## 3 Methodology
### Deep Neural Networks
#### 3.2.2 Recurrent Neural Network and LSTM
\begin{array}{l}\sum\limits_{i=1}^{N}\left(e^{-\frac{RUL_{{g_{i }}-RUL_{{g_{i}}}}}{13}}-1\right),RUL_{{g_{i}}}-RUL_{{y_{i}}}<0,\\ \sum\limits_{i=1}^{N}\left(e^{\frac{RUL_{{g_{i}}-RUL_{{y_{i}}}}}{10}}-1\right),RUL_{{{\hat{y}}_{i}}}-RUL_{{y_{i}}}\geq 0.\end{array} \right. \tag{24}\]

## 4 Implementation

The flowchart of the proposed scheme of RUL prognostics is presented in Figure 8. Firstly, the raw C-MAPSS dataset is processed to select proper input data using the variance threshold, and the corresponding data are normalized by the z-score normalization method. Next, the preselected data are transformed into a one-dimensional health indicator. The processed datasets are split into the training set and the testing set. Furthermore, the CNN-LSTM-NN hybrid networks to be built. Then the network parameters are initialized, which includes the number of hidden layers, the number of neurons, batch size, and so forth. The hybrid scheme takes the training set as the input, and the true RUL of the training set are used as the target outputs. The Adam optimizer is utilized to optimize the training network, with the learning rate set at 0.001 to achieve stable convergence. The number of training epochs is 500. The batch-size is set as 250. Figure 7: Penalty score function. Finally, the testing set is input to the training model for the RUL prognostics, and the evaluation values are obtained. ## 5 Experiments

In this section, we present our experimental setup, which includes the detailed dataset values and a brief introduction of various prognostics models. To empirically evaluate the effectiveness of the proposed method in addressing equipment RUL prognostics, we conducted a series of experiments on the dataset and compared it with several existing methods. First, we introduce the C-MAPSS dataset. Then we discuss the experimental results along with the analysis. ### C-MAPSS Dataset

The dataset used to verify the proposed model is from NASA [44]. The NASA Commercial Modular Aero-Propulsion System Simulation(C-MAPSS) dataset of turbofan engine degradation simulations is a widely used benchmark dataset [12]. The dataset consists of multiple multivariate time series (shown in Table 1). The dataset is supposed to be from the same type of fleet engines. Each engine starts with a different state of initial wear and manufacturing variation, which is unrevealed to the researchers. This wear and variation are considered normal, i.e., they are not considered a fault condition. Three operational settings are given in the dataset. However, we only consider the effect of sensor measurement data in this paper. The measurement data is polluted by sensor noise. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **FD001** & **FD002** & **FD003** & **FD004** \\ \hline
**Train trajectories** & 100 & 260 & 100 & 249 \\
**Test trajectories** & 100 & 259 & 100 & 248 \\
**Sensor measurements** & 21 & 21 & 21 & 21 \\
**Maximum cycle** & 362 & 378 & 525 & 543 \\
**Minimum cycle** & 128 & 128 & 145 & 128 \\
**Mean cycle** & 206 & 206 & 247 & 245 \\
**Fault conditions** & 1 & 1 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Information on the C-MAPSS dataset. Figure 8: Flowchart of the proposed scheme. The engine is normally running at the beginning of each time series, and a fault occurs at some point during the series. For the training set, the fault grows in magnitude until system malfunction occurs. For the testing set, the time series ends some time before system failure. The purpose is to prognosticate the numbers of remaining operational cycles before failure in the testing set, i.e., the number of operational cycles after the last cycle in which the engine will continue to run normally. In addition, true RUL values for the testing set data are given. The dataset contains 4 sub-datasets with different numbers of time series, shown in Table 1. Moreover, the dataset consists of training sets and testing sets.