LL2_kappa: Compute part of kappa log-likelihood

Description

Compute part of log-likelihood involving kappa (or kappa_q) for numerical optimization

Usage

LL2_kappa(
  kappa,
  Amat,
  Fmat,
  Gmat,
  GFinvG,
  OplusW,
  u,
  v,
  C1 = 1/(4 * pi),
  Q = NULL
)

Value

Value of log-likelihood at logkappa

Arguments

kappa: Value of kappa for which to compute log-likelihood
Amat: Mesh projection matrix
Fmat: Matrix used in computation of SPDE precision
Gmat: Matrix used in computation of SPDE precision
GFinvG: Matrix used in computation of SPDE precision
OplusW: Sparse matrix containing estimated values of RHS of trace in part 2 of log-likelihood. In common smoothness model, represents the sum over q=1,...,Q.
u: Vector needed for part 3 of log-likelihood
v: Vector needed for part 3 of log-likelihood
C1: For the unit variance case, \(\tau^2 = C1/\kappa^2\), where \(C1 = 1/(4\pi)\) when \(\alpha=2\), \(\nu=1\), \(d=2\)
Q: Equal to the number of networks for the common smoothness model, or NULL for the network-specific smoothness model

Details

This is the function to be maximized in order to determine the MLE for \(\kappa\) or the \(\kappa_q\)'s in the M-step of the EM algorithm in spatial model. In the model where \(\kappa_q\) can be different for each network \(q\), the optimization function factorizes over the \(\kappa_q\)'s. This function computes the value of the part of the optimization function pertaining to one of the \(\kappa_q\)'s.