## 1 Introduction  
Prognostic health management (PHM) is a recent discipline supporting the realization of predictive maintenance in complex production systems. It is based on several condition monitoring (CM) techniques, e.g., vibration analysis, acoustic emissions analysis, and oil analysis, which are able to provide useful information regarding the health condition of monitored equipment in real time. The PHM process is usually divided into four main steps [1; 2; 3], i.e., data collection, signal processing or feature extraction, diagnostics or health assessment, and prognostics. In previous research by the authors [4], an effort was made to collect, in a unique reference framework, the models and approaches proposed to date in the literature for feature extraction, diagnostics, and prognostics. The aim was to provide readers with a wide range of possible solutions for implementing PHM in all its parts, depending on the specific application and objectives. According to this PHM approach,data are collected through appropriate sensors installed on critical components and stored in a suitable device. Due to the high sampling frequency allowed by advanced data acquisition tools, and potential measurement errors and noise, signals that are collected by sensors do not provide direct comprehension of the health state of equipment. Therefore, signals are first analysed in time, frequency, and/or time-frequency domains, in order to extract relevant features that highlight a clear distinction among different health conditions of the equipment. Subsequently, the relationship between the extracted features and the corresponding health status of the system is established (diagnostics). Finally, based on the fault modes, the degradation rates and the failure thresholds (FT) identified during the diagnostics, the remaining useful life (RUL) of the monitored equipment is predicted (prognostics) as the length of time between the current time and the time at which the FT is estimated to be exceeded. In the literature, a large number of methods that are related to machine learning (ML), artificial intelligence (AI), and statistical learning theory (SLT) are available to conduct any of these processes [5; 6; 7; 8; 9]. However, most applications: (1) are oriented either to diagnostics or prognostics, which makes it difficult to practically understand the relationship between the two different tasks [10]; and, (2) adopt a supervised learning approach, i.e., require many training data corresponding to component health conditions for model construction [11]. Unfortunately, conducting lab tests, e.g., accelerated testing, is expensive from both economic and time consumption points of view; in addition, there exist fault behaviours that cannot be known a priori, especially for new machinery produced on order. These reasons limit the implementation of PHM to real industrial contexts. An evolution of the PHM approach that is described above is presented in [12]. According to these authors, a machinery health prognostics program consists of four technical processes: data acquisition, health indicator (HI) construction, health stage (HS) division, and remaining useful life (RUL) prediction. Data acquisition deals with the collection of CM data through one or more sensors installed on the equipment. HI construction deals with the extraction of a proper HI, whose trend reproduces the degradation behaviour of the system. HS division deals with the division of the whole lifetime of the system into two or more stages, based on varying degradation trends of the HI. Finally, RUL prediction deals with the estimation of the time that is left before the FT is reached. The main difference between the two approaches lies in the consideration of the variable time. In the first approach, features are extracted for each of the available health/fault conditions and diagnostics identifies the different fault classes that are based on those features.