# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 1 Introduction  
The identification and localisation of damage in one-, two-, or three-dimensional structures through their mode shapes are widespread in the Structural Health Monitoring (SHM) community. It is well known that by inserting a localised discontinuity, the mode shapes diverge from their usual deflection path [1]. This principle has been extensively applied to investigate damage-induced variations in the mode shapes slope or curvature for both 1-dimensional beam-like [2] and 2-dimensional plate-like [3, 4] structures. In this regard, a review of classic approaches can be found in Reference [5]. These approaches consider modal curvatures [6], mode shape rotations [7], and/or several other Damage-Sensitive Features (DSFs) based on the structure's eigenvectors. In particular, mode shapes-based methods are preferred over other modal parameters for damage localisation, since they inherently have the spatial resolution needed for this specific task [8, 9]. However, even having established that the mode shapes and derived quantities can be exploited as a reliable and spatially dense DSFs, it remains to define how these features can be used for anomaly detection. The changes in the mode shapes can be detected, e.g., through a Machine Learning (ML) process, trained exclusively on the mode shapes extracted from the current state ofthe structure. This can be applied both to a known pristine condition or to an already-damaged structure, since the basis of outlier detection is to identify variations from the configuration \"as it is\" [10]. Indeed, no method can actually detect \"damage\", but rather its effects on the structural properties [11]. That is to say, the damage detection algorithms are inherently subject to false positives as they could mislabel noisy data for actual deviations. This problem is very commonly encountered with measurements from real-life applications. The economic, social, and safety concerns of structural false alarms must not be neglected, as they are one of the major reasons which still hampers the perceived usefulness of SHM and continuous monitoring, limiting their spreading. For example, for any fixed- or rotary-wing aircraft, an onboard Health and Usage Monitoring System (HUMS) would be useless, if not even dangerous, if constantly sending false alarms. These are all major practical issues; the latter one, for instance, can be arguably considered the main factor behind the industry reluctance to apply SHM systems at large scale in the last two decades [12], and the reduction of false alarms is still a matter of research as of 2020 [13]. The main concept of this work departs from the consideration that structural damages are rare events. Therefore, assuming a normal distribution for damage occurrences is both statistically unprincipled and potentially inaccurate with all the resulting risks. Thus, Rare Event Modelling (REM) should be preferably applied. For scalar values, this can be done following the well-known Extreme Value Theory (EVT), also known as Extreme Value Statistics (EVS) [14]. As the name suggests, the EV framework--a part of the more general theory of the order statistics--deals with the statistics of extremes. This represents an ideal framework for data-driven thresholding [13], not only for damage assessment but generally for several applications of novelty detection (as reviewed in Reference [15]). The EVT framework for novelty detection has been firstly proposed by [PERSON] [16], especially for applications with biomedical signals [17]. For SHM purposes, the EVT was firstly introduced by [PERSON] et al. [18] and applied on time series. [PERSON] et al. [19] further deepened these studies considering ARX models in the frequency domain. [PERSON] and colleagues subsequently used the EVS framework for the analysis of delaminated composite panels, with the use of wavelets [20] and time-reversal acoustics [21]. [PERSON] et al.