## 3 Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
### An Adaptive Modeling Framework  
Based on the kriging method, we propose a kriging-based adaptive surrogate modeling framework (Figure 5) in which three main parts are involved, i.e., the kriging model, support points, and model validation. In the framework, the performance of the kriging surrogate is improved continuously by adding support points in loops. The common approach for selecting the support point is sequential, that is, just one support point (i.e., the point at which the prediction error is largest) is chosen in each loop [24]. To be more efficient, we try to identify a larger sample set of support points in each loop. In this strategy, the points at which the performance of the kriging surrogate is lower have a larger probability of being selected as support points. Concerning the overall accuracy of the surrogate model, an additional issue should be considered, namely, that the selected support points may be close to one another, which can lead to a redundant effect on surrogate performance improvements. To be more effective, the optimization of the support point set is necessary in each loop. At the end of each loop, the accuracy of the kriging model and the validation of the surrogate should be examined, based on which we can decide if we continue the next loop or not. Overall, in this framework, three issues are of major concern: (i) selecting support points (i.e., to generate the initial support point set in an efficient way); (ii) optimizing the support point set (i.e., to remove the inclusion of support points that are close to one another); (iii) validating the surrogate model (i.e., determining whether to terminate the loop or not). From Equations (16) and (17), the kriging prediction gives both the mean value \(\overset{\alpha}{\mathbf{y}}(\mathbf{x})\) and the error variance \(\sigma_{\mathbf{y}}^{2}\). The error variance reveals the accuracy of the predicted mean value. Here, we take the prediction error variance as the measure for the selection of support points. That is, more support points should be selected from the regions where the error variance is larger. This means that the larger the prediction error variance is, the more likely the point is to be selected as the support point. Therefore, the original sample set of support points should satisfy a probability distribution whose density monotonically increases with the prediction error variance. To this end, the reject sampling technique is used [27] to generate the initial set of support points. Then, the clustering method is used to eliminate the inclusion of support points that close to one another. Considering that the k-means method is easy to trap in local optimum [28], the bisecting k-means method [29] combined with the iterative self-organizing data technique algorithm (ISODATA) is used to implement the optimization. The steps for generating the support point set in each loop are listed as:  
(i). Generate the original sample set of support points \(X^{*}\) (with the scale of \(n^{*}\)) using the reject sampling technique. (ii). Divide the original support point set into \(K\) clusters using the bisecting k-means method. (iii). Set the lower limit of distance between two nearby clusters as \(d_{min}\); set the lower limit of number of clusters as \(N_{s,min}\); set the upper limit of standard deviation of each cluster as _Sigma_; set the largest number of iterations as \(N_{iter}\); denote the number of clusters as \(N_{c}\) (the initial value is \(K\)). (iv). Calculate the scale of each cluster and the standard deviation of the samples in each cluster. (v). Find the clusters whose standard deviations are larger than _Sigma_ and whose scales are two times larger than \(n_{s,min}\); split the cluster into two new clusters with their two center points located at \(\pm\)_Sigma_ to the original center point. (vi). Calculate the distance between any two of the clusters; find the two clusters between which the distance is less than \(d_{min}\) and merge them afterwards (only one merging is allowable for each cluster). (vii).