Matern SPDE model object for INLA

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

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 fmesher::fm_mesh_1d(), fmesher::fm_mesh_2d(), fmesher::fm_mesh_3d(), or fmesher::fm_collect() object.
alpha: Fractional operator order, $0<\alpha\leq 2$ supported. ($\nu=\alpha-d/2$)
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 $\tau$.
B.kappa: Matrix with specification of log-linear model for $\kappa$.
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.
spde: An spde model object
theta: Parameters in the model's internal scale

Value

An inla.spde2 object.

Details

This method constructs a Matern SPDE model, with spatial scale parameter $\kappa(u)$ and variance rescaling parameter $\tau(u)$.

$$(\kappa^2(u)-\Delta)^{\alpha/2}(\tau(u) $$$$ x(u))=W(u)$$

Stationary models are supported for $0 < \alpha \leq 2$, with spectral approximation methods used for non-integer $\alpha$, with approximation method determined by fractional.method.

Non-stationary models are supported for $\alpha=2$ 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 $B^\tau_0$, $B^\tau_k$, $B^\kappa_0$, and $B^\tau_k$ 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.

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

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 <- fm_rcdt_2d_inla(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)

# \donttest{
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$loc),
        color.palette = col.pal
    )
}
#> Error: The `rgl` argument of `plot.fm_mesh_2d()` was deprecated in fmesher
#> 0.1.0 and is now defunct.
#> ℹ Please use `plot_rgl()` instead.
# }
result.field <- inla.spde.result(result, "field", spde)
plot(result.field$marginals.range.nominal[[1]])