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Simulate log-Gaussian Cox processes on metric graphs

Source: R/graph_lgcp.R
graph_lgcp_sim.Rd

This function simulates point patterns from a log-Gaussian Cox process (LGCP) driven by Whittle-Matérn Gaussian random fields on metric graphs. The intensity function is modeled as \(\lambda(s) = exp(\beta + u(s))\), where \(\beta\) is an intercept parameter and \(u(s)\) is a Gaussian field with Whittle-Matérn covariance.

Usage

graph_lgcp_sim(n = 1, intercept = 0, sigma, range, alpha, graph)

Arguments

n

Integer. Number of replicate point patterns to simulate. Default is 1.

intercept

Numeric scalar or vector. Mean value(s) of the log-intensity field. Can be a constant or vary spatially if provided as a vector of length equal to the number of mesh nodes.

sigma

Numeric. Marginal standard deviation parameter of the Gaussian field.

range

Numeric. Practical correlation range parameter of the Gaussian field.

alpha

Numeric. Smoothness parameter of the Whittle-Matérn field. Currently supports values 1 and 2, corresponding to exponential and Matérn-3/2 covariance functions respectively.

graph

A metric_graph object with a built mesh. The graph must have an existing mesh (built with graph$build_mesh()) and computed FEM matrices (computed with graph$compute_fem()).

Value

If n = 1, returns a list with components:

u

Numeric vector of the simulated Gaussian field values at mesh nodes

edge_number

Integer vector of edge numbers where points were simulated

edge_loc

Numeric vector of locations along edges (normalized coordinates)

If n > 1, returns a list of length n, where each element is a list with the above components.

Details

The function implements a two-step simulation procedure:

  1. Simulate the Gaussian random field u(s) using finite element methods

  2. Generate point locations using acceptance-rejection sampling from the intensity \(\lambda(s) = exp(\beta + u(s))\)

The Gaussian field is characterized by the SPDE: \((\kappa^2 - \Delta)^{\alpha/2} \tau u = \mathcal{W}\) where \(\kappa, \tau\) are derived from the range and sigma parameters, and \(\mathcal{W}\) is white noise.

See also

lgcp_graph for fitting LGCP models, precompute_lgcp_graph for efficient model fitting

Examples

if (FALSE) { # \dontrun{
# Create a metric graph and build mesh
graph <- metric_graph$new()
graph$build_mesh(h = 0.1)
graph$compute_fem()

# Simulate a single point pattern
lgcp_data <- graph_lgcp_sim(
  intercept = -1, 
  sigma = 0.5, 
  range = 2, 
  alpha = 2, 
  graph = graph
)

# Simulate multiple replicates
lgcp_replicates <- graph_lgcp_sim(
  n = 10,
  intercept = -1, 
  sigma = 0.5, 
  range = 2, 
  alpha = 2, 
  graph = graph
)

# Plot the simulated intensity
graph$plot_function(X = exp(lgcp_data$u), vertex_size = 0)
} # }