# Remaining Useful Life Prediction of Cutting Tools Using an Inverse Gaussian Process Model
## 2 Methods  
### Performance Degradation Modeling Based on Inverse Gaussian Process  
In this paper, tool degradation is assumed to follow an inverse Gaussian process, as follows:  
\[y(t)\sim\text{IG}(\mu\Lambda\left(t\right),\ \lambda[\Lambda(t)]^{2}, \tag{1}\]  
where \(\mu\) is related to degeneration rate, \(\lambda\) presents the fluctuation of the degradation process, and \(\Lambda(t)\) is a monotone increasing function. If the \(\mu\) and \(\lambda\) are known, the tool degradation process Y(\(t\)) has independent increments. In addition, Y(\(t\)) is also monotonically increasing. For a given failure threshold \(\omega\), the failure time T of the tool can be defined as the first passage of time when Y(\(t\)) exceeds the threshold \(\omega\). Accordingly, the probability density function (PDF) and cumulative distribution function (CDF) of failure time T can be obtained as follows [37]:  
\[F_{\text{T}}(t)=\text{P}(t<t)=\text{P}(y(t)>\omega)\]  
\[=\Phi\Bigg{[}\sqrt{\frac{\lambda}{\omega}}\bigg{(}\Lambda(t)-\frac{\omega}{\mu }\bigg{)}\Bigg{]}-\exp\bigg{(}\frac{2\lambda\Lambda(t)}{\mu}\bigg{)}\Phi\bigg{[} -\frac{\lambda}{\omega}\bigg{(}\Lambda(t)+\frac{\omega}{\mu}\bigg{)}\Bigg{]}, \tag{2}\]  
and  
\[f_{\text{T}}(t)=\sqrt{\frac{\lambda}{\omega}}\Lambda^{\prime}(t) \Phi\Bigg{[}\sqrt{\frac{\lambda}{\omega}}\bigg{(}\Lambda(t)-\frac{\omega}{\mu }\bigg{)}\Bigg{]}-\frac{2\lambda}{\mu}\Lambda^{\prime}(t)\exp\bigg{(}\frac{2 \lambda\Lambda(t)}{\mu}\bigg{)}\Phi\bigg{[}-\frac{\lambda}{\omega}\bigg{(} \Lambda(t)+\frac{\omega}{\mu u}\bigg{)}\bigg{]}\]  
\[+\sqrt{\frac{\lambda}{\omega}}\Lambda^{\prime}(t)\exp\bigg{(}\frac{2\lambda \Lambda(t)}{\mu}\bigg{)}\Phi\bigg{[}-\frac{\lambda}{\omega}\bigg{(}\Lambda(t) +\frac{\omega}{\mu}\bigg{)}\bigg{]}, \tag{3}\]  
Due to the variability in the raw materials and the dynamic working conditions, the degradation rate \(\mu\) itself can vary from unit to unit. We assume that the degradation rate \(\mu\) can be modeled as a random effect, which follows a certain distribution to account for this aspect. The typical model for the random effects for \(\mu\) in the IG process includes the truncated normal distribution and gamma distribution. In this study, we assume that \(1/\mu\) follows a normal distribution \(N\big{(}\alpha_{\mu},\sigma_{\mu}^{-2}\big{)}\), considering the two parameters correspond to the mean and variance of degradation rate respectively. The model parameters of normal distribution have definite physical meaning, so it is convenient to quantify the subjective information such as expert information. Then according to the total probability formula, the CDF of residual life Tr, considering the random degradation rate, can be expressed as follows [38]:  
\[\text{F}_{\text{Tr}}(t)=\int\text{F}_{\text{T}}(t|1\,/\mu)f_{1/\mu}(x)dx\]  
\[=\text{E}_{1/\mu}\Bigg{[}\Phi\Bigg{(}\sqrt{\frac{\lambda}{\omega}}\bigg{(} \Lambda(t)-\frac{\omega}{\mu}\bigg{)}\Bigg{)}\Bigg{]}+\text{E}_{1/\mu}\Bigg{[} \exp\bigg{(}\frac{2\lambda\Lambda(t)}{\mu}\bigg{)}\Phi\Bigg{(}-\sqrt{\frac{ \lambda}{\omega}}\bigg{(}\Lambda(t)+\frac{\omega}{\mu}\bigg{)}\bigg{)}\Bigg{]}, \tag{4}\]  
In order to simplify the course of the derivation for the RUL distribution, two lemmas are given [16; 39]:  
**Lemma 1. **_If \(Y\sim N\big{(}\alpha_{1},\sigma_{1}^{2}\big{)}\), and \(a\), \(b\in R\), then the following holds:_  
\[\mathrm{E}_{Y}[\Phi(a+bY)]=\Phi\Big{[}(a+b\alpha_{1})/\left(1+b^{2}\sigma_{1}^{2 }\right)\Big{]}, \tag{5}\]  
**Lemma 2. **_If \(Z\sim N\big{(}\alpha_{2},\sigma_{2}^{2}\big{)}\), and \(A\), \(B\), \(C\in R\), then the following holds:_  
\[\mathrm{E}_{Z}[\exp(AZ)\Phi(B+CZ)]=\exp(A\alpha_{2}+\frac{A^{2}\sigma_{2}^{2}} {2})\Phi\left(\frac{B+C\alpha_{2}+AC\sigma_{2}^{2}}{\sqrt{1+C^{2}\sigma_{2}^{2 }}}\right), \tag{6}\]  
Based on Lemmas 1 and 2, we can calculate (4) explicitly.