Investigations on the sensitivity of sparsity measures to the sparsity of impulsive signals

• Published in: Mechanical systems and signal processing 2022-10, Vol.178, p.109315, Article 109315
• Main Authors: Wang, Dong, Liu, Jie, Sun, Shilong, Shen, Changqing, Hou, Bingchang, Yan, Tongtong, Peng, Zhike
• Format: Article
• Language: English
• Subjects: Change point analysis; Domains; Exact solutions; Feature extraction; Impulses; Kurtosis; Localized faults; Machine condition monitoring; Machinery condition monitoring; Mathematical analysis; Repetitive transients; Sensitivity; Signal processing; Smoothness; Sparsity; Sparsity measures; Success
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2022.109315

•The sensitivity of sparsity measures to sparsity of impulsive signals is investigated.•Analytical expressions of sparsity measures of impulsive signals are derived.•Theoretical change points of the aforementioned analytical expressions are derived.•Gini index has the most stable sensitivity to sparsity of impulsive signals. In the domain of machine condition monitoring, quantification of repetitive transients caused by localized faults is a trending research topic. In recent years, besides kurtosis and negative entropy, other sparsity measures, such as Gini index, smoothness index, l-m mean, Box-Cox sparsity measures, and their generalization coined as the sum of weighted normalized square envelope, have been studied to quantify repetitive transients for machine condition monitoring. Although some experimental and theoretical studies about sparsity measures have been reported, theoretical investigations on the sensitivity of sparsity measures to the sparsity of impulsive signals are still not considerably explored. The aim of this paper is to provide theoretical supports for revealing the sensitivity of some sparsity measures in the domain of machine condition monitoring to the sparsity of impulsive signals. The first contribution of this paper proves analytical expressions of some sparsity measures of impulsive signals generated from the Bernoulli distribution at different success probabilities. Here, each of impulsive signals consists of some non-impulses (zeros) and impulses (ones) produced by several Bernoulli trials at a certain success probability in a successive order over some period of time. Hence, at different success probabilities, the number of impulses (ones) generated from the Bernoulli distribution can be increased from one to many so that sparse signals (a few ones and lots of zeroes) and dense signals (a few zeros and lots of ones) can be respectively simulated. The second contribution of this paper is to study a piecewise model based change-point analysis to derive theoretical change points of the aforementioned analytical expressions so as to provide strong supports for revealing the sensitivity of some sparsity measures to the sparsity of impulsive signals. The last contribution of this paper is to use derived theoretical change points to show that the Gini index has the most stable sensitivity to the sparsity of impulsive signals. Theoretical results of this paper can be easily extended to explain the sensitivity of sparsity measures