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

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.

Climate change both facilitates and inhibits invasive plant ranges in New England

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Exploiting Maximum Entropy method and ASTER data for assessing debris flow and debris slide susceptibility for the Giampilieri catchment (north‐eastern Sicily, Italy)

“…Merow et al (2013) illustrate the MaxEnt model architecture as requiring presence only (PO) data and a set of predictors distributed across a regularly gridded space. In analogy to the aforementioned article but in a landslide susceptibility framework, the algorithm initially estimates the density of landslide occurrences within the landscape and generates landslide relative occurrence rates (ROR; Fithian and Hastie, 2013) per each cell. This quantity can be seen as the ratio between probability density of covariates across locations within the considered geographic space where the landslide is present and the probability density of covariates across the entire geographic space, thus obtaining insights on the relative proneness to fail of a given cell compared to another.…”