## 2 Methods
### Degradation Process Modelling and RUL Prediction  
The occurrence of a failure in a component/system is often the result of a degradation process that is hidden in the condition monitoring data. However, parameters collected through condition monitoring often represent undirect degradation indicators, whose relationship with the latent degradation of the system is unknown. State space models (SSMs) are often adopted to deal with this issue [17]. These models describe the stochastic process describing the degrading progression of the occurrence of a failure in a system. An SSM comprises two parts: the state equation and the observation equation. The first reflects the evolution of the failure, which is usually unmeasurable and, thus, called the latent degradation condition. The observation equation reflects the relationship between the latent degradation condition and the indirect degradation indicator. Mathematically, given an unobservable state process \(\{x_{t}\}_{t\geq 0}\) and an observation series \(\left\{y_{t}\right\}_{t\geq 0}\), the SSM is completely specified by the initial state distribution \(\pi(x_{0})\) and the conditional probability density function \(\pi\left(y_{t}|x_{t}\right)\) for \(t\geq 1\)[26]:  
\[\left\{ \begin{array}{c}\pi\left(y_{t}|x_{04},y_{1:t-1}\right)=\pi\left(y_{t} |x_{t}\right)\\ \pi\left(x_{t}|x_{0:t-1},y_{1:t-1}\right)=\pi\left(y_{t}|x_{t-1}\right)\end{array}  \right.=\left\{ \begin{array}{c}y_{t}=f(x_{t})+v_{t}\\ x_{t}=g(x_{t-1})+w_{t}\end{array} \right. \tag{16}\]  
where the first equation of the system represents the observation equation, while the second represents the state equation; \(x_{t}\) is the unobserved state of the system at time \(t\); \(y_{t}\) is the observation at time \(t\); and, \(v_{t}\) and \(w_{t}\) are the process and measurement noises, respectively, which are independent of each other. Given an SSM describing the latent degradation process of a system, the main task is to make an inference of the unobserved health state and predict the future state based on CM data. Several empirical models may be used to establish the state evolution equation. Among these, dynamic Bayesian methods provide a unified framework for state estimation of the stochastic process [27; 28; 29]. The choice of the most suitable type of SSM depends on the nature of the system dynamics and noise source. The most commonly adopted are the Kalman filter (KF), for linear dynamics and Gaussian noise; the extended Kalman Filter (EKF), for non-linear dynamics and Gaussian noise; and, particle filtering (PF), when both the dynamic evolution of the degradation and the noise are non-linear. In addition to the choice of the specific degradation process models, there are other key aspects in RUL prediction. The first regards the definition of a proper HI that can be used to track the fault progression [30]. This can be one of the extracted features, a combination of the extracted features, or a correlation coefficient between vibration signals that corresponds to different component conditions [31]. It is important that the HI be as monotonic as possible, i.e., it should decrease in time as the degradation increases. To this purpose, a dynamic smoothing/filtering technique can be applied to eliminate anomalous peaks and dips in the HI values that could negatively affect the analysis [32]. The second aspect regards the definition of the time to start the prediction (TSP) [33], which is usually subjectively determined or based on statistical methods. Finally, to compute the RUL, the failure threshold (FT) has to be set. Indeed, the RUL is computed as the time difference between the current time and the time at which the HI is expected to reach the pre-fixed failure threshold. The threshold can be set based on expert judgments, similar components, prior knowledge about the component, or a different algorithm. It is important to note that the RUL is a probabilistic quantity and, therefore, confidence values of the prediction must always be determined.