## 1 Introduction
The aim is to provide a framework that is similar to those applied for batch processes, in which: relevant features are extracted online as data are collected; a monotonic HI is built during the algorithm's execution; and, the information collected during the anomaly detection and data partitioning algorithms are used for both the degradation modelling and RUL prediction. As shown in Figure 1, the proposed methodology comprises two main blocks: in the first block, which is completely online, anomaly detection and ADP algorithms are combined in order to detect changes in the data structure and form data clouds corresponding to different HS of the degradation process. In the second step, conducted offline, the degradation model for the identified failure is built and a failure threshold (FT) is set. Finally, the so-built degradation model is updated online, as a certain HS is reached for the real-time RUL estimation. The novel aspect of the proposed methodology is the use of the ADP algorithm for HS division, which makes the procedure unsupervised and implementable \"from scratch\". Indeed, in the literature, there exist some examples of online and/or real-time applications of PHM [19; 20; 21]. However, in such cases, a priori offline analysis is always conducted, not only for the FT identification, but also for diagnostics. In contrast, the proposed methodology does not rely on pre-built classification models. Similar data are automatically grouped in the same clusters and, when an anomaly is detected, the methodology is able to decide whether the following data can form another cluster. The creation of  
Figure 1: The proposed methodology. a new cluster certainly requires human intervention to understand what the new cluster corresponds to. However, since all similar data are grouped into the same cluster, the knowledge of what has happened at a specific moment (that is, which condition the observation corresponds to), allows for the automatic labelling of all observations belonging to the same cluster, facilitating a subsequent offline, supervised, and more accurate analysis. For a clear comprehension of the proposed methodology, the algorithms chosen for anomaly detection, data partitioning, and degradation modelling are described in the first part of the paper. Subsequently, the methodology is applied to a critical component of an automatic machinery operating in a real industrial context. The component under analysis is made of four suction cups, which rotate and push the product forward. When two suction cups become detached, then the component is considered not to perform its function properly, i.e., it is considered to have failed. Suction pressure has been identified by experts as the best indicator for recognizing whether one or more suction cups became detached. The following assumptions have been made: (1) one only sensor is used for signal collection, (2) at the beginning, no prior knowledge is available regarding the current condition of the monitored system, and (3) no historical data is used for training models. The remainder of this paper is organized, as follows. Section 2 is dedicated to the mathematical formulations of the algorithms adopted for anomaly detection, HS division, and degradation modelling. These algorithms are named autonomous anomaly detection (AAD) and autonomous data partitioning (ADP), which were in proposed by [PERSON] in [22; 23], and are described in Sections 2.1 and 2.2, respectively, and state space models (SSMs), which are described in Section 2.3. In Section 3, the proposed methodology is introduced, explaining how the described methods can be integrated and applied to a real case. In Section 4, the case study for validating the proposed methodology is presented. Finally, Section 5 is dedicated to the discussion of the main issues that emerged during the application and the future directions of the research.