\) times to generate random data (\(U_{1}\), \(U_{2}\),..., \(U_{k}\)). Then construct the empirical PDF of area metric (\(f_{u,n}(x)\)) using \(U\) values. In this paper, the Pearson system [19] is used to construct empirical PDFs of the \(U\) values. Table 2 shows the calculated type II error of experimental conditions I and II. The type II error of experimental condition II is lower than that of experimental condition I. The calculated results show that type II error can be minimized if more test data are sampled at the experimental condition where the difference between the PDF of predicted results and the mother distribution of the experimental data is larger than other conditions.