## 3 Advancements and Alternatives to Modal-Based Damage Identification
### Non-Modal Damage Sensitive Features
#### 3.3.2 Time Series Based Damage Sensitive Features  
Auto-Regressive (AR) models and their common adaptations (ARMA and ARMAX) are very versatile tools that can be used instead of modal parameters to extract DSFs such as residual errors or AR model parameters, provided that a suitable model order is selected [158]. Given the present paper's aim and scope, the reader is advised to inquire the following publications for a more detailed account of AR type models' theoretical base [159; 160]. However, it is important to note that their damage sensitivity arises from the assumption of stationarity, which allows for the detection of nonlinear behaviour in the vibration data by inducing changes in the residual errors and/or model parameters [161]. Residual errors are assumed to be Gaussian and are assessed using Gaussian-based outlier detection techniques such as MSD [162; 163] or Gaussian mixture models [164]. Multiple sensor locations may be used for the purposes of damage localisation; however, prior to fitting the models, the sensor data must be normalised in order to remove variable external condition influences such as traffic and temperature, so that all signals have similar statistical characteristics. The common method of normalising sensor data for the purpose of AR model fitting is to apply Equation (11) to the sensor data, where the normalised data \((x_{norm})\) is determined from the time series' mean (\(\mu_{i}\)) and standard deviation (\(\sigma_{i}\)) applied to each time step \((x_{i}(t))\). This also produces a much more stationary signal that is suitable for AR type models. \[x_{norm}=\frac{x_{i}(t)-\mu_{i}}{\sigma_{i}} \tag{11}\]  
[PERSON] [165] developed a novel ARMAX model that can identify changes in both stiffness and mass separately using output-only data by applying the ARMAX model to the standard equation of motion. Two sets of ARMAX models were constructed for the verification process; one for a baseline state and another for the damaged state. The percentage differences of the two model's ARMAX coefficients are taken as damage features. Their methodology was verified by two numerical models; one consisting of a 4-DOF mass/spring system, and the other being the IASC-ASCE benchmark problem [166] of a multi-story structure with incremental damage applied. The results show that the proposed methodology could consistently detect, locate, and quantify changes in mass and stiffness, independently of one another. This is due to the fact that the mass and stiffness are separated into the denominator and the numerator in the developed model. However, a real-world application is required to determine its true robustness, particularly under varying environmental conditions. [PERSON] [167] proposed another time-series-based damage detection model that incorporates environmental data and a full numerical model of the structure that extrapolates expected behaviour so that the assessed vibration response can be deemed normal or abnormal. The method is known as Polynomial Chaos Nonlinear Auto-Regressive with eXogenous input (PC-NARX) [168]. The Polynomial Chaos Expansion (PCE) allows the parameters to be characterised as random variables, for example; acceleration time histories are represented by their Probability Density Function (PDF) parameters, so that measured vibration data can be handled as a set of random variables. Thisspeeds up subsequent runtimes considerably, as large acceleration data sets can be reduced to a few representative values. Likewise, this methodology caters for the inclusion of structural uncertainty, by allowing structural properties and dimensions to be included in the numerical model as PDF variables. The PC-NARX then predicts the dynamic response of a structure through a process called meta-modelling. Validation of the PC-NARX was conducted by comparing its predictions and simulation speed to that of an FE model. A normalised one-step-ahead prediction error of 0.0074% was obtained by PC-NARX with a simulation speed 100 times faster than that of the FE model.