## 4 k-LSTM-GFT Model for RUL Estimation  
When sensor data are available, typically, it is possible to develop an RUL prediction model whose predictions may be exploited to optimize the scheduling of maintenance actions. The effectiveness of this process closely depends on the accuracy of predictions and on a mechanism to account for RUL prediction uncertainties, since no model is perfect. For this purpose, we exploit the k-LSTM-GFT model developed by [9]. This approach is depicted in Figure 2 and is clarified in the following paragraphs.  
The k-LSTM-GFT combines a reliability approach, the generalized fault tree (GFT) analysis, with an ensemble of k-fold LSTM (k-LSTM). The GFT is a data-driven generalization of the traditional fault tree analysis (FTA) proposed by [46] that graphically represents  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline  & **FD001** & **FD002** & **FD003** & **FD004** \\ \hline Training instances & 100 & 260 & 100 & 249 \\ Testing instances & 100 & 259 & 100 & 248 \\ Operating conditions & 1 & 6 & 1 & 6 \\ Fault models & 1 (HPC) & 1 (HPC) & 2 (HPC and fan) & 2 (HPC and fan) \\ \hline \hline \end{tabular}
\end{table}
Table 1: C-MAPSS datasets description.  
a failure event by a data-driven fault tree, whose basic events are derived from the categorization of the continuous sensor data [47]. The fault tree structure is obtained automatically from a training process, and enables the qualitative analysis of failures and estimates the failure probability based on the observed sensor data. Furthermore, during the training, a probability threshold \(p_{t}\) is determined. This consists of a probability where it is recommended to take proactive action since the equipment or tool is near failure. Its value is the one that minimizes maintenance costs during the training process [46]. Additionally, the failure probability derived from GFT is integrated as a feature into the k-LSTM model, which estimates the RUL of the assets.  
Figure 2 illustrates the architecture of the k-LSTM-GFT model. It fuses the GFT approach and LSTM by using the failure probability estimated by the GFT model and the probability of BEs as features for the LSTM. At the top of the figure, the input data are represented. They consist of normalized sensor data and the failure probability estimated by the GFT. These inputs are combined into a matrix with \(n_{F}\) columns, representing the number of features (including the failure probability estimated by the GFT), and \(n_{W}\) rows, representing the values from previous time steps. As we are working with the subset FD001, we chose \(n_{W}=30\), which corresponds to the minimum number of working cycles until failure, for an engine in the test subset [11].  
Figure 2: Architecture of the k-LSTM-GFT [9].  
The training data are divided into \(k=5\) folds, each represented by a different color. In this setup, each fold is used as validation data, while the remaining folds serve as training data, resulting in the development of five distinct models. The final RUL prediction is generated by averaging the outputs of these individual models. As depicted at the bottom of Figure 2, the deep learning (DL) model comprises an LSTM layer, followed by three dense layers, each incorporating a dropout rate of 0.2. For more comprehensive details regarding the k-LSTM-GFT model, please refer to [9].