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.

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.

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.

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.