Matern SPDE model object with PC prior for INLA

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

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.

Usage

inla.spde2.pcmatern(mesh, ...)

# S3 method for class 'inla_mesh_3d'
inla.spde2.pcmatern(mesh, ...)

# Default S3 method
inla.spde2.pcmatern(mesh, ...)

inla.spde2.pcmatern3d(
  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 inla.mesh() or inla.mesh.1d() object.
...: Further arguments passed to submethods.
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.

Value

An inla.spde2 object.

Details

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).

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

Author

Finn Lindgren finn.lindgren@gmail.com