Rational approximations of stationary anisotropic Gaussian Matern random fields

rspde.anistropic2d computes a Finite Element Method (FEM) approximation of a Gaussian random field defined as the solution to the stochastic partial differential equation (SPDE): $$C(h) = \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)}(\sqrt{h^T H^{-1}h})^\nu K_\nu(\sqrt{h^T H^{-1}h})$$, based on a SPDE representation of the form $$(I - \nabla\cdot(H\nabla))^{(\nu+1)/2}u = c\sigma W$$, where $c>0$ is a constant. The matrix $H$ is defined as $$\begin{bmatrix} h_x^2 & h_xh_yh_{xy} \\ h_xh_yh_{xy} & h_y^2 \end{bmatrix}$$

Usage

rspde.anistropic2d(
  mesh,
  nu = NULL,
  nu.upper.bound = 2,
  rspde.order = 1,
  prior.hx = NULL,
  prior.hy = NULL,
  prior.hxy = NULL,
  prior.sigma = NULL,
  prior.precision = NULL,
  prior.nu = NULL,
  prior.nu.dist = "lognormal",
  nu.prec.inc = 0.01,
  type.rational.approx = "chebfun",
  shared_lib = "detect",
  debug = FALSE,
  ...
)

Arguments

mesh: Spatial mesh for the FEM approximation.
nu: If nu is set to a parameter, nu will be kept fixed and will not be estimated. If nu is NULL, it will be estimated.
nu.upper.bound: Upper bound for the smoothness parameter $\nu$. If NULL, it will be set to 2.
rspde.order: The order of the covariance-based rational SPDE approach. The default order is 1.
prior.hx: A list specifying the prior for the parameter $h_x$ in the matrix $H$. This list may contain two elements: mean and/or precision, both of which must be numeric scalars. The precision refers to the prior on $\log(h_x)$. If NULL, default values will be used. The mean value is also used as starting value for hx.
prior.hy: A list specifying the prior for the parameter $h_y$ in the matrix $H$. This list may contain two elements: mean and/or precision, both of which must be numeric scalars. The precision refers to the prior on $\log(h_x)$. If NULL, default values will be used. The mean value is also used as starting value for hy.
prior.hxy: A list specifying the prior for the parameter $h_x$ in the matrix $H$. This list may contain two elements: mean and/or precision, both of which must be numeric scalars. The precision refers to the prior on $\log((h_{xy}+1)/(1-h_{xy}))$. If NULL, default values will be used. The mean value is also used as starting value for hxy.
prior.sigma: A list specifying the prior for the variance parameter $\sigma$. This list may contain two elements: mean and/or precision, both of which must be numeric scalars. The precision refers to the prior on $\log(\sigma)$. If NULL, default values will be used. The mean value is also used as starting value for sigma.
prior.precision: A precision matrix for $\log(h_x), \log(h_y), \log((h_{xy}+1)/(1-h_{xy})), \log(\sigma)$. This matrix replaces the precision element from prior.kappa, prior.sigma, prior.gamma, and prior.rho respectively. For dimension 1 prior.precision must be a 4x4 matrix. For dimension 2, $\rho$ is a vector of length 2, so in this case prior.precision must be a 5x5 matrix. If NULL, a diagonal precision matrix with default values will be used.
prior.nu: a list containing the elements mean and prec for beta distribution, or loglocation and logscale for a truncated lognormal distribution. loglocation stands for the location parameter of the truncated lognormal distribution in the log scale. prec stands for the precision of a beta distribution. logscale stands for the scale of the truncated lognormal distribution on the log scale. Check details below.
prior.nu.dist: The distribution of the smoothness parameter. The current options are "beta" or "lognormal". The default is "lognormal".
nu.prec.inc: Amount to increase the precision in the beta prior distribution. Check details below.
type.rational.approx: Which type of rational approximation should be used? The current types are "chebfun", "brasil" or "chebfunLB".
shared_lib: String specifying which shared library to use for the Cgeneric implementation. Options are "detect", "INLA", or "rSPDE". You may also specify the direct path to a .so (or .dll) file.
debug: Logical value indicating whether to enable INLA debug mode.
...: Additional arguments passed internally for configuration purposes.

Value

An object of class inla_rspde_spacetime representing the FEM approximation of the space-time Gaussian random field.

Examples

library(fmesher)
n_loc <- 2000
loc_2d_mesh <- matrix(runif(n_loc * 2), n_loc, 2)
mesh_2d <- fm_mesh_2d(loc = loc_2d_mesh, cutoff = 0.03, max.edge = c(0.1, 0.5))
aniso_model <- rspde.anistropic2d(mesh = mesh_2d)