corr_complex: Complex Gaussian processes
In cmvnorm: The Complex Multivariate Gaussian Distribution

Description Usage Arguments Details Author(s) References Examples

Various utilities for investigating complex Gaussian processes

corr_complex(z1, z2 = NULL, distance.function = complex_CF, means =
NULL, scales = NULL, pos.def.matrix = NULL)
complex_CF(z1,z2, means, pos.def.matrix)
scales.likelihood.complex(pos.def.matrix, scales, means,  zold, z,
               give_log = TRUE, func = regressor.basis)
interpolant.quick.complex(x, d, zold, Ainv, scales = NULL, pos.def.matrix = NULL,
               means=NULL, func = regressor.basis, give.Z = FALSE,
               distance.function = corr_complex, ...)

`z, z1, z2`	Points in C^n
`distance.function`	Function giving the (complex) covariance between two points in C^n
`means, pos.def.matrix, scales`	In function `complex_CF()`, the mean and covariance matrix of the distribution whose characteristic function is used to give the covariance matrix; `scales` is used to specify the diagonal of `pos.def.matrix` if the off-diagonal elements are zero
`zold, d, give_log, func, x, Ainv, give.Z,...`	Direct analogues of the arguments in `interpolant()` and `scales.likelihood()` in the emulator package

Function complex_CF() returns a (slightly reparameterized) characteristic function of a complex Gaussian distribution. The covariance is given by

[omitted, see PDF]

where x-x' is interpreted as the distance between two observations, mu is the mean of the distribution (which is in general a complex vector), and B a positive-definite matrix.
Function corr_complex() is the complex analogue of corr.matrix(). It returns a matrix with entry (i,j) equal to the covariance of the process at observation i and observation j, or cov(eta(x_i),eta(x_j)). The elements are calculated by complex_CF().

This function includes only a single method, that of nested calls to apply(). I could not figure out how to generalize method 1 of corr.matrix() to the complex case.
Function scales.likelihood.complex() is a complex version of scales.likelihood() which takes a positive definite matrix and a mean. The formula used is

[omitted, see PDF]

. Here and elsewhere, A^* means the complex conjugate of the transpose.
Function interpolant.quick.complex() is a complex version of interpolant.quick().

[omitted, see PDF]

This is the complex version of Oakley's equation 2.30 or Hankin's equation 5.

More details are given in the package vignette.

Robin K. S. Hankin

Hankin, R. K. S. 2005. “Introducing BACCO, an R bundle for Bayesian Analysis of Computer Code Output”, Journal of Statistical Software, 14(15)
J. Oakley 1999. Bayesian uncertainty analysis for complex computer codes, PhD thesis, University of Sheffield.

complex_CF(c(1,1i),c(1,-1i),means=c(1i,1i),pos.def.matrix=diag(2))

V <- latin.hypercube(7,2,complex=TRUE)

cm <- c(1,1+1i)                     # "complex mean"
cs <- matrix(c(2,1i,-1i,1),2,2)   # "complex scales"
tb <- c(1,1i,1-1i)                     # "true beta"

A <- corr_complex(V,means=cm,pos.def.matrix=cs)
Ainv <- solve(A)
z <- drop(rcmvnorm(n=1,mean=regressor.multi(V) %*% tb, sigma=A))


betahat.fun(V,Ainv,z)    # should be close to 'tb'

#scales.likelihood.complex(cs,cm,V,z)   # log-likelihood evaluated true parameters



interpolant.quick.complex(x=0.1i+V[1:3,],d=z,zold=V,Ainv=Ainv,pos.def.matrix=cs,means=cm)