## 2 Stochastic Subspace Identification Method
The dimension of \([\Delta_{1}]\) can, in general, be employed to estimate the system order or number of poles. However, in practical work, the partial diagonal terms of the singular-valuematrix \([\Delta]\) may be nonzero, produced by noise from the procedure of data acquisition and numerical truncation. Through the elimination of partial matrix \([\Delta_{2}]\), consisting of the smaller singular values, a minimum realization is obtained that results in a minimum order system representing the structural system. In Equation (5), we can, therefore, choose the number of effective singular values to obtain the minimum order realization through the SVD analysis of \([\Omega]\). From Equations (2) and (5), with appropriate partitioning of \([\alpha]\) and \([\hat{X}_{i}]\), the following equations can be written:  
\[ \begin{array}{l}[\alpha]=[U_{1}][\Delta_{1}]^{s}\\ [\hat{X}_{i}]=[\Delta_{1}]^{t}[V_{1}]^{T}\end{array}  \tag{6}\]  
where \(s+t=1\). Indeed, one possible choice is \([\alpha]=[U_{1}][\Delta_{1}]^{1/2}\) and \([\hat{X}_{i}]=[\Delta_{1}]^{1/2}[V_{1}]^{T}\), which appears to make both \([\alpha]\) and \([\hat{X}_{i}]\) balanced. However, poor computational efficiency may occur, caused by relatively large dimensions of \([\Omega]\). In this paper, we construct the data matrix \([\Omega][\Omega]^{T}\) composed of \([\Omega]\), and perform the SVD analysis of \([\Omega][\Omega]^{T}\) to determine the order of a structural system to be identified. It can be shown that the eigenvalue of \([\Omega][\Omega]^{T}\) is the square roots of the eigenvalues of \(([\Omega][\Omega]^{T})([\Omega][\Omega]^{T})\), and that the corresponding eigenvectors of \([\Omega][\Omega]^{T}\) are the same as those of \(([\Omega][\Omega]^{T})([\Omega][\Omega]^{T})\). The dimension \(li\times j\) of \([\Omega]\) can be reduced to the dimension \(li\times li\) of \(([\Omega][\Omega]^{T})([\Omega][\Omega]^{T})\), where \(j>>li\). Based on the above, the efficiency of modal estimation can be improved, and system order can be determined through the SVD analysis of \(([\Omega][\Omega]^{T})\big{(}[\Omega][\Omega]^{T})\). In addition, to further improve the efficiency of the SSI method, we consider extracting the predictive-state matrices \([\hat{X}_{extract1}]\) and \([\hat{X}_{extract2}]\) with a recursive relationship directly from the original predictive-state matrix \([\hat{X}_{i}]\), as described next. From Equation (2), the predictive-state matrix \([\hat{X}_{i}]\) can be obtained from the observation matrix \([\alpha]\) in the following:  
\[ \begin{array}{l}[\hat{X}_{i}]=[\alpha]^{\oplus}[\Omega]\\ =\left[\begin{array}{cc}\hat{\pi}_{i}&\hat{\pi}_{i+1}\end{array}\right. \cdots\qquad\hat{\pi}_{i+j-3}\quad\hat{\pi}_{i+j-2}\qquad\hat{\pi}_{i+j-1}\end{array}  \right]_{2n\times j}\end{array} \tag{7}\]  
From the measured stationary responses at \(n\) stations on a structure under test, we define a system matrix \([A]\), such that  
\[[A][\hat{X}_{extract1}]=[\hat{X}_{extract2}] \tag{8}\]  
where \([\hat{X}_{extract1}]\) is a predictive-state matrix of measured response from \([\hat{X}_{i}]\), and \([\hat{X}_{extract2}]\) is a predictive-state matrix of time-delayed response from \([\hat{X}_{i}]\) as follows  
\[ \begin{array}{l}[\hat{X}_{extract1}]=\left[\hat{\pi}_{i}\,\hat{\pi}_{i+1}\, \,\cdots\,\hat{\pi}_{i+j-3}\,\hat{\pi}_{i+j-2}\right]_{2n\times j-1}\\ [\hat{X}_{extract2}]=\left[\hat{\pi}_{i+1}\,\,\cdots\,\hat{\pi}_{i+j-3}\,\hat{ \pi}_{i+j-2}\,\hat{\pi}_{i+j-1}\right]_{2n\times j-1}\end{array}  \tag{9}\]  
Therefore, following almost the same procedure as used in Equation (7), the system matrix \([A]\) can be obtained through the least-squares method. By extracting the predictive-state matrixes \([\hat{X}_{extract1}]\) and \([\hat{X}_{extract2}]\) with a recursive relationship directly from the original predictive-state matrix \([\hat{X}_{i}]\), we can then avoid the step of reevaluating the predictive-state matrix at the next-time moment in the conventional SSI method, which can further improve the computational efficiency of the SSI method.