inla.spde2.matern: Matern SPDE model object for INLA

In andrewzm/INLA: Functions which allow to perform full Bayesian analysis of latent Gaussian models using Integrated Nested Laplace Approximaxion

Description

Create an inla.spde2 model object for a Matern model.

Usage

inla.spde2.matern(mesh,
                  alpha = 2,
                  param = NULL,
                  constr = FALSE,
                  extraconstr.int = NULL,
                  extraconstr = NULL,
                  fractional.method = c("parsimonious", "null"),
                  B.tau = matrix(c(0,1,0),1,3),
                  B.kappa = matrix(c(0,0,1),1,3),
                  prior.variance.nominal = 1,
                  prior.range.nominal = NULL,
                  prior.tau = NULL,
                  prior.kappa = NULL,
                  theta.prior.mean = NULL,
                  theta.prior.prec = 0.1,
                  n.iid.group = 1)

Arguments

mesh	The mesh to build the model on, as an inla.mesh or inla.mesh.1d object.
alpha	Fractional operator order, 0<α≤q 2 supported. (ν=α-d/2)
param	Parameter, e.g. generated by param2.matern.orig
constr	If TRUE, apply an integrate-to-zero constraint. Default FALSE.
extraconstr.int	Field integral constraints.
extraconstr	Direct linear combination constraints on the basis weights.
fractional.method	Specifies the approximation method to use for fractional (non-integer) alpha values. 'parsimonious' gives an overall approximate minimal covariance error, 'null' uses approximates low-order properties.
B.tau	Matrix with specification of log-linear model for τ.
B.kappa	Matrix with specification of log-linear model for κ.
prior.variance.nominal	Nominal prior mean for the field variance
prior.range.nominal	Nominal prior mean for the spatial range
prior.tau	Prior mean for tau (overrides prior.variance.nominal)
prior.kappa	Prior mean for kappa (overrides prior.range.nominal)
theta.prior.mean	(overrides prior.*)
theta.prior.prec	Scalar, vector or matrix, specifying the joint prior precision for theta.
n.iid.group	If greater than 1, build an explicitly iid replicated model, to support constraints applied to the combined replicates, for example in a time-replicated spatial model. Constraints can either be specified for a single mesh, in which case it's applied to the average of the replicates (ncol(A) should be mesh$n for 2D meshes, mesh$m for 1D), or as general constraints on the collection of replicates (ncol(A) should be mesh$n * n.iid.group for 2D meshes, mesh$m * n.iid.group for 1D).

Details

This method constructs a Matern SPDE model, with spatial scale parameter κ(u) and variance rescaling parameter τ(u).

(kappa^2(u)-Delta)^(alpha/2) (tau(u) x(u)) = W(u)

Stationary models are supported for 0 < α ≤q 2, with spectral approximation methods used for non-integer α, with approximation method determined by fractional.method.

Non-stationary models are supported for α=2 only, with

• log tau(u) = B.tau_0(u) + sum_{k=1}^p B.tau_k(u) theta_k

• log kappa(u) = B.kappa_0(u) + sum_{k=1}^p B.kappa_k(u) theta_k

The same parameterisation is used in the stationary cases, but with B^τ_0, B^τ_k, B^κ_0, and B^τ_k constant across u.

Integration and other general linear constraints are supported via the constr, extraconstr.int, and extraconstr parameters, which also interact with n.iid.group.

Value

An inla.spde2 object.

Author(s)

Finn Lindgren [e-mail]

See Also

inla.mesh.2d, inla.mesh.create, inla.mesh.1d, inla.mesh.basis, inla.spde2.generic

Examples

n = 100
field.fcn = function(loc) (10*cos(2*pi*2*(loc[,1]+loc[,2])))
loc = matrix(runif(n*2),n,2)
## One field, 2 observations per location
idx.y = rep(1:n,2)
y = field.fcn(loc[idx.y,]) + rnorm(length(idx.y))

mesh = inla.mesh.create(loc, refine=list(max.edge=0.05))
spde = inla.spde2.matern(mesh)
data = list(y=y, field=mesh$idx$loc[idx.y])
formula = y ~ -1 + f(field, model=spde)
result = inla(formula, data=data, family="normal")

## Plot the mesh structure:
plot(mesh)

if (require(rgl)) {
  col.pal = colorRampPalette(c("blue","cyan","green","yellow","red"))
  ## Plot the posterior mean:
  plot(mesh, rgl=TRUE,
       result$summary.random$field[,"mean"],
       color.palette = col.pal)
  ## Plot residual field:
  plot(mesh, rgl=TRUE,
       result$summary.random$field[,"mean"]-field.fcn(mesh$loc),
       color.palette = col.pal)
}

result.field = inla.spde.result(result, "field", spde)
plot(result.field$marginals.range.nominal[[1]])

andrewzm/INLA documentation built on May 10, 2019, 11:12 a.m.