## 2 Stochastic Subspace Identification with Uncertainty
### Calculation of Uncertainty  
The pool of candidate modes obtained by SSI contains real modes but also spurious ones, which should be identified. The real modes have the peculiarity of appearing frequently within a specific frequency range according to the change in the model order. Considering this feature, a stabilization diagram plotting the frequencies of the candidate modes according to the change in the model order is used to sort out the real modes. However, there are no clear criteria about the setting of the specific frequency range nor the number of occurrences that will decide whether a candidate mode within this specific frequency range is a real mode. [PERSON], [PERSON] and [PERSON] [14] proposed a method for estimating the uncertainty bounds of modal parameters obtained from SSI based upon first-order sensitivity. Later, [PERSON] and [PERSON] [15] presented a technique for efficiently calculating these bounds. This technique makes it possible to obtain not only the modal parameters that can usually be derived from SSI but also the uncertainty of each of these parameters. Let \(\sigma_{f_{k}}\), \(\sigma_{\zeta_{k}}\) and \(\sigma_{\phi_{k,m}}\) be the standard deviation of the natural frequency \(f_{k}\), and let the standard deviation of the damping ratio be \(\tilde{\zeta}_{k}\), and the standard deviation of the \(m\)-th component \(\phi_{k,m}\) (\(m=1,\ \cdots,n_{y}\)) of the mode shape be \(\phi_{k}\). Since the uncertainty of the modal parameters tends to be small in the case of a real mode, this uncertainty can be exploited as an index discriminating real and spurious modes. In addition, similar modes exhibit similar modal parameters as well as similar uncertainty. Therefore, the uncertainty can also be used as an index for deciding whether two different candidate modes can be assumed to be an identical mode. Among the uncertainties in modal parameters, the uncertainty in mode shapes can be defined as the uncertainty of MAC. [PERSON], [PERSON] and [PERSON] [18] presented a method for calculating uncertainty of MAC in a strict manner. Another method is to use a Gaussian distribution approximation based on the Central Limit Theorem as shown below. Of these two methods, the former method provides a more accurate uncertainty of MAC. The distribution of the MAC index of a mode can be obtained from the average and standard deviation of each component of the mode shape. The variability of \(\phi_{k,m}\) is derived from the normal distribution of complex numbers \(\tilde{\phi}_{k,m}\sim CN(\phi_{k,\ m},\ \sigma_{\phi_{k,m}}^{2})\). The altered mode shape cluster \(\tilde{\phi}_{k}\) of \(\phi_{k}\) can be obtained if this process is applied to all the components. The distribution of the MAC index can be drawn by calculating the MAC indices between the as-obtained \(\tilde{\phi}_{k}\) and \(\phi_{k}\). This study drew the distribution of the MAC index by finding 1000 altered mode shapes. Figure 1 compares the cumulative probability distribution (in blue) for the MAC indices of mode shape \(\phi_{k}\) and altered mode shape \(\tilde{\phi}_{k}\) and the cumulative probability distribution (in red) of the normal distribution with average \(\mu_{MAC_{k}}\) and standard deviation \(\sigma_{MAC_{k}}\). When the cumulative probability is less than 50%, the sample distribution and the normal distribution are seen to be in relatively good agreement. However, the curves deviate from each other when the cumulative probability becomes higher than 50%. Nevertheless, since the subsequent process uses only that part of the MAC index distribution with cumulative probability lower than 50%, the discrepancy observed above does not constitute a problem. Figure 1a relates to a case with high average and low standard deviation, and Figure 1b to a case with low average and high standard deviation. Both cases show that the MAC index distribution can be approximated by a normal distribution.