# Optimization of Design Parameters in LSTM Model for Predictive Maintenance
## 4 A Numerical Experiment  
### An Experimental Design  
We designed a numerical experiment to evaluate the performance of the solution procedure suggested in this paper as follows. We used a dataset gathered from a repository hosted by NASA. Specifically, we considered the IMS bearing dataset containing vibration sensor data obtained from a performance degradation experiment of four different bearing components of machinery [37]. The bearing is an essential element of machinery which has simple dynamics for analysis and reasonable lifetime for assessment. Each IMS bearing dataset consists of three sets recorded during different timelines with 10 min intervals, which is summarized in Table 2. Among them, we utilized the information of Bearing 1 belonging to Set No. 2 to formulate a predictive maintenance model based on LSTM. It contains 984 files generated at each time point recorded over about 7 days. Figure 7: Calculation steps in MPTM operator. Each file in the dataset stores 20,480 sensor measurements recorded at one time point. Then, the feature value of a time point was acquired by processing all 984 sensor measurements using a specific equation for calculating the feature value. Figure 8 illustrates an example of the pre-processing of vibration data to calculate the RMS value at all time points, where vibration data recorded at each time point were transformed into the RMS value at that time point. As a result, 984 RMS values were calculated from 984 files included in dataset No. 2. Including the RMS tackled in the above example, we considered a total of 7 features for processing vibration data at a certain time point as a candidate for the HI [38,39]. Descriptions and equations for calculating each feature are described in Table 3, where \(x_{i}\) is the \(i\)-th measurement and \(\overline{x}\) is the mean of all measurements. Similar to the RMS, each feature was calculated for 984 time slots based on vibration data of 984 files. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Features** & **Descriptions** & **Equations for Calculating Features** \\ \hline Root Mean Square (RMS) & Square root of mean square & \(RMS_{t}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}\) \\ Peak-to-Peak & Difference between positive and negative peak & \(PtPt_{t}=\max(x_{i})-\min(x_{i})\) \\ Kurtosis & Steepness of distribution of samples & \(Kurtosis_{t}=\frac{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}{\left( \frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}\right)^{2}}-3\) \\ Skewness & Asymmetry of distribution of samples & \(Skewness_{t}=\frac{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{3}}{\left( \frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}\right)^{2}}\) \\ Crest Factor (CF) & Extremeness of positive peak compared to other samples & \(CF_{t}=\frac{\max(x_{i})}{\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}}\) \\ Waveform Factor (WF) & Coefficient affecting shape of vibration or wave & \(WF_{t}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}\) \\ Waveform Factor Entropy (WFE) & Entropy of WF to acquire robust values & \(WFE_{t}=\frac{1}{M}\sum\limits_{i=0}^{M-1}WFE_{t-i}*\log(WF_{t-i})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description and equation for calculating features. Figure 8: Pre-processing of vibration data. The definitions of normal and fault classes used to train the LSTM for detecting fault occurrence were sequences recorded in the time period (300, 450) and (700, 900), respectively, which was the same as the reference model used in [3]. To train the LSTM for monitoring, 70% of sequences belonging to each class were used as a training set, while the rest were used as a test set for validation of the result. Additionally, we committed 100 epochs of training for both LSTMs for prediction and monitoring, with 10 epochs of patience as an early stopping condition, which means that the training procedure may be terminated before 100 epochs if there is no improvement of the performance metric in 10 epochs.