Compound complex Wishart distributions

Density and random generation for the compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) laws (Ferreira, Nascimento and Frery, 2022). They describe the PolSAR return as a random sum \(S = \sum_{i=1}^N Z_i\) of independent (unscaled) complex Wishart matrices \(Z_i \sim \mathcal{W}^C_m(\Sigma, L)\) with \(E(Z_i) = L\Sigma\), where the number of scatterers \(N\) follows a zero-truncated Poisson law with mean parameter \(\lambda\) (CTPCW) or a geometric law on \(\{1, 2, \ldots\}\) with success probability \(p\) (CGCW). Their densities are $$f(S) = \sum_{k=1}^{\infty} P(N = k) \frac{|S|^{kL - m}}{|\Sigma|^{kL}\Gamma_m(kL)} \exp\{-\mathrm{tr}(\Sigma^{-1}S)\},$$ which are able to describe multimodal intensities. The series is evaluated in log-scale and truncated when the terms become negligible.

Usage

dctpcw(x, lambda, Sigma, L, log = FALSE, kmax = 500L)

rctpcw(n, lambda, Sigma, L, output = c("array", "vec"))

dcgcw(x, p, Sigma, L, log = FALSE, kmax = 500L)

rcgcw(n, p, Sigma, L, output = c("array", "vec"))

Arguments

x

PolSAR observations: a complex array m x m x n, a single Hermitian matrix or an n x m(m+1)/2 matrix (see as_polsar_vec()).

lambda

Mean parameter of the zero-truncated Poisson law (\(\lambda > 0\)).

Sigma

Hermitian positive definite m x m matrix (the mean).

L

Number of looks of each Wishart component (\(L > m - 1\)).

log

Logical; if TRUE the log-density is returned.

kmax

Maximum number of terms of the series.

n

Number of observations.

output

Format of the generated sample: "array" (m x m x n) or "vec" (n x m(m+1)/2).

p

Success probability of the geometric law (\(0 < p < 1\)).

Value

dctpcw()/dcgcw() return (log-)densities; rctpcw()/rcgcw() return the simulated sample, with the simulated numbers of components in the attribute "N".

References

Ferreira, J. A., Nascimento, A. D. C. and Frery, A. C. (2022). PolSAR models with multimodal intensities. Remote Sensing, 14(20), 5083. doi:10.3390/rs14205083

See also

CompoundWishart for the R6 interface and EM estimation.

Examples

Sigma <- polsar_example_sigma("forest")
set.seed(1)
S <- rctpcw(3, lambda = 2, Sigma = Sigma, L = 4)
attr(S, "N")
#> [1] 1 2 2
dctpcw(S, lambda = 2, Sigma = Sigma, L = 4, log = TRUE)
#> [1] -4.110388 -2.837832 -3.709631
S2 <- rcgcw(3, p = 0.6, Sigma = Sigma, L = 4)
dcgcw(S2, p = 0.6, Sigma = Sigma, L = 4, log = TRUE)
#> [1] -7.2704475 -0.7213841 -6.2591386