https://github.com/cran/spate
Tip revision: 20b4d4bea90274d57f50b28b72478ce190dc4c34 authored by Fabio Sigrist on 25 January 2015, 00:00:00 UTC
version 1.4
version 1.4
Tip revision: 20b4d4b
matern.spec.Rd
\name{matern.spec}
\alias{matern.spec}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{
Spectrum of the Matern covariance function.
}
\description{
Spectrum of the Matern covariance function. Note that the spectrum is
renormalized, by dividing with the sum over all frequencies so that
they sum to one, so that
\eqn{\sigma^2} is the marginal variance no matter how many
wavenumbers are included.
}
\usage{
matern.spec(wave, n, ns=4, rho0, sigma2, nu = 1, norm = TRUE)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
\item{wave}{
Spatial wavenumbers.
}
\item{n}{
Number of grid points on each axis. n x n is the total number of spatial points.
}
\item{ns}{Integer indicating the number of cosine-only terms. Maximally
this is 4.}
\item{rho0}{
Range parameter.
}
\item{sigma2}{
Marginal variance parameter.
}
\item{nu}{
Smoothness parameter of the Matern covariance function. By default this equals 1 corresponding to the Whittle covariance function.
}
\item{norm}{
logical; if 'TRUE' the spectrum is multiplied by n*n so that after
applying the real Fourier transform 'real.FFT' one has the correct normalization.
}
}
\details{
The Matern covariance function is of the form
\deqn{\sigma^2 2^(1-\nu) \Gamma(\nu)^{-1} (d/\rho_0)^{\nu} K_{\nu}(d/\rho_0)}
with 'd' being the Euclidean distance between two points and K_nu(.)
a modified Bessel function. Its spectrum is given by
\deqn{2^{\nu-1} \nu ((1/\rho_0)^(2\nu)) (\pi*((1/\rho_0)^2 + w)^(\nu + 1))^{-1}}
where 'w' is a spatial wavenumber.
}
\value{
Vector with the spectrum of the Matern covariance function.
}
\author{
Fabio Sigrist
}
\examples{
n <- 100
spec <- matern.spec(wave=spate.init(n=n,T=1)$wave,n=n,rho0=0.05,sigma2=1,norm=TRUE)
sim <- real.fft(sqrt(spec)*rnorm(n*n),n=n,inv=FALSE)
image(1:n,1:n,matrix(sim,nrow=n),main="Sample from a Gaussian process
with Matern covariance function",xlab="",ylab="",col=cols())
}