## 3 Development of an HS-ANN Hybrid Model  
### Sampling of Training Data of ANN Model  
The data used for developing an ANN model is divided into two parts: the training data for training the model and the test data for verifying or testing the performance of the trained model. The training data should be sampled to represent the characteristics of the population. Otherwise, the model would not perform well for the cases that had not been encountered during the training. For example, if a variable of the training data consists of only relatively small values, the model would not perform well for large values of the variable because the model has not experienced the large values \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Author** & **Correlation Coefficient** & **Number of Data** & **Remarks** \\ \hline [PERSON] [27] & 0.92 ([PERSON] _et al_. [3]) & 579 & \
\begin{tabular}{c} Empirical formula, Equation (3) \\ in this paper \\ \end{tabular} \\ \hline [PERSON] _et al_. [3] & 0.91 & 609 & Including data of [PERSON] _et al_. [28] \\ \hline [PERSON] and [PERSON] [6] & 0.902 to 0.952 & 641 &
\begin{tabular}{c} Including data of \\ low-crested structures \\ \end{tabular} \\ \hline [PERSON] _et al_. [9] & 0.906 to 0.936 & 554 & ANN-PCA hybrid models \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation coefficients of different empirical formula or ANN models. and vice versa. Therefore, in general, the size of the training data is taken to be larger than that of the test data. In the previous studies of [PERSON] _et al._[3] and [PERSON] and [PERSON] [6], however, only 100 randomly sampled data were used for training the models, which is much smaller than the total number of data, 579 or 641. This might be one of the reasons why the ANN models do not show superior performance compared with the empirical formula (see Table 1). To overcome this problem, the stratified sampling method is used in this study to sample 100 training data as in the previous studies while using the remaining 479 data to test the model. The key idea of stratified sampling is to divide the whole range of a variable into many sub-ranges and to sample the data so that the probability mass in each sub-range becomes similar between sample and population. Since a number of variables are involved in this study, the sampling was done manually. There are two kinds of statistical tests to evaluate the performance of stratified sampling, _i.e._, parametric and non-parametric tests. Since the probability mass function of each variable in this study does not follow the normal distribution, the chi-square (\(\chi^{2}\)) test is used, which is one of the non-parametric tests. The test is fundamentally based on the error between the assumed and observed probability densities [29]. In the test, each of the range of the \(n\) observed data is divided into \(m\) sub-ranges. In addition, the number of frequencies (\(n_{i}\)) of the variable in the \(i\)th sub-range is counted. Furthermore, the observed frequencies (\(n_{i},\ i=1\) to \(m\)) and the corresponding theoretical frequencies (\(e_{i},\ i=1\) to \(m\)) of an assumed distribution are compared. As the total sample points \(n\) tends to \(\infty\), it can be shown [30] that the quantity, \(\sum_{i=1}^{m}{(n_{i}-e_{i})^{2}/e_{i}}\), approaches the \(\chi^{2}\) distribution with \(f=m-1-k\) degree of freedom, where \(k\) is the number of parameters in the assumed distribution. \(k\) is set to zero for non-normal distribution. The observed distribution is considered to follow the assumed distribution with the level of significance \(\sigma\) if  
\[\sum_{i=1}^{m}{\frac{{(n_{i}-e_{i})^{2}}}{{e_{i}}}}<c_{1-\sigma,f}, \tag{4}\]  
where \(c_{1-\sigma,f}\) indicates the value of the \(\chi^{2}\) distribution with \(f\) degree of freedom at the cumulative mass of \(1-\sigma\). In this study, a 5% level of significance is used. Table 2 shows the input and output variables in the ANN model.