The scaled complex Wishart distribution

Density and random generation for the scaled complex Wishart law \(\mathcal{W}_m(\Sigma, L)\), the reference model for multilook PolSAR covariance (coherency) matrices in homogeneous areas: $$f(Z) = \frac{L^{mL} |Z|^{L-m}}{|\Sigma|^L \Gamma_m(L)} \exp\{-L\, \mathrm{tr}(\Sigma^{-1} Z)\},$$ where \(\Gamma_m(L) = \pi^{m(m-1)/2}\prod_{i=0}^{m-1}\Gamma(L-i)\) and \(E(Z) = \Sigma\).

Usage

dwishart_sar(x, Sigma, L, log = FALSE)

rwishart_sar(n, 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()).

Sigma

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

L

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

log

Logical; if TRUE the log-density is returned.

n

Number of observations.

output

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

Value

dwishart_sar() returns a vector of (log-)densities; rwishart_sar() returns the simulated sample.

Details

Random matrices are generated with the complex Bartlett decomposition, so non-integer numbers of looks \(L > m - 1\) are allowed.

References

Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. The Annals of Mathematical Statistics, 34(1), 152–177.

Lee, J.-S., Grunes, M. R. and Kwok, R. (1994). Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution. International Journal of Remote Sensing, 15(11), 2299–2311.

See also

WishartSAR for the R6 interface.

Examples

Sigma <- polsar_example_sigma("forest")
set.seed(1)
Z <- rwishart_sar(3, Sigma, L = 4)
dim(Z)
#> [1] 3 3 3
dwishart_sar(Z, Sigma, L = 4, log = TRUE)
#> [1] 11.42539 14.85491 13.54976