A full generalization of the Gini index for bearing condition monitoring

The classic Gini index (GI) is generalized recently by using nonlinear weight sequences as sparsity measures for sparse quantification and machine condition monitoring. The generalized GIs with different weight parameters are more robust to random transients. However, they show insufficient performance in discriminating repetitive transients under noise contamination. To overcome this shortage, this paper proposes a two-parameter generalization method to tune not only the weight parameter but also the norm order, allowing for a full generalization of the classic GI to quantify transient features and leading to new statistical indicators which are named fully generalized GIs (FGGIs). Mathematical derivations show that FGGIs satisfy at least four of the six typical attributes of sparsity measures and that those with weight parameter equal to one satisfy at least five sparse attributes, proving that they are a new family of sparsity measures. Numerical simulations demonstrate that FGGIs can monotonically evaluate the sparseness of the signals and that the FGGIs with appropriate parameters exhibit improved performance in resisting random transient interferences and discriminating noise-contaminated repetitive transients compared to traditional sparsity measures. The performance of FGGIs in the condition monitoring of rolling element bearings is validated using two different run-to-failure experiment datasets, including a gradual failure and a sudden failure. The results show that increasing the norm order can improve the capability of FGGIs to characterize transient fault features, allowing more accurate trending of bearing health conditions, and therefore achieving better condition monitoring performance than the traditional sparsity measures.

Automated summary

The classic Gini index has been generalized using nonlinear weight sequences to quantify sparsity and monitor machine condition. The generalized GIs with different weight parameters are more robust to random transients, but they struggle to distinguish repetitive transients under noise contamination. To address this issue, this paper proposes a two-parameter generalization method to tune both the weight parameter and the norm order, resulting in fully generalized GIs (FGGIs) that can quantify transient features. Mathematical derivations show that FGGIs satisfy multiple attributes of sparsity measures, proving that they are a new family of sparsity measures. Numerical simulations demonstrate that FGGIs can evaluate signal sparsity and exhibit improved performance in resisting random transients and discriminating noise-contaminated repetitive transients compared to traditional sparsity measures. The performance of FGGIs in the condition monitoring of rolling element bearings is validated using two different run-to-failure experiment datasets, showing that increasing the norm order improves the capability of FGGIs to characterize transient fault features and achieve better condition monitoring performance.