Beta distribution: Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the beta distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the beta distribution.

Usage

Se_beta(alpha, beta)
re_beta(alpha, beta, delta)
hce_beta(alpha, beta, delta)
ae_beta(alpha, beta, delta)

Value

The functions Se_beta, re_beta, hce_beta, and ae_beta provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the beta distribution and $\delta$.

Arguments

alpha: The strictly positive shape parameter of the beta distribution ($\alpha > 0$).
beta: The strictly positive shape parameter of the beta distribution ($\beta > 0$).
delta: The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Author

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.

Details

The following is the probability density function of the beta distribution: $$ f(x)=\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1-x\right)^{\beta-1}, $$ where $0\leq x\leq1$, $\alpha > 0$ and $\beta > 0$, and $\Gamma(a)$ denotes the standard gamma function.

References

Gupta, A. K., & Nadarajah, S. (2004). Handbook of beta distribution and its applications. CRC Press.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Beta distributions. Continuous univariate distributions. 2nd ed. New York, NY: John Wiley and Sons, 221-235.

Examples

Run this code

# Computation of the Shannon entropy
Se_beta(2, 4)
delta <- c(1.2, 3)
# Computation of the Rényi  entropy
re_beta(2, 4, delta)
# Computation of the Havrda and Charvat entropy
hce_beta(2, 4, delta)
# Computation of the Arimoto entropy
ae_beta(2, 4, delta)
# A graphic presentation of the Havrda and Charvat entropy (HCE)
library(ggplot2)
delta <- c(0.2, 0.3, 0.5, 0.8, 1.2, 1.5, 2.5, 3, 3.5)
hce_beta(2, 1.2, delta)
z <- hce_beta(2, 1.2, delta)
dat <- data.frame(x = delta , HCE = z)
p_hce <- ggplot(dat, aes(x = delta, y = HCE)) + geom_line()
plot <- p_hce + ggtitle(expression(alpha == 2~~beta == 1.2))

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