Hybrid Whittle-Matern SPDE model for INLA / inlabru (alpha = 2)

Creates an INLA cgeneric latent model for the hybrid Whittle-Matern SPDE $$Y(s) = \beta_X^T L^{-1} X(s) + \tau^{-1} L^{-1} W(s),$$ where $L = \kappa^2 - \Delta$ and the deterministic mean component $\beta_X^T L^{-1} X$ is part of the latent model itself: the cgeneric stores the centred field $u(s) = \tau^{-1} L^{-1} W(s)$ and returns $\mu(s) = \beta_X^T L^{-1} X(s)$ via the mu callback. The coefficients beta_X, together with tau and kappa, are estimated jointly as cgeneric hyperparameters – this is the genuine fully Bayesian fit, not a plug-in approximation.

Usage

rspde.hybrid.matern(
  mesh,
  X,
  prior.tau = NULL,
  prior.kappa = NULL,
  prior.beta_x = NULL,
  start.ltau = NULL,
  start.lkappa = NULL,
  start.beta_x = NULL,
  ...,
  separate_kappa_mu = FALSE,
  prior.kappa_mu = NULL,
  start.lkappa_mu = NULL,
  debug = FALSE,
  shared_lib = "detect"
)

Arguments

mesh: An fmesher mesh (1d or 2d).
X: Numeric vector or matrix of covariate fields evaluated at the mesh nodes; one column per covariate.
prior.tau: A list with mean and prec (Gaussian prior on log(tau)). Defaults: mean = 0, prec = 0.1.
prior.kappa: A list with mean and prec (Gaussian prior on log(kappa)). Defaults: mean derived from the mesh range, prec = 0.1.
prior.beta_x: A list with mean (vector of length p) and prec (vector of length p). Independent Gaussian priors on the regression coefficients. Defaults: mean = 0, prec = 0.001 (uninformative).
start.ltau, start.lkappa: Starting values for log(tau) and log(kappa). Default to the prior means.
start.beta_x: Numeric vector of starting values for beta_X. Default: zeros.
...: Additional arguments passed to the underlying model constructor.
separate_kappa_mu: Logical. If TRUE, the deterministic mean operator uses its own range parameter kappa_mu, estimated as an extra hyperparameter, rather than sharing kappa with the random field. Default: FALSE.
prior.kappa_mu: A list with mean and prec (Gaussian prior on log(kappa_mu)). Only used when separate_kappa_mu = TRUE. Defaults to the same prior as prior.kappa.
start.lkappa_mu: Starting value for log(kappa_mu). Only used when separate_kappa_mu = TRUE. Defaults to the prior mean.
debug: Passed to INLA.
shared_lib: Which shared lib to use for the cgeneric. See rspde.matern() for details.

Value

An INLA cgeneric latent model that can be used in INLA formulae as f(idx, model = <model>).

Details

Currently the implementation is restricted to the case alpha = 2 (which is the main case of practical interest: in 2d this is the Matern with nu = 1).

The cgeneric hyperparameters are $$\theta = (\log\tau,\; \log\kappa,\; \beta_{X,1}, \ldots, \beta_{X,p}).$$ The model stores the FEM C and G matrices and the covariate matrix X and (re)computes $Q = \tau^2\, L\, C^{-1}\, L$ and $\mu = L^{-1}(X \beta_X)$ at every INLA call. The mean computation is a single sparse Cholesky solve done with Eigen.

Examples

if (FALSE) { # \dontrun{
library(INLA)
x <- seq(0, 1, length.out = 81)
mesh <- fmesher::fm_mesh_1d(x)
X <- cbind(sin(2 * pi * x), cos(2 * pi * x))
hyb <- rspde.hybrid.matern(mesh = mesh, X = X)
# ... use in formula: y ~ -1 + f(idx, model = hyb)
} # }