Some further thoughts about spectral kurtosis, spectral L2/L1 norm, spectral smoothness index and spectral Gini index for characterizing repetitive transients. (August 2018) - Wang, Dong
Graphical abstract: The relationship between spectral kurtosis, spectral L2 / L1 norm, spectral Lp / Lq norm, the reciprocal of spectral smoothness index and spectral Gini index for characterizing repetitive transients, especially bearing fault signals. Highlights: Extraction of repetitive transients is formulated as some optimization problems. Spectral Gini index is formally defined. Two geometrical explanations for spectral Gini index are made. The relationship between spectral L 2/ L 1 norm and spectral Gini index is clarified. It is mathematically shown that spectral statistical parameters are affected by outliers. Abstract: Thanks to the great efforts made by Antoni (Mech. Syst. Signal Pr. 20 (2006) 282–307), spectral kurtosis has been recognized as a milestone to characterize repetitive transients, especially bearing fault signals caused by bearing defects. The basic idea of spectral kurtosis is to use kurtosis to quantify an analytic bearing fault signal constructed from band-pass filtering and Hilbert transform. In our previous study (Mech. Syst. Signal Pr. 104 (2018) 290–293), we mathematically showed that spectral kurtosis can be decomposed into squared envelope and squared L2 / L1 norm. Based on this finding, we defined spectral L2 / L1 norm and then extended spectral L2 / L1 norm to spectral Lp / Lq norm. Moreover, when p  = 1 and q  = 0, we mathematically showed that spectral L1 / L0 norm is the reciprocal of spectral smoothness index. Here, being similar with the functionality of kurtosis, smoothness index introduced by Bozchalooi and Liang (J. Sound Vib., 308 (2007) 246–267) has been recognized as another attractive and important statistical parameter to characterize repetitive transients. Hence, the mathematical connection between spectral kurtosis and the reciprocal of spectral smoothness index was well established. Further, we derived an analytical expression of spectral Lp / Lq norm when complex Gaussian noises were considered as an input to spectral Lp / Lq norm. Consequently, spectral Lp / Lq norm was able to be normalized by the analytical expression. In this paper, we give some further thoughts about spectral kurtosis, spectral L2 / L1 norm, the reciprocal of spectral smoothness index and spectral Gini index for characterizing repetitive transients. Firstly, we formulate extraction of repetitive transients as maximization of spectral Lp / Lq norm. Most existing fault detection algorithms derived from spectral kurtosis can be naturally extended to maximize spectral Lp / Lq norm. Secondly, we formally define spectral Gini index and then mathematically clarify its relationship with spectral L2 / L1 norm. Moreover, we calculate spectral Gini index for complex Gaussian noises so as to normalize and redefine spectral Gini index. Thirdly, the relationship between spectral kurtosis, spectral L2 / L1 norm, spectral Lp / Lq norm, the reciprocal of spectral smoothness index and spectral Gini index for characterizing repetitive transients is revealed. Finally, we mathematically show that each of spectral kurtosis, spectral L2 / L1 norm, the reciprocal of spectral smoothness index and spectral Gini index is a monotonically increasing function of the maximum of squared envelope, which indicates that spectral kurtosis, spectral L2 / L1 norm, the reciprocal of spectral smoothness index and spectral Gini index are affected by outliers. Based on this finding, experimental comparisons are conducted to show that the reciprocal of spectral smoothness index and spectral Gini index are less sensitive to outliers and they are more preferable to be used in most existing algorithms instead of maximization of spectral kurtosis for extraction of repetitive transients, especially bearing fault signals.