Investigations on improved Box-Cox sparsity measures for machine condition monitoring

Sparsity measures are commonly utilized as health indicators for machine condition monitoring. Recently, with the assistance of Box-Cox transformation, kurtosis and negative entropy have been smoothly extended to form Box-Cox sparsity measures (BCSMs). However, traditional BCSMs do not generate sparsity measures that outperform negative entropy, which means it is meaningless to some extent. Therefore, in this paper, traditional BCSMs are further extended to develop more robust sparsity measures. Firstly, inspired by the limited weight range of the Gini index, the traditional BCSMs are extended to the case of $\lambda <0$ by a two-parameter Box-Cox transformation. Then, by examining the decomposition forms of $L2/L1$ norm and Hoyer measure, the advantage of directly applying the Box-Cox transformation to the sparsity measure is discovered. Thus, the improved BCSMs (IBCSMs) are naturally proposed by performing the classical Box-Cox transformation on the BCSMs with $\lambda \ge -1$. Subsequently, three key properties of the proposed sparsity measures are analyzed through three numerical experiments. Finally, the proposed sparsity measures are deployed as health indicators to characterize the degradation process of three bearings. Numerical and experimental results demonstrate the salient advantages of the proposed IBCSMs in incipient fault detection.