## 1 Introduction
### Diagnostics and Prognostics Using Hidden Markov Model (HMM)  
The hidden Markov model (HMM) has been applied to forecast health status based on observations in Refs. [10; 11; 12; 13]. HMM has also been a vital modeling tool in the field of bioinformatics [14], speech [15], image [16], and handwriting recognition [17], as well as financial time series predictions [18]. Scholars have lately identified and explored HMM also for its diagnostic and prognostic (DP) capabilities. In an early attempt at HMM for diagnostics, [PERSON] et al. [19] introduced HMM as part of a hybrid model with a neural network providing prior temporal correlations to develop a probabilistic framework for antenna health diagnosis. The Markov model's ability to capture probabilistic changes in the degradation parameters was then realized by [PERSON] and [PERSON] in Ref. [20]. They used a discrete-time Markov chain to estimate the probable transitions through seven degradation states to assess the lubricant condition to support maintenance decision making. Given the apparent gain in using Markov-based modeling for degrading systems, researchers started exploring the tool for mechanical system diagnosis and prognosis (DP). Figure 1 shows the count of research articles published relating to HMM-based DP of mechanical components, namely, bearings, rotating shaft, drill bits, wind turbines, turbofans, boilers, etc. Two pertinent review works are included in this count. The first review by [PERSON] et al. [21] on rotating machinery prognostics discussed Kalman filter, particle filter, stochastic filter, artificial neural network, HMM, etc. In addition, the authors briefly reviewed HMM with its computational limitations and application to remaining useful life (RUL) prediction. The second review by [PERSON] et al. [22] discussed HMM in the context of the systematic evaluation of HMM variants and their applications over the years. However, the review does not deal with any studies on applications of HMM in the field of diagnosis and prognosis. There is an increasing interest in applying HMM to fault diagnosis and failure prognosis over the past decade [22]. However, HMM for lubricating oil degradation modeling has not yet been fully explored. The recent studies by [PERSON] et al. [23] and [PERSON] et al. [24] on lubricating oil RUL prediction used wear debris level as the lubricating oil degradation feature and provided a physical model for the relationship between oil degradation and particle contamination. A vector autoregressive model [23] was considered as an observation process to generate input for the three-state HMM. The authors of that study focused on a healthy portion of the data and classified the states as healthy, unhealthy, and failed. This experience and assumption-based discrete state classification may not provide adequate maintenance decision support in a practical scenario, more so for relatively new technologies with limited data history. A detailed review by [PERSON] et al. [25] listed out the various diagnostic and prognostic approaches for lubricating oil condition monitoring, but again, it cited few studies on the use of HMM. [PERSON] and [PERSON] [26] presented a prognostic model for dependent failure modes of a system. The authors considered a standard three-state HMM model with Marshall-Olkin bivariate exponential distribution for the joint distribution of the dependent failure modes. Their model, though quite powerful, cannot track the low degradation level increments. A review of the data-driven approaches by [PERSON] et al. [27] can be referred to for work relating to HMM for prognostics. The authors in their article had also reviewed the different variants of HMM, including hidden semi-Markov model (HSMM), hidden hierarchical Markov model (HHMM), factorial hidden Markov model (FHMM), etc. for numerous applications. However, their work did not address HMM's application in lubricating oil diagnosis and prognosis. A recent work by [PERSON] et al.