Theoretical Investigations on Kurtosis and Entropy and Their Improvements for System Health Monitoring

System health monitoring as the basis of prognostics and health management (PHM) aims to explore health indices/features for PHM to do condition monitoring, perform abnormal detection, and provide degradation trends for prognostic models. Kurtosis and negative entropy are two classic and popular indices to measure the sparsity of impulsive transients, and they are prone to be affected by impulsive noise. The purpose of this article is twofold. Theoretically, new propositions and their proofs are proposed to illustrate how kurtosis and negative entropy can help to characterize impulsive transients and why they are affected by impulsive noise. Next, a weighted residual regression-based index is proposed to relieve the sensitivity of kurtosis and entropy to impulsive noise and to provide monotonic trends for gear and bearing degradation assessment. Theoretical results show that kurtosis and negative entropy are changed with the length of nonimpulsive regions of impulsive transients. Experimental results demonstrate that the proposed method has better incipient fault detection ability and monotonic trending ability than kurtosis and negative entropy for bearing and gear health monitoring and degradation assessment.