## 3 Advancements and Alternatives to Modal-Based Damage Identification
### Advancements to Environmental & Operational Challenges
#### 3.1.2 Pattern Recognition Methods  
One drawback of many data normalisation techniques is the requirement of environmental or operational data which may not be available. In this case, Principle Component Analysis (PCA) may potentially be employed by considering the first principle components as representative of environmental-induced variation. If these principle components are removed, one should be left with a dataset representing the true behaviour of only the bridge. [PERSON] et al. (2013) implemented this technique using simulated linear data and, later, picewise linear data ([PERSON] et al., 2014), by increasing the PCA model order to account for the nonlinear response to sub-zero temperatures. The premise of this technique lies in utilising PCA's dimensionality reduction capabilities in a novel way. It is considered that environmental variables (temperature, wind, humidity, water level, etc.) or operational variables (vehicle speed and mass distribution) will each increase the dimensionality of the measured vibration signal. The number of dimensions can be identified via PCA by counting the non-zero eigenvalues of the covariance matrix ([PERSON] et al., 2014), which represent the variables' unique, shared erroneous variance ([PERSON] et al., 2014). Once identified, one can remove the environmental and operational influences, without the need to physically measure them. This method may be slightly subjective and has been shown to be sensitive to noise, but can be considered as an alternative for linear problems when no recorded environmental data is available. Similar to PCA, Factor Analysis (FA) has been shown to reveal linear dependencies and correlations between multiple variables in a dataset [107], for example, natural frequencies. However, instead of directly measuring the overall variance like PCA, FA assumes that there are a smaller number of underlying, unobservable factors called \"latent common factors\" (\([\xi]\) in Equation (7)) which are assumed to be independent with zero mean and unit variance that describe the variables' influence on the data. The number of latent common factors is limited in number to the number of variables in \([X]\), and each one attains a \"factor load\" \([\Lambda]\) which quantifies their influence and allows for their removal. Each factor load is equivalent to a principal component multiplied by the square root of the corresponding singular value. Once the latent common factors \([\xi]\) and factor loads \([\Lambda]\) are removed from the measured variables \([X]\), the remaining data are known as \"unique factors\" \([\varepsilon]\), which should be a diagonal matrix of independent variables that are unaffected by environmental and operational effects, and as such, can be used for damage detection purposes [108]. As this is a supervised machine learning methodology, sufficient training data is required for a successful application that captures all expected daily and seasonal variations. [PERSON] et al. [109] successfully implemented this technique in their assessment of output-only vibration-based damage detection under changing environmental conditions. \[[X]=\ [\Lambda][\xi]+[\varepsilon] \tag{7}\]  
[PERSON] & [PERSON] [110] advanced the theory of FA in SHM to cater for non-linearities by incorporating linear FA into a Neural Network (NN) type format that allows multiple linear factor models within each measured variable. Both [PERSON] & [PERSON] [110] and [PERSON] [108] assessed non-linear FA on the Z-24 Bridge and concluded that it is a feasible damage detection methodology and that it is less susceptible to over-fitting-related issues when compared to other non-linear methods. However, it should be noted that although FA assumes that the latent common factors are independent of each other and Gaussian in nature, this is not always the case in reality, as it is known that temperature and humidity are correlated and not all environmental and operational variables are normally distributed.