## 3 PCAWF- and ARPCAWF-Based OMA Methods and Their Limitations
### Theory of PCAWF  
When a linear structure is time-varying, its vibration response signals under operating conditions (unmeasured ambient loads) are nonstationary; the transformation matrix and PC matrix obtained from the PCA model are also time-varying. The basic principle of PCAWF is that the newly added data and the historical data matrix are assigned different weighting factors. The newly added data  
Figure 1: Active and passive control of vibration response based on PCA.  
should have a higher weight to better characterize the newest state. The historical data matrix should be assigned a lower weight to reduce the influence of these data on the newest processing data matrix. There are many different ways of assigning the weighting factors, such as the radiation weighting factor [38], tissue weighting factor [39], and normalized weighting factor [40] techniques. In this paper, we describe the exponential forgetting factor as an example. Figure 2 illustrates the principle of the exponential forgetting factor algorithm.  
Specifically, in Figure 2, \(\widehat{X}^{(i)}\in\mathbb{R}^{N\times i}\) is the weighted data. The blue curves represent the weight of each column of data, ranging from 0 to 1. As new data are added, the number of data columns increases and the weight of old data decreases. The range of the exponential forgetting factor \(\mu\) is [0, 1] and the newest time data vector is assumed to have the unit weight. The previous data are then weighted according to \(\widehat{X}^{(i+1)}=[\mu\widehat{X}^{(i)},\vec{x}(i+1)]\). Thus, the weighted data matrix for the PCAWF model is as follows:  
\[\begin{array}{l}\widehat{\mathbf{X}}^{(1)}=[\vec{x}(1)]\\ \widehat{\mathbf{X}}^{(2)}=[\mu\widehat{\mathbf{X}}^{(1)},\vec{x}(2)]=[\mu \vec{x}(1),\vec{x}(2)]\\ \widehat{\mathbf{X}}^{(3)}=[\mu\widehat{\mathbf{X}}^{(2)},\vec{x}(3)]=[\mu^{2 }\vec{x}(1),\mu\vec{x}(2),\vec{x}(3)]\\ \vdots\\ \widehat{\mathbf{X}}^{(i)}=[\mu\widehat{\mathbf{X}}^{(i-1)},\vec{x}(i)]=[\mu ^{(i-1)}\vec{x}(1),\mu^{(i-2)}\vec{x}(2),\mu^{(i-3)}\vec{x}(3),\cdots,\vec{x} (i)]\\ \widehat{\mathbf{X}}^{(i+1)}=[\mu\widehat{\mathbf{X}}^{(i)},\vec{x}(i+1)]=[ \mu^{(i)}\vec{x}(1),\mu^{(i-1)}\vec{x}(2),\mu^{(i-2)}\vec{x}(3),\cdots,\vec{x} (i+1)]\\ \vdots\\ \widehat{\mathbf{X}}^{(T)}=[\mu\widehat{\mathbf{X}}^{(T-1)},\vec{x}(T)]=[\mu ^{(T-1)}\vec{x}(1),\mu^{(T-2)}\vec{x}(2),\mu^{(T-3)}\vec{x}(3),\cdots,\mu^{(T- i-1)}\vec{x}(i+1),\cdots,\vec{x}(T)]\end{array} \tag{5}\]  
where \(i\) is the time step, \(\vec{x}(i)\) is the data vector from time step \(i\), and \(\widehat{\mathbf{X}}^{(i)}\) is the data matrix of the first \(i\) samples.  
The PCAWF model takes its transformation matrix and PC matrix from the weighted data matrix \(\widehat{\mathbf{X}}^{(i)}\) rather than the original data matrix \(\mathbf{X}^{(i)}\). A larger \(\mu\) can bring the average weight closer between the new data and old data. If \(\mu=1\), then \(\widehat{\mathbf{X}}^{(i)}=\mathbf{X}^{(i)}\) and the PCAWF model reduces to the PCA model. Furthermore, smaller values of \(\mu\) better reflect the newest state of the system and indicate that the algorithm has good tracking performance. However, the value of \(\mu\) should not be too small, otherwise it will increase the sensitivity of the estimated parameters. The forgetting factor \(\mu\) should be selected using prior knowledge of the time-varying speed of LTV structures. A fast time-varying speed requires a large forgetting factor. Similar to the window size in a moving window approach, the forgetting factor is a tuning parameter that varies depending on how quickly the normal process changes [31].  
Figure 2: Weight reduction process for old data.