# Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Properties of FCRE

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Example**

**1.**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

**Example**

**2.**

**Proposition**

**3.**

**Proof**

**of Proposition 3.**

**Proposition**

**4.**

**Proof**

**of**

**Proposition**

**4.**

**Example**

**3.**

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**5.**

**Example**

**4.**

## 3. Fractional Conditional Cumulative Residual Entropy (FCCRE) and Some Properties

**Definition**

**1.**

**Proposition**

**6.**

**Proof**

**of**

**Proposition**

**6.**

**Example**

**5.**

**Proposition**

**7.**

**Proof**

**of**

**Proposition**

**7.**

**Example**

**6.**

**Proposition**

**8.**

**Proof**

**of**

**Proposition**

**8.**

**Example**

**7.**

**Proposition**

**9.**

**Proof**

**of**

**Proposition**

**9.**

**Proposition**

**10.**

**Proof**

**of**

**Proposition**

**10.**

**Example**

**8.**

**Proposition**

**11.**

**Proof**

**of Proposition 11.**

**Definition**

**2.**

**Proposition**

**12.**

**Proof**

**of Proposition 12.**

## 4. Empirical Fractional Conditional Cumulative Residual Entropy

**Proposition**

**13.**

**Proof**

**of Proposition 13.**

**Proposition**

**14.**

**Proof**

**of Proposition 14.**

## 5. Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Mackay, D.J.C. Information Theory, Inference & Learning Algorithms; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Zhou, R.; Cai, R.; Tong, G. Applications of Entropy in Finance: A Review. Entropy
**2013**, 15, 4909–4931. [Google Scholar] [CrossRef] - Santamaria, I.; Erdogmus, D.; Principe, J.C. Entropy minimization for supervised digital communications channel equalization. IEEE Trans. Signal Process.
**2002**, 50, 1184–1192. [Google Scholar] [CrossRef] - Kullback, S. Information Theory and Statistics; Wiley: Hoboken, NJ, USA, 1959. [Google Scholar]
- Mason, W. The Stereotypes of Classical Transfer Theory. J. Polit. Econ.
**1956**, 64, 492–506. [Google Scholar] [CrossRef] - Hung, L.Y.; Myers, R.C.; Smolkin, M. Holographic Calculations of Rényi Entropy. J. High Energy Phys.
**2012**, 2011, 118–131. [Google Scholar] [CrossRef] [Green Version] - Rao, M.; Chen, Y.; Vemuri, B.C.; Wang, F. Cumulative residual entropy. A new measure of information. IEEE Trans. Inf. Theory
**2004**, 50, 1220–1228. [Google Scholar] [CrossRef] - Asadi, M.; Zohrevand, Y. On the dynamic cumulative residual entropy. J. Stat. Plan. Inference
**2007**, 137, 1931–1941. [Google Scholar] [CrossRef] - Sunoj, S.; Linu, M. Dynamic cumulative residual Rényi’s entropy. Statistics
**2012**, 46, 41–56. [Google Scholar] [CrossRef] - Rajesh, G.; Sunoj, S. Some properties of cumulative Tsallis entropy of order α. Stat. Pap.
**2016**, 60, 933–943. [Google Scholar] [CrossRef] - Psarrakos, G.; Navarro, J. Generalized cumulative residual entropy and record values. Metrika
**2013**, 76, 623–640. [Google Scholar] [CrossRef] - Rajesh, G.; Abdul-Sathar, E.; Nair, K.M.; Reshmi, K. Bivariate extension of dynamic cumulative residual entropy. Stat. Methodol.
**2014**, 16, 72–82. [Google Scholar] [CrossRef] - Baratpour, B.S.; Bami, Z. On the discrete cumulative residual entropy. J. Iran. Stat. Soc.
**2012**, 2, 203–215. [Google Scholar] - Park, S.; Rao, M.; Shin, D.W. On cumulative residual Kullback–Leibler information. Stat. Probabil. Lett.
**2012**, 82, 2025–2032. [Google Scholar] [CrossRef] - Gómez-Villegas, M.A.; Main, P.; Navarro, H.; Susi, R. Assessing the effect of kurtosis deviations from Gaussianity on conditional distributions. Appl. Math. Comput.
**2013**, 219, 10499–10505. [Google Scholar] [CrossRef] - Wu, X.; Moo, P.W. Joint Image/Video Compression and Encryption via High-Order Conditional Entropy Coding of Wavelet Coefficients. In Proceedings of the IEEE International Conference on Multimedia Computing & Systems, Florence, Italy, 7–11 June 1999. [Google Scholar]
- Abe, S.; Rajagopal, A.K. Nonadditive conditional entropy and its significance for local realism. Phys. A
**2001**, 289, 157–164. [Google Scholar] [CrossRef] [Green Version] - Porta, A.; Castiglioni, P.; Bari, V.; Bassani, T.; Marchi, A.; Cividjian, A.; Quintin, L.; Di Rienzo, M. K-nearest-neighbor conditional entropy approach for the assessment of the short-term complexity of cardiovascular control. Physiol. Meas.
**2012**, 34, 17–33. [Google Scholar] [CrossRef] - Keshmiri, S. Conditional Entropy: A Potential Digital Marker for Stress. Entropy
**2021**, 23, 286. [Google Scholar] [CrossRef] [PubMed] - Marcelo, U.R. Entropies based on fractional calculus. Phys. Lett. A
**2009**, 373, 2516–2519. [Google Scholar] - Machado, J.T. Fractional Order Generalized Information. Entropy
**2014**, 16, 2350–2361. [Google Scholar] [CrossRef] [Green Version] - Yudong, Z.; Xiaojun, Y.; Carlo, C.; Rao, R.V.; Wang, S.; Philips, P. Tea Category Identification Using a Novel Fractional Fourier Entropy and Jaya Algorithm. Entropy
**2016**, 18, 77. [Google Scholar] - Mao, X.; Shang, P.J.; Wang, J.; Yin, Y. Fractional cumulative residual Kullback-Leibler information based on Tsallis entropy. Chaos Solitons Fractals
**2020**, 139, 110292. [Google Scholar] [CrossRef] - Dong, K.; Zhang, X. Multiscale fractional cumulative residual entropy of higher-order moments for estimating uncertainty. Fluct. Noise Lett.
**2020**, 19, 2050038. [Google Scholar] [CrossRef] - Di Crescenzo, A.; Kayal, S.; Meoli, A. Fractional generalized cumulative entropy and its dynamic version. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 102, 105899. [Google Scholar] [CrossRef] - Xiong, H.; Shang, P.J.; Zhang, Y.L. Fractional cumulative residual entropy. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 78, 104879. [Google Scholar] [CrossRef] - Rao, B.L.S.P. Asymptotic Theory of Statistical Inference. J. Am. Stat. Assoc.
**1988**, 151, 564–565. [Google Scholar] - Pollard, D. Convergence of stochastic processes. Economica
**1958**, 52, 276–280. [Google Scholar] - Kuo, K.L.; Song, C.C.; Jiang, T.J. Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. J. Multivar. Anal.
**2017**, 157, 115–123. [Google Scholar] [CrossRef] - Di Crescenzo, A.; Longobardi, M. On cumulative entropies. J. Stat. Plan. Inference
**2009**, 139, 4072–4087. [Google Scholar] [CrossRef] - Park, S.; Noughabi, H.A.; Kim, I. General cumulative Kullback–Leibler information. Commun. Stat.
**2018**, 47, 1551–1560. [Google Scholar] [CrossRef] - Kayal, S. On generalized cumulative entropies. Probabil. Eng. Inf. Sci.
**2016**, 30, 640–662. [Google Scholar] [CrossRef]

