## 2 Stochastic Subspace Identification Method  
The analysis of the stochastic subspace identification (SSI) method is based on the framework of the state-space model. To treat the measurement data with the SSI method, this method is derived from the continuous-time domain to discrete time. Because the SSI method can be used to process the output-only system, which is different from the deterministic state space, we consider the input to be a stationary random process that can be expressed as a random, discrete-time, state-space equation. Since the identification process of the SSI method can be implemented from the output measurement data only, the ambient excitation is assumed to be white noise input,without considering external force input. Therefore, the external force and the noise can be combined as white noise. To apply the measurement data to the SSI method, we can construct a Hankel matrix \([H]\) composed of the measurement data, from which the relationship between the different measurement channels and different sampling times are as follows:  
\[[H]_{2 li\times j}=\left[ \begin{array}{cccc}\underline{y}_{0}&\underline{y}_{1}& \cdots&\underline{y}_{j-1}\\ \underline{y}_{1}&\underline{y}_{2}&\cdots&\underline{y}_{j}\\ \vdots&\vdots&\ddots&\vdots\\ \underline{y}_{i-1}&\underline{y}_{j}&\cdots&\underline{y}_{i+j-2}\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ \underline{y}_{i}&\underline{y}_{i+1}&\cdots&\underline{y}_{i+j-1}\\ \underline{y}_{i+1}&\underline{y}_{i+2}&\cdots&\underline{y}_{i+j+1}\\ \vdots&\vdots&\ddots&\vdots\\ \underline{y}_{2i-1}&\underline{y}_{2i}&\cdots&\underline{y}_{2i+j-2}\\ \end{array} \right]_{2 li\times j}=\left[ \begin{array}{c}[Y_{p}]_{(li)\times j }\\ \cdots\cdots\cdots\cdots\cdots\\ \left[Y_{f}\right]_{(li)\times j}\\ \end{array} \right]=[\alpha]_{2 li\times 2n}[\hat{X}]_{2n\times j} \tag{1}\]  
where the upper half of this matrix is called \"the past\" and denoted \([Y_{p}]\), and the lower half of the matrix is called \"the future\" and is denoted \([Y_{f}]\)[16]. A conditional mean for Gaussian processes can be completely described by its covariance. Since the shifted data matrices are also defined as covariance, the projection can be calculated directly. Note that the state matrix estimated by Kalman filter, and the state-space model can be constructed by the measured output vector used to estimate the predictive-state matrix \([\hat{X}_{i}]\). The projection matrix \([\Omega]\) can be expressed as a product of the observability matrix \([\alpha]\) and the predictive-state matrix \([\hat{X}_{i}]\) of the Kalman filter in the following [18]:  
\[[\Omega] =E\Big{(}[Y_{f}][Y_{p}]^{T}\Big{)}\Big{(}E\Big{(}[Y_{p}][Y_{p}]^{T} \Big{)}\Big{)}^{\oplus}[Y_{p}] \tag{2}\] \[=[\alpha]_{li\times 2n}[\hat{X}_{i}]_{2n\times j}\]  
where  
\[[\alpha]_{li\times 2n}=\left[ \begin{array}{c}[C]\\ [C][A]\\ \ldots\\ [C][A]^{i-1}\end{array} \right]_{li\times 2n} \tag{3}\] \[[\hat{X}_{i}]_{2n\times j}=\left[ \begin{array}{cc}\hat{x}_{i}& \hat{x}_{i+1}&\cdots&\hat{x}_{i+j-1}\\ \end{array} \right]_{2n\times j} \tag{4}\]  
\([C]\) is the output/observation matrix; \(\oplus\) is Moore-Penrose pseudoinverse; \([A]\) is the system matrix; \(E[\cdot]\) is the expectation operator. In the first line of Equation (2), the first four matrices in the product introduce the covariance between channels at different time delays, and the last matrix in this product defines the conditions. By using the SVD analysis and choosing the effective singular-value number, \([\Omega]\) can be expressed in minimum order realization as:  
\[[\Omega]= [U][\Delta][V]^{T}\] \[=\left( \begin{array}{cc}[U_{1}]&[U_{2}]\\ \end{array} \right)\Big{(} \begin{array}{cc}[\Delta_{1}]&0\\ 0&[\Delta_{2}]\approx 0\\ \end{array} \right)\left( \begin{array}{c}[V_{1}]^{T}\\ [V_{2}]^{T}\\ \end{array} \right) \tag{5}\] \[\approx[U_{1}]_{li\times 2n}[\Delta_{1}]_{2n\times 2n}[V_{1}]^{T}_{2n \times j}\]  
where \([U]\) and \([V]\) are both unitary matrixes, and \([\Delta]\) is a matrix containing singular values.