# Early Warning Signals for Bearing Failure Using Detrended Fluctuation Analysis
## 4 Results and Discussion  
In this paper, a novel approach is proposed to extract the hidden statistical characteristics of the vibration measurements of bearings. As part of the run-to-failure experiment on several bearings, several data sets of horizontal and vertical vibrations signals were collected. DFA can help in understanding the statistical and fractal characteristics of the vibration signals by calculating their scaling exponents (\(\alpha\)). It is argued that the dynamics of the scaling exponent could provide an early warning for bearing failure. Evidently, bearing vibrations are non-stationary under normal and faulty operating conditions [31]. The non-stationarity in the time series manifests itself in a time-varying mean and variance over different segments of the series. Since DFA is one of the robust methods to evaluate the long-range correlation in non-stationary data, it was applied to two segments of the vibration signals (horizontal and vertical), as shown in Figure 5. The fluctuation functions in Figure 5b,d have two linear regions with different slopes at the short-range (10-50 samples) and long-range (100-500 samples) scales. This behavior of the fluctuation function is known as the crossover phenomenon [32]. It may result from the sinusoidal trends in the vibrations signals. Moreover, these trends could be the culprit of the proven non-stationarity in the vibration measurements. In the horizontal vibration signal, the scaling exponents at the short range (\(\alpha_{1}\)) and long range (\(\alpha_{2}\)) were 0.72 and 0.42, respectively. The calculated scaling exponents show that the horizontal signal possessed long-range correlation at the short range, and it was anti-correlated at the long-range scale. On the other hand, the vertical vibration signal had scaling exponents of 0.41 and 0.27 at the short-range and long-range scales, respectively. This signal had anti-correlated behavior on both scales. Even though all the scaling exponents were below 1, similar to the stationary time series (\(\alpha<1\)), it is evident that the vibrations signals were non-stationary as a result of the existing trends and the clear crossover phenomenon. After examining the long-range correlations in the two segments (2560 samples each) of the horizontal and vertical vibration signals, it is important to study the long-range correlations in the complete run-to-failure vibration signals. It would be interesting to see how the evolution of different stages of bearing failure was reflected in scaling exponent dynamics. To that end, the scaling exponents were computed in the complete horizontal and vertical run-to-failure vibration signals from data set 1_1, as shown in Figures 6a and 7a. Each signal consisted of 2800 segments (2560 samples) and total duration of 7.77 h (28,000 s). Figure 5: The 2560-sample bearing vibration signals (data set 1_1) and their corresponding fluctuation functions: (**a**) the bearing vibration signal (2560 samples or 100 ms) on the horizontal axis; (**b**) the fluctuation function (\(F(n)\)) vs. the box size (n) for the vibration signal on the horizontal axis; (**c**) the bearing vibration signal (2560 samples or 100 ms) on the vertical axis; (**d**) the fluctuation Figure 6. smaller time series consisting of 500 segments (5000 s) each, except the last one, which had 300 segments. Then, the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) were estimated for each segment inside these series. The average fluctuation functions inside each of these 6 smaller time series are shown in Figures 6b and 7b. In both figures, the results show a persistent increase in the average scaling exponents (\(\alpha_{1}\)) at the short range by moving from the first time series (pink) to the last one (green). Although the average scaling exponents (\(\alpha_{2}\)) show a decreasing trend, it will be shown later that the scaling exponent (\(\alpha_{2}\)) behaves differently in other data sets.