SpatialBernoulli package

SpatialBernoulli.jl is a package to define a spatially correlated Bernoulli variable Y using a latent Gaussian construction.

Model

The Spatial Bernoulli model is defined as follows:

• Y ~ SB(C_Y, λ)

• (X_{Y,1}, …, X_{Y,D}) ~ N(0, C_Y)

• For all spatial locations s:

  • Y_s = 1 if X_{Y,s} ≤ Φ⁻¹(λ_s)
  • Y_s = 0 otherwise

Here:

• C_Y is the covariance matrix of the latent Gaussian field
• λ = (λ_s)_{s=1,…,D} is the vector of marginal probabilities
• Φ denotes the cumulative distribution function (CDF) of the standard normal distribution

This model is used in the paper to be inserted later to model precipitation occurrence across a large region.

Features

The package provides several methods, including:

• Model definition
• Probability density functions (pdf) and their logarithm (logpdf), computed using bivariate Gaussian integrals with MvNormalCDF.jl
• Maximum likelihood estimation (not recommended for high-dimensional samples)
• Maximum pairwise likelihood estimation with bivariate integrals computed using MvNormalCDF.jl
• Fast maximum pairwise likelihood estimation with bivariate integrals computed using the approximation of Tsay (2023)

Example

The documentation also provides a simulated example.

Below are examples of generation from the Spatial Bernoulli model with:

• a constant marginal probability λ in space, and
• an exponential covariance function C_Y(h) = exp(-h / ρ)