# Modeling of Distorted Degradation Data Based on Oil Analysis
## 1 Introduction  
Prognostics and Health Management (PHM) focuses on condition monitoring and the extrapolation of degradation trends based on recent observations [1, 2, 3]. Among the established techniques for monitoring the state of machinery, oil analysis is the most popular. Oil monitoring is mainly concerned with the analysis of sampled oil to detect wear-related problems [4, 5, 6, 7]. Shell in its advisory report indicates that oil contamination causes approximately 70% of in-service failures in diesel engines, of which 50% are the result of wear-related problems [8]. The wear is complex, including multiple friction pairs, such as wet clutches, transmission gears, sealing rings, bearings, and confluence planetary gear trains. The wear describes the degradation of mechanical systems to some degree. However, unless the wear is directly observable, such as in the case of a brake pad, it is difficult to quantify it in general. For most mechanical systems, wear is not directly observable and can only be assessed via other measured condition information data such as the concentrations of metal elements. The concentration of metal within oil samples is a good indicator of wear and has been widely used for wear evaluation [8, 9, 10, 11, 12]. [PERSON] [8] used metal concentrations to describe the deterioration of a maintained plant. [PERSON] et al. [9] considered the metal concentration as a potential failure indicator and then proposed a linear regression model to determine a linear course of metal particle generation. [PERSON] et al. [10] used the Wiener process to model metal concentrations and thus optimized the planned maintenance interval of the power shift steering transmission. However, in practical scenarios, many factors such as oil top-ups, wear particle removal, and different measurement methods can disturb the observation of the metal concentration, thereby causing the observed data to deviate from the actual physical degradation. This distortion (deviation) indicates that it is infeasible to directly use the observed data to evaluate the wear condition. In other words, it is necessary to recover the actual physical degradation curve based on the observed data. [PERSON] et al. [13] considered the influence of oil consumption and replenishment and developed an analytical approach to the wear rate of internal combustion engines based on spectrometric oil analysis. [PERSON] et al. [14] used oil volume correction to transform the observed data into absolute wear values. [PERSON] et al. [15] considered the effect of the time-varying wear coefficient. A model of wear debris concentration was built based on [PERSON]'s method with different wear coefficients in corresponding wear stages. However, the literature mentioned above assumes that the refilling is carried out at predetermined maintenance intervals [13; 14; 15; 16] and does not consider dynamic, non-periodic oil top-ups driven by real-time operational conditions. Moreover, the above models were established based on particular mechanical structures such as an internal combustion engine, a continuous flow stirred tank reactor (CFSTR), or a gearbox, thereby restricting their applications. Considering the disturbance caused by non-periodic oil top-ups, a piecewise linear method was employed to address the resulting variations in oil spectrum analysis [17; 18]. The Hermite interpolation approach was used to model the wear characteristics [19]. The grey model and AR time series model were also adopted [20]. However, these methods rely on subjective judgments to identify oil top-up points and use local observed information to correct the data. This indicates that these methods not only lack objectivity but also do not make full use of existing observation information. This paper focuses on rectifying observed degradation data distorted by non-periodic oil top-ups.