# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 2 Hypothesis Test for Validity Check  
It is desirable to employ a sufficient amount of experimental data for a validity check. However, in the design of engineered products, it is extremely challenging to increase the sample size for the validity check due to high cost and lengthy required test time. Validity check with limited experimental data raises two issues. The first issue is how to incorporate the evidence of limited experimental data collected from different experimental settings (testing conditions) into a single measure that quantifies overall mismatch. The second issue is how to determine whether a particular value of the single measure is acceptable for the validity check, while considering the uncertainty in the single measure due to a small sample size of experiments. A hypothesis testing-based approach called \"hypothesis test for validity check\" (HTVC) [12] was devised to address the issues in the field. In HTVC, the null hypothesis (\(H_{0}\)) is the claim that the computational model is valid, while the alternative hypothesis (\(H_{\text{a}}\)) is the claim that the computational model is not valid. \(H_{0}\) can be rejected only if a metric for the validity check exceeds a critical value. Otherwise, \(H_{0}\) cannot be rejected. HTVC employs the u-pooling method for the validity check. One advantage of the u-pooling method is that it can integrate evidence collected from all experimental data under various experimental settings (e.g., environmental loading and boundary conditions) into a single mismatch metric. In the u-pooling method, the u-value (\(u_{i}\)) obtained from the \(i\)th experimental setting can be obtained by transforming every datum (\(x_{i}\)) with the prediction distribution (\(F_{i}\)) of a computational response. \[u_{i}=F_{i}(x_{i}) \tag{1}\]  
With the assumption that the datum (\(x_{i}\)) is distributed in accordance with the corresponding prediction distribution (\(F_{i}\)), the \(u\)-value (\(u_{i}\)) follows a uniform distribution of [0, 1]. This is called the probability integral transform theorem in statistics [14]. All observations from different experimental settings can be transformed into a single universal probability scale. The \(u\)-value can be back-transformed through a suitable distribution function \(G\). \[y_{i}=G^{-1}(u_{i}) \tag{2}\]The appropriate distribution function of \(G\) was found in [15]. When a uniform distribution is chosen for \(G\), cumulative distribution functions (CDF) of \(y_{i}\) can be obtained with responses from experiments. The mismatch between the prediction distribution and the empirical distribution can be quantified by integrating the area between the CDF of the uniform distribution (\(F_{\text{uni}}(u)\)) and the empirical CDF (\(F_{u}(u)\)) of \(u\)-values corresponding to the experimental data. \[U=\text{area}(F_{\text{uni}},F_{u})=\int_{0}^{1}\big{|}F_{\text{uni}}(u)-F_{u}( u)\big{|}du,0\leq U\leq 0.5 \tag{3}\]  
where \(U\) is the area metric. As the similarity between the predicted CDF and the empirical CDF is increased, the calculated area metric is decreased. For example, if the model is valid and enough experimental data exists, the area metric gets close to zero. If infinite experimental data are used for the validity check, we can reject the null hypothesis unless the area metric is zero. It is because epistemic uncertainty in the area metric due to limited experimental data can be eliminated theoretically. However, in practice, uncertainty exists in the area metric due to limited sampling, although the model is valid (i.e., predicted and experimental results follow an identical distribution). As shown in Figure 2, uncertainty in the area metric can be characterized for a given size (\(n\)) of experimental data. The minimum value of the area metric gets close to zero as the number of experimental data increases infinitely [12].