Cumulative Permuted Fractional Entropy and its Applications

Fractional calculus and entropy are two essential mathematical tools, and their conceptions support a productive interplay in the study of system dynamics and machine learning. In this article, we modify the fractional entropy and propose the cumulative permuted fractional entropy (CPFE). A theoretical analysis is provided to prove that CPFE not only meets the basic properties of the Shannon entropy but also has unique characteristics of its own. We apply it to typical discrete distributions, simulated data, and real-world data to prove its efficiency in the application. This article demonstrates that CPFE can measure the complexity and uncertainty of complex systems so that it can perform reliable and accurate classification. Finally, we introduce CPFE to support vector machines (SVMs) and get CPFE-SVM. The CPFE can be used to process data to make the irregular data linearly separable. Compared with the other five state-of-the-art algorithms, CPFE-SVM has significantly higher accuracy and less computational burden. Therefore, the CPFE-SVM is especially suitable for the classification of irregular large-scale data sets. Also, it is insensitive to noise. Implications of the results and future research directions are also presented.