## 3 Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
### An Adaptive Modeling Framework
If \(N_{c}\) < \(K\)/2, turn to step (iv); if \(N_{c}\) > \(2K\), turn to step (vi); if the number of iterations reaches \(N_{iter}\) or the iteration approaches to convergence, move to step (viii). (viii) Gather the center points of the clusters and take these center points as the optimized support point set \(X^{**}\). Taking the support point set \(X^{**}\) as the input, we get the corresponding output \(Y^{**}\) through high-fidelity simulations or experiments. Then, add \(Z^{**}\) = [\(X^{**}\) \(Y^{**}\)] to the experiment set and the updated DoE can then be used to construct a new kriging model with the higher performance. There are two tasks in the model validation. One is to judge if the loop terminates or not. The other is to check the effectiveness of the kriging surrogate. To this end, three parts are included in the validation. First, considering the overall accuracy of the kriging model, a cross-validation method [30] is adopted. The cross validation divides the experiment  
Figure 5: Kriging-based adaptive surrogate modeling framework. set (i.e., DoE) into two groups. One group is used to construct the kriging model and the other is used as the validation set. Based on the error measures, the accuracy of model is evaluated depending on the two groups. Instead of establishing an independent validation set, the cross-validation technique is more efficient and feasible. Here, we utilize the random sampling for the cross validation: (i) sample \(n\) points from the experiment set (scale \(N\)) that are taken as the validation set, and the residual (\(N\)-\(n\)) points are used to construct the kriging model; (ii) predict the output for validation points using the constructed kriging model and measure the prediction errors; (iii) repeat the sampling, prediction, and error measuring; (iv) calculate the statistical error. To assure the reliability of cross validation, the sample scale \(n\) should be small enough and the number of repeats should be large enough. Therefore, we set \(n=1\) with \(N\) being the number of repeats. The coefficient of determination (RD, Equation (18)), accounting for the overall accuracy of the prediction, and the mean error percent (ME, Equation (19)), accounting for the overall deviation of the prediction, are used as the statistical error measures. The larger DR and smaller AME implies better overall performance of the kriging model. \[\text{RD}_{i}^{2}=1-\frac{SSE}{SST} \tag{18}\]  
\[\text{ME}_{i}=\sum_{p=1}^{N}|y_{i}(\mathbf{x}^{p})-\hat{y}_{i}(\mathbf{x}^{p })|/\sum_{p=1}^{N}|y_{i}(\mathbf{x}^{p})| \tag{19}\]  
where \(i\) means the \(i\)-th element of output vector \(\mathbf{y}\), \(y\) represents the observation result (i.e., true value), \(\hat{y}\) is the prediction of \(y\) (i.e., estimated value), \(\mathbf{x}\) is the sampled validation point, the upper index \(p\) means the \(p\)-th repeat, and \(n\) is the total number of repeats. Second, considering the local error of the kriging model, we take the support point set generated in the loop as another validation set for local validation. The roots of the mean square error (RMSE, Equation (20)) and mean absolute error (MAE, Equation (21)) are used to measure the local performance of the kriging model:  
\[\text{RMSE}=\sqrt{\frac{1}{m}\sum_{j=1}^{m}\left(\hat{y}_{j}-y_{j}\right)^{2}} \tag{20}\]  
\[\text{MAE}=\frac{1}{m}\sum_{j=1}^{m}\left|\hat{y}_{j}-y_{j}\right| \tag{21}\]  
where \(m\) is the number of validation points (i.e., the scale of the support point set in each loop). The scale of the support point set is generally small and insufficient to check the overall accuracy of the kriging model. Most points in the support point set have a larger prediction error variance, which is helpful to validate the local performance of the kriging surrogate. Third, the kriging method, as a mathematical approach, does not reveal the nature of the engineering or physical issues. Therefore, it is necessary to validate the performance of the surrogate in engineering practice.