**Figure 1.**The relationship among fractional cumulative residual entropy, fractional order $q$, and probability $P$ for typical examples of the uniform distribution on $[0,1]$.

**Figure 2.**The fractional cumulative residual entropy of constant sequence and uniform distribution on $[0,1]$.

**Figure 3.**The fractional cumulative residual entropy of the uniform distribution, exponential distribution, and the distribution $F\left(W\right)$.

**Figure 4.**The fractional cumulative residual entropy of uniform distribution on $[0,1]$ and exponential distribution $\lambda =E\left({U}^{2}\right)/2E\left(U\right)$.

**Figure 5.**The fractional cumulative residual entropy of exponential distribution with $\lambda =0.5$ and exponential distribution $\lambda =E\left({U}^{2}\right)/2E\left(U\right)$.

**Figure 6.**The fractional cumulative residual entropy and empirical fractional cumulative residual entropy of the uniform distribution on $[0,1]$.

**Figure 7.**Fractional conditional cumulative residual entropy ${\epsilon}_{q}\left(U|Z\right),{\epsilon}_{q}\left(V|Z\right)$, and ${\epsilon}_{q}\left(U+V|Z\right)$.

**Figure 8.**The fractional conditional cumulative residual entropy of ${\epsilon}_{q}\left(U|V\right)$.

**Figure 9.**The fractional conditional cumulative residual entropy of ${\epsilon}_{q}\left(U|V\right)$ and ${\epsilon}_{q}\left(Z|V\right)$.

**Figure 10.**The fractional conditional cumulative residual entropy of ${\epsilon}_{q}\left(U|\eta \right)$ and the $q$ power of conditional cumulative residual entropy of ${\left[\epsilon \left(U|\eta \right)\right]}^{q}$.

**Figure 12.**The fractional conditional cumulative residual entropy of aero-engine gas path time series.

q = 0.1 | q = 0.2 | q = 0.3 | q = 0.4 | q = 0.5 | q = 0.6 | q = 0.7 | q = 0.8 | q = 0.9 | q = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|

CCRE | 0.0026 | 0.0027 | 0.0029 | 0.0030 | 0.0031 | 0.0033 | 0.0035 | 0.0036 | 0.0038 | 0.0040 |

q-order FCCRE | 0.5752 | 0.3308 | 0.1903 | 0.1094 | 0.0629 | 0.0362 | 0.0208 | 0.0120 | 0.0069 | 0.0040 |

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**MDPI and ACS Style**

Dong, K.; Li, S.; Li, D.
Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy. *Fractal Fract.* **2022**, *6*, 400.
https://doi.org/10.3390/fractalfract6070400

**AMA Style**

Dong K, Li S, Li D.
Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy. *Fractal and Fractional*. 2022; 6(7):400.
https://doi.org/10.3390/fractalfract6070400

**Chicago/Turabian Style**

Dong, Keqiang, Shushu Li, and Dan Li.
2022. "Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy" *Fractal and Fractional* 6, no. 7: 400.
https://doi.org/10.3390/fractalfract6070400