## 3 Health Score Correction Based on the Reliability Model
### Reliability Indicators  
Reliability indicators include cumulative failure probability \(F(t)\), failure probability density \(f(t)\), reliability \(R(t)\), and failure rate \(\lambda(t)\). Their definitions are shown below:  
Cumulative failure probability \(F(t)\) is defined as the probability that a device fails to perform its specified function (i.e., experiences a failure) under specified conditions before a given time \(t\). It is also referred to as uncertainty. The cumulative failure probability failure distribution curve serves as the foundation for describing the random failure patterns of equipment and conducting reliability analysis. Its definition is expressed as follows:  
\[F(t)=P(T\leq t) \tag{20}\]  
where \(T\) represents the time of equipment failure. It is important to note that \(t\) is a generalized concept and is not limited solely to the service time of the equipment. Instead, it depends on the specific object of analysis and the application scenario. When the reliability analysis focuses on equipment in the transportation sector, the operational mileage is often used as a unit for reliability analysis [2 ].  
Failure probability density \(f(t)\) is defined as the probability that a device fails to perform its specified function under specified conditions at a unit moment. Assuming \(f(t)\) is smooth, if the sample size is continuously increased while the interval length is reduced,  
Figure 1: Flowchart of health score correction based on the reliability model.  
the frequency of samples falling within the interval covering a specific point will decrease and tend toward zero as the interval length shrinks. However, the ratio of frequency to interval length will gradually converge to a fixed value, which is the failure probability density \(f(t)\). The curve formed by all such \(f(t)\) values is called the failure probability density curve, representing the likelihood of the random variable occurring within a unit interval. Therefore, the definition of \(f(t)\) is expressed as follows:  
\[f(t)=\frac{dF(t)}{dt} \tag{21}\]  
Reliability \(R(t)\) is defined as the probability that a device can perform its specified function under specified conditions within a given time \(t\), and it can be expressed as:  
\[R(t)=P(T>t) \tag{22}\]  
where \(T\) represents the time of equipment failure. From the definition, it can be seen that reliability \(R(t)\) and cumulative failure probability \(F(t)\) are complementary. Therefore, \(R(t)\) can be derived from \(F(t)\) as follows:  
\[R(t)=1-F(t) \tag{23}\]  
Failure Rate \(\lambda(t)\) is defined as the probability that a device which has not failed by time \(t\) under specified conditions, will fail in the unit time interval immediately following \(t\). It describes the dynamic characteristics of the likelihood of failure, and its definition is expressed as follows:  
\[\lambda(t)=\lim_{\Delta t\to 0}\frac{P(\,t<T\leq t+\Delta t|T>t)}{\Delta t} \tag{24}\]  
Based its definition, failure rate \(\lambda(t)\) can be derived from failure probability density \(f(t)\) and reliability \(R(t)\) as follows:  
\[\lambda(t)=\frac{f(t)}{R(t)} \tag{25}\]