Model 5: Skew-Normal Weighted Niche

1 Ecological Motivation

This model combines two extensions:

1. Skewness (from Model 4): the niche can be asymmetric along each environmental axis.
2. Background weighting (from Model 2): the non-uniform distribution of environments in $M$ is modelled via a KDE $\hat{g}(\mathbf{x})$.

The result is a model that estimates an asymmetric fundamental niche while accounting for the fact that presence records are more likely in environments that are common in $M$ (the use-availability DGP).

2 Data-Generating Process

The DGP is the same as Model 2, but with a skew-normal kernel replacing the Gaussian:

$$p(\mathbf{x}_i \mid \text{presence in } M) = \frac{f_{\text{SN}}(\mathbf{x}_i;\mu,\Sigma,\alpha)\;\hat{g}(\mathbf{x}_i)} {\int_M f_{\text{SN}}(\mathbf{x};\mu,\Sigma,\alpha)\;\hat{g}(\mathbf{x})\,d\mathbf{x}}, \tag{1}$$

where $f_{\text{SN}}$ is the skew-normal density (Model 4, Eq. 1).

3 Mathematical Specification

3.1 Negative Log-Likelihood

$$-\ell^*(\theta) = \frac{1}{2}\sum_{i=1}^{n} q_i - \sum_{i=1}^{n}\log\Phi(z_i) - \sum_{i=1}^{n}\log\hat{g}(\mathbf{x}_i) + n\cdot\operatorname{logsumexp}_j\!\left[ \log\hat{g}(\mathbf{y}_j) + \log\Phi(z_j) - \tfrac{1}{2}q_j \right] + \text{penalty}, \tag{2}$$

where:

• $q_i = \|L^{-1}(\mathbf{x}_i - \mu)\|^2$ (squared Mahalanobis distance at presence point $i$),
• $q_j$ is the same at background point $j$,
• $z_i = \alpha^\top\operatorname{diag}(\sigma)^{-1}(\mathbf{x}_i - \mu)$ is the skew score at presence point $i$,
• $z_j$ is the skew score at background point $j$,
• $\hat{g}(\cdot)$ is the Gaussian KDE of the background,
• the penalty term is as in Model 4, Eq. (4).

3.2 Structure of the Denominator

The denominator integrand (inside the logsumexp) has three log-scale terms:

$$a_j = \underbrace{\log\hat{g}(\mathbf{y}_j)}_{\text{availability}} + \underbrace{\log\Phi(z_j)}_{\text{skewness}} - \underbrace{\tfrac{1}{2}q_j}_{\text{Mahalanobis}}. \tag{3}$$

This is exactly the log of $\hat{g}(\mathbf{y}_j) \cdot \Phi(z_j) \cdot \exp(-\frac{1}{2}q_j)$, which is proportional to $\hat{g} \cdot f_{\text{SN}}$ evaluated at background point $j$ — the discrete approximation to the denominator integral in Eq. (1).

3.3 Relationship to Models 2 and 4

Component	Model 2 (weighted)	Model 4 (skew-normal)	Model 5 (this)
Mahalanobis	$\checkmark$	$\checkmark$	$\checkmark$
$\log\Phi(z)$	—	$\checkmark$	$\checkmark$
KDE weighting	$\checkmark$	—	$\checkmark$
Background data	Required	Not used	Required

4 Implementation Call Chain

optimize_niche(likelihood = "skew_normal_weighted")
  └─ niche_obj.cpp::create_niche_obj_ptr()
       └─ loglik_niche_math_skew_normal_weighted_eigen()  [C++, lines 590–656]
            ├─ build_L_cov()                               [theta → mu, sigma, L_cov]
            ├─ alpha = Map(theta + 2*p + n_v, p)           [line 615]
            ├─ Mahalanobis q1 (occ), q2 (background)
            ├─ sum_log_phi_occ = sum_log_pnorm(occ)        [line 624]
            ├─ z_den = skew_z(diff_den, sigma, alpha)      [line 632]
            ├─ log_phi_den = log_pnorm(z_den)              [lines 633–634]
            ├─ a_j = log_w_den + log_phi_den - 0.5*q2      [line 640]
            ├─ logsumexp(a)                                  [lines 642–644]
            ├─ prior_penalty_value(has_alpha=true)           [lines 646–647]
            └─ neg_log = 0.5*sum_q1 - sum_log_phi_occ
                         - log_w_occ + n*lse + penalty       [lines 649–653]

4.1 Key Code (loglik_niche_math_cpp.cpp, lines 590–656)

The C++ kernel:

1. Reconstructs $L_{\text{cov}}$ and maps $\alpha$ (lines 611–615).
2. Computes occurrence-side Mahalanobis distances and $\sum_i\log\Phi(z_i)$ (lines 620–624).
3. Computes background-side Mahalanobis distances $q_j$ and skew scores $z_j$ (lines 626–634).
4. Evaluates the denominator integrand $a_j = \log\hat{g}_j + \log\Phi(z_j) - \frac{1}{2}q_j$ (line 640).
5. Applies the logsumexp trick (lines 642–644).
6. Adds the ridge + alpha penalty (lines 646–647).
7. Returns the guarded negative log-likelihood (lines 649–655).

4.2 R Entry Point

optimize_niche(env_occ, env_m, likelihood = "skew_normal_weighted",
               prior_log_sigma_lambda = 1.0,
               prior_alpha_lambda = 1.0)

5 When To Use

• When both asymmetric tolerances and non-uniform environmental availability are expected.
• As a model-selection alternative to "weighted" (Model 2): fit both and compare via AIC/BIC. If the skewness parameters are close to zero, the simpler Model 2 is preferred.
• For species with known thermal or precipitation asymmetries in regions where the accessible area has heterogeneous environments.

6 Limitations

1. Parameter count. Adds $p$ skewness parameters over Model 2. Requires larger sample sizes for reliable estimation.

2. Patil & Ord drift (shared with Model 2). The ridge prior on $\log\sigma$ is essential; do not set $\lambda = 0$.

3. Unimodal. The skew-normal is still unimodal. For bimodal niches, mixture extensions are needed (not implemented).

4. Numerical stability. The product $\Phi(z_j) \cdot \hat{g}_j$ can be very small for background points far from the niche centre. The logsumexp trick (lines 642–644) handles this by working entirely in log space.

References