## 2 Methods
### RQ Decomposition  
The output-only stochastic methods aim to estimate system matrices directly from measured acceleration response. To this end, output measurements are re-structured in order to construct Block Hankel matrix of SSI algorithm. Hankel matrix provides an enhanced form to take advantages of its appealing properties in linear algebra. Hankel matrix is a full-rank block matrix that transforms measurement row vector \((Y)\) into block matrix which is identical along matrix's anti-diagonals. Hankel matrix is partitioned into past and future input data as shown in Equation (1). It is efficient to set the size of Hankel matrix into \(2i+j+1\) columns where \(i\) is number of rows and \(j\) is number of columns. In the first step, some characteristic values of extended observability matrix and system-order are calculated. For the second step, system matrices are calculated from the extracted observability matrix. Differences among subspace algorithms are in the way the system matrices are calculated from the observability matrices. In SS-DATA, first, state sequences are extracted from the extended observability matrix and then system matrices are derived solving a set of least squares equations [42]. A unifying theorem is adapted to determine extended observability matrix and system-order by calculating oblique projection. The theorem is based on rearranging the extracted response measurements \((Y_{k})\) data into Hankel block matrix. \[Y_{0|2i-1}=\left[ \begin{array}{cccc}y_{0}&y_{1}&\ldots&y_{j-1}\\ \vdots&\vdots&\ddots&\vdots\\ \dfrac{y_{i-1}}{y_{i}}&y_{i}&\ldots&y_{i+j-2}\\ \hline y_{i}&y_{i+1}&\ldots&y_{i+j-1}\\ \vdots&\vdots&\ddots&\vdots\\ y_{2i-1}&y_{2i}&\ldots&y_{2i+j-2}\end{array} \right]=\left[ \begin{array}{c}Y _{0|2i-1}\\ \hline Y_{0|2i-1}\end{array} \right]=\left[ \begin{array}{c}Y_{p}\\ \hline Y_{f}\end{array} \right] \tag{1}\]  
where \(p\) and \(f\) subscripts, respectively, denote the past and future sequences of the output data. \(RQ\) decomposition is used to calculate oblique projection of Hankel matrix. The resultant of \(R\) and \(Q\) matrices are upper triangular and orthogonal matrices, respectively. \[Y_{0|2i-1}=\left[ \begin{array}{c}Y_{p}\\ \hline Y_{f}\end{array} \right]=\left[ \begin{array}{c}Y_{0|i-1}\\ \hline Y_{ii}\\ Y_{i+1|2i-1}\end{array} \right]\stackrel{{ RQ}}{{\rightarrow}} \left[ \begin{array}{cccc}R_{11}&0&0&0\\ R_{12}&R_{22}&0&0\\ R_{13}&R_{23}&R_{33}&0\\ R_{14}&R_{24}&R_{34}&R_{44}\end{array} \right]\left[ \begin{array}{c}Q_{1}{}^{ T}\\ Q_{2}{}^{T}\\ Q_{3}{}^{T}\\ Q_{4}{}^{T}\end{array} \right], \tag{2}\]  
Since \(R\) is a triangular matrix the computational burden to estimate corresponding system matrices are dramatically reduced by using this decomposition technique. \[O_{i}=Y_{p}/Y_{f}=\left[R_{Y_{p}}\right]\left[R_{Y_{f}}\right]^{T}, \tag{3}\]  
Figure 1: Diagram of the stochastic subspace identification (SSI)-DATA algorithm. where \(Y_{p}^{+}\) and \(Y_{f}^{-}\) are obtained by shifting the borders of the future and past data block down. \[O_{i-1}=Y_{f}^{-}/Y_{p}^{+}, \tag{4}\]  
Principle angle and direction between two subspaces and can be determined using SVD. The cosines of the principle angles \((U\) and \(V)\) are denoted by the singular values \((S)\). \[W_{1}O_{i}W_{2}=USV^{T}, \tag{5}\]  
where \(W_{1}\) and \(W_{2}\) are weighting matrices of the singular angles to estimate system order. The system order \((n)\) is a vector formed by the diagonal elements of the singular value matrix. Three methods are defined for determination of the weighting matrices including:  
* Principle component (PC);
* Unweighted principle component (UPC); and
* Canonical variate analysis (CVA). Assigning two weighting matrices of \(W_{1}\) and \(W_{2}\) allows to draw a proper state-space basis of an identified model. SVD is one of the most powerful matrix decomposition techniques from linear algebra.