## 1 Introduction
### Diagnostics and Prognostics Using Hidden Markov Model (HMM)
[28] also recommended using the healthy portion of the oil data, due to the unavailability of optimal segmentation techniques, for the nonstationary time series. Figure 1: Trend showing the number of publications on HMM-based diagnostics and prognostics over the past three decades, focusing on mechanical components such as bearings, shafts, turbofans, and turbines. There exists a research gap on optimal state-space identification in the context of fault diagnosis and failure prognosis. Based on existing research explorations, the limitations of HMM-based lubricating oil diagnosis and prognosis may be listed as follows:  
1. The state space is prespecified based on experience, assumption, or data segmentation;
2. There is no possible update to the state space based on the trend of the degradation data second item;
3. The number of parameters is limited;
4. The analysis is based purely on the healthy portion of the oil data with minimal inherent nonlinearity. Though advanced variants of HMM have been widely explored in the fields of speech [15] and handwriting recognition [17], music recognition and classification [29] profile analysis in structural biology [30], etc., studies that focus on their application to diagnostic and prognostic case studies are still lacking. In this study, we propose a state-based lubricating oil diagnosis and prognosis using the hierarchical Dirichlet process (HDP)-HMM to overcome the limitations of the standard HMM. HDP-HMM is a nonparametric Bayesian approach that solves the identification of the number of states, state updating, and the limitation on the number of parameters. Our work here provides a practical LCM approach, compared to other nonparametric Bayesian methods, i.e., multi-output Gaussian process regression (MO-GPR), which we have explored earlier in Ref. [31]. Our previous work based on the MO-GPR method to predict lubricating oil RUL faced the following limitations:  
* RUL prediction requires one or more historical degradation time-series patterns;
* Nonlinearity and nonmonotonicity of degradation trends affect the RUL prediction accuracy;
* Degradation states and state evolution trends cannot be extracted and estimated. The HDP-HMM overcomes all of these above limitations by estimating the degradation states even for nonlinear trends without the availability of any historical data sets. In machine learning, HDP is a nonparametric Bayesian method to cluster data, which uses the Dirichlet process for clustering. The Dirichlet process is a generalization of the Dirichlet distribution, which is defined as a distribution over distributions of N outcomes. Therefore, any draw from a Dirichlet distribution is a probability distribution in itself. The Dirichlet process also follows the same characteristic, except that the number of observations, N, can reach infinity. Initially, a speaker diarization problem was solved by [PERSON] et al. using HDP-HMM [32]. The authors used a sticky HDP-HMM to segmentize the audio recording without prior information on the speakers' number. The sticky HDP-HMM resolved the over-segmentation problem posed by the standard HDP-HMM. The authors in Ref. [32] further developed the blocked Gibbs sampler to estimate the state sequence considering a nonparametric emission distribution. The emission or output probability for hidden states, \(Z_{T}\), is defined as \(\text{P}(y_{i}\in A)|\,|Z_{t}=z_{t}\rangle\), where \(y_{i}\) is the observation at time \(t\). The latest work by [PERSON] and [PERSON] [33] discusses bearing health status identification using HDP-HMM. The HDP-HMM is improved by replacing the ergodic topology of transition from the emission state to all other states, with the left-to-right model having state transition to itself and to one other unique state only. The proposed work aims at estimating the hidden number of states for a given oil degradation trajectory using sticky HDP-HMM. It uses HDP-HMM to model unknown state space learned from degradation data.