# A Data-Driven Fault Tree for a Time Causality Analysis in an Aging System
## 2 Time Causality Analysis Methods  
Time causality analysis is a causal interference over time where the temporary dependency between events over the stochastic process is captured and represented using analytical methods. The time causality analysis can achieve the fault prognosis task by providing the expert with the essential knowledge regarding the fault evolution and the change in its causality structure over time [11]. [PERSON] distinguishes the model-based and data-driven methods to address the fault prognosis issue. In what follows, a brief literature review of each prognostic method is discussed, and their strengths and limitations are highlighted to clarify the research gap [7]. The model-based method for time causality analysis relies heavily on human expertise to describe the system's behaviour over time in degraded conditions [12]. [PERSON], [PERSON] [13] address the drawback of the system downtime due to fault evolution in complex industrial process by expert knowledge enrichment. First, the time-delayed mutual information (TDMI) is employed to model the fault causality in the form of a time-delayed signed digraph (TD-SDG) mode. Then, a general fault prognosis strategy is used to optimize the system's downtime based on TD-SDG and the principal component analysis (PCA) technique. [PERSON], [PERSON] [14] propose an enriched fault tree for Active Assisted Living Systems (AALS). The fault tree basic events are ranked according to the degree of their importance based on the expert prior knowledge and the imprecise failure probabilities of those basic events. [PERSON], [PERSON] [15] combine the domain knowledge with the extracted knowledge from the database to build an enriched fault tree. First, the expert constructs the fault tree skeleton, which represents the main causality structure for the fault event. Then, some extracted patterns from the database that may depict unknown combinations of root causes are deployed to enrich the initial fault tree. [PERSON], [PERSON] [16] integrate the bond graph modelling technique with the Bayesian network to predict the faults in a high-speed train traction system. The bond graph represents the system structure that is mainly constructed based on expert prior knowledge, while the Bayesian network enriches the expert prior knowledge represented by the bond graph through discovering the hidden causal relations. Indeed, the model-based time causality approach can provide interpretable and relatively accurate models that can be built from the first principle of the system's faults. It is mainly applicable on simple systems with well-known causes where human knowledge about the faults, their occurrence and development is clear. Its limited implementation in complex systems has been overcome by enriching those models based on data-driven techniques, in which the unseen events are discovered and added to the model's prior knowledge. However, forming the model skeleton prior to knowledge by the expert in complex systems to identify the principal causality structure of the faulty situation and combining and positioning the extracted hidden fault knowledge from the data in the constructed model is a challenging task. Unlike the model-based methods, the data-driven time causality method explores the data using machine learning (ML) techniques and does not impose a model to predict the behaviour of a complex system [17]. The ML data-driven methods build unbiased models and are able to deal with noisy and correlated variables [18]. [PERSON], [PERSON] [19] proposed a methodology to predict the remaining useful time (RUL) using the Wavelet Packet Decomposition of the vibration signal and Fast Fourier Transform. The pre-processed signals are treated as input features to learn the Artificial Neural Network (ANN) that predicts the RUL. [PERSON], [PERSON] [20] implemented the long short-term memory (LSTM) neural network rather than relying on feature engineering and an ANN for fault prognosis in aircraft turbofan engines.