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An Entropy Formulation Based on the Generalized Liouville Fractional Derivative
Rui A C Ferreira1, J Tenreiro Machado2
1Grupo Física-Matemática, Faculdade de Ciências, Universidade de Lisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
Researchers developed a novel entropy formula inspired by the Liouville fractional derivative. This new entropy measure was applied to the Dow Jones Industrial Average time series, also generalizing Jensen-Shannon divergence.
Area of Science:
- Information Theory
- Fractional Calculus
- Financial Mathematics
Background:
- Entropy quantifies uncertainty in probability distributions.
- Fractional calculus extends differentiation and integration to non-integer orders.
- Financial time series analysis often involves complex, non-linear dynamics.
Purpose of the Study:
- To introduce a new entropy formula based on the Liouville fractional derivative.
- To apply the novel entropy definition to analyze the Dow Jones Industrial Average.
- To generalize Jensen-Shannon divergence and examine its behavior with fractional orders in financial time series.
Main Methods:
- Derivation of a new entropy formula incorporating the Liouville fractional derivative.
- Application of the proposed entropy measure to the Dow Jones Industrial Average (DJIA) dataset.
- Generalization of the Jensen-Shannon divergence for fractional orders.
Main Results:
- A novel fractional entropy formula was successfully formulated.
- The new entropy measure provided insights into the Dow Jones Industrial Average's distribution.
- The generalized Jensen-Shannon divergence demonstrated a dependency on fractional order for the time series.
Conclusions:
- The proposed fractional entropy offers a new tool for analyzing probability distributions.
- The Liouville fractional derivative provides a promising framework for extending information-theoretic measures.
- Fractional calculus can offer valuable perspectives in financial time series analysis.
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