## 1 Introduction
### Lubricating Condition Monitoring (LCM)  
The degradation data from oil wear condition monitoring (CM) can provide information on the health status of both the oil and the machine itself. In the literature, wear-based health status identification has been widely explored for several industrial applications. To assess machine health status, LCM is primarily based on a subjective evaluation of wear debris quantity, size, morphology, and chemical structure that requires specific domain expertise, thereby limiting the utility and application of LCM [1]. In addition, the wear debris concentration (WDC) in lubricating oil is affected by oil changes and replenishments, resulting in negative spikes in the degradation signal [2].  
Various researchers have harnessed the importance of wear state identification for effective diagnosis, prognosis, and health management. One of the studies by [PERSON] et al. [3] proposed a wear state identification model considering wear states as degradation features. Two states, i.e., normal and abnormal wear, were classified using a binary-class method, and a support vector data description method was used to study the evolution of these states. [PERSON] et al. [4] also proposed a model to identify wear states using multiple sensor information fusion with the binary class method for the normal and abnormal states. The wear state identification approach of [PERSON] et al. [5] combined wear rate and wear mechanism using the mean shift algorithm. The sample data were clustered, and a chart was prepared for wear rate and wear mechanisms to define and classify the traits of each wear state. This wear state identification method was further extended by considering the severity of wear and the wear rate and wear mechanism by [PERSON] et al. [6].  
An adaptive mean shift algorithm with a binary class method was then proposed to model wear-state evolution. Run-in, quasi-stationary, and break-down phases were considered for wear data quantification and segregation based on the debris particle size. The quasi-stationary phase provided information on the dominant wear mode, whereas wear particle quantity governed the transition from the quasi-stationary phase to the breakdown phase [7]. In another attempt, three operational states based on flow in dual filters were proposed by [PERSON] et al. [8]. [PERSON] et al. [9] that predicted the remaining useful life of lubricating oil, considering the first passage time approach for degradation trajectory and continuous-time Markov chain for the dynamic operating environment. The study also assumed that the number of operating states is known. The work of these researchers above highlights the limitations of the existing approaches: predefined state space, inability to accurately estimate the state space, binary classification model forcing the system to dwell in only two states, the nonlinearity of degradation signal, which complicates the analysis, and the necessity for data segregation/clustering with the mean shift algorithm for any state classification. The \"system-based\" wear state identification models may provide effective diagnosis, but they cannot be used for prognostics and health management. Given that the wear state cannot be directly observed, the wear model may not accurately identify the health status in the context of LCM.  
The use of data-driven state-space models provides an effective solution to this problem. For example, the state-space model of oil degradation data traverses through a discrete set of states with a certain set of transition probabilities. Thus, the oil degradation data provides an observation sequence for an unobservable stochastic process or hidden states.