## 1 Introduction
In the second approach, the extracted HI describes the whole degradation process leading to a particular fault and the HS division divides this time interval into stages, in which the fault is increasingly severe. The existence of both the HI and the HS makes it easier to compute the RUL of the system, which can, therefore, be defined as the difference between the current time and the time at which the HI will reach a pre-set threshold. The second approach offers the opportunity to implement the PHM online and in real time, from the very beginning of the functioning of a system. On one hand, this might fill the existing gap between the theory and industrial needs, transforming a supervised approach into an unsupervised one that does not require either test labs or time-consuming offline analysis. Indeed, some tasks of the PHM can be performed directly at the edge of machinery, in an online way. The edge device can be connected to a cloud, in which relevant information supporting the online tasks, e.g., relevant features and the failure threshold, can be collected and shared among similar components of different machinery. On the other hand, this might offer the opportunity to overcome the limits that are imposed by offline methods when facing non-stationary signals and dynamic environments. Indeed, PHM can be applied to streaming data, not only allowing for the attainment of real-time feedback on the machinery health conditions, but also allowing models to update their structure based on actual data; that is, the models are able to incrementally learn from actual data. As real-time and online applications of PHM require the use of models handling streaming data [13], the areas of anomaly/fault detection and dynamic degradation modelling are of significant interest. Under the hypothesis that anomalies in streaming data may represent incipient faults or may be indicative of some changes in the system behaviour, anomaly detection algorithms can be successfully applied to fault detection in online applications. In particular, within the empirical data analytics (EDA) framework [14], which is a non-parametric and data-driven methodology for data analysis, anomaly detection relies on the recursive density estimation (RDE), which allows for the recursive calculation of the density of the data as they are available and the detection of differences between new and previous data. In addition, autonomous data partitioning (ADP) is also proposed within the EDA framework, which uses the RDE to group data into entities, named clouds. Data clouds are similar to clusters, but with an arbitrary shape and do not require the definition of prototypes or a membership function, which is a non-parametric function and reflects the real distribution of the data [15]. Degradation modelling essentially involves the development of a probability model that is able to describe the degradation phenomenon over time. Stochastic processes, e.g., Wiener, gamma, and inverse gamma processes, are suitable to model the randomness in degradation processes due to the unobserved environmental factors that influence the degradation process [16]. Stochastic processes can also be described by state space models (SSMs), which consider both the latent degradation process and evolution of the failure [17]. These models allow for dynamic state estimation through Bayesian techniques, which update their parameters, as new data are available and have been shown to be highly effective in online applications [18]. In this paper, we combine feature extraction, HI construction, anomaly detection, HS division, and RUL prediction in a unique methodology, which is unsupervised and suitable for online and real-time implementations of PHM.