log_lik: Log-Likelihood Function for Spatial Model

In AgePopDenom: Model Fine-Scale Age-Structured Population Data using Open-Source Data

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
)

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.

Value

A numeric scalar. The computed log-likelihood value.

Note

This function requires a helper function, compute_cov, to compute the covariance matrix based on spatial parameters.

AgePopDenom documentation built on April 12, 2025, 1:41 a.m.