## 3 Advancements and Alternatives to Modal-Based Damage Identification
### Advancements to Machine Learning Methodologies for Damage Indentification  
As detailed for most modal-based damage features, their drawback predominantly stems from their susceptibility to environmental and operational influences; however, given the recent advancements in removing environmental and operational influences, as detailed in Section 3.1, more reliable damage feature sets can now be extracted. The final step is to implement a suitable outlier detection algorithm to separate damaged parameters from those of which are undamaged. Mahalanobis Squared-Distance (MSD) [131] is a common Gaussian-based multivariate outlier detection technique used in SHM that's based on detecting divergences from the mean and covariance matrix of a training dataset. The MSD is determined as per Equation (8), where \(\{X\}_{\zeta}\) is the potential outlier, \(\{\overline{X}\}\) is the mean of the training data, and \([\Sigma]\) is the covariance matrix of the training data. The success of the MSD is dependent on clean Gaussian distributed data of a sufficient sample size and/or number of variables, so that a reliable threshold value can be determined. \[D_{\zeta}=(\{X\}_{\zeta}-\{\overline{X}\})^{T}[\Sigma]^{-1}(\{X\}_{\zeta}-\{ \overline{X}\}) \tag{8}\]  
An example of the MSD in use in bridge SHM can be seen in [PERSON]'s [132] modal comparative study, where multiple damage features (frequencies, MAC, COMAC. and modal damping) were extracted for the first four modes from vibration data induced by a moving vehicle on a steel truss bridge subjected to four progressively induced damage scenarios. Initially, each feature was assessed individually per mode using the MSD. This provided mixed results, with damping ratios exhibiting almost no discernible damage sensitivity for any mode or damage scenario, while the other features managed slightly better, but exhibited little consistently across all modes. [PERSON] & [PERSON] [132] went on to demonstrate how the results can improve if a number of modal parameters are considered simultaneously per mode, and even further so if combined across all four modes. This method of feature stacking increases the dimensionality of the damage feature assessed and allows the utilisation of MSD's multivariate capacity. However, to successfully use MSD in this way requires a standardisation of the various damage features so that their covariance matrices are comparable in scale. For this, the authors used an adapted MSD algorithm known as the Mahalanobis-Taguchi System (MTS) [133]. The MTS is an efficient supervised learning method that sorts and removes variables in a multidimensional pattern recognition problem by using signal-to-noise ratios and orthogonal arrays to quantify each variable's contribution. This can be particularly beneficial when using multiple modal parameters for damage detection that exhibit inconsistent damage sensitivity when individually assessed. Unknown or damaged data is first normalised using the healthy data's mean and standard deviation. Although it has not seen much use in the SHM field as of yet, it has been widely used in medical diagnosis studies that consider a large number of variables. One comparative study [134] used a cancer diagnosis problem to assess MTS's performance to that of a Neural Network's and found that the MTS classification accuracy was higher than that of the NN overall, whilst also being much faster and having identified three variables for removal. A further advancement to the standard MSD is to pair it with the Minimum Covariance Determinate (MCD) estimator [99] that mitigates the uncertainty regarding the presence of outliers in training data. It achieves this by using an iterative process of subdividing the data into two groups and calculating their determinate in order to refocus the covariance matrix on the mode of the total dataset instead of the mean. The assumption of Gaussian distribution still holds for the data, but allows outliers to be ignored.