## 2 Methodology
### Reliability Modeling of Marine Oil System
Detect the inflection point of the cumulative contribution (maximizing the curvature criterion): \(\kappa(k)=\frac{|C^{n}(k)|}{(1+C^{n}(k)^{2})^{3/2}}\)  
The inflection point \(k^{*}\) corresponds to the maximum curvature \(\kappa(k)\). Features ranked 1 to \(k^{*}\) are selected as final covariates. The corresponding feature parameter type is used as a covariate in the training and calculation of the proportional risk model. Reliability Analysis of Individual Components  
The proportional hazards model (PHM) can be used to establish a mapping relationship between the equipment's operational state information and reliability, allowing for the reliability analysis of individual equipment. It has been widely applied in engineering. Normal proportional hazards model (NPHM)  
If the normal distribution is chosen as the failure distribution function, the proportional hazards model becomes the normal proportional hazards model (NPHM), with the following expression:  
\[h(t\mid Z)=h_{0}(t)e^{BZ} \tag{13}\]  
where \(h(t\mid Z)\) is the hazard function (or hazard rate) given the covariate \(Z\). \(h_{0}(t)\) is the baseline hazard function, representing the hazard rate in the absence of any covariate influence. \(X\) is the covariate influencing the hazard rate, usually a vector. \(\beta\) is the model parameter, indicating the effect of the covariate on the hazard rate. If the exponential distribution is chosen as the failure distribution function, the proportional hazards model becomes the exponential proportional hazards model (EPHM) expressed as:  
\[h(t\mid Z)=\lambda e^{BZ} \tag{14}\]  
where \(\lambda\) is the baseline hazard rate (a constant). \(e^{BZ}\) represents the effect of the covariate on the hazard rate. \(t\) is time. 3. The Weibull proportional hazards model (WPHM) \[h(t,Z)=\frac{\beta}{\eta}\Big{(}\frac{t}{\eta}\Big{)}^{\beta-1}e^{\gamma Z}\] where \(t\) is time, \(\beta\) is the shape parameter, \(\eta\) is the scale parameter, \(Z\) is a vector of covariates that reflects the operational state of the equipment, i.e., \(Z=\cdot[Z_{1},Z_{2},Z_{3},\ldots,Z_{n}]^{T}\), and \(\gamma\) is the regression parameter, \(\gamma=[\gamma_{1},\gamma_{2},\gamma_{3},\ldots,\gamma_{n}]\), which represents the extent to which the response covariates \(Z\) influence the equipment's failure rate. The reliability function from the initial time to time t in the proportional hazards model can be expressed as:  
\[R(t\mid Z)=e^{-\int_{0}^{t}h(s,Z)ds} \tag{15}\]  
This function describes the probability that the system or equipment will continue to operate without failure up to time t, given the impact of the covariates on the system's failure rate. For different components, the fault distribution function is not the same. To determine the exact fault distribution type of each component, the historical full-life-cycle experimental records of the components are used as the dataset; employ Maximum Likelihood Estimation (MLE) to compute the log likelihood value \(\ell(\hat{\theta})\) for each model and use the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for model selection. The calculation process is as follows:  
Assume we have a set of independent failure time data \(T=\{t_{1},t_{2},\ldots,t_{N}\}\), where each \(t_{i}\) represents the failure time of the \(i\)-th component, and they are independent and identically distributed (i.i.d.). For different distributions, we formulate their probability density functions (PDFs) and derive their respective log-likelihood functions to estimate model parameters. The NPHM (Normal Distribution), EPHM (Exponential Distribution), and WPHM (Weibull Distribution) have the following probability density functions \(f(t)\) and cumulative distribution functions \(F(t)\).