inla.spde2.pcmatern {INLA} | R Documentation

## Matern SPDE model object with PC prior for INLA

### Description

Create an `inla.spde2` model object for a Matern model, using a PC prior for the parameters.

### Usage

    inla.spde2.pcmatern(
      mesh,
      alpha = 2,
      param = NULL,
      constr = FALSE,
      extraconstr.int = NULL,
      extraconstr = NULL,
      fractional.method = c("parsimonious", "null"),
      n.iid.group = 1,
      prior.range = NULL,
      prior.sigma = NULL
    )

### Arguments

`mesh` | The mesh to build the model on, as an `fmesher::fm_mesh_2d()` object, or other `fmesher` mesh supporting `fmesher::fm_dof()`, `fmesher::fm_manifold_dim()` and `fmesher::fm_dof()`, such as `fmesher::fm_mesh_1d()`, `fmesher::fm_mesh_3d()`, or `fmesher::fm_collect()`.
`alpha` | Fractional operator order, `0<\alpha\leq 2` supported, for `\nu=\alpha-d/2>0`.
`param` | Further model parameters. Not currently used.
`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.
`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).
`prior.range` | A length 2 vector, with `⁠(range0,Prange)⁠` specifying that `P(\rho < \rho_0)=p_\rho`, where `\rho` is the spatial range of the random field. If `Prange` is `NA`, then `range0` is used as a fixed range value.
`prior.sigma` | A length 2 vector, with `⁠(sigma0,Psigma)⁠` specifying that `P(\sigma > \sigma_0)=p_\sigma`, where `\sigma` is the marginal standard deviation of the field. If `Psigma` is `NA`, then `sigma0` is used as a fixed range value.

### Details

This method constructs a Matern SPDE model, with spatial range `\rho` and standard deviation parameter `\sigma`. In the parameterisation

`(\kappa^2-\Delta)^{\alpha/2}(\tau`

`x(u))=W(u)`

the spatial scale parameter `\kappa=\sqrt{8\nu}/\rho`, where `\nu=\alpha-d/2`, and `\tau` is proportional to `1/\sigma`.

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`.

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

The joint PC prior density for the spatial range, `\rho`, and the marginal standard deviation, `\sigma`, and is

``

`\pi(\rho, \sigma) =`

`\frac{d \lambda_\rho}{2} \rho^{-1-d/2} \exp(-\lambda_\rho \rho^{-d/2})`

`\lambda_\sigma\exp(-\lambda_\sigma \sigma)`

where `\lambda_\rho` and `\lambda_\sigma` are hyperparameters that must be determined by the analyst. The practical approach for this in INLA is to require the user to indirectly specify these hyperparameters through

`P(\rho < \rho_0) = p_\rho`

and

`P(\sigma > \sigma_0) = p_\sigma`

where the user specifies the lower tail quantile and probability for the range (`\rho_0` and `p_\rho`) and the upper tail quantile and probability for the standard deviation (`\sigma_0` and `\alpha_\sigma`).

This allows the user to control the priors of the parameters by supplying knowledge of the scale of the problem. What is a reasonable upper magnitude for the spatial effect and what is a reasonable lower scale at which the spatial effect can operate? The shape of the prior was derived through a construction that shrinks the spatial effect towards a base model of no spatial effect in the sense of distance measured by Kullback-Leibler divergence.

The prior is constructed in two steps, under the idea that having a spatial field is an extension of not having a spatial field. First, a spatially constant random effect (`\rho = \infty`) with finite variance is more complex than not having a random effect (`\sigma = 0`). Second, a spatial field with spatial variation (`\rho < \infty`) is more complex than the random effect with no spatial variation. Each of these extensions are shrunk towards the simpler model and, as a result, we shrink the spatial field towards the base model of no spatial variation and zero variance (`\rho = \infty` and `\sigma = 0`).

The details behind the construction of the prior is presented in Fuglstad, et al. (2016) and is based on the PC prior framework (Simpson, et al., 2015).

### Value

An `inla.spde2` object.

### Author(s)

Finn Lindgren finn.lindgren@gmail.com

### References

Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2016) Constructing Priors that Penalize the Complexity of Gaussian Random Fields. arXiv:1503.00256

Simpson, D., Rue, H., Martins, T., Riebler, A., and Sørbye, S. (2015) Penalising model component complexity: A principled, practical approach to constructing priors. arXiv:1403.4630

### See Also

`fmesher::fm_mesh_2d_inla()`, `fmesher::fm_rcdt_2d_inla()`, `fmesher::fm_mesh_1d()`, `fmesher::fm_basis()`, `inla.spde2.matern()`, `inla.spde2.generic()`

### Examples

    ## Spatial interpolation
    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_mesh_2d_inla(loc, max.edge = 0.05, cutoff = 0.01)
    spde <- inla.spde2.pcmatern(mesh,
        prior.range = c(0.01, 0.1), prior.sigma = c(100, 0.1)
    )
    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)
    par(mfrow = c(2, 1))
    plot(result.field$marginals.range.nominal[[1]],
        type = "l", main = "Posterior density for range"
    )
    plot(inla.tmarginal(sqrt, result.field$marginals.variance.nominal[[1]]),
        type = "l", main = "Posterior density for std.dev."
    )
    par(mfrow = c(1, 1))

    ## Spatial model
    set.seed(1234234)

    ## Generate spatial locations
    nObs <- 200
    loc <- matrix(runif(nObs * 2), nrow = nObs, ncol = 2)

    ## Generate observation of spatial field
    nu <- 1.0
    rhoT <- 0.2
    kappaT <- sqrt(8 * nu) / rhoT
    sigT <- 1.0
    Sig <- sigT^2 * inla.matern.cov(
        nu = nu,
        kappa = kappaT,
        x = as.matrix(dist(loc)),
        d = 2,
        corr = TRUE
    )
    L <- t(chol(Sig))
    u <- L %*% rnorm(nObs)

    ## Construct observation with nugget
    sigN <- 0.1
    y <- u + sigN * rnorm(nObs)

    ## Create the mesh and spde object
    mesh <- fm_mesh_2d_inla(loc,
        max.edge = 0.05,
        cutoff = 0.01
    )
    spde <- inla.spde2.pcmatern(mesh,
        prior.range = c(0.01, 0.05),
        prior.sigma = c(10, 0.05)
    )

    ## Create projection matrix for observations
    A <- fm_basis(mesh = mesh, loc = loc)

    ## Run model without any covariates
    idx <- 1:spde$n.spde
    res <- inla(y ~ f(idx, model = spde) - 1,
        data = list(y = y, idx = idx, spde = spde),
        control.predictor = list(A = A)
    )

    ## Re-run model with fixed range
    spde.fixed <- inla.spde2.pcmatern(mesh,
        prior.range = c(0.2, NA),
        prior.sigma = c(10, 0.05)
    )

    res.fixed <- inla(y ~ f(idx, model = spde) - 1,
        data = list(y = y, idx = idx, spde = spde.fixed),
        control.predictor = list(A = A)
    )

[Package INLA version 26.08.22 Index]