## 4 AdaBoost Regressor-Based Seismic Vulnerability Analysis Method
### A Selection and Brief Introduction of the AdaBoost Regressor  
For a given engineering structure, its seismic response depends on the ground motion and the material parameters. A proper regressor is supposed to be determined such that the relationship between seismic vulnerability parameters and the PGA or other material parameters can be represented. Nowadays, there are lots of widely used machine learning models, which can meet the requirements of this work. AdaBoost is an ensemble learning method that improves prediction accuracy by iteratively adjusting weights so that multiple weak learners (such as decision trees) are combined into a single strong learner. Compared with single models (e.g., linear regression, support vector machines, etc.), it is better adapted to complex seismic response patterns. Concerning some commonly used ML models, Gaussian processes (GPs) perform well on small data sets and can provide uncertainty estimates. However, the computational complexity of GPs is higher, especially in the case of multiple predictions, and the computational overhead is significantly higher than that of AdaBoost. GPs depend on the selection of kernel functions, so different kernel functions have a great influence on the model, which increases the difficulty of model optimization. In contrast, AdaBoost adaptively adjusts the weight of the weak learners, which can ensure accuracy while maintaining a high computational efficiency. Neural networks (NNs) can learn complex nonlinear maps, but its model training process requires a large amount of data, and the complexity of hyperparameter tuning is high. At our data size, NNs do not show significant performance advantages in computational cost. Since AdaBoost achieves a better balance between model stability and computational efficiency, this study uses the AdaBoost regressor as the surrogate model.  
The AdaBoost regressor is one of the commonly used regressors that are of a satisfactory predictive value. It is a meta-estimator that begins by fitting a regressor on the original dataset and then fits additional copies of the regressor on the same dataset but where the weights of instances are adjusted according to the error of the current prediction. As such, subsequent regressors focus more on difficult cases and provide a more accurate prediction [59]. The AdaBoost regressor is a typical learning model with boosting. The core principle of the boosting method is depicted in Figure 10. Specifically, the key hyperparameters, including the number of estimators, the learning rate, and the base regressor are carefully tuned to optimize performance. The training process follows an iterative approach: Firstly, an individual estimator is trained from the initial training set, and then the distribution (weight) of the training samples is adjusted according to the performance of the individual estimator, so that the training samples that the previous individual estimatorsthat made mistakes or produced large errors receive more attention in the following period. Then, the next individual estimator is trained based on the adjusted sample distribution (weight). Repeat this until the number of individual estimators reaches a predetermined value, and finally a strong estimator is built by combining the base estimators. Compared to alternative methods such as GPs and NNs, AdaBoost offers notable computational advantages. Unlike GP, which has a cubic complexity (O(n\({}^{3}\))), making it less scalable, AdaBoost scales linearly (O(nT), where \(T\) is the number of estimators), allowing for efficient training even on large datasets. Additionally, while NNs require extensive hyperparameter tuning and GPU acceleration, AdaBoost can be effectively trained on a standard CPU, making it a computationally efficient alternative. Empirical results demonstrate that AdaBoost achieves a favorable trade-off between prediction accuracy and computational cost, making it well suited for seismic vulnerability analysis.