rspde.spacetime: Space-Time Random Fields via SPDE Approximation

In rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations

Space-Time Random Fields via SPDE Approximation

Description

rspde.spacetime computes a Finite Element Method (FEM) approximation of a Gaussian random field defined as the solution to the stochastic partial differential equation (SPDE):

d u + \gamma(\kappa^2 + \kappa^{d/2}\rho\cdot\nabla - \Delta)^\alpha u = \sigma dW_C

where C is a Whittle-Matérn covariance operator with smoothness parameter \beta and range parameter \kappa. This function is designed to handle space-time random fields using either 1D spatial models or higher-dimensional FEM-based approaches.

Usage

rspde.spacetime(
  mesh_space = NULL,
  mesh_time = NULL,
  space_loc = NULL,
  time_loc = NULL,
  drift = TRUE,
  alpha,
  beta,
  prior.kappa = NULL,
  prior.sigma = NULL,
  prior.rho = NULL,
  prior.gamma = NULL,
  prior.precision = NULL,
  graph_dirichlet = TRUE,
  bounded_rho = TRUE,
  bound_rho = NULL,
  shared_lib = "detect",
  debug = FALSE,
  ...
)

Arguments

mesh_space	Spatial mesh for the FEM approximation, or a metric_graph object for handling models on metric graphs.
mesh_time	Temporal mesh for the FEM approximation.
space_loc	A vector of spatial locations for mesh nodes in 1D spatial models. This should be provided when mesh_space is not specified.
time_loc	A vector of temporal locations for mesh nodes. This should be provided when mesh_time is not specified.
drift	Logical value indicating whether the drift term should be included. If FALSE, the drift coefficient \rho is set to zero.
alpha	Integer smoothness parameter \alpha.
beta	Integer smoothness parameter \beta.
prior.kappa	A list specifying the prior for the range parameter \kappa. This list may contain two elements: mean and/or precision, both of which must be numeric scalars (numeric vectors of length 1). The precision refers to the prior on \log(\kappa). If NULL, default values will be used. The mean value is also used as starting value for kappa.
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.rho	A list specifying the prior for the drift coefficient \rho. This list may contain two elements: mean and/or precision, both of which must be numeric scalars if dimension is one, and numeric vectors of length 2 if dimension is 2. The precision applies directly to \rho without log transformation. If NULL, default values will be used. Will not be used if drift = FALSE. The mean value is also used as starting value for rho.
prior.gamma	A list specifying the prior for the weight \gamma in the SPDE operator. 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(\gamma). If NULL, default values will be used. The mean value is also used as starting value for gamma.
prior.precision	A precision matrix for \log(\kappa), \log(\sigma), \log(\gamma), \rho. 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.
graph_dirichlet	For models on metric graphs, use Dirichlet vertex conditions at vertices of degree 1?
bounded_rho	Logical. Should rho be bounded to ensure the existence, uniqueness, and well-posedness of the solution? Defaults to TRUE. Note that this bounding is not a strict condition; there may exist values of rho beyond the upper bound that still satisfy these properties. When bounded_rho = TRUE, the rspde_lme models enforce bounded rho for consistency. If the estimated value of rho approaches the upper bound too closely, we recommend refitting the model with bounded_rho = FALSE. However, this should be done with caution, as it may lead to instability in some cases, though it can also result in a better model fit. The actual bound used for rho can be accessed from the bound_rho element of the returned object.
bound_rho	A positive number specifying the bound for rho. If NULL, the default bound will be used.
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.

rSPDE documentation built on Jan. 26, 2026, 9:06 a.m.