The “inside cell” ratio

Let $$C_s = \{c _1 , c_2 , …d_n \}$$ be the
set of cells at scale $$s$$ where the occurrences of a node X where found. The $$C _{s−1} = \{d_ 1 , d_ 2 , …d _k \}$$ is
the corresponding set of cells at an upper scale (ancestor of $$s$$) where the occurrences of a node X where found.

Note that the ratio:
$$r_s = \frac{\#C_{s-1}}{\#C_s}$$

gives us an indicator of how the occurrences are dispersed in the space.

If $$r_s$$ is low means that the
spatial distribution is constrained in a region as small as the unit area of the upper scale while if $$r_s$$ is close to 1 it tells us that the occurrences are as spatially distributed as the cells in the upper scale.
The method can be applied recursively to the sucessive scales to obtain a list of ratios $$r_1 , r_2 , ..r_s ,..$$ that can be fitted in model to estimate geographic extensions.

A model for presence-only data

This is the set-up of a conditional auto-logistic regressive model (CAR) for predicting species presence using a sample signal and a presence-only data.

Set-up

Let $$Sp$$ be a species and $$Y$$ and $$X$$ two random variables corresponding to the events of: $$Sp$$ is in location $$x_i$$ and: location $$x_i$$ has been sampled. (The variable $$X$$ and $$x_i$$ are not related)

$$Y$$ and $$X$$ are consider to be independent binary (Bernoulli) variables conditional to the latent processes $$S$$ and $$P$$ respectively.

Latent processes

These variables are modeled as this:

$$logit(S_k) = \beta_s^t d_s(x_k) + \psi_k +O_k$$

$$logit(P_k) = \beta_p^t d_p(x_k) + \psi_k + O_k$$

Where $$O_k$$ is an offset term, $$d_p(x_k), d_s(x_k)$$ are the covariates for p and s respectively; and $$\psi_k$$ is modeled as a Gaussian Markov Random Field.

Common Gaussian Markov Field (CGMRF)

This process is modeled as this:

$$\psi_k = \phi_k + \theta_k$$

$$\phi_k | \phi_{-k}, \mathbb{W},\tau^2 \sim N \left( \frac{\sum_{i=1}^{K} w_{ki} \phi_i}{\sum_{i=1}^{K}w_{ki}}, \frac{\tau^2}{\sum_{i=1}^{K}w_{ki}}\right)$$
$$\theta_k \sim N\left(0, \sigma^2\right)$$

$$\tau^2, \sigma^2 \sim^{iid} Inv.Gamma(a,b)$$

Where $$\mathbb{W}$$ is the adjacency matrix of the lattice, $$\theta_k$$ is an independent noise term with constant variance. $$sigma^2$$ and $$\tau^2$$ are independent and identically distributed hyperparameters sampled from an inverse gamma distribution.

The corresponding Directed acyclic Graph can be seen in the next figure.

Implementation

A current implementation of this model can be found here:
Of particular interest is the file: joint.binomial.bymCAR.R where you can find the joint sample between line 113 and 149.