Fractional Order Generalized Information

Abstract

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.

Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy

“…Various fractional entropies have been proposed in the literature [26][27][28][29][30]. We show that a solution of Equation (1) is bounded by the Tsallis entropy under some conditions.…”

Integer and fractional-order entropy analysis of earthquake data series

“…The generalized entropy (6) does not obey some of the Khinchin axioms except for α = 0 [32]. In this case it leads to the classical Shannon entropy (4).…”

“…Recently, the concepts of information content and entropy of order α ∈ R were proposed [32,54], inspired on fractional calculus. These generalizations of the classical definitions are given by: (6) where D α (·) is the derivative of order α and Γ (·) and ψ (·) represent the gamma and digamma functions, respectively.…”

Fractional Differential Texture Descriptors Based on the Machado Entropy for Image Splicing Detection

“…At present, information theory is generalized in view of fractional calculus and has gained new applications in engineering and physics [19][20][21][22][23]. Machado has recently introduced a novel formula for entropy by utilizing fractional calculus as follows [23]:…”