Cumulative Permuted Fractional Entropy and its Applications

Area of Science:

• Mathematics
• Information Theory
• Machine Learning

Background:

• Fractional calculus and entropy are key mathematical tools for analyzing system dynamics and machine learning.
• Existing entropy measures may not fully capture the complexity of intricate systems.

Purpose of the Study:

• To propose and theoretically validate a novel entropy measure, cumulative permuted fractional entropy (CPFE).
• To demonstrate the efficacy of CPFE in measuring system complexity and uncertainty for data classification.
• To integrate CPFE into support vector machines (SVMs) for enhanced classification performance.

Main Methods:

• Theoretical analysis to establish CPFE's properties, ensuring compliance with Shannon entropy axioms.
• Application of CPFE to discrete distributions, simulated, and real-world datasets.
• Development and evaluation of CPFE-SVM, a novel classification algorithm integrating CPFE with SVMs.

Main Results:

• CPFE satisfies fundamental entropy properties and exhibits unique characteristics.
• CPFE effectively quantifies the complexity and uncertainty of complex systems, enabling reliable classification.
• CPFE-SVM demonstrates superior accuracy and reduced computational cost compared to five state-of-the-art algorithms.
• CPFE-SVM excels in classifying irregular, large-scale datasets and exhibits noise insensitivity.

Conclusions:

• CPFE is a robust measure for system complexity and uncertainty.
• CPFE-SVM offers a powerful and efficient solution for classifying complex and irregular data.
• The proposed methods hold significant potential for advancing machine learning applications in diverse fields.