# Time Series RUL Estimation of Medium Voltage Connectors to Ease Predictive Maintenance Plans
## 4 The Autoregressive Integrated Moving Average (ARIMA) Approach for RUL Prediction  
The ARIMA model was developed by [PERSON] and [PERSON] in 1970 [22]. The projections generated by means of the ARIMA model can be understood as the integration of the latest observations combined with the long-term historical tendency [26], being an intuitively reasonable model to describe many practical time series [22]. ARIMA is a type of time-domain model that is often used to fit and forecast time series that exhibit temporal correlation. ARIMA models have been applied to forecast time series of climate-related variables [26], economic variables [27], or pandemics [28], as well as to determine the RUL of batteries [1] and aircraft engines [29], among many other applications. ARIMA models are suitable to describe both stationary and nonstationary time series data. The properties of stationary time series are independent of the observation time, so the time series that present seasonality or trends are nonstationary, since seasonality and trend influence the time series at different times. However, cyclic time series without trend or seasonality are stationary. ARIMA models are based on autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) models [1]. ARIMA models take into account a linear combination of the current value, past values, nonseasonal differences, and lagged forecast errors of a time series degradation data for predicting the future response. The differences are used for removing nonstationarity, since stationary time series do not depend on the observation time. The AR or autoregressive term represents the regression variable based on the past values, whereas the MA or moving average term performs a linear combination of the regression errors [20]. The AR(\(p\)) model of order \(p\) can be viewed as a discrete set of time-lags linear equations,  
\[\hat{R}_{t}=\alpha_{1}R_{t-1}+\ldots+\alpha_{p}R_{t-p}, \tag{4}\]  
where \(R_{t}\) and \(\hat{R}_{t}\) denote the measured and predicted values of the electrical resistance at time \(t\), respectively. The MA (\(q\)) term of order \(q\) can be expressed as,  
\[\hat{R}_{t}=\varepsilon_{t}+\gamma_{1}\varepsilon_{t-1}+\ldots+\gamma_{q} \varepsilon_{t-q}, \tag{5}\]  
where the error terms are given by,  
\[\varepsilon_{t}=R_{t}-\hat{R}_{t} \tag{6}\]  
Figure 2: Evolution of the resistance of a power connector according to the IEC 61238-1-1 standard [25]. When neglecting the nonseasonal difference terms with order \(d\), the ARIMA (_p,0,q_) is obtained, which, in fact, is an ARMA (_p,q_) model, which can be described as,  
\[\hat{R}_{t}=\alpha_{1}R_{t-1}+\ldots+\alpha_{p}R_{t-p}+\varepsilon_{t}+\gamma_{ 1}\varepsilon_{t-1}+\ldots+\gamma_{q}\varepsilon_{t-q}. \tag{7}\]  
A nonseasonal ARIMA (_p,d,q_) model includes \(p\) autoregressive terms, \(d\) nonseasonal differences, and \(q\) lagged forecast errors. The ARIMA (_p,d,q_) is as an ARMA (_p,d_) model of \(\Delta^{d}R_{t}\), that is,  
\[\hat{R}_{t}=\alpha_{1}\Delta^{d}R_{t-1}+\ldots+\alpha_{p}\Delta^{d}R_{t-p}+ \varepsilon_{t}+\gamma_{1}\varepsilon_{t-1}+\ldots+\gamma_{q}\varepsilon_{t- q}, \tag{8}\]  
where \(\Delta^{d}R_{t}=\left(1-L\right)^{d}R_{t}\), and \(L\) is the lag operator, i.e., \(LR_{t}=R_{t-1}\) and \(L^{d}R_{t}=R_{t-d}\), so \(R_{t}\) is obtained from \(\Delta^{d}R_{t}\) by performing \(d\) successive integrations. Therefore, ARIMA models can be understood as an improved version of ARMA models, which are suitable for stationary and nonstationary time series. In the case of nonstationary time series, the _d_-order nonseasonal differences terms are required to convert nonstationary into stationary time series. It is noted that in most applications, \(d\) is 0 or 1 [1]. The ARIMA (_p,d,q_) problem consists of determining the \(\alpha_{i}\) and \(\gamma_{j}\) coefficients from the time series data of \(\Delta^{d}R_{t}\).