## 3 Damage Detection Algorithm  
The chosen feature, the TEC coefficients, has been used to train a machine learning (ML) algorithm, obtaining a model of the structural response of interest on unaltered conditions. The algorithm applied is essentially the one that is described in [21], being the proposed DSF, and not the ML procedure utilised, the novelty of this paper; the procedure is only briefly restated for completeness. As for any ML approach, two phases exist--training and testing--where the former can be executed offline, while the latter may be performed online. During the training phase, the pattern of the DSF coming from the baseline model is 'learnt' by the algorithm, which builds a statistical model out of it. This model is then used as a comparison for the same feature when extracted from the structural response under unknown conditions. A metric of distance between any test case and the reference model is then utilised to discern whether these conditions under inspection correspond to a structural change that is substantial enough to be linked with damage. This threshold value, as will be better explained later, is derived via statistical means. This classic statistical framework is well described in [59]. Signal pre-processing on the acceleration THs was executed, as indicated in [22]. For \(s\) acquisition channels, \(n_{tr}\) realisation and thus \(n_{tr}\)'s THs for training the ML algorithm, it is then possible to extract \(n_{tr}\) feature vectors, i.e., \(\epsilon^{(i)}\in\mathbb{R}^{s\cdot(L-1)\times 1}\), where \(i=1,\ldots,n_{tr}\) represents the \(i\)-th vectors of \(L\) CCs, where \(L\) has been arbitrarily chosen and set equal for all of the alternatives tested here (both numerically and experimentally) to enforce comparability; note that the size \(L-1\) derives from the first CC, \(\mathbf{c}^{(i)}[0]\), being always discarded to mitigate the DC component and other input-specific effects [20]. The following hypotheses have been made. First, the cepstral features are assumed to follow a Gaussian multi-variate distribution (which is reasonable for a large enough training set and very commonly used for ML); therefore, all \(\mathbf{c}^{(i)}\) are considered as being practically uncorrelated and thus independent identically distributed (i.i.d.) vectors. This is necessary to completely define the 'normality' model (i.e., the baseline model of the unaltered reference situation) by means of its sample covariance matrix \(\mathbf{S_{tr}}\) and the sample mean vector \(\mathbf{m_{tr}}\) alone. These can be computed, respectively, as  
\[\mathbf{S_{tr}}=\frac{1}{n_{tr}}\sum\
olimits_{i=1}^{n_{tr}}\Big{(}\mathbf{c}^{(i )}-\mathbf{m_{tr}}\Big{)}\Big{(}\mathbf{c}^{(i)}-\mathbf{m_{tr}}\Big{)}^{T} \tag{10}\]  
and  
\[\mathbf{m}_{tr}=\frac{1}{n_{tr}}\sum\
olimits_{i=1}^{n_{tr}}\mathbf{c}^{(i)} \tag{11}\]  
The Squared Mahalanobis Distance (SMD) of a \(d\)-dimensional point \(\mathbf{x}\) from the baseline model is used as a damage metric; \(\mathbf{x}\) corresponds to one element of the test set, i.e., to a single test feature vector \(\overline{\mathbf{c}}\). The physical meaning behind the use of SMD is that features coming from more different structural conditions will be more distant in the feature space; the advantages of this approach, which is the favourite (and often the default) option for outlier detection, are well-known and properly described in [21]. Basically, it relies on the sample covariance matrix \(\mathbf{S_{tr}}\); this allows for accounting for feature variability under confounding factors, such as temperature variation, wind speed, or operational conditions, as long as the given samples are measured during these different, non-damage-related external conditions. The SMD can be analytically defined as  
\[\mathcal{D}^{2}(\mathbf{x})=\left(\mathbf{x}-\mathbf{m}_{tr}\right)^{T} \mathbf{S_{tr}}^{-1}(\mathbf{x}-\mathbf{m}_{tr}) \tag{12}\]  
and the procedure can be easily extended and generalized to multiple elements in the test set.