Box-Cox sparse measures: A new family of sparse measures constructed from kurtosis and negative entropy

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Sparse measures have attracted lots of interests from many fundamental research domains to be as objective functions of signal processing algorithms, health indices of degradation modeling and input features to machine learning algorithms. Among them, kurtosis and negative entropy are the most two popular sparse measures to characterize the sparsity of signals. For example, kurtosis and negative entropy are used in machine condition monitoring to quantify the sparsity of repetitive transients caused by localized rotating machine faults and to indicate an onset of early rotating faults. When kurtosis and negative entropy are decomposed into the sum of weighted normalized square envelope, the main difference between kurtosis and negative entropy is whether the logarithm transformation is applied to normalized square envelope to form a weight. In this paper, Box-Cox transformation as generalized power transformation is introduced to generalize the weights used in kurtosis and negative entropy and subsequently a new family of sparse measures, coined as Box-Cox sparse measures (BCSM), are proposed. The only parameter in the proposed BCSM is a transformation parameter λ⩾0. The contributions of this paper are summarized as follows. Firstly, this paper provides new propositions for intuitive sparse attributes of the proposed BCSM, which theoretically prove that the proposed BCSM satisfies all six intuitive sparse attributes. Secondly, in numerical and experimental studies, it is shown that (1) the proposed BCSM converges when the length of a signal increases; (2) only when a distribution is quite sparse, the proposed BCSM with λ>1 can indicate the sparsity of the distribution. Being different form the performance of the proposed BCSM with λ>1, the proposed BCSM with 0⩽λ⩽1 can steadily indicate that a distribution is getting sparser, which indicates that negative entropy (λ=0) is the best choice among all the proposed sparse measures and it is better than kurtosis (λ=1) to quantify the sparsity of repetitive transients caused by rotating faults for machine condition monitoring; (3) the proposed BCSM with 0⩽λ⩽1 is more effective in monitoring bearing and gear health conditions than the proposed BCSM with λ>1. Thirdly, the proposed BCSM of a complex Gaussian signal is investigated to provide a theoretical baseline for machine condition monitoring. Finally, the proposed BCSM can be applied to any situations, where sparse measures are needed.