Fit Stage 2 spatial model via MLE — fit_spatial

Estimates beta (6x1), A (6x6 lower triangular), and rho (6x1) by maximizing the Gaussian likelihood from Eq. (7) of Russell et al. (2020), extended to handle partial observations. NOAA sites observe components 1-3 and ADCIRC sites observe components 4-6 of the 6-dimensional parameter vector.

Usage

fit_spatial_model(
  stage1,
  dat,
  W_tap,
  D,
  start = NULL,
  method = "L-BFGS-B",
  control = list(maxit = 1000, trace = 1),
  check_gradient = FALSE
)

Arguments

stage1: An evfuse_stage1 object from fit_gev_all or fit_gev_detrended.
dat: An evfuse_data object.
W_tap: Tapered covariance matrix from taper_W.
D: Distance matrix from compute_distances.
start: Optional named list of starting values for beta, A, rho.
method: Optimization method for optim (default "L-BFGS-B").
control: Control list passed to optim.
check_gradient: If TRUE, verify analytic gradient against numerical finite differences at the starting values before optimizing.

Value

A list with components:

beta: Estimated mean vector (6x1).
A: Estimated lower triangular matrix (6x6).
rho: Estimated range parameters (6x1).
Sigma: Estimated covariance matrix at observed sites.
optim_result: Raw output from optim.
grad_check: If check_gradient = TRUE, the output of verify_gradient.

Details

Uses an analytic gradient derived from the coregionalization structure (see Sections 2, 5, 6 of the gradient notes). The key identity is that dSigma/dA_ab and dSigma/drho_k are rank-1 Kronecker products, enabling O(n_obs^2) gradient computation per parameter instead of O(n_obs^3).