## 2 Materials and Methods
### Damage-Sensitive Features
#### 2.1.1 Transmissibility-Based Features  
Transmissibility is defined as the amplitude of the frequency-response function of an input-output pair of dynamic sensors. Typically, a transmissibility value that exceeds 1 indicates amplification and occurs when the input frequency lies in the vicinity of a characteristic frequency of the dynamic subsystem defined by the locations of the sensor pair. In buildings, peaks in transmissibility are linked to the stiffness and mass distribution of the structure above the reference (lowest) sensor. Thus, assuming that the mass does not change significantly in the short term, a change in the transmissibility indicates stiffness  
Figure 1: DSFs are extracted from vibration-based monitoring data and allow for detection of the onset of damage and the quantification of its severity over time. In addition, multiple sensors enable the definition of substructures through sensor pairs and consequently convey information regarding the location of damage. reduction, as encountered both during nonlinearity (which can be temporary or recoverable) as well as due to residual damage. In buildings, especially those featuring stiff floor diagrams such as reinforced concrete slabs, each floor can be considered a separate degree of freedom (DOF) and thus, transmissibility input-output pairs can be constructed for each pair of floors. As seismic excitation is ground-borne, the input DOF in this investigation is chosen as the lower floor of the pair. The use of transmissibility for damage detection has been proposed before [55; 72; 73], as stiffness reduction in turn provokes a reduction in the frequency of fundamental modes of the substructure defined by the input-output sensor pair, for which transmissibility is calculated using Equation (1):  
\[T_{io}(\mathbf{\omega_{m}})=\frac{P_{oo}(\mathbf{\omega_{m}})}{P_{ii}(\mathbf{\omega_{m}})}\, \tag{1}\]  
where \(P_{ik}\) denotes the spectral density of an acceleration signal \(\bar{x}_{k}\), while the subscripts \(i\), \(o\) refer to the input and output signal, respectively, and \(\mathbf{\omega_{m}}\) is the frequency band of interest. In the context of purely data-driven assessment, three DSF are formulated to track the change in transmissibility between a healthy _reference_ state and the current (unknown) building state. The first DSF, \(Tmax_{io,m}\), returns the frequency value that maximizes the transmissibility in a given bandwidth \(\mathbf{\omega_{m}}\):  
\[Tmax_{io,m}=\underset{\omega\in\mathbf{\omega_{m}}}{\text{argmax}}\,T_{io}(\mathbf{ \omega_{m}})\, \tag{2}\]  
where \(T_{io}\) is the transmissibility between the input signal (\(\bar{x}_{i}\)) and the output signal (\(\bar{x}_{o}\)), as defined in Equation (1). The DSF can then, for instance, be taken as the relative difference between the reference and the new state. While \(Tmax_{io,m}\) only involves the peak in transmissibility, the frequency centroid, \(Tcnt_{io,m}\), relates to information from the entire frequency bandwidth, \(\mathbf{\omega_{m}}\), as defined in Equation (3):  
\[Tcnt_{io,m}=\frac{\sum_{i\in m}\omega_{i}\,T_{io}(\omega_{i})}{\sum_{i\in m} \,T_{io}(\omega_{i})}. \tag{3}\]  
where \(\mathbf{\omega_{m}}\) and \(T_{io}\) have already been defined. Again, the change in \(Tcnt_{io,m}\) can be evaluated as the relative difference between a reference and a target (current) state. A third transmissibility-based DSF is defined as the vector colinearity, which is used to quantify the similarity between the _reference_ transmissibility and the one measured during a strong ground motion.