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 `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`. For fmesher::fm_collect meshes, the constraint is applied separately for each sub-mesh.
`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

### 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`.

### 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

### See Also

`fmesher::fm_mesh_2d_inla()`, `fmesher::fm_rcdt_2d_inla()`, `fmesher::fm_mesh_1d()`, `fmesher::fm_basis()`, `inla.spde2.pcmatern()`, `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 <- 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)

    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 26.08.22 Index]