  Machine health management is one of the main research contents of PHM
technology, which aims to monitor the health states of machines online and
evaluate degradation stages through real-time sensor data. In recent years,
classic sparsity measures such as kurtosis, Lp/Lq norm, pq-mean, smoothness
index, negative entropy, and Gini index have been widely used to characterize
the impulsivity of repetitive transients. Since smoothness index and negative
entropy were proposed, the sparse properties have not been fully analyzed. The
first work of this paper is to analyze six properties of smoothness index and
negative entropy. In addition, this paper conducts a thorough investigation on
multivariate power average function and finds that existing classical sparsity
measures can be respectively reformulated as the ratio of multivariate power
mean functions (MPMFs). Finally, a general paradigm of index design are
proposed for the expansion of sparsity measures family, and several newly
designed dimensionless health indexes are given as examples. Two different
run-to-failure bearing datasets were used to analyze and validate the
capabilities and advantages of the newly designed health indexes. Experimental
results prove that the newly designed health indexes show good performance in
terms of monotonic degradation description, first fault occurrence time
determination and degradation state assessment.
