# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 2 Gaussian Processes and Extreme Function Theory
### Test of the Performances of the Procedure  
To quantitatively estimate the performances of the described approach, this was tested on numerically simulated and experimental datasets with known damage conditions. For this purpose, three datasets--training, validation, and test--are defined. The training and the validation datasets, needed to preliminary set and validate the normality model offline, are made up of functions drawn from the undamaged conditions, while the test set includes both damaged and undamaged states. For clarity's sake, the several steps described in the previous Subsection are graphically depicted in Figure 2.  
In comparison to a preliminary version of the algorithm described in Reference [28] and tested for simple cantilever beams, the following enhancements have been included. Firstly, a data augmentation strategy was applied to overcome the limited amount of available measured data (numerical data were generated with the same scarcity as for the real data from the experimental case study). Dealing with the feature of transverse displacements, the latent function \(f(x)\) of Equation (6) is the resulting mean of all the original mode shapes and the resulting standard deviation indicates how data are spread out from the average. Both the mean and the standard deviation were evaluated for each channel, to consider the distribution for the specific measurement point and not of the whole dataset. After generating the desired number of noise-free copies, white Gaussian noise was added by iteratively reducing the Signal-to-Noise-Ratio (SNR) until the artificially simulated data fell completely within the set limits. In this case, the range was arbitrarily chosen to be \(\pm 2\sigma\) and not to cover the whole \(\pm 3\sigma\) range, since this restriction strengthens the \"normality\" condition (i.e., undamaged conditions, not far from the median value). It should be noted that data coming from the augmentation process are involved in the training and validation steps, while only original data are tested to assess the efficiency of the algorithm. Figure 2: The flow chart of the complete procedure. As a second improvement, a step of k-fold cross-validation was added, to avoid overfitting of model parameters estimations. In contrast to the usual validation procedure, the \(k\)-th holdout fold subset was used to train the Gaussian Process and estimate its hyperparameters, while keeping the remaining \(k-1\) folds for the CDF estimation and model testing. This inversion in the common paradigm of using the larger part of data for training step is due to the need to take into consideration only the lower 10% of the available data to estimate the minima form of CDF as defined in Equation (13). Subsequently, the training (TR), validation (VA), and test (TS) subsets were assembled from the given d structural configurations considered as the \"normality\" model (generally \(d=1\), the integer measurements data set) and \(d\)* structural configurations considered as damaged, each with \(n\) actual observations and \(n^{*}\) artificial copies. Table 1 describes the procedure settings, as applied for all the numerical and the experimental case studies for consistency. As said, the test set contains both undamaged (\(\mathbf{\varphi}_{und}\)) and damaged (\(\mathbf{\varphi}_{dam}\)) mode shapes. When a \(\mathbf{\varphi}_{und}\) is classified as damaged, it is recorded as a false positive; on the other hand, misclassified \(\mathbf{\varphi}_{dam}\) are considered as false negatives. Otherwise, as pictorially described in Figure 3, the results are labelled as true positives and true negatives.