# A Data-Driven Fault Tree for a Time Causality Analysis in an Aging System
## 1 Introduction  
The aging of a system is characterized by the progressive deterioration of its initial performance over time, including--among other factors--the occurrence of faults that adversely affect the system's reliability [1]. Causality analysis methods aim to diagnose the fault event through identifying, isolating and quantifying the effect of the root causes on the system performance so that the appropriate maintenance actions can be performed to restore the system to good condition [2]. The future fault behaviour and its drawback on the system's performance are essential in order to optimize the maintenance decision-making [3]. [PERSON] et al. [4] proposed a comprehensive survey of real-time fault diagnosis methods that are mainly categorized into model-, signal-, and knowledge-based techniques. The fault prognosis task provides a model that depicts the progression of a specific failure mode from its inception until the time of failure [5]. The time causality analysis builds a prognostic model that captures the fault causality behaviour over time [6]. A prognostic model may use a mathematical expression that quantifies the fault causality evolution or a graphical representation that depicts the changes in the causality structure over time [7]. Both the event-based and the data-driven methods are commonly deployed to provide relevant fault prognostic models. The event-based method requires the involvement of experts from different fields with detailed prior knowledge about the fault time causality. However, this knowledge could be biased and reflects only the expert opinions about the fault development [8]. On the other hand, the data-driven method can directly extract the fault evolution knowledge from the data, which is unbiased knowledge and reflects the fault causality. However, it lacks the interpretability and the expert knowledge representation to identify fault hierarchical causality over time [9]. [PERSON] and [PERSON] [10] have developed a data-driven fault tree method for causality analysis which addresses the lack of interpretability of the data-driven model and overcomes the model-based limitation regarding the expert prior knowledge. The methodvisualizes the fault causality architecture of a simple system using one-level fault tree that consists of three layers. The condition layer identifies the fault root causes and their coverage ranges within the dataset. The pattern layer arranges the root causes in the form of interpretable conjunctions. The solution layer combines some selected patterns that depict the fault event. Although the proposed tree is interpretable for the expert, the model hides the fault hierarchical cause-and-effect relations in a complex system. Moreover, it reflects the fault causality in a static way without considering the influence of a system's aging on the change in the fault causality structure over time. From a practical point of view, human experts look for models that are able to explain and represent the fault causality structure in addition to having prediction capability. Ensuring that the fault and its impact and consequences are well represented to human experts guarantees optimal preventive maintenance actions. Another challenge in a complex system with regard to data-driven fault prognosis models is graphically modeling the deterioration and performance degradation. Consequently, the fault causality structure can be changed over a system's life. Therefore, these complex systems need models that are able to capture these changes in an interpretable manner. This is a crucial feature that helps anticipate the impacts of a fault and provides more precise knowledge about the processes that will be affected in the future by a currently occurring fault. In this paper, an interpretable time causality analysis (ITCA) methodology is developed to address the problem of fault prognosis in an aging system using a data-driven fault tree model.