## 2 Materials and Methods
### Modeling for the Descriptive Time Series Analysis  
The ARIMA model, which is an integrated model of the autoregressive (AR) and moving average (MA) models, was applied to the modeling process to develop the time series analysis [17]. The model was fitted to the time series data to predict future points (i.e., downtime) in the series. Recall that the ARIMA model is applied when the data show substantiation of non-stationarity in the sense of the mean, where an initial differencing stage can be used one or more times to eliminate the non-stationarity (i.e., fault trend) of the mean function. Therefore, it predicts the future value using a linear combination of past observations of a specific variable to be predicted by composing autocorrelation using time series modeling. That is, early observations affect later observations. Additionally, the MA error will affect the predicted value during the prediction phase. Thus, the AR model uses lags of the dependent variable as independent variables. However, the MA model uses past errors that follow a white noise distribution as explanatory variables. If \(y\) is denoted as the d-th difference, \(y^{\prime}_{t}\) is the differenced series. Thus, the general forecasting model can be expressed as Equation (1). This model assumes stationarity and is analyzed after  
Figure 1: A research flowchart for descriptive time series analysis for LINAC maintenance records. Failure instances (\(n=359\)) were collected over time, and an ARIMA model was adopted for predictive modeling. AR, autoregression; MA, moving average; \(p\), term for the number of AR; d, term for the integration of AR and MA; \(q\), the number of lagged forecast residual errors for MA; RMSE, root mean square error; MAPE, mean absolute percentage error. preprocessing, such as log transformation and differencing for forecasting or prediction (Figure 1 and Equation (1)) [17; 18]. \[y_{t}^{\prime}=\varphi_{0}+\varphi_{1}y_{t-1}^{\prime}+\varphi_{2}y_{t-2}^{ \prime}+\cdots+\varphi_{p}y_{t-p}^{\prime}+\theta_{1}\epsilon_{t-1}+\theta_{2} \epsilon_{t-2}+\cdots+\theta_{q}\epsilon_{t-q}+\epsilon_{t} \tag{1}\]  
where \(y\) regresses on itself lagged by the \(n\)-th period; \(\varphi_{i}(i=1,\ldots,p)\) and \(\theta_{j}(\mathrm{j}=1,\ldots,q)\) are defined by the weights for the AR and MA parameters; \(\varphi_{1}\) and \(\theta_{1}\) are therefore the coefficients of the first AR and MA terms, respectively; and \(\epsilon_{t}\) is a residual term with mean zero and variance \(\sigma_{e}^{2}\). In Equation (1), the error term (\(\epsilon\)) reflects the previous state at present. This implies that the MA model estimates the rate of change using autocorrelated errors. Thus, the resulting integrated model, the ARIMA model, is expressed as the ARIMA (\(p\), \(d\), \(q\)) for the AR, MA, and autoregressive moving average (ARMA). \(p\) is the order of AR terms, \(\mathrm{d}\) is the number of nonseasonal differences needed for stationarity, and \(q\) is the order of MA terms. To select the best ARIMA model for the fitting process through estimation, we used information criteria to determine the potential AR/MA orders [19]. Here, parameter combinations for the ARIMA (\(p\), \(\mathrm{d}\), \(q\)) were chosen for the seasonal ARIMA as follows (\(0<p=\mathrm{d}=q<2\)). For example, an ARIMA (0, 0, 0) model is white noise, which means that the errors are uncorrelated across time. An ARIMA (1, 0, 0) model is a first-order AR model, which is a stationary and autocorrelated series. It can be predicted as a multiple of its previous value with a constant (\(y_{t}^{\prime}=\varphi_{0}+\varphi_{1}y_{t-1}^{\prime}\)). For the ARIMA (0, 0, 1), the MA model processes means identically as the infinite sum of exponentially weighted past observations of the process. For model estimation, the Akaike information criterion (AIC) was used to seek a simple model with lower orders [20; 21]. During this evaluation process, the ARIMA model parameters were found in various combinations using \(p\), \(\mathrm{d}\), and \(q\).