6 Remote Detection of Abnormal Behavior in Mechanical Systems 65 Table 6.1 Bhattacharyya distance and p-value for all seismic accelerometer features calculated for 278–282 Hz SAX-RMS (g2/Hz) SeisX-Power (g2) SeisXKurtosis SeisY-RMS (g2/Hz) SeisY-Power (g2) SeisYKurtosis SeisZ-RMS (g2/Hz) SeisZ-Power (g2) SeisZKurtosis 0 m Bhattacharya Dist. 0.816 0.589 0.484 0.673 0.450 0.429 1.67 1.37 1.15 P-Value 0.122 0.126 0.00370 0.0577 0.0474 0.00487 0.115 0.109 0.000105 7 m Bhattacharya Dist. 0.610 0.449 0.206 1.09 0.835 0.277 0.864 0.648 0.544 P-Value 0.0940 0.153 0.0375 0.0267 0.0421 0.0146 0.0525 0.0609 0.00951 16 m Bhattacharya Dist. 0.260 0.0495 1.34 0.0416 0.0317 0.267 0.915 0.796 0.218 P-Value 0.0778 0.301 0.0153 0.913 0.959 0.0732 0.313 0.326 0.297 From this set of features, SeisZ-Kurtosis is selected as the optimal feature because it has the largest Bhattacharyya distance in the group of features with p-value >0.05 of a fault feature as the distance between the sensor and the motor is increased. For this application, “high-performing” is defined using the Bhattacharyya distance and the p-value from a one-way ANOVA test. The parameters that define the feature space that features are selected from include sensor type, frequency range and spectral variables (RMS, power and kurtosis). The p-value from a one-way ANOVA test is used as a preliminary metric for identifying features that show a statistically significant difference between healthy and faulty data sets. A p-value greater than 0.05 indicates that, for the selected feature, there is no evidence for variation between healthy and faulty measurements (null hypothesis validation) and the opposite validation is true for p-values less than 0.05. The Bhattacharyya distance, which has been shown to be a good metric for feature selection in classification problems [17, 20, 21], is used as a secondary measure for ranking features with a null hypothesis rejection. The Bhattacharyya distance, BD(x, y), between two distributions x and y is given by: BD(x,y) = 1 4 ln 1 4 σ 2 x / σ 2 y + σ 2 y / σ 2 x +2 + 1 4 μx −μy 2 / σ 2 x +σ 2 y (6.2) The Bhattacharyya distance represents the separation between two distributions, based on the mean (μ) and standard deviation (σ) of each. Hence, in this application, features with the highest Bhattacharyya distance between healthy and faulty data sets are used for relating fault feature observability to distance. Table 6.1 includes the Bhattacharyya distance and p-value for each feature calculated for the seismic accelerometer over the frequency range of 278–282 Hz. Using the procedure described above and outlined in Fig. 6.5, the seismic Z-Axis kurtosis feature was selected as the optimal feature within this set. This feature (SeisZ-Kurtosis) will be used to explain the results from this work. 6.6 Results The results from this study are drawn from comparing healthy, faulty and motor off distributions for the SeisZ-Kurtosis feature over the frequency range of 278–282 Hz. The power spectrum for Z direction at the various test locations can be seen in Fig. 6.6. At the local measurement location, the spectral magnitude of healthy and faulty response is separated from environment noise, with an increase at the inner ball pass frequency in the faulty data. This peak is still detectable at 7 m, but the amplitude falls closer to the environment noise. At 16 m, the faulty operating case is barely distinguishable from the background noise. A tight frequency range around a predictable frequency for feature detection is critical to separate out background noise from healthy and faulty operating cases. While the SeisZ-Kurtosis feature outperformed a subset of the feature space that was studied in this work, it is likely that there are other features outside of the feature set shown in Table 6.1 that have equal or better performance metrics. Beyond the procedure that is outlined for the feature selection process, consideration was also given to the expected versus observed behavior to avoid anomalies. For example, referencing Table 6.1, the SeisX-Kurtosis feature is considered an anomaly because it does not show a consistent feature trend as the sensors are moved from 0 m to 16 m.
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