## 2 Materials and Methods
### Extraction of Damage Indicators  
The DSFs, presented in the previous Section 2.1, are either extracted from AVs or on the basis of the GM signal. In the context of continuous monitoring, the separation of the GM-induced response from AVs requires temporal delimitation of the continuously acquired GM signal. Therefore, in this work, a triggering condition based on the ratio between short-time average and long-time average (STA/LTA) of absolute acceleration values, recorded at the base of a building, is used [39; 80], as indicated in the top subplot of Figure 2. Then, the GM signal is split into shorter time windows of a fixed pre-defined window length in order to track the evolution of nonlinearity and pick up temporary shifts in the evaluated DSFs (see bottom subplot of Figure 2). As the majority of DSFs involve discrepancies between reference (typically \"healthy\") and newly acquired data, it is essential to adopt appropriate metrics for comparison. Due to the uncertainties that underlie monitoring data, primarily due to noise contamination, probabilistic metrics offer a more complete representation of the observed discrepancies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DSF** & **Domain** & **Signal** & **Nonlinearity** &
\begin{tabular}{c} **Residual** \\ **Damage** \\ \end{tabular} & **Localization** & **Equation** \\ \hline \(TAC\) & f & AV \& GM & ✓ & ✓ & (✓) & (4) \\ \(Tmax\) & f & AV \& GM & ✓ & ✓ & (✓) & (2) \\ \(Tcnt\) & f & AV \& GM & ✓ & ✓ & (✓) & (3) \\ \(WLC\) & tf & AV \& GM & ✓ & ✓ & ✓ & (6) \\ \(CohAC\) & f & GM & ✓ & & & (4) \\ \(CE\) & tf & GM & ✓ & & ✓ & (9) \\ \(TS\) & tf & GM & ✓ & & ✓ & (10) \\ \(ME\) & tf & AV \& GM & ✓ & ✓ & & (7) \\ \(MER\) & tf & AV \& GM & ✓ & ✓ & ✓ & (8) \\ \(KPRX\) & t & AV \& GM & ✓ & ✓ & ✓ & (11) \\ \(LFIT\) & t & GM & ✓ & & ✓ & (12) \\ \(\Delta f_{n}\) & f & AV & & ✓ & & \\ \(PAD\) & f & AV & & ✓ & ✓ & (15) \\ \(MAC\) & f & AV & & ✓ & & \\ \(\Delta MSC\) & f & AV & & ✓ & ✓ & (14) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the damage-sensitive features (DSFs) reviewed in Section 2.1. When comparing a single DSF value with a reference distribution, the Mahalonobis distance [81] can be adopted, which reflects a point-to-distribution metric, and is derived using Equation (16):  
\[d_{M}=\sqrt{(\mathbf{x}-\mathbf{\mu})^{T}\Sigma^{-1}(\mathbf{x}-\mathbf{\mu})}\,, \tag{16}\]  
where \(\mathbf{x}\) is the vector of DSFs characterizing a new state, \(\mathbf{\mu}\) is the vector containing the mean of the reference (\"healthy\") distributions of the DSFs and \(\Sigma\) is the corresponding covariance matrix. The traditional formulation of the Mahalonobis distance (see Equation (16)) combines multiple DSFs and adopts a Euclidian distance that is agnostic to the direction of a change. However, the adoption of a signed distance (which reflects the direction of a change) may be important for certain DSFs; for instance, in the case of \(KPRX\) and \(Tcnt\), an increase in stiffness is unlikely to result from nonlinearity. Therefore, a slightly differentiated formulation of \(d_{M}\) is proposed, per the normalized distance for a single DSF, as presented in Equation (17)  
\[d_{MS}(Dl^{e})=\frac{Dl^{e}-\mu(Dl^{h})}{\sigma(Dl^{h})}\,, \tag{17}\]  
Figure 2: Extraction of damage indicators from vibration-based monitoring data. A triggering algorithm based on the ratio between short-time average and long-time average (STA/LTA) is used to frame the GM signal, which is then decomposed into shorter windows for computation of the associated DIs. where \(\mu\) and \(\sigma\) reflect the mean and standard-deviation, respectively, and the superscripts \(e\) and \(h\) refer to the newly acquired and reference (\"healthy\") data, respectively. A second probabilistic distance metric can be defined on the basis of splitting the acquired records into shorter time-windows (as shown in Figure 2), or by considering multiple windows of AV.