PoissonRatioUQ: An R package for band ratio uncertainty quantification

TL;DR

This work addresses uncertainty quantification for ratios of Poisson intensities in spatially binned remote-sensing data by developing a permanental-process–based Bayesian framework that infers $Z(s)=\frac{\lambda_a(s)}{\lambda_b(s)}$ and propagates uncertainty to a quantity of interest $T(s)$ through a forward relation $Z=(mT+z_0)^p$. The core approach uses a permanental process with intensity $\lambda(s)=\frac{1}{2}\left(f_1(s)^2+f_2(s)^2\right)$, where $f_i(s)\sim\mathcal{GP}(0,k)$, enabling a MAP estimate via Representer Theorem and a Laplace approximation to obtain a tractable posterior. The paper details the associated generalized Beta Prime distributions that describe the posterior of intensity ratios and transformed quantities, and provides an R package, PoissonRatioUQ, with utilities for generalized Beta Prime calculations, permanental-process estimation, ratio estimation, and uncertainty scoring (CRPS, HPD). Demonstrations on toy ratio problems show accurate recovery of $Z(x)$ and stable uncertainty quantification, while forward-model nonlinearities illustrate propagation of estimation error into $T$. The package also highlights practical features, such as univariate and spatial analyses, and outlines future work to improve scalability and enable unbinned-data extensions, making the approach applicable to large remote-sensing datasets and similar band-ratio problems.

Abstract

We introduce an R package for Bayesian modeling and uncertainty quantification for problems involving count ratios. The modeling relies on the assumption that the quantity of interest is the ratio of Poisson means rather than the ratio of counts. We provide multiple different options for retrieval of this quantity for problems with and without spatial information included. Some added capability for uncertainty quantification for problems of the form $Z=(mT+z_0)^{p}$, where $Z$ is the intensity ratio and $T$ the quantity of interest, is included.

Figures (4)

• Figure 3.1: Example retrieval and timing study using the permanental process model to retrieve the ratio function in Eq. \ref{['eq:timingstudy_eqn']}
• Figure 3.2: Estimation results from two realizations of the Poisson data for the nonlinear transformation between the ratio and quantity of interest given by Eq. \ref{['eq:qoi_nonlin']} along with estimated 95% HPD intervals.
• Figure 4.1: Comparison: Empirical distribution, CDF, and quantiles from rbetaprime() compared to analytic expressions. Plots generated with 1000 random draws from the distribution.
• Figure 4.2: Demonstration of the algorithm for calculating the highest-posterior density sets on a mixture of Gaussians. The algorithm is able to capture the bimodal nature of the distribution.