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Finite-Sample Equivalence in Statistical Models for Presence-Only Data.

William Fithian1, Trevor Hastie1

  • 1Department of Statistics, Stanford University, 390 Serra Mall, Stanford, California 94305-4065, USA.

The Annals of Applied Statistics
|December 11, 2014
PubMed
Summary

Statistical modeling of presence-only data uses methods like inhomogeneous Poisson process (IPP) and Maxent. We show IPP and Maxent yield identical density estimates, unlike logistic regression, and propose a unified approach.

Keywords:
Poisson process modelsPresence-only datacase-control samplinglogistic regressionmaximum entropyspecies modeling

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Area of Science:

  • Ecology
  • Statistical Modeling
  • Biodiversity Research

Background:

  • Presence-only data modeling is crucial in ecology.
  • Methods like inhomogeneous Poisson process (IPP), Maxent, and logistic regression are widely used.
  • Relationships between these methods are increasingly recognized.

Purpose of the Study:

  • To clarify the theoretical relationships between IPP, Maxent, and logistic regression for presence-only data.
  • To explain why IPP intensity is more suitable than occurrence probability for presence-only inference.
  • To propose a unified modeling framework.

Main Methods:

  • Comparative analysis of statistical models for presence-only data.
  • Focus on inhomogeneous Poisson process (IPP), maximum entropy (Maxent), and logistic regression.
  • Development of infinitely weighted logistic regression.

Main Results:

  • IPP and Maxent provide identical density estimates.
  • Logistic regression yields different estimates, especially in misspecified models.
  • Infinitely weighted logistic regression is equivalent to IPP in finite samples.

Conclusions:

  • A unified framework based on exponential family density estimation connects IPP, Maxent, and logistic regression.
  • Infinitely weighted logistic regression offers a flexible