Space-Time Random Fields via SPDE Approximation

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.

Examples

library(INLA)
library(MetricGraph)
#> This is MetricGraph 1.4.1
#> - See https://davidbolin.github.io/MetricGraph for vignettes and manuals.
#> 
#> Attaching package: ‘MetricGraph’
#> The following object is masked from ‘package:stats’:
#> 
#>     filter
graph <- metric_graph$new()
#> Starting graph creation...
#> LongLat is set to FALSE
#> The current tolerances are:
#> 	 Vertex-Vertex 0.001
#> 	 Vertex-Edge 0.001
#> 	 Edge-Edge 0
#> Creating edges...
#> Setting edge weights...
#> Computing bounding box...
#> Setup edges and merge close vertices
#> Computing the relative positions of the edges...
#> Snap vertices to close edges
#> Total construction time: 1.40 secs
#> Creating and updating vertices...
#> Storing the initial graph...
#> Computing the relative positions of the edges...
graph$build_mesh(h = 0.1)
graph$compute_fem()

# Define the time locations
time_loc <- seq(from = 0, to = 10, length.out = 11)

# Create the model
model <- rspde.spacetime(mesh_space = graph,
                         time_loc = time_loc,
                         alpha = 2,
                         beta = 1)