Sample from the full conditional of the Fourier coefficients.

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Description

Sample from the full conditional of the Fourier coefficients.

Usage

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sample.four.coef(w=NULL,wFT=NULL,spec=NULL,Gvec=NULL,tau2=NULL,par=NULL,n,T,
                 NF=n*n,indCos=(1:((n*n-4)/2)*2+3),ns=4,nu=1,dt=1)

Arguments

w

Observed data or latent process w (depending on which data model is used) in an T x n*n matrix with columns and rows (points on a grid stacked into a vector) corresponding to time and space, respectively.

wFT

Vector of length T*n*n containing the real Fourier transform of 'w'.

spec

Spectrum of the innovations \hat{ε} in a vector of length n*n. If 'spec' is not given, it is constructed based on 'par'.

Gvec

The propagator matrix G in vector format obtained from 'get.G.vec'. If 'Gvec' is not given, it is constructed based on 'par'.

tau2

Measurement error variance tau2. If 'NULL'; tau2=par[9].

par

Vector of parameters for the SPDE in the following order: rho_0, sigma^2, zeta, rho_1, gamma, alpha, mu_x, mu_y, tau^2. If 'spec' and 'Gvec' are given, 'par' will not be used.

n

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

T

Number of points in time.

NF

Number of Fourier functions used.

indCos

Vector of integers indicating the position cosine terms in the 1:NF real Fourier functions. The first 'ns' cosine wavenumbers in 'wave' are not included in 'indCos'.

ns

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

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.

Value

A T x n*n matrix with a sample from the full conditional of latent process α.

Author(s)

Fabio Sigrist

Examples

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##Specifications for simulated example
n <- 50
T <- 4
par <- c(rho0=0.1,sigma2=0.2,zeta=0.5,rho1=0.1,gamma=2,alpha=pi/4,muX=0.2,muY=-0.2,tau2=0.01)
spateSim <- spate.sim(par=par,n=n,T=T,seed=4)
w <- spateSim$w
##Sample from full conditional
Nmc <- 50
alphaS <- array(0,c(T,n*n,Nmc))
wFT <- real.fft.TS(w,n=n,T=T)
for(i in 1:Nmc){
  alphaS[,,i] <- sample.four.coef(wFT=wFT,par=par,n=n,T=T,NF=n*n)
}
##Mean from full conditional
alphaMean <- apply(alphaS,c(1,2),mean)
xiMean <- real.fft.TS(alphaMean,n=n,T=T,inv=FALSE)

par(mfrow=c(2,4),mar=c(1,1,1,1))
for(t in 1:4) image(1:n,1:n,matrix(w[t,],nrow=n),xlab="",ylab="",col=cols(),
                    main=paste("w(",t,")",sep=""),xaxt='n',yaxt='n')
for(t in 1:4) image(1:n,1:n,matrix(xiMean[t,],nrow=n),xlab="",ylab="",col=cols(),
                    main=paste("xiPost(",t,")",sep=""),xaxt='n',yaxt='n')

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