# Early Warning Signals for Bearing Failure Using Detrended Fluctuation Analysis
## 4 Results and Discussion
In addition, the plots of the scaling exponent (\(\alpha_{1}\)) for the data sets are shown in Figures 8-10. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Data Set** & **Axis** & **Range 1** & **Kendall’s Tau of \(\alpha_{1}\)** & **Range 2** & **Kendall’s Tau of \(\alpha_{2}\)** \\ \hline \multirow{3}{*}{1\_1} & Horizontal & 10–50 & 0.67 & 100–500 & \(-\)0.53 \\  & Vertical & 10–50 & 0. Figure 8: The scaling exponents (\(\alpha_{1}\)) of non-overlapping 2560-sample segments of the vibration signals toward bearing failure: (**a**) data set 1_2 (horizontal); (**b**) data set 1_2 (vertical); (**c**) data set 1_3 (horizontal); (**d**) data set 1_3 (vertical); (**e**) data set 1_4 (horizontal); (**f**) data set 1_4 (vertical); (**g**) data set 1_5 (horizontal); (**h**) data set 1_5 (vertical); (**i**) data set 1_6 (horizontal); (**j**) data set 1_6 (vertical); (**k**) data set 1_7 (horizontal); (**l**) data set 1_7 (vertical). Figure 9: The scaling exponents (\(\alpha_{1}\)) of non-overlapping 2560-sample segments of the vibration signals toward bearing failure: (**a**) data set 3_1 (horizontal); (**b**) data set 3_1 (vertical); (**c**) data set 3_2 (horizontal); (**d**) data set 3_2 (vertical); (**e**) data set 3_3 (horizontal); (**f**) data set 3_3 (vertical). The increase in the scaling exponent (\(\alpha_{1}\)) is clear from the positive [PERSON]'s tau coefficients (\(\tau>0\)) for 33 out of the 34 vibration signals. Moreover, 22 vibration signals had a significant rise in the [PERSON]'s tau coefficient with \(\tau\geq 0.5\). That provides strong evidence for the existence of critical slowing down in the vibration signals toward their failure. On the other hand, as mentioned earlier, the scaling exponents (\(\alpha_{2}\)) at the long range do not possess a consistent behavior among the different data sets. Some vibration signals have an increasing trend and other signals have a decreasing trend. Practically, a large portion of machine breakdowns originates from bearing failures; therefore, it is of great importance to monitor bearing condition. The main goals of bearing monitoring are decreasing maintenance cost and maintaining a smooth and continuous running of the system. As part of predictive maintenance, the proposed method can be integrated as a new detection technique in bearing condition monitoring through its vibration measurements. Figure 10: The scaling exponents (\(\alpha_{1}\)) of non-overlapping 2560-sample segments of the vibration signals toward bearing failure: (**a**) data set 2_1 (horizontal); (**b**) data set 2_1 (vertical); (**c**) data set 2_2 (horizontal); (**d**) data set 2_2 (vertical); (**e**) data set 2_3 (horizontal); (**f**) data set 2_3 (vertical); (**g**) data set 2_4 (horizontal); (**h**) data set 2_4 (vertical); (**i**) data set 2_5 (horizontal); (**j**) data set 2_5 (vertical); (**k**) data set 2_6 (horizontal); (**l**) data set 2_6 (vertical); (**m**) data set 2_7 (horizontal); (**n**) data set 2_7 (vertical). Moreover, ball bearings are mostly used in low- and medium-speed applications, such as industry, automotive, agriculture, etc. So, the proposed method has the potential to be used in different rotating equipment in the industrial sector (electric motors, pumps, compressors, assembly lines, and elevators) and the automotive sector (engines, steering wheel, driveshaft, driveline, and gearboxes).