Model 1: Presence-Only Gaussian Niche

1 Ecological Motivation

The fundamental ecological niche of a species (sensu Hutchinson 1957) is the set of environmental conditions under which the species can maintain a viable population. When the niche is modelled as a multivariate normal density $f(\mathbf{x};\mu,\Sigma)$ in $p$-dimensional environmental space, the problem reduces to estimating the centroid $\mu$ and the shape/orientation matrix $\Sigma$ from observed presence records.

The presence-only model is the simplest formulation: it treats the occurrence records $\{\mathbf{x}_1,\dots,\mathbf{x}_n\}$ as an i.i.d. sample from the normal density and estimates $(\mu,\Sigma)$ by maximum likelihood.

2 Mathematical Specification

2.1 Density

The multivariate normal density in $p$ dimensions is

$$f(\mathbf{x};\mu,\Sigma) = (2\pi)^{-p/2}\,|\Sigma|^{-1/2}\, \exp\!\left[-\tfrac{1}{2} (\mathbf{x}-\mu)^\top\Sigma^{-1}(\mathbf{x}-\mu)\right]. \tag{1}$$

This is Equation 2 of Jimenez et al. (2019), written for a single observation.

2.2 Log-Likelihood

Given $n$ presence records, the log-likelihood is

$$ \ell(\mu,\Sigma) = -\frac{n}{2}\log|\Sigma| -\frac{1}{2}\sum_{i=1}^{n} (\mathbf{x}_i-\mu)^\top\Sigma^{-1}(\mathbf{x}_i-\mu) + \text{const.} \tag{2} $$

The constant $-\frac{np}{2}\log(2\pi)$ does not depend on $(\mu,\Sigma)$ and is dropped in the implementation because it does not affect the optimum.

2.3 Negative Log-Likelihood (Minimised Objective)

The optimizer minimises the negative log-likelihood:

$$-\ell(\mu,\Sigma) = \frac{n}{2}\log|\Sigma| + \frac{1}{2}\sum_{i=1}^{n} q_i, \qquad q_i = \|L^{-1}(\mathbf{x}_i-\mu)\|^2, \tag{3}$$

where $\Sigma = LL^\top$ (Cholesky factorisation) and $q_i$ is the squared Mahalanobis distance.

3 Parameterisation

3.1 Theta Layout

The unconstrained parameter vector $\theta$ has length $2p + p(p-1)/2$:

$$\theta = \bigl[\underbrace{\mu_1,\dots,\mu_p}_{p},\; \underbrace{\log\sigma_1,\dots,\log\sigma_p}_{p},\; \underbrace{v_1,\dots,v_{p(p-1)/2}}_{\text{C-vine}}\bigr].$$

Block	Length	Constraint	Meaning
$\mu$	$p$	Unconstrained	Niche centroid
$\log\sigma$	$p$	$\sigma_k > 0$ via $\exp$	Marginal standard deviations
$v$	$p(p-1)/2$	Unconstrained	C-vine partial correlations

3.2 Covariance Reconstruction

The covariance matrix is reconstructed as $\Sigma = L_{\text{cov}}\,L_{\text{cov}}^\top$, where

$$L_{\text{cov}} = \operatorname{diag}(\sigma)\;\cdot\;L_{\text{corr}},$$

and $L_{\text{corr}}$ is the lower-triangular Cholesky factor of the correlation matrix $R$, built from the C-vine parameterisation of Lewandowski, Kurowicka & Joe (2009). The LKJ prior with shape $\eta$ penalises extreme correlations when $\eta > 1$.

4 Implementation Call Chain

optimize_niche(likelihood = "presence_only")
  └─ niche_obj.cpp::create_niche_obj_ptr()
       └─ loglik_niche_math_presence_only_eigen()    [C++, lines 65–95]
            ├─ build_L_cov()                          [theta → mu, sigma, L_cov]
            ├─ L_cov.triangularView.solve(diff)       [Mahalanobis via forward solve]
            └─ 0.5 * n * log_det + 0.5 * sum_q       [Eq. 3]

4.1 Key Code (loglik_niche_math_cpp.cpp, lines 65–95)

The C++ kernel:

1. Reconstructs $L_{\text{cov}}$ from $\theta$ via build_L_cov() (lines 40–59).
2. Computes $Y = L_{\text{cov}}^{-1}(X - \mu)$ by triangular forward solve (line 84).
3. Sums squared column norms $\sum_i \|y_i\|^2$ (line 85).
4. Computes $\log|\Sigma| = 2\sum_k \log L_{\text{cov}}(k,k)$ (lines 87–89).
5. Returns 0.5 * n * log_det + 0.5 * sum_q (line 92), guarded by OPTIM_PENALTY for NaN/Inf (line 93).

4.2 R Entry Point

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

This function reconstructs $L_{\text{cov}}$ via cvine_cholesky() in R, then calls loglik_niche_presence_only_cpp() (the Rcpp-exported wrapper).

5 Analytical Properties

Property	Value
MLE for $\mu$	Sample mean $\bar{\mathbf{x}}$
MLE for $\Sigma$	Sample covariance $\frac{1}{n}\sum_i(\mathbf{x}_i-\bar{\mathbf{x}})(\mathbf{x}_i-\bar{\mathbf{x}})^\top$
Likelihood shape	Unimodal (convex negative log-likelihood)
Identifiability	Full: $\mu$ and $\Sigma$ are both identifiable
Background data	Not used
Dropped constants	$(2\pi)^{-np/2}$ — does not affect $\hat\theta$

6 When To Use

• Sanity check: The MLE has a closed form, so numerical results should match the sample mean and covariance exactly (up to optimiser tolerance).
• Warm start: optimize_niche(warm_start = TRUE) uses the presence-only MLE as the initial point for the more complex weighted models.
• No background available: When the accessible area $M$ is unknown or the background sample is unreliable.

7 Limitations

1. Ignores environmental availability. The estimate reflects the observed distribution of presences, which is confounded by the distribution of environments in $M$. If $M$ over-represents certain environments, $\hat\mu$ will be biased toward those environments.

2. No background correction. Unlike the "weighted" and "ip_weighted" models, no KDE of $g(\mathbf{x})$ is used. The resulting niche estimate is the realised niche (or a biased estimate of it), not the fundamental niche.

3. Normality assumption. The niche is assumed to be ellipsoidal. For species with multi-modal or strongly skewed niches, consider "skew_normal" or "skew_t".

References

Jimenez, L., J. Soberón, J. A. Christen, and D. Soto. 2019. “On the Problem of Modeling a Fundamental Niche from Occurrence Data.” Ecological Modelling 397: 74–83. https://doi.org/10.1016/j.ecolmodel.2019.01.020.