## 3 ARIMA-ML Framework
### Pre-Processing Module  
In signal processing, data pre-processing is an essential first step. In the ARIMA-ML framework, first, a moving-average operation (not to be confused with the MA model) with a suitable window size (e.g., 2 in this study) is used to smooth the raw signals and remove some noisy data points. In addition, while adopting segmentation in the TS, it is easier to achieve local stationarity for a short time period because the strong nonlinearity of the TS signal can be relieved by fitting local linear models for each segment. Moreover,  
Figure 1: The proposed two-stage ARIMA-ML framework. normalization (standardization) resolves the sensor-to-sensor scale variations of the TS due to different loading, calibration, or resolution conditions. [PERSON] et al. [2] and [PERSON] et al. [3] indicated the effectiveness of performing segmentation and normalization. The segmented TS should maintain sufficient information of the structural properties (e.g., stiffness) where each segment contains at least one reciprocating loading cycle, e.g., the three sample segments in Figure 2. While conducting sliding window, two patterns, namely (1) _non-overlapping_ (Non-OL) and (2) _overlapping_ (OL), can be applied as discussed in the validation experiments. The segmentation using OL increases the amount of data analogous to data augmentation methods, which are, in general, useful for training ML algorithms, especially under limited data regimes. To minimize the variance of the local time period, instead of normalizing the whole signal, it is normalized within each segment. Let \(y_{j}^{i}\) be the \(j\)-th segment in the \(i\)-th record of a collection of TS signal records, and the normalized signal, \(\tilde{y}_{j^{\prime}}^{i}\), is obtained as follows,  
\[\tilde{y}_{j}^{i}=\left(y_{j}^{i}-\mu_{j}^{i}\right)\Big{/}\sigma_{j}^{i}, \tag{17}\]  
where \(\mu_{j}^{i}\) and \(\sigma_{j}^{i}\) are, respectively, the mean and standard deviation of \(y_{j}^{i}\). After normalization, \(I\) differencing is conducted for each segment. The selection of \(I\) is based on the demand (least operations to achieve stationarity) and varies from case to case. Usually, \(I=1\) or 2 can achieve acceptable results [16], and \(I=1\) is used in this study. Even though SSND is conducted, stationarity of the processed data is not guaranteed. Moreover, manually examining all segments is time-consuming and impractical. Thus, an automated filtering of the non-stationary segmentation is introduced through using the Augmented Dickey-Fuller (ADF) test [27], which uses _hypothesis testing_ to determine the statistical relationship in the data between two interpretations, namely the _null hypothesis_ and _alternative hypothesis_. The former usually represents a commonly accepted fact or default assumption, and the latter is the opposite. They are mutually exclusive, i.e., if one is true, the other must be false. The decision for accepting/rejecting the null hypothesis is based on whether the \(p\)-value (calculated probability) exceeds a designated threshold (significant level). Herein, the null hypothesis is accepted by the presence of a unit root (a stochastic trend in a TS, and its presence implies a systematic \"unpredictable\" pattern of the TS) in univariate TS data, which represents the non-stationarity condition. On the other hand, the alternative hypothesis is test-dependent and represents stationarity or trend-stationarity (a TS is trend-stationary if an underlying linear or nonlinear trend, e.g., a function solely given in terms of time, can be removed while leaving a stationary process [16]). Figure 2: Illustration of the segmentation operation using sliding windows (overlapping (OL) for Segment #1 and Non-OL for Segments # 2 and 3) for a sample TS signal. If the \(p\)-value from the ADF test performed after SSND on a segment is larger or equal to 0.05, the null-hypothesis is accepted, which means that the segment is non-stationary, and it is subsequently discarded.