# Guided Wave-Based Monitoring of Evolution of Fatigue Damage in Glass Fiber/Epoxy Composites
## 4 Statistical Multivariate Outlier Analysis  
In this study, the experimental tests are under controlled laboratory condition, and the guided waves signals are not affected by various uncertainties. However, in real structural applications with operational environmental effects, the guided wave signals will being inevitably influenced by environmental conditions such as temperature and loads, and contaminated by measurement noise. Thus a statistical analysis is needed to consider these various uncertainties. Since outlier analysis have been demonstrated as a robust unsupervised learning pattern recognition tool for damage detection [40; 41; 42; 43; 44], it is employed in this study to analyze the WPT features obtained from the response guided wave signals under different fatigue states.  
In statistics, an outlier is an observation that is significantly different from the rest of the population and the outlier is believed to be generated by an alternative mechanism, for example, damage in structures [40; 41]. Statistical outlier analysis can be considered as a special class of pattern recognition, which classifies the abnormal data or state from the normal ones. To determine whether an observed data is an outlier, discordancy tests are usually performed. In the case of multivariate data, the discordancy test is based on deviation statistics and given by  
\[D_{\xi}=(\left\{\mathbf{x}_{\xi}\right\}-\left\{\mathbf{\breve{x}}\right\})^ {T}[\Sigma]^{-1}(\left\{\mathbf{x}_{\xi}\right\}-\left\{\mathbf{\breve{x}} \right\}) \tag{9}\]  
where \(D_{\xi}\) is the Mahalanobis squared distance (MSD), \(\mathbf{x}_{\xi}\) is the measured data corresponding to the potential outlier, and \(\mathbf{\breve{x}}\) and \(\Sigma\) are the mean vector and covariance matrix of the samples, respectively. Superscript \(T\) indicates transpose. The latter two values may be calculated with or without the potential outlier depending upon whether inclusive or exclusive measures are preferred [40]. In this study,the statistics is exclusively computed from the signal features, i.e., WP energy vectors defined in Equation (8) in the pristine state, without including potential outliers, i.e., WP energy vectors in the fatigue states.  
In order to consider the various uncertainties on guided wave signals and construct a suitable mean vector and a covariance matrix, the guided wave signals in pristine state are copied 500 times and each copy is subsequently corrupted with amplitude fluctuation and measurement noise as described in Equation (10).  
\[S^{c}(t)=(1+e)\times S^{e}(t)+\sigma \tag{10}\]  
where \(S^{e}(t)\) is the experimentally obtained guided wave signals and \(S^{c}(t)\) is the corrupted wave signals, \(e\) is an random number to reflect the amplitude fluctuation caused by environmental conditions, and \(\sigma\) is the measurement noise which is usually modeled as Gaussian white noise. In this study, we assume that \(e\) comes from a uniform distribution \([e_{\text{min}},e_{\text{max}}]\), and the level of measurement noise is described by signal-to-noise ratio (SNR). Then WPT is applied to these copies to obtain the signal features. For determining whether an observation of signal features or its MSD is an outlier, a proper threshold value \(D_{th}\) should be calculated. If the MSD is less than \(D_{th}\), it can be considered that the change of response guided waves is introduced by environmental conditions or measurement noise; otherwise, it is because of fatigue damage occurrence and accumulation.