## 1 Introduction  
During the lifecycle, structures are usually subjected to different types of loading stages, from service conditions to extreme events, e.g., earthquakes, tsunamis, and hurri-canes, which cause different degrees and types of structural damage leading to downtimes, losses, injuries, and sometimes mortalities. Therefore, structural health monitoring (SHM) has become an important domain of research and applications in structural engineering during the last few decades. Several approaches and damage criteria have been developed for SHM, and many of them utilize vibration-based damage detection, and have received considerable attention in the context of a statistical pattern recognition paradigm [1]. This paradigm includes four major steps: (1) operational evaluation, (2) structural data collection, (3) damage-sensitive feature extraction, and (4) statistical model development for classification, where engineers use the collected signals from sensors to determine the location, type, and extent of the damage and make decisions (e.g., repair strategy) about the health status of the structures. Damage feature extraction is an open subject in statistical pattern recognition. Many studies focus on how to locate, extract, and interpret the damage feature [2; 3; 4; 5]. Since the turn of this century, time series (TS) modeling of vibration signals using a family of auto-regressive (AR) models was found to be effective in damage detection and has been used to capture damage features in structures. According to [PERSON] et al. [6], past studies using AR series models are grouped into: (1) residual-based and (2) coefficient-based models, where the former identifies damage through the residuals computed from the difference between the measured and fitted data, and the latter uses the coefficients of the fitted model as damage features. The residual-based approach is founded on the facts that a well-fitted model trained by undamaged data is not a good fit for damaged data and that residuals increase with the increase of the damage. [PERSON] & [PERSON] [7] constructed a two-stage prediction model combining AR and AR with exogenous (ARX) models for fitting acceleration signals. They defined a damage-sensitive feature related to the standard deviation of the residual error using the AR-ARX model. [PERSON] et al. [8] designed a wireless unit that included implementation of the fast Fourier transform and TS modeling. In this unit, the ARX input model was used to fit the absolute acceleration and a damage feature related to the ratio of the standard deviations of the residuals of the ARX model obtained from the undamaged and damaged structural accelerations was also used. A damage state is identified if this ratio exceeds a threshold value. Subsequently, [PERSON] et al. [9] improved this algorithm using a normalized relative inter-story acceleration instead of a single floor acceleration, making the algorithm more robust and capable of detecting minor damage patterns. Other studies [10; 11; 12; 13] modified the processing procedure and format of the residuals for simulated or experimental data. The coefficient-based modeling has been shown to be more effective in many TS studies. [PERSON] et al. [2] proposed an algorithm using an auto-regressive moving-average (ARMA) model for the vibration signals with a damage feature expressed as a function of the first three AR coefficients, and localization indices introduced based on the AR coefficients. The algorithm detected the existence of all damage patterns, including minor ones. [PERSON] et al. [3] selected the first three regressive coefficients as the feature vector and used a Gaussian mixture model and Mahalanobis distance to detect, quantify, and localize the damage. [PERSON] [14] proposed a substructuring approach that combined ARMA with an exogenous (ARMAX) model to identify localized damage in any story of a shear building with a limited number of sensors. Recently, [PERSON] et al.