A Review of Fractional Order Entropies

Abstract

Fractional calculus (FC) is the area of calculus that generalizes the operations of differentiation and integration. FC operators are non-local and capture the history of dynamical effects present in many natural and artificial phenomena. Entropy is a measure of uncertainty, diversity and randomness often adopted for characterizing complex dynamical systems. Stemming from the synergies between the two areas, this paper reviews the concept of entropy in the framework of FC. Several new entropy definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. However, FC is not yet well disseminated in the community of entropy. Therefore, new definitions based on FC can generalize both concepts in the theoretical and applied points of view. The time to come will prove to what extend the new formulations will be useful.

Keywords: entropy; fractional calculus; information theory.

Figure 1

The values of $S_{\alpha }^{\left(AS\right)}$, $S_{\alpha }^{\left(U\right)}$, $S_{\alpha }^{\left(Y\right)}$, $S_{\alpha }^{\left(M\right)}$, $S_{\alpha }^{\left(J\right)}$, $S_{\alpha }^{\left(K\right)}$ and $S_{\alpha }^{\left(FM\right)}$ versus $\alpha \in [0,1]$ for the (a) Poisson, (b) Gaussian, (c) Lévy and (d) Weibull distributions.

Figure 2

The values of $S_{q,\alpha }^{\left(RCJ\right)}$, $S_{q,\alpha }^{\left(M{L}_1\right)}$ and $S_{q,\alpha }^{\left(M{L}_2\right)}$ versus $\alpha \in [-0.6,0.6]$ and $q\in [1.2,2.2]$ for the (a–c) Poisson, (d–f) Gaussian, (g–i) Lévy and (j–l) Weibull distributions.

Figure 3

The entropy of the DJIA stock index for daily closing values in the time period from 1 January 1987 up to 24 November 2018, with one-day sampling interval: (a) $S_{\alpha }^{\left(AS\right)}$, $S_{\alpha }^{\left(U\right)}$, $S_{\alpha }^{\left(Y\right)}$, $S_{\alpha }^{\left(M\right)}$, $S_{\alpha }^{\left(J\right)}$, $S_{\alpha }^{\left(K\right)}$ and $S_{\alpha }^{\left(FM\right)}$ versus $\alpha \in [0,1]$; (b–d) $S_{q,\alpha }^{\left(RCJ\right)}$, $S_{q,\alpha }^{\left(M{L}_1\right)}$ and $S_{q,\alpha }^{\left(M{L}_2\right)}$ versus $\alpha \in [-0.6,0.6]$ and $q\in [1.2,2.2]$.

Figure 4

The map of co-occurrence of the authors’ keywords in the 170 papers extracted from Scopus for constructing Table 2. The minimum value of co-occurrence of each keyword is 3. The clusters are represented by $\mathcal{C}=\{{\mathcal{C}}_1,\cdots ,{\mathcal{C}}_6\}$.