# Automated Modal Analysis Using Stochastic Subspace Identification and Field Monitoring Data
## 3 Proposed Procedure
### Matrix Construction and Decomposition  
Admittedly, the size of the Hankel matrix is based on fundamental frequency, and \(2i\geq 1/f_{0}\Delta t\) is the minimal value to take into account for the Nyquist frequency. The choice of size of the Hankel matrix can affect the resultant decomposition performed by SVD. Theoretically, a non-centering (non-zero-mean) Hankel matrix does not inherently  
Figure 1: The proposed three-step procedure for implementing SSI for automated OMA.  
destabilize SVD; however, large mean values can dominate the decomposition, leading to reduced sensitivity or poorer low-rank approximations. Moreover, singular vectors and singular values often align with the dominant direction, which may include mean values if the Hankel matrix is not centering. A similar phenomenon can be found even if the responses of only a few modes are not zero-mean. For example, Figure 3a demonstrates singular values decomposed using different sizes of Hankel matrix. The singular values are unclearly decomposed when \(i=31\) and \(i=36\). Obviously, an inappropriate size choice makes the lagged vectors non-centering and eventually disturbs the distribution of singular values.  
Figure 2: The demonstration of data preprocessing for implementing SSI for automated OMA. (**a**) picker provided by normalized Arias intensity; (**b**) picker provided by normalized AIC; (**c**) Fourier spectrum and basic modal information.  
On the contrary, the phenomenon can be mitigated once a zero-mean modal response is secured, especially the fundamental mode. Considering that vibration modes are represented in singular values as pairs, the normalized differences between singular pairs are defined as  
\[\sigma_{\text{diff},p}=2\cdot\frac{\sigma_{2p}-\sigma_{2p+1}}{\sigma_{2p}+\sigma _{2p+1}}\;p=0,\;1,\;2,\ldots \tag{13}\]  
Figure 3b-d show the normalized differences between the first three singular pairs after implementing SSI-Data. The largest differences can be observed around the bottom left where \(2i\geq 1/f_{0}\Delta t\) is unsatisfied. Clearly, they wave across the size of the Hankel matrix, indicating that only some numbers of rows can bring about clear decomposition with very few differences. Notably, similar observations can be found after implementing SSI-Cov, although this is as significant as implementing SSI-Data. Figure 3e shows normalized differences between singular pairs.  
Figure 3: The demonstration of matrix construction for implementing SSI for automated OMA. (**a**) normalized singular values; (**b**) \(\sigma_{\text{diff}}\) for first mode; (**c**) \(\sigma_{\text{diff}}\) for second mode; (**d**) \(\sigma_{\text{diff}}\) for third mode; (**e**) \(\sigma_{\text{diff}}\) for first mode and integer multiples of fundamental periods.  
differences between the first singular pair accompanied by the integer multiples of the fundamental periods as  
\[i=\text{nint}\bigg{(}\frac{p}{2f_{0}\Delta t}\bigg{)}\ p=1,\ 2,\ 3,\ldots \tag{14}\]  
where \(\text{nint}()\) is an operator of finding the nearest integer. Evidently, the minimal value is consistent with past studies [43; 44]. It is apparent that the suggested values result in clearer decomposition, since the lagged vectors cover complete cycles, which are considered zero-mean. Therefore, Equation (14) is proposed for choosing the size of the Hankel matrix when implementing SSI and for use in the stability diagram, instead of consecutive numbers.