# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 2 Methods
### Fragility Analysis  
Fragility analysis refers to a computational scheme that evaluates the conditional probability (P) of the seismic demand (D) exceeding the capacity (C) of the structure for a given ground motion intensity measure (IM) level (Equation (23)). Both the seismic demand and structural capacity are commonly expressed in terms of an engineering demand parameter (EDP). \[\text{P}=\text{P}(\text{D}\geq\text{C}|\text{IM}) \tag{23}\]  
The seismic fragility of rocking-based systems is derived by the appropriate combination of the rocking overturn probability (\(\text{P}_{\text{Ov}}\)) and the probability of developing a finite rocking rotation without overturning occurrence (\(\text{P}_{\text{FR}}\)), using the total probability theorem (Equation (24)) [31]. It is noteworthy that rocking initiation is omitted since it is determined by the seismic motion acceleration level (Equation (19)). \[\text{P}(\text{D}\geq\text{C}|\text{IM})=\text{P}_{\text{Ov}}+(1-\text{P}_{ \text{Ov}})\cdot\text{P}_{\text{FR}} \tag{24}\]  
Figure 1: Examined rocking frame. Excluding the overturning cases, the probability (\(\mathrm{P_{FR}}\)) is calculated by the following equation:  
\[\mathrm{P_{FR}}=\Phi\left(\frac{\ln(\mathrm{S_{d}}/\mathrm{S_{c}})}{\sqrt{\beta_{ \mathrm{D}|\mathrm{IM}}^{2}}}\right) \tag{25}\]  
where \(\Phi\) is the cumulative distribution function of the standard normal distribution, \(\mathrm{S_{c}}\) denotes the median value of the capacity associated with a predefined limit state, and \(\beta_{\mathrm{D}|\mathrm{IM}}\) expresses the seismic demand uncertainties conditioned on the IM. The dependency between the median seismic demand and an IM is expressed as follows [72]:  
\[\mathrm{S_{d}}=\mathrm{a}(\mathrm{IM})^{\mathrm{b}} \tag{26}\]  
where \(\mathrm{a}\) and \(\mathrm{b}\) are the seismic demand model coefficients. The above parameters correspond to the linear regression coefficients between the IM values and the corresponding seismic demands (\(\mathrm{d_{i}}\)) in logarithmic space. The uncertainty introduced in the probabilistic seismic demand model (PSDM) is expressed as the dispersion of the median demand, which is defined by the logarithmic standard deviation of the model:  
\[\beta_{\mathrm{D}|\mathrm{IM}}\cong\sqrt{\frac{\sum\left(\ln(\mathrm{d_{i}})- \ln(\mathrm{a}(\mathrm{IM})^{\mathrm{b}})\right)^{2}}{\mathrm{N}-2}} \tag{27}\]  
For the estimation of the probability of rocking overturn (\(\mathrm{P_{Ov}}\)), the problem should be treated as a categorical one. Therefore, the data should be grouped into nonoverturned and overturned data. In this study, the maximum likelihood estimator (MLE) approach is adopted in order to estimate the overturning vulnerability function parameters optimally (mean \(\mu\) and standard deviation \(\sigma\)) [73; 74]. The following expression gives the maximum likelihood function \(\mathrm{L}\):  
\[\mathrm{L}=\prod_{\mathrm{j}=1}^{\mathrm{m}}\mathrm{P_{j}}^{\gamma_{j}}\left( 1-\mathrm{p_{j}}\right)^{1-\gamma_{j}} \tag{28}\]  
where \(\mathrm{z_{j}}\) is a parameter of the binomial distribution, \(\mathrm{p_{j}}\) is the collapse probability of a certain IM level, \(\mathrm{m}\) is the number of IM values, and \(\mathrm{\Pi}\) denotes a product of all IM values.