# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 2 Hypothesis Test for Validity Check
The statistical moments of the probability density functions (PDFs) of the area metric in in Figure 2 are summarized in Table 1. Details on how to construct the PDFs in Figure 2 are explained in [12]. HTVC evaluates the validity of the computational model using an upper-tailed test with a predetermined rejection region. 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. \[U>D_{n}(\alpha) \tag{4}\]  
where \(D_{n}(\alpha)\) is the critical value of the area metric; \(\alpha\) is the significance level for a given size (\(n\)) of experimental data. For example, \(D_{18}(0.05)\) is \(0.137\), when the significance level is set to \(0.05\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(n\) & Mean & Standard Deviation & Skewness & Kurtosis \\ \hline
6 & 0.1296 & 0.0535 & 1.1326 & 4.2585 \\
12 & 0.0914 & 0.0389 & 1.2337 & 4.7938 \\
18 & 0.0741 & 0.0319 & 1.2177 & 4.6966 \\
30 & 0.0569 & 0.0247 & 1.4871 & 6.3728 \\
60 & 0.0407 & 0.0178 & 1.3245 & 5.3782 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical moments of the PDFs of the area metric (all rows follow a [PERSON]’s type I distribution). Figure 2: Probability density functions (PDFs) of the area metric according to experimental data sizes. The significance level is referred to as type I error (i.e., the probability that a predicted model is rejected although it is valid). The type I error can be calculated. \[\text{Type I error}=\int_{D_{n}(x)}^{\infty}f_{u,n}(x)dx \tag{5}\]  
where \(f_{u,n}(x)\) is the PDF of the area metric for a given size (\(n\)) of experimental data. If the calculated area metric falls in the rejection region, the null hypothesis should be rejected. Otherwise, the null hypothesis should not be rejected since it is still plausible. In the model validation, type II error can be defined as the risk (probability) that a computational model is not rejected when it is invalid. In [12], HTVC does not consider type II error. The type II error needs to be considered for validating computational models of safety-related and mission-critical systems. As a simple approach, we can consider employing higher type I error (significance level) since an increase of type I error results in a decrease of type II error. However, doing so causes an increase in the validation cost (i.e., more experiments are required for the validity check). To this end, this study attempted to devise an effective experiment design method that can reduce type II error while maintaining type I error at an acceptable level. To the best of our knowledge, this study is the first attempt to address type II errors in validity check for model validation. The focus of this paper is not on data-driven models but on physics-based models in computational mechanics. The physics-based models are typically governed by [PERSON]'s laws of motion and laws of thermodynamics and fluid dynamics. Examples are to predict the friction of rubber materials and fatigue crack of solder joints in engineered products. The physical testing of the engineered products is very costly, while a couple of samples are available in the early stage of product development. It should be noted that the number of samples used in validating physics-based models is very small (typically less than ten samples) compared to the number of samples in validating data-driven models (maybe more than hundreds or thousands of samples). In adaptive control of nonlinear systems, the acquisition of a number of additional data online may be feasible. For example, [PERSON] et al. [16] used a radial basis function neural networks (RBFNN)-based controller to approximate the functions of uncertain nonlinear systems. The focus was on how to deal with uncertainty in model parameters. In [17], an adaptive neural network (NN)-based observer was implemented to nonlinear systems subjected to disturbances. The meaning of \"adaptive\" in [16; 17] can be identical to the meaning of \"adaptive\" in this paper.