spate (version 1.7.5)

matern.spec: 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 \(\sigma^2\) is the marginal variance no matter how many wavenumbers are included.

Usage

matern.spec(wave, n, ns=4, rho0, sigma2, nu = 1, norm = TRUE)

Value

Vector with the spectrum of the Matern covariance function.

Arguments

wave

Spatial wavenumbers.

n

Number of grid points on each axis. n x n is the total number of spatial points.

ns

Integer indicating the number of cosine-only terms. Maximally this is 4.

rho0

Range parameter.

sigma2

Marginal variance parameter.

nu

Smoothness parameter of the Matern covariance function. By default this equals 1 corresponding to the Whittle covariance function.

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.

Author

Fabio Sigrist

Details

The Matern covariance function is of the form $$\sigma^2 2^(1-\nu) \Gamma(\nu)^{-1} (d/\rho_0)^{\nu} K_{\nu}(d/\rho_0)$$ with 'd' being the Euclidean distance between two points and K_nu(.) a modified Bessel function. Its spectrum is given by $$2^{\nu-1} \nu ((1/\rho_0)^(2\nu)) (\pi*((1/\rho_0)^2 + w)^(\nu + 1))^{-1}$$ where 'w' is a spatial wavenumber.

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())