Generalized relative entropy: New look at Rényi entropy and its exploration from complexity measures to sparsity measures with applications in machine condition monitoring.

• The "bilateral reduction" effect of Rényi entropy is theoretically studied and proved. • Relative entropy is extended by adding order α to generalized relative entropy. • Generalized relative entropy satisfies six sparse criteria. • Conversion between complexity measures and sparsity measures is realized. • New sparse measure cluster is proposed to enrich machine condition monitoring. Complexity measures and sparsity measures are capable of detecting and characterizing underlying dynamic changes in machine condition monitoring. Previous studies have developed Shannon entropy and Kullback-Leibler (KL) divergence and their generalization, namely Rényi entropy and Rényi divergence, but their theoretical properties as a function of sparsity and their transitions from complexity measures to sparsity measures are not fully explored and discussed. This paper continues conducting theoretical and experimental investigations on Rényi entropy and Rényi divergence and exploring their theoretical properties and transitions to enrich the domain of complexity and sparsity measures and their applications to machine condition monitoring. Specifically, this paper theoretically proves and verifies that Rényi entropy has a similar "bilateral reduction" effect exhibited by other complexity measures including correlation dimension, approximate entropy, fuzzy entropy and sample entropy. Inspired by relative entropy, which is a special case of KL divergence and a sparsity measure, and the relationship between KL divergence and Rényi divergence, in this paper, we attempt to generalize relative entropy by adding an order of α , to convert a series of complexity measures from Rényi entropy to a series of sparsity measures, which answers how to realize transitions from complexity measures to sparsity measures. More importantly, this paper is helpful to provide a guideline for correct uses of Rényi entropy, Rényi divergence, and generalized relative entropy for the quantification of impulsive transients caused by machine faults for machine condition monitoring. [ABSTRACT FROM AUTHOR]