Finite-sample equivalence in statistical models for presence-only data

TL;DR

This paper clarifies the relationships among IPP, Maxent, and logistic regression models for presence-only data, showing their equivalences and differences, and introduces a new weighted logistic regression method that aligns with IPP estimates.

Contribution

It demonstrates the equivalence of IPP and Maxent models, highlights the finite-sample differences with logistic regression, and proposes infinitely weighted logistic regression as a practical, exact alternative.

Findings

IPP and Maxent produce identical density estimates.

Logistic regression generally yields different estimates, especially with model misspecification.

Infinitely weighted logistic regression matches IPP estimates exactly in finite samples.

Abstract

Statistical modeling of presence-only data has attracted much recent attention in the ecological literature, leading to a proliferation of methods, including the inhomogeneous Poisson process (IPP) model, maximum entropy (Maxent) modeling of species distributions and logistic regression models. Several recent articles have shown the close relationships between these methods. We explain why the IPP intensity function is a more natural object of inference in presence-only studies than occurrence probability (which is only defined with reference to quadrat size), and why presence-only data only allows estimation of relative, and not absolute intensity of species occurrence. All three of the above techniques amount to parametric density estimation under the same exponential family model (in the case of the IPP, the fitted density is multiplied by the number of presence records to obtain a fitted intensity). We show that IPP and Maxent give the exact same estimate for this density, but logistic regression in general yields a different estimate in finite samples. When the model is misspecified - as it practically always is - logistic regression and the IPP may have substantially different asymptotic limits with large data sets. We propose ``infinitely weighted logistic regression,'' which is exactly equivalent to the IPP in finite samples. Consequently, many already-implemented methods extending logistic regression can also extend the Maxent and IPP models in directly analogous ways using this technique.