# Fault Detection in Industrial Equipment through Analysis of Time Series Stationarity
## 3 Materials and Methods
### Data Preparation  
In order to make the dataset less extensive, and to reduce noise at the same time, downsampling was performed, changing the period to 5 min instead of the original 1 min--i.e., the mean of each period of 5 min was taken from the original dataset. Thus, the dataset \begin{table}
\begin{tabular}{c c c c} \hline
**Variable** & **VCP 1** & **VCP 2** & **VCP 3** \\ \hline AF & 516k; 517k; 508k & 487k; 517k; 511k & 512k; 517k; 497k \\ Mean & 0.69; 1.03; 1.47 & 2.13; 2.04; 1.89 & 1.03; 1.85; 1.54 \\ STD & 0.84; 0.57; 0.54 & 0.47; 0.54; 0.67 & 0.46; 0.81; 0.76 \\ Min & \(-\)0.05; \(-\)0.049; \(-\)0.05 & \(-\)0.07; \(-\)0.04; \(-\)0.04 & \(-\)0.04; \(-\)0.01; \(-\)0.05 \\
25\% & 0.56; 0.69; 1.21 & 1.96; 1.87; 1.55 & 0.81; 1.53; 1.07 \\
50\% & 0.67; 0.85; 1.52 & 2.25; 2.12; 1.99 & 0.96; 1.81; 1.36 \\
75\% & 0.84; 1.20; 1.85 & 2.49; 2.37; 2.34 & 1.11; 2.03; 1.74 \\ Max & 5.34; 23.90; 11.58 & 25.11; 19.89; 10.03 & 2.60; 9.88; 20.69 \\ \hline \end{tabular}  
* AF—absolute frequency, STD—standard deviation, Min—minimum, Max—maximum VCP \(n\)—vibration of chip pump \(n\).  
\end{table} 
Table 2: Statistical data related to 2017, 2018, and 2019. Vibration values, in mm/s, for the three chip pumps. \begin{table}
\begin{tabular}{c c c c} \hline
**Year** & **VCP1** & **VCP2** & **VCP3** \\ \hline
2017 & 8964 (1.71\%) & 37,721 (7.18\%) & 13,130 (2.50\%) \\
2018 & 8026 (1.53\%) & 7916 (1.51\%) & 8220 (1.56\%) \\
2019 & 16,893 (3.21\%) & 14,555 (2.77\%) & 28,027 (5.33\%) \\ \hline \end{tabular}
\end{table} 
Table 3: Missing values from 2017, 2018, and 2019 data. was reduced to one fifth of its original size; approximately 100k rows instead of approximately 500k. This will not affect the information that is needed for the present research, since any high-frequency components possibly lost are not necessary for the stationarity analysis. Maximum and minimum values may be affected, but again they are not relevant for this goal. Additionally, many maximum and minimum values may be outliers caused by reading errors or possible external interference in the sensors, or other extreme values which can be smoothed without affecting the result of this analysis. Table 4 shows a summary of the statistical parameters of the dataset after downsampling. After dataset reduction, it is clear that the average values remained largely unchanged, but the maximum values were affected. After further analysis, some periods were identified as candidates for subsequent analysis, given their small percentage of missing values and their temporal extent. The following ranges were selected:  
* Range 1 (R1): 5 November 2017-20 January 2018;
* Range 2 (R2): 4 February 2018-27 October 2018;
* Range 3 (R3): 4 November 2018-16 February 2019. These periods are of interest because of their low number of missing data samples. Table 5 shows a summary of the statistical parameters of these intervals. As the table shows, there were still some missing points in the second interval, so they were imputed using linear interpolation. Moreover, it is also possible to notice a difference between the values of absolute frequency observed in the second interval due to its longer time span. As shown in the previous Table 6, after applying linear interpolation to the second time interval, the values hardly changed. \begin{table}
\begin{tabular}{c c c c} \hline
**Var** & **VCP1** & **VCP2** & **VCP3** \\ \hline AF & 103,549; 103,734; 101,886 & 97,883; 103,752; 102,354 & 102,715; 103,692; 99,658 \\ Mean & 0.69; 1.03; 1.47 & 2.14; 2.04; 1.89 & 1.03; 1.85; 1.54 \\ SD & 0.28; 0.56; 0.53 & 0.88; 0.52; 0.65 & 0.45; 0.80; 0.74 \\ Min & \(-\)0.05; \(-\)0.05; \(-\)0.05 & \(-\)0.05; \(-\)0.04; \(-\)0.04 & \(-\)0.02; \(-\)0.01; \(-\)0.01 \\ 25\% & 0.58; 0.72; 1.21 & 2.00; 1.89; 1.56 & 0.83; 1.55; 1.07 \\ 50\% & 0.67; 0.85; 1.51 & 2.25; 2.12; 2.00 & 0.96; 1.82; 1.36 \\ 75\% & 0.84; 1.12; 1.84 & 2.47; 2.36; 2.34 & 1.08; 2.00; 1.73 \\ Max & 4.53; 21.28; 5.81 & 24.41; 6.63; 7.45 & 2.55; 5.94; 17.25 \\ \hline \end{tabular}
\end{table}