## 2 Materials and Methods
### Damage Detection  
The detection task, given the transformed strain data, is broken down to determine an appropriate threshold, which can reliably separate pristine samples from damaged ones. With the assumption that the transformed data shows high values at features spatially close to a damage and otherwise values close to zero, the following damage detection strategy is implemented: Given a matrix of transformed benchmark residual data \(\mathbf{\Delta}^{R}\), the row-wise (i.e., feature-wise) median is calculated as  
\[\text{median}\Big{(}\mathbf{\Delta}^{R}\Big{)}=\mathbf{\bar{\Delta}}^{R}, \tag{5}\]  
where \(\mathbf{\bar{\Delta}}^{R}\in\mathbb{R}^{L}\). The median is advantageous here because it is robust against possible bad prediction results in the benchmark data, i.e., high residual values. These can occur due to the nature of the generated data, which covers a broad range of physics-based, yet still hypothetical, load cases. Some of these might operate in quite unique value ranges and are therefore more difficult to predict. The result of Equation (5) is further generalized by computing the arithmetic mean over all sample medians  
\[\text{mean}\Big{(}\mathbf{\bar{\Delta}}^{R}\Big{)}=\widetilde{\mu}^{R}, \tag{6}\]  
where \(\widetilde{\mu}^{R}\) represents the lowest bound for a decision threshold \(t\).  
A major challenge in the given classification task is to account for the discrepancy between numerical model and the real structure. A straightforward approach to address this is to introduce a known, pristine expert sample in the form of a residual \(\delta^{E}\in\mathbb{R}^{J}\) from the available experimental data. In an application scenario, such an expert sample could be easily acquired during the initial setup of the SHM application with the target structure. With this sample available, its arithmetic mean over all features is computed as  
\[\text{mean}\Big{(}\delta^{E}\Big{)}=\mu^{E}, \tag{7}\]  
which yields a lower bound \(\mu^{E}\) for the test data. Since the goal is to identify outliers, it is sensible to compute the mean, which is more sensitive in this regard.  
A transformed sample of a pristine structure is expected to have a mean value that is close to \(\widetilde{\mu}^{R}\) or \(\mu^{E}\). In the worst acceptable case, it might possibly have a value slightly higher than \(\mu^{E}\). The inclusion of such a sample is controlled by a tolerance parameter \(\gamma\in[0,1]\). The threshold \(t\) is then computed as  
\[t=\text{max}\Big{\{}\widetilde{\mu}^{R},\mu^{E}\Big{\}}\cdot(1+\gamma). \tag{8}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Grid Searched Values** & **Final Value** \\ \hline Loss function of boosting process & - & absolute error \\ Maximum number of trees & \(\{200,300\}\) & 300 \\ Maximum depth of each tree & \(\{\text{None},40,60\}\) & 40 \\ Maximum number of leaves for each tree & \(\{31,40\}\) & 40 \\ Minimum number of samples per leaf & \(\{30,50\}\) & 50 \\ L2 regularization parameter & \(\{0,0.5\}\) & 0.5 \\ \hline \end{tabular}
\end{table}
Table 1: Non-default parameter values used for the scikit-learn implementation of the histogram-based gradient boosting regression tree, together with the values used in the parameter grid search.  
The edge-case, where the mean of an expert residual \(\mu^{E}\) is smaller then that of the lowest bound of the benchmark data \(\widetilde{\mu}^{R}\), is covered by the maximum value function. The detection of damage in a given test sample \(\delta\in\mathbb{R}^{J}\) ultimately comes down to the decision rule  
\[\text{mean}(\delta)>t. \tag{9}\]  
If the condition (9) is fulfilled, the SHM application labels the test sample as damaged, or otherwise as pristine. The mean of the transformed test sample \(\text{mean}(\delta)\) can also be interpreted as a damage index.