spate (version 1.5)

innov.spec: Spectrum of the innovation term epsilon.

Description

Spectrum of the innovation term epsilon.

Usage

innov.spec(wave,n,ns=4,rho0,sigma2,zeta,rho1,alpha,gamma,nu=1,dt=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

Number of real Fourier functions that have only a cosine and no sine term. 'ns' is maximal 4.

rho0

Range of the Matern covariance funtion for the innovation term epsilon

sigma2

Marginal variance of the Matern covariance funtion for the innovation term epsilon

zeta

Damping parameter

rho1

Range parameter of the diffusion term

alpha

Parameter that determines the direction of anisotropy in the diffusion term

gamma

Parameter that determines the amount of anisotropy in the diffusion term

nu

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

dt

Temporal lag between two time points. By default, this equals 1.

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.

Value

Vector with the spectrum of the integrated innovation term epsilon hat.

Examples

n <- 100
spec <- innov.spec(wave=spate.init(n=n,T=1)$wave,n=n,rho0=0.05,sigma2=0.5,zeta=0.5,
                   rho1=0.05,alpha=pi/4,gamma=2,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 the integrated
stochastic innovation",xlab="",ylab="",col=cols())