matern.spec: Spectrum of the Matern covariance function. In spate: Spatio-Temporal Modeling of Large Data Using a Spectral SPDE Approach

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 σ^2 is the marginal variance no matter how many wavenumbers are included.

Usage

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

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.

Details

The Matern covariance function is of the form

σ^2 2^(1-ν) Γ(ν)^{-1} (d/ρ_0)^{ν} K_{ν}(d/ρ_0)

with 'd' being the Euclidean distance between two points and K_nu(.) a modified Bessel function. Its spectrum is given by

2^{ν-1} ν ((1/ρ_0)^(2ν)) (π*((1/ρ_0)^2 + w)^(ν + 1))^{-1}

where 'w' is a spatial wavenumber.

Value

Vector with the spectrum of the Matern covariance function.

Fabio Sigrist

Examples

 ```1 2 3 4 5``` ```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()) ```

Example output

```Loading required package: mvtnorm
```

spate documentation built on May 29, 2017, 11:37 a.m.