# Health Monitoring of Civil Infrastructures by Subspace System Identification Method:
## Vibration-Based Damage Detection (VDD)
### Time-Domain (TD) Methods
#### 3.2.2 Auto-Regressive Moving Average (ARMA) Methods  
Auto-regressive moving average (ARMA)-based methods outperformed the purely statistical methods. The auto regressive (AR) part aims to model linear function time-history and the moving average (MA) part aims to determine the moving average of the time-series [71]. The general structure of the AR-based models depends on which of the A(q), B(q), C(q), D(q), and F(q) polynomials are used in the model. Equation (1) shows the general structure of AR-based models. \[A(q)y(t)=\frac{B(q)}{F(q)}u(t)+\frac{C(q)}{D(q)}e(t) \tag{1}\]  
Table 1 shows the most common ARMA models that were used in the system identification of civil engineering structures. Several studies have been conducted while using autoregressive-based methods for the identification of damage in structures. Table 2 presents a review of applied ARMA methods in SHM and modal analysis with a focus on the test methods, damage features, and the incorporated pattern recognition techniques. The structures range from a very simple model of a mass spring system to more elaborate real world structures, as shown in Table 2. In many cases, the coefficients of the incorporated AR-based algorithm are directly used as damage features. For modal analysis, the extracted system parameters \begin{table}
\begin{tabular}{c c c} \hline \hline
**Used Polynomials** & **Name of Model** \\ \hline A(q), B(q) & ARX \\ A(q), B(q), C(q) & Auto-Regressive Moving Average with eXogenous variable (ARMAX) \\ A(q), C(q) & ARMA \\ \(A(q)\), \(B(q)\), \(D(q)\) & Auto-Regressive Auto-Regressive with eXogenous input (ARARX) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Common auto-regressive moving average (ARMA) models shown with their associated polynomials. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method/Reference** & **Test Model** & **Damage Feature** & **Pattern Recognition** \\ \hline AR [72] & Progressive failure test of retrofitted reinforced concrete bridge column & AR coefficients & X-bar control chart \\ \hline AR-ARX [73] & Laboratory-scale 8- DOF mass-spring system & Residual error between prediction and reference signal & Multi-paradigm statistical pattern recognition \\ \hline ARMAV [74] & Numerical model of a simply supported beam and a real-size steel-quake structure & Changes in natural frequency & Confidence interval and normal distribution of random variables \\ \hline AR-ARX [75] & A simulated three-story frame structure & Residual errors & Sequential Probability Ratio Test (SPRT) \\ \hline VFAR [76] & Numerical simulations of a six-story shear building and a three-span pre-stressed box-grider bridge & Natural frequency and damping ratio & - \\ \hline VBAR [77] & Numerical model of a lumped mass system and experimental model of a steel cantilever beam & Natural frequency & - \\ \hline ARMAV [78] & Steel-quake structure & Natural frequencies and damping ratio & - \\ \hline TV-ARMAV [79] & Supported beam with a moving mass & Natural frequencies and mode shapes and damping ratio & - \\ \hline ARMA [80] & Analytical and experimental results of the ASCE benchmark structure & First three AR components & Multi-paradigm statistical signal processing approach combined with the pattern classification \\ \hline ARMA [81] & IASC-ASCE four-story experimental benchmark structure, Z24 bridge and the Malaysia-Singapore Second Link Bridge & Normalized AR coefficient & T-test statistical control chart \\ \hline ARMA [82] & Simulated ASCE benchmark structure & GMM & Mahalanobis criterion function \\ \hline ARIMA [83] & Data from Malaysia-Singapore Second Link bridge & Coefficients of the ARIMA model & Statistical outlier detection \\ \hline TARMA [84] & Laboratory-scale steel beam with a moving mass & Natural frequencies & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Autoregressive-based methods applied in the structural health monitoring (SHM) of civil engineering structures. are utilized for the evaluation of dynamic behavior in a structure.