Gini Indices II and III: Two new Sparsity Measures and Their Applications to Machine Condition Monitoring

Abstract

Machine condition monitoring (MCM) uses signal processing and machine learning methods to analyze monitoring data and perform timely condition-based maintenance. Since monitoring data usually have a sparsity property, sparsity measures (SMs) are naturally considered to quantify the sparsity of signals and they serve as the objective functions of many signal processing and machine learning methods. Although Gini index, kurtosis, smoothness index, negative entropy, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lp</i> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lq</i> norm have been considerably investigated for MCM, the design of new SMs for enhancing MCM is rarely reported. In this article, based on the ratio of different quasi-arithmetic means (RQAM), two new SMs, coined as Gini index Ⅱ (GI2) and Gini index Ⅲ (GI3), are designed. New proofs show that the GI2 and GI3 satisfy all six sparsity attributes. Subsequently, the GI2 and GI3 of the square envelope of Gaussian white noise are theoretically investigated and their theoretical values are, respectively, equal to 2/3 and 1/3, which can be used as baselines for machine abnormality detection. Once GI2 and GI3 exceed the baselines, abnormal health conditions can be detected without needing historical data and prior fault knowledge. Finally, simulated and experimental case studies showed that the proposed GI2 and GI3 have competitive performance with Gini index and that they are better than kurtosis, negative entropy, and smoothness index, in characterizing the sparsity of signals. This article demonstrates that the RQAM is a potential framework to design new SMs.