ffbs.spectral {spate}

Forward Filtering Backward Sampling algorithm in the spectral space of the SPDE.

Description

Forward Filtering Backward Sampling algorithm for sampling from the joint full conditional of the coefficients \alpha and for evaluation of the log-likelihood.

Usage

ffbs.spectral(w=NULL,wFT=NULL,spec=NULL,Gvec=NULL,tau2=NULL,par=NULL,n,T,lglk=FALSE,
              BwSp=TRUE,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{\epsilon} 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.

lglk

Logical; if 'TRUE' the value of the log-likelihood is returned as well.

BwSp

Logical; if 'TRUE' a sample from the full conditional of \alpha is returned.

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 list with entries (depending on whether 'lglk' are 'BwSp' are 'TRUE' or 'FALSE'):

simAlpha

A T x n*n matrix with a sample from the full conditional of latent process \alpha,

ll

The evaluated log-likelihood,

Author(s)

Fabio Sigrist