log_lik: Log-Likelihood Function for Spatial Model

Description

Computes the log-likelihood for a spatial statistical model with a covariance structure determined by parameters including spatial decay and variance.

Usage

log_lik(
  par,
  p1,
  p2,
  d1,
  d2,
  y,
  u_dist,
  n_x,
  tau2_1 = 1,
  tau2_2 = 1,
  age_param_data
)

Value

A numeric scalar. The computed log-likelihood value.

Arguments

par

A numeric vector of parameters to estimate. The vector contains:

par[1:p1]: Coefficients for fixed effects in dataset 1 ($\beta_1$).
par[(p1 + 1):(p1 + p2)]: Coefficients for fixed effects in dataset 2 ($\beta_2$).
par[p1 + p2 + 1]: Spatial decay parameter ($\gamma$).
par[p1 + p2 + 2]: Log of the variance parameter ($\sigma^2$).
par[p1 + p2 + 3]: Log of the range parameter ($\phi$).

p1

An integer. The number of fixed-effect parameters in dataset 1.

p2

An integer. The number of fixed-effect parameters in dataset 2.

d1

A numeric matrix. Design matrix for dataset 1 used to model the mean structure.

d2

A numeric matrix. Design matrix for dataset 2 used to model the mean structure.

y

A numeric vector. Observed response variable, including both datasets.

u_dist

A numeric matrix. Distance matrix for spatial locations.

n_x

An integer. The number of unique spatial locations.

tau2_1

Variance parameter for first process (default = 1)

tau2_2

Variance parameter for second process (default = 1)

age_param_data

A numeric matrix or vector. Additional parameters specific to age-based modeling.

Details

The log-likelihood is computed as: $$ -0.5 \left[ \log(\det(M)) + (y - \mu)^T M^{-1} (y - \mu) \right] $$ where:

$M$ is the covariance matrix, computed using compute_cov.
$\mu$ is the mean structure, determined by the design matrices d1, d2 and coefficients $\beta_1, \beta_2$.

The covariance matrix $M$ is computed using spatial parameters ($\gamma, \sigma^2, \phi$) and the distance matrix u_dist.