## 4 AdaBoost Regressor-Based Seismic Vulnerability Analysis Method
### AdaBoost Regressor-Based Seismic Vulnerability Analytical Procedures  
On the basis of the aforementioned basic theory, the seismic vulnerability analysis method based on the AdaBoost regressor was proposed and the steps are summarized in Figure 11. There are three main stages for the overall vulnerability assessment:  
(1) The structural seismic responses with various uncertainties are obtained by the time history calculations for the simulation model;  
(2) The mappings from the PGA to the mean \(\mu\) and standard deviation \(\sigma\) of various responses are formed. Then, AdaBoost regressors with a single variable as an input are used to replace the mappings. Considering the material strength, the seismic vulnerability curves can be obtained according to the mapping of predicted responses and PGA;  
(3) Further, the mappings from uncertain parameters (PGA, elastic modulus, etc.) to the mean \(\mu\) and standard deviation \(\sigma\) of various responses with other uncertain parameters are formed. AdaBoost regressors with multiple variables as inputs are also employed to replace the mappings. Considering the material strength, the seismic vulnerability curves can be obtained according to the mapping from uncertain parameters to predicted responses.  
These three steps combine into the vulnerability analysis method, and a more detailed illustration is introduced in the following parts.  
Figure 10: The core principle of the boosting algorithm.  
4.1. Part A: Sample Collecting Based on Simulation Part B: Seismic Vulnerability Based on the Single-Variable-Input AdaBoost Regressor  
As mentioned before, the obtained seismic vulnerability curve quantifies a structural failure probability and considers the uncertainties related to material parameters and ground motions. It should be mentioned that other types of uncertainties, including geometric size, construction quality, and simulation precision, are not considered in this work. They can be accounted for in the future in the same way as indicated in this paper. Those parameters considered generally obey the distributions such as normal distributions and logarithmic normal distributions, so they can be obtained by random sampling. Firstly, the material parameters and PGA can be sampled from the distributions by Latin hypercube sampling (LHS), then \(M\) groups of parameter combinations can be obtained. Each group of parameter combinations includes the parameters with the determined number. Secondly, to consider the ground motion uncertainty, \(N\) sets of ground motions that meet the requirements of site condition and design are selected. Specifically, a design spectrum provided by relevant codes or standards [49; 50] should be first selected according to the site condition (i.e., soil type and characteristic period) and the seismic fortification intensity (i.e., PGA). Then, \(N\) sets of ground motions can be selected by spectral matching using the mean spectrum after scaling them. For each parameter combination and each ground motion, the simulation model can generate one seismic response; hence, there are \(M\times N\) time history computations that need to be conducted, after which \(M\times N\) samples can be acquired along with the computations.