Negative log-likelihood (M-restricted, math scale, Cholesky version)

Computes the negative log-likelihood of the multivariate normal niche model restricted to the set of existing environments $\mathbf{E}(t;G)$ (Jimenez et al. 2019, Eq. 3–5). The density at an occurrence point is normalised by the sum of the density over all background points in M:

Usage

loglik_niche_math_cpp(theta, env_occ, env_m, 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.
env_m: Data frame or matrix ($n_m \times p$) of background environmental values from the accessible area M.
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

A scalar numeric value: the (negative) log-likelihood.

Details

$$ -\log\mathcal{L} = \frac{1}{2}\sum_{i=1}^{n} (\mathbf{x}_i - \mu)^\top \Sigma^{-1}(\mathbf{x}_i - \mu) + n \cdot \log\!\left( \sum_{\mathbf{y} \in M} \exp\!\left[-\frac{1}{2} (\mathbf{y} - \mu)^\top \Sigma^{-1}(\mathbf{y} - \mu) \right] \right) $$

The $|\Sigma|^{-1/2}$ factor cancels between numerator and denominator (Eq. 3), so it does not appear in the objective. The logsumexp is computed with the max-shift trick for numerical stability.

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.

Examples

# \donttest{
theta <- start_theta(example_env_occ_2d)
ll <- loglik_niche_math_cpp(theta,
  env_occ = example_env_occ_2d,
  env_m = example_env_m_2d,
  eta = 1, neg = TRUE
)
print(ll)
#> [1] 662.9334
# }