Inferring the presence and abundance of rare waterbirds species from scarce data

Abundance data are used in ecology for species monitoring and conservation. These count data often display several specific characteristics like numerous missing data, high variance, and a high proportion of zeros, particularly when monitoring rare species. We present a model that aims to impute missing data and estimate the effect of covariates on species presence and abundance. It is based on the log-normal Poisson model, which offers more flexibility in the variance of counts than a Poisson model. A latent variable is added for the overrepresentation of zeros in the data. The imputation of missing data is made possible by assuming that the latent variance matrix has low rank and the inclusion of covariates. \ We demonstrate the identifiability in the presence of missing data. Since maximum likelihood inference is intractable, we use a variational expectation-maximization algorithm to infer the parameters. We provide an estimate of the asymptotic variance of the estimators and derive prediction intervals for the imputations, an estimate of the temporal trend, and a procedure for detecting a potential change in this trend. \ We evaluate our imputations and associated prediction intervals using artificially degraded monitoring data set. We conclude with an illustration on a monitoring waterbirds data set.

💡 Research Summary

This paper addresses the challenging problem of estimating presence and abundance of rare waterbird species when monitoring data are sparse, contain many zeros, and suffer from missing observations. The authors propose a novel statistical framework that extends the Poisson‑Log‑Normal (PLN) model by incorporating a zero‑inflation component and a low‑rank latent Gaussian structure, resulting in what they call the Zero‑Inflated PLN with Principal Component Analysis (ZI‑PLN‑PCA).

The model works as follows. For each site‑year pair (i, j), a Bernoulli variable Uᵢⱼ indicates species presence, with logit(πᵢⱼ)=xᵢⱼᵀγ linking covariates to presence probability. Conditional on presence (Uᵢⱼ=1), the count Yᵢⱼ follows a Poisson distribution with mean λᵢⱼ=exp(xᵢⱼᵀβ+Zᵢⱼ). The latent term Zᵢⱼ captures temporal dependence across years for the same site. It is modeled as Zᵢ = C Wᵢ, where Wᵢ∼N(0,I_q) and C is a p × q loading matrix; thus the covariance Σ=C Cᵀ is constrained to low rank, providing a parsimonious representation of inter‑annual correlations.

Identifiability is rigorously examined. The authors prove that, provided every pair of years is jointly observed at least once (condition Q =