# Early Warning Signals for Bearing Failure Using Detrended Fluctuation Analysis
## 4 Results and Discussion
For the horizontal and vertical vibration signals, the means and standard deviations of the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) are shown in Table 3. In the horizontal vibration signal, the average scaling exponent (\(\alpha_{1}\)) changed from 0.43 (anticorrelated) at the beginning of the experiment to 0.98 (long-range correlated) before the bearing failure. The change in the scaling exponent (\(\alpha_{1}\)) at the short range could be related to a shift in the mechanical system dynamics as part of the run-to-failure experiment. Similarly, in the vertical vibration signal, the average scaling exponent (\(\alpha_{1}\)) had a gradual increase with time, starting from 0.18 in the first time series and increasing up to 0.47 in the last time series. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Axis**} & **Scaling** & \multirow{2}{*}{**0–5000 s**} & **5000–** & **10,000–** & **15,000–** & **20,000–** & **25,000–** \\  & **Exponent** & & **10,000 s** & **15,000 s** & **20,000 s** & **25,000 s** & **28,000 s** \\ \hline \multirow{2}{*}{H} & \(\alpha_{1}\) (10–50) & 0.43 \(\pm\) 0.10 & 0.74 \(\pm\) 0.08 & 0.90 \(\pm\) 0.07 & 1.02 \(\pm\) 0.04 & 1.01 \(\pm\) 0.05 & 0.98 \(\pm\) 0.09 \\  &  
In Figure 6c,d and Figure 7c,d, the plots show all the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) over time toward bearing failure for the horizontal and vertical vibration signals. The increase in the scaling exponent (\(\alpha_{1}\)) is clear in scaling exponent samples and its smoothed version (red-color line). Using the [PERSON]'s tau (\(\tau\)) coefficient, the trend in the scaling exponent (\(\alpha_{1}\)) can be quantified by analyzing the statistical association between the scaling exponent and time. The calculated Kendall's tau coefficients of the scaling exponent (\(\alpha_{1}\)) were 0.67 and 0.74 in the horizontal and vertical vibration signals, respectively. Based on the significant values of the Kendall's tau before bearing failure, it appears that this rise was related to accelerated failure conditions and the proximity to bearing failure. Tracking the Kendall's tau over time could be an efficient tool in the preventative maintenance field. Usually, complex and dynamic systems undergo hidden changes as they approach a major critical transition. These changes are known as the critical slowing down phenomenon where the system requires more time to recover from any small disturbance [33]. The slower recovery can be examined in the system state by calculating the lag-1 autocorrelation or the scaling exponent. As the system is approaching the critical transition, it shows a persistent and gradual increase in the autocorrelation and the scaling exponent [34,35]. The critical slowing down could be the culprit of the change in the scaling exponent (\(\alpha_{1}\)) before bearing failure. Therefore, the [PERSON]'s tau of the scaling exponent at the short-range scale could provide a reliable measure of the proximity to bearing failure. To validate the proposed method, DFA was applied on several vibration signals from 17 data sets to calculate the scaling exponents over all the segments. Then, the trend in the scaling exponent for  
Figure 7: The vertical bearing vibration signal (data set 1_1) and its corresponding fluctuation functions and scaling exponents: (**a**) the vertical vibration signal; (**b**) the average fluctuation function vs. the box size over several segments of the vertical vibration signal; (**c**) the scaling exponents (\(\alpha_{1}\)) inside 2560-sample windows of the vertical vibration signal toward the bearing failure; (**d**) the scaling exponents (\(\alpha_{2}\)) inside 2560-sample windows of the vertical vibration signal toward the bearing failure. each vibration signal was examined by calculating the Kendall's tau coefficient. Table 4 illustrates the Kendall's tau of the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) for the 17 data sets with a total of 34 horizontal and vertical vibration signals.