Correlation dimension and approximate entropy for machine condition monitoring: Revisited

Abstract

The sparsity of signals is of great concern in various research domains. In mechanical systems and signal processing, repetitive transients are the symptoms of localized gear and bearing faults and they are sparse signals. During the recent years, sparsity measures, such as kurtosis and Shannon entropy, have been thoroughly studied to quantify repetitive transients for machine condition monitoring. Spectral kurtosis and spectral negative Shannon entropy are two typical examples of sparsity measures for machine condition monitoring. Besides sparsity measures, complexity measures including correlation dimension (CD) and approximate entropy (AE) have been experimentally studied during the recent years. However, theoretical investigations on these two complexity measures for machine condition monitoring are seldom reported. This paper aims to fill this research gap and propose some new theorems and proofs to show that CD and AE have a “bilateral reduction” effect, which is a proper measure of entropy. Specifically, CD with any dimension and AE with smaller dimension become smaller when a signal is getting sparser or more deterministic, which is significantly different from sparsity measures that are monotonically increasing when a signal is getting from more deterministic to sparser. This new discovery is able to help readers fully understand the main difference between sparsity measures and complexity measures. In view of this discovery, it is suggested that the concept of blind fault component separation should be used to separate low-frequency periodic components (a deterministic signal) from high-frequency repetitive transients (a sparse signal) before complexity measures are used for machine condition monitoring. This suggestion aims to avoid the uncertainty of machine condition monitoring caused by low-frequency periodic components and high-frequency repetitive transients.