Health Assessment for Multi-component Systems Based on Proportional Hazards Model and Dynamic Bayesian Network

As a key task of prognostics and health management, the health assessment of systems depends on both age and covariate processes. Cox’s proportional hazards model is an effective tool for system health assessment, capable of capturing both the failure rate and the effect of the covariate process. However, most existing literature assumes the system failure as a whole, which exhibits certain limitations when dealing with the failure interactions among components in complex systems. This paper develops a method that combines a dynamic Bayesian network with a proportional hazards model, where a dynamic Bayesian network is utilized to characterize the failure dependency among components failure, and the joint effect of age and covariate processes on component failure is quantified by the proportional hazards model. The integration of two models facilitates the health assessment of complex multi-component systems. Finally, the effectiveness of the proposed model is showcased through a numerical case study.


Introduction
System health assessment plays a crucial role in ensuring availability and reducing operational costs.Its primary task is to derive the current and future health status or failure time of system.Typically, changes in system health depend not only on its own age but also the effect of covariate processes [1].Proportional hazards model (PHM) can effectively represent the impact of covariate processes on the system health and have been widely applied in fields such as electronics, aeronautics and equipment.
Existing studies on PHMs focus on covariate processes representation, such as Markov process [2], Gamma process [3] ,Wiener process [4] and so on.These studies often assume that the system follows a simple failure process, which is not suitable for complex multi-component systems.Complex systems contain intricate structures and many internally correlated components, and the interaction of failures among components is beyond the capability of PHM.
Bayesian network (BN) has become an effective tool for probabilistic modeling and causal inference over a set of random variables.Literature survey on BN reliability modeling can be found in Cai [5].Combining Bayesian networks with PHM can better describe the operating condition of complex systems.Kraisangka [6] has established a Bayesian network interpretation of proportional hazards models.Li [7] used BN and PHM to establish a reliability assessment model for systems with common cause failures.This paper combines the strength of Bayesian network and PHM and establish a dynamic Bayesian network-proportional hazards model (DBN-PHM), to achieve health assessment for complex system by integrating covariate processes and failure interactions among components.

Proportional Hazards Model
It is considered in proportional hazards models that system failure depends on both system age t and covariate process X={Xt |t＞0}.The failure rate is described as follows: where h0(t) is the baseline hazard function, capturing the failure process with age; φ(Xt) is the link function that quantifies the covariate impact on failure rate.There are various forms of Eq.( 1).The most popular one is Weibull PHM that contains the Weibull baseline failure rate and the exponential link function, which is given by: where η and β represent the scale and shape parameters in the Weibull distribution, respectively, γ is the coefficient for the time-dependent covariate X that can be either discrete-state or continuous stochastic processes.Without loss of generality, X is assumed to be discrete with state space Ω={0, 1, …, M}, the assumption is also applicable for continuous-state covariate by appropriate discretization method.Assume that the system current age is t and the covariate state is i∈Ω, then the survival function of system and state transition of covariate at time t+Δt is given as follows: It can be seen that the state transition in Eq.(3) and Eq.( 4) are conditional probabilities, which correspond to the temporal CPTs in DBN between two time slices.Inspired by this, the equivalent representation by DBN can be given by Fig. 1, in which node St denotes system failure and node Xt is covariate variable at time t.And the temporal CPT of St→St+Δt and Xt→Xt+Δt in Fig. 1 can be translated from Eq.(3) and Eq.( 4), as shown in Table 1 and Table 2.It should be noted that the probability above is dependent on both t and time interval Δt, which is often non-stationary.Therefore, it is assumed that Δt is sufficiently small and the transition will not occur within Δt.Thus Eq.( 4) can be simplified as:

Extension to DBN-PHM
Taking advantage of DBN in multivariate dynamic modeling, the DBN form of PHM in Section 2.2 can be further extended to DBN-PHM, as shown in Fig. 2. Nodes are added to represent component failure of system and hierarchically capture failure interactions among them, such as {X1, X4, F1, F2, S}.Meanwhile more covariate nodes, like {X2, X3}, can also be added and linked with failure nodes as in Fig. 1, respectively.In this way, the BN at each time slice is responsible for multi-component failure modeling, and the temporal covariate effect between two neighbor time slices is captured by PHM.Furthermore, the integration of PHM and DBN can also simplify the parameter learning of CPTs.As previously mentioned, the transition probability is often time-dependent, thus the CPT learning needs observation data of nodes at each time slice, which is hard to afford.But the PHM parameter can be estimated with less data [8], and then translated to CPTs through method in Section 2.2.

Case Study
In this section, the DBN-PHM in Fig. 2 is taken as a numerical example, the parameter setting is given as follows, where the CPTs among node{X1, X2, X3, F1, F2, S} is given from Table 3 to Table 6, and the temporal CPT of covariate node X2 and X3 on node X4 is given as a Weibull PHM form, as shown in Eq.( 6).
The reliability of system node S at each time slice can be derived through DBN inference.The reliability curves with and without covariate effect are displayed as Fig. 3.It can be seen that the covariate processes accelerate system failure, which demonstrate the rationality and effectiveness of our model.1.5 , , exp 1.5 0.8 900 900

Conclusions
This paper proposes an approach to integrate the proportional hazards model and dynamic Bayesian network, which combines the strengths of dynamic failure dependency modeling among components and the characterization of one variable on the hazard rate of another.Therefore, the model is more in line with the operational conditions of complex systems in practical engineering.There are some important conclusions: (1) The proportional hazards model can be equivalently represented using a dynamic Bayesian network; (2) The CPTs between the failure node and covariate node in DBN can be interpreted by the survival and state transition function using PHM, meanwhile the parameter estimation is simplified with less data; (3) The DBN-PHM can be extended by introducing more failure and covariate nodes to describe complex failure interaction among components.

Figure 3 .
Figure 3.The reliability curve of system node S.

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2.2.Bayesian Network Interpretation of PHMBN is a probabilistic graphical model capable of modeling the joint probability distribution over a finite set of random variables.A BN consists of a directed acyclic graph (DAG) that reflects the causal connections among variables, which are represented by nodes.And directed arcs indicate relationships among nodes, quantified by conditional probability tables (CPTs).Dynamic Bayesian Network (DBN) extends BN by incorporating the time dimension to address the dynamic nature of random variables.In a DBN, a series of time slices is employed, with each slice containing a BN.Temporal probabilistic dependencies between variables in different time slices are established through temporal arcs.

Table 5 .
CPT of S.