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Space-Time Random Fields via SPDE Approximation

Source: R/inla_rspde_spacetime.R
rspde.spacetime.Rd

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): $$d u + \gamma(\kappa^2 + \rho\cdot\nabla - \Delta)^\alpha u = \sigma dW_C$$ 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,
  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.

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)
#> 
#> Attaching package: ‘MetricGraph’
#> The following object is masked from ‘package:stats’:
#> 
#>     filter
graph <- metric_graph$new()
#> Starting graph creation...
#> LongLat is set to FALSE
#> The current tolerances are:
#> 	 Vertex-Vertex 0.001
#> 	 Vertex-Edge 0.001
#> 	 Edge-Edge 0
#> Setup edges and merge close vertices
#> Snap vertices to close edges
#> Total construction time: 1.41 secs
graph$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)