R: Matern SPDE model object for INLA

inla.spde2.matern {INLA}

R Documentation

Matern SPDE model object for INLA

Description

Create an inla.spde2 model object for a Matern model. Use inla.spde2.pcmatern instead for a PC prior for the parameters.

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,
  ...
)

inla.spde2.theta2phi0(spde, theta)

inla.spde2.theta2phi1(spde, theta)

inla.spde2.theta2phi2(spde, theta)

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).
`...`	Additional parameters for special uses.

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_klog 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_klog 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.

Functions

inla.spde2.theta2phi0: Convert from theta vector to phi0 values in the internal spde2 model representation
inla.spde2.theta2phi1: Convert from theta vector to phi1 values in the internal spde2 model representation
inla.spde2.theta2phi2: Convert from theta vector to phi2 values in the internal spde2 model representation

Author(s)

Finn Lindgren finn.lindgren@gmail.com

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]])

[Package INLA version 21.06.11 Index]