Description Usage Arguments Details Author(s) References Examples

Various utilities for investigating complex Gaussian processes

1 2 3 4 5 6 7 8 | ```
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 |

`distance.function` |
Function giving the (complex) covariance
between two points in |

`means, pos.def.matrix, scales` |
In function |

`zold, d, give_log, func, x, Ainv, give.Z,...` |
Direct analogues of the
arguments in |

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.mattrix()`

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
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)
``` |

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