## 4 Selection of Seismic _IMs_ for the PCA Algorithm
### PCA Algorithm  
The PCA algorithm was first proposed by [PERSON] [29] in 1901 and was then extended to random variables by [PERSON] [30] in 1933; it is currently a commonly used data dimensionality reduction method. PCA utilizes a method that converts multiple variables into several principal components through dimensionality reduction techniques. These principal components are new variables that are composed of linear combinations of the original variables and that retain most of the information from the original variables. Moreover, the PCA algorithm does not require the data distribution to be taken into account. Assuming that there is a total of \(n\) pulse-like ground motions and the \(p\) seismic _IMs_ are selected, i.e., each seismic _IM_ has \(n\) data, therefore, a seismic _IM_ matrix of \(n\times p\) dimensions can be formed:  
\[ \begin{array}{l l l l}\textbf{IM}&=&\left[\begin{array}{cccc}IM_{1,1}&IM_{1,2 }&\cdots&IM_{1,p}\\ IM_{2,1}&IM_{2,2}&\cdots&IM_{2,p}\\ \vdots&\vdots&&\vdots\\ IM_{n,1}&IM_{n,2}&\cdots&IM_{n,p}\\ \textbf{IM}_{1}&\textbf{IM}_{2}&\cdots&\textbf{IM}_{p}\\ \end{array}\right]\\ &=&\left[\begin{array}{cccc}IM_{1,1}&IM_{1,2}&\cdots&IM_{1,p}\\ IM_{2,1}&IM_{2,2}&\cdots&IM_{2,p}\\ \vdots&\vdots&&\vdots\\ IM_{n,1}&IM_{n,2}&\cdots&IM_{n,p}\\ \end{array}\right]\\ \end{array}  \tag{2}\]  
where \(IM_{i,j}\) is the \(j\)-th \((j=1,2,\cdots,p)\) seismic _IM_ of the \(i\)-th \((i=1,2,\cdots,n)\) ground motion, and \(\textbf{IM}_{j}\) is the \(j\)-th vector of seismic _IM_, \(\textbf{IM}_{j}=\left[IM_{1,j}&IM_{2,j}&\cdots&IM_{n,j}\right]^{T}\)\((j=1,2,\cdots,p)\). Due to the differing measurement units of various _IMs_, there may be significant differences in the values of some _IMs_. Therefore, in accordance with Equation (3), each seismic _IM_ vector is standardized to eliminate differences in both the magnitude order and dimensions between each seismic _IM_:  
\[IM_{i,j}^{\prime}=\frac{IM_{i,j}-\mu_{j}}{\sqrt{s_{j}}} \tag{3}\]  
where \(M_{i,j}^{\prime}\) is the \(j\)-th seismic _IM_ value of the \(i\)-th ground motion after standardization processing; \(\mu_{j}\) is the mean value of the \(j\)-th seismic _IM_ vector, \(\mu_{j}=\sum\limits_{i=1}^{n}IM_{i,j}/n\); \(s_{j}\) is the sample variance of the \(j\)-th seismic _IM_ vector, \(s_{j}=\sum\limits_{i=1}^{n}\left(IM_{i,j}-\mu_{j}\right)^{2}/(n-1)\). After undergoing standardized processing, the _IM_ matrix in Equation (2) is transformed into:  
\[\textbf{{IM}}^{\prime}=\left[\textbf{{IM}}^{\prime}_{1}\quad\textbf{{IM}}^{\prime }_{2}\quad\cdots\quad\textbf{{IM}}^{\prime}_{p}\right] \tag{4}\]  
where \(\textbf{{IM}}^{\prime}_{j}\) is the vector of _j_-th seismic _IM_ value after standardized processing, \(\textbf{{IM}}^{\prime}_{j}=\left[\textbf{{IM}}^{\prime}_{1,j}\quad\textbf{{IM} }^{\prime}_{2,j}\quad\cdots\quad\textbf{{IM}}^{\prime}_{n,j}\right]^{T}\). According to the standardized _IM_ matrix \(\textbf{{IM}}^{\prime}_{j}\), the correlation coefficient matrix, \(R\), can be obtained as:  
\[\textbf{{R}}=\left[ \begin{array}{cccc}r_{1,1}&r_{1,2}&\cdots&r_{1,p}\\ r_{2,1}&r_{2,2}&\cdots&r_{2,p}\\ &\vdots&\\ r_{p,1}&r_{p,2}&\cdots&r_{p,p}\end{array} \right] \tag{5}\]  
where \(r_{l,m}\) is the correlation coefficient between the _l_-th and _m_-th seismic _IM_, \(r_{l,m}=\sum\limits_{k=1}^{n}\left(\textbf{{IM}}^{\prime}_{k,l}-\textbf{{IM}} ^{\prime}_{k,m}\right)/\left(n-1\right)\left(l,m=1,2,\cdots,p\right)\). If we calculate the eigenvalues \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{p}\) and corresponding eigenvectors \(\textbf{{\mu}}_{1},\textbf{{\mu}}_{2},\cdots\textbf{{\mu}}_{p}\) of the correlation coefficient matrix \(R\), where \(\textbf{{\mu}}_{l}=\left[\textbf{{\mu}}_{l,1}\quad\textbf{{\mu}}_{l,2}\quad \cdots\quad\textbf{{\mu}}_{l,p}\right]^{T}\) (\(l=1,2,\cdots,p\)), then, new \(p\) variables can be obtained:  
\[\left\{