Negative log-likelihood (presence-only, math scale)

Computes the negative log-likelihood of a multivariate normal niche model using only presence records, with no background correction (Jimenez et al. 2019, Eq. 2 without the M-restricted denominator).

Usage

loglik_niche_math_presence_only(theta, env_occ, eta = 1, neg = TRUE, ...)

Arguments

theta

Numeric vector of unconstrained parameters (math scale): \([\mu, \log\sigma, v]\) of length \(2p + p(p-1)/2\).

env_occ

Data frame or matrix (\(n \times p\)) of environmental values at presence points.

eta

Numeric scalar, shape parameter for the LKJ C-vine prior on the correlation matrix (default 1 = uniform over correlations).

neg

Logical. If TRUE (default), returns the negative log-likelihood (suitable for minimisation).

...

Additional arguments (ignored; for compatibility).

Value

Scalar numeric: the (negative) log-likelihood.

Details

The minimised objective is: $$ -\log\mathcal{L} = \frac{n}{2}\log|\Sigma| + \frac{1}{2}\sum_{i=1}^{n} (\mathbf{x}_i - \mu)^\top \Sigma^{-1}(\mathbf{x}_i - \mu) $$ The \((2\pi)\) normalisation constant is dropped (does not affect the optimum).

Internally, \(\Sigma = L L^\top\) is reconstructed from theta via cvine_cholesky, and the Mahalanobis distances are computed via a triangular solve \(L^{-1}(\mathbf{x}_i - \mu)\).

References

Jimenez, L., Soberon, J., Christen, J. A., & Soto, D. (2019). On the problem of modeling a fundamental niche from occurrence data. Ecological Modelling, 397, 109823.

See also

[optimize_niche()] for multi-start fitting, [loglik_niche_math_cpp()] for the M-restricted model.

Examples

# \donttest{
theta <- start_theta(example_env_occ_2d)
ll <- loglik_niche_math_presence_only(theta, example_env_occ_2d)
# }