# Automated Modal Analysis Using Stochastic Subspace Identification and Field Monitoring Data
## 3 Proposed Procedure
### System Realization  
After the matrix construction and decomposition, the structural system can be realized through singular values and vectors, including the modal parameters identified from the state (or system) matrix. In this step, the most important thing is the selection of the system order, \(n_{e}\), and the perfect one is \(n_{e}=2n\) in the case of simple structures. As stated in Section 2, this can be generally estimated according to the magnitudes of the singular values and SNR. However, this is always difficult and impractical in the case of complex structures or field applications. For example, \(n_{e}\) can be estimated as  
\[\sigma_{n_{e}}\geq C_{1}\cdot\sigma_{1}\geq\sigma_{n_{e}+1} \tag{15}\]  
where \(C_{1}\) is the criterion of this (first) approach and can be determined according to the ratio between the magnitudes of the modal response and the noise in the sense of percentage. Alternatively, \(n_{e}\) can also be estimated according to the ratio between the magnitudes of the total response and the noise, as  
\[\sigma_{n_{e}}\geq C_{2}\cdot\sum_{p}\sigma_{p}\geq\sigma_{n_{e}+1} \tag{16}\]  
where \(C_{2}\) is the criterion of this (second) approach in the sense of percentage. These approaches lack a clear definition of SNR and how it can be determined. Therefore, different approaches are proposed to estimate system order and are discussed in this study.  
The third approach is to detect the drop in the distribution of singular values, leveraging the gap between informative and noisy data. To facilitate this approach, consecutive differences are calculated and scaled using the latter singular values as  
\[\sigma_{p}^{d}=\frac{\sigma_{p}-\sigma_{p+1}}{\sigma_{p}}\ p=1,\ 2,\ 3,\ldots \tag{17}\]  
The significant drop can be determined by the largest values. In some cases, the largest gap may not happen between informative and noisy data, so a conservative one can be select as  
\[n_{e}=\text{max}\Bigg{\{}p\in\mathbb{N}|\frac{\sigma_{p}^{d}}{\text{max}( \sigma_{p}^{d})}>C_{3}\Bigg{\}} \tag{18}\]  
where \(C_{3}\) is the criterion of this (third) approach in the sense of percentage. [PERSON] originally proposed this approach with different steps, including taking logarithms, shifting, differentiating, and normalizing [50]. Equations (17) and (18) provide simplified steps with the same results.  
The consecutive differences for a series of singular pairs display sequential up-and-down values, along with the number of singular values. Lower consecutive differences indicate that two singular values are paired, and higher consecutive differences represent that they are not. On the opposite side, the singular values extracted from noisy data typically show irregular behavior in the consecutive differences. The last approach is to take advantage of this phenomenon and attempt to identify the unique boundary between informative and noisy data. Starting from the consecutive differences in Equation (17), a decay function is introduced as  
\[\rho_{q}=\sum_{p=1}^{q}\frac{1}{p}\sigma_{p}^{d} \tag{19}\]  
Through a similar concept from AIC, the system order can be selected as  
\[n_{e}=\max\bigl{\{}q\in\mathbb{N}\big{|}\rho_{q}>\rho_{r},\ r=q+1,\ q+2,\ q+3, \ldots\bigr{\}} \tag{20}\]  
The above equation finds, firstly, that the decay function is in steady descent, indicating that the consecutive differences after \(n_{e}\) exhibit no sequential up-and-down values along with the number of singular values. Most importantly, the last (fourth) approach has no additional criterion to be decided.  
Once the system order is determined, Equations (8) and (9) can be used to estimate the state (or system) matrix and output matrix characterizing the LTI structural system. Note that identifying the modal parameters from the state (or system) matrix requires the application of eigenvalue decomposition (ED) to the state (or system) matrix. A stability diagram and MAC are still needed to move out unstable and spurious poles, as shown in Section 2.2.