Density, distribution function, quantile function and random generation for the \(\mathcal{G}_I^0(\alpha, \gamma, L)\) distribution, the multiplicative model for multilook SAR intensities with reciprocal gamma backscatter and gamma speckle (Frery et al., 1997).
Usage
dg0i(x, alpha, gamma, L, log = FALSE)
pg0i(q, alpha, gamma, L, lower.tail = TRUE, log.p = FALSE)
qg0i(p, alpha, gamma, L, lower.tail = TRUE, log.p = FALSE)
rg0i(n, alpha, gamma, L)Arguments
- x, q
Vector of quantiles.
- alpha
Roughness parameter (\(\alpha < 0\)).
- gamma
Scale parameter (\(\gamma > 0\)).
- L
Number of looks (\(L > 0\)).
- 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
dg0i() gives the density, pg0i() the distribution function,
qg0i() the quantile function and rg0i() generates random deviates.
Details
The density is $$f(z) = \frac{L^L \Gamma(L - \alpha)}{\gamma^{\alpha} \Gamma(-\alpha) \Gamma(L)} \frac{z^{L - 1}}{(\gamma + L z)^{L - \alpha}}, \quad z > 0,$$ with roughness \(\alpha < 0\), scale \(\gamma > 0\) and number of looks \(L > 0\). If \(Z \sim \mathcal{G}_I^0(\alpha, \gamma, L)\) then \(-\alpha Z / \gamma\) follows a Snedecor F distribution with \(2L\) and \(-2\alpha\) degrees of freedom, which is used to compute the distribution and quantile functions exactly.
References
Frery, A. C., Muller, H.-J., Yanasse, C. C. F. and Sant'Anna, S. J. S. (1997). A model for extremely heterogeneous clutter. IEEE Transactions on Geoscience and Remote Sensing, 35(3), 648–659.
Nascimento, A. D. C., Cintra, R. J. and Frery, A. C. (2010). Hypothesis testing in speckled data with stochastic distances. IEEE Transactions on Geoscience and Remote Sensing, 48(1), 373–385.
See also
G0I for the R6 interface (estimation and entropy).
Examples
set.seed(1)
z <- rg0i(5, alpha = -1.5, gamma = 2, L = 8)
dg0i(z, alpha = -1.5, gamma = 2, L = 8)
#> [1] 0.41574253 0.19938366 0.44241797 0.28971117 0.04380056
pg0i(c(0.5, 1, 2), alpha = -1.5, gamma = 2, L = 8)
#> [1] 0.08289216 0.29852294 0.58466711
qg0i(c(0.1, 0.5, 0.9), alpha = -1.5, gamma = 2, L = 8)
#> [1] 0.5416068 1.6194935 6.9285395