The generalized Gamma distribution for SAR intensities

Density, distribution function, quantile function and random generation for the generalized Gamma (Stacy) law with power $\alpha > 0$, shape $\phi > 0$ and inverse-scale $\mu > 0$: $$f(z) = \frac{\alpha \mu^{\alpha\phi}}{\Gamma(\phi)} z^{\alpha\phi - 1} \exp\{-(\mu z)^\alpha\}, \quad z > 0.$$ For $\alpha = 1$ it reduces to the Gamma law with shape $\phi$ and rate $\mu$; in SAR applications $\phi$ plays the role of the number of looks.

Usage

dggamma(x, alpha, phi, mu, log = FALSE)

pggamma(q, alpha, phi, mu, lower.tail = TRUE, log.p = FALSE)

qggamma(p, alpha, phi, mu, lower.tail = TRUE, log.p = FALSE)

rggamma(n, alpha, phi, mu)

Arguments

x, q: Vector of quantiles.
alpha: Power parameter ($\alpha > 0$).
phi: Shape parameter ($\phi > 0$).
mu: Inverse-scale parameter ($\mu > 0$).
log, log.p: Logical; if TRUE, probabilities/densities are given as logarithms.
lower.tail: Logical; if TRUE (default) probabilities are $P(X \le x)$, otherwise $P(X > x)$.
p: Vector of probabilities.
n: Number of observations.

Value

Density, distribution function, quantile function or random deviates.

References

Li, H.-C., Hong, W., Wu, Y.-R. and Fan, P.-Z. (2011). On the empirical- statistical modeling of SAR images with generalized gamma distribution. IEEE Journal of Selected Topics in Signal Processing, 5(3), 386–397.

Examples

dggamma(c(0.5, 1, 2), alpha = 1.5, phi = 2, mu = 1)
#> [1] 0.2633207 0.5518192 0.3546345
rggamma(3, alpha = 1.5, phi = 2, mu = 1)
#> [1] 2.148747 1.881254 1.194354