Compressed multi-scale entropy and its application in mechanical fault diagnosis

Multi-scale entropy (MSEn) is a powerful tool for measuring the complexity of time series, and it has been widely recognized in fault diagnosis. However, most of the current multi-scale entropy methods ignore information aliasing caused by the coarse-graining process, which makes it difficult to maintain the information reliability of the original data and thus affects the analysis results. In response to the aforementioned issue, this paper first analyzes the information aliasing of the classical MSEn and then proposes the compressed multi-scale entropy (CoMSEn) for mechanical fault diagnosis. In the proposed CoMSEn, the random Gaussian matrix is introduced to achieve transformation domain projection, and the results under different compression rates are treated as the coarse-graining results under different scales to achieve multi-scale analysis. Furthermore, to eliminate the influence of randomness of the Gaussian matrix, the mean value of the coarse-grained sequence sample entropy obtained through multiple compressions is adopted as the final result. Benefiting from the random Gaussian matrix satisfying the restricted isometric property with high probability under the assumption that the original signal is sparse in the specific transform domain, the coarse-grained sequences obtained under different compression rates can preserve the essential discriminative information of the raw data and effectively suppress the information aliasing caused by the linear down-sampling in classical MSEn, thereby providing a theoretical basis for the entropy calculation results' reliability. The effectiveness of the proposed method is verified utilizing the CWRU rolling bearing dataset and an actual reciprocating compressor valve dataset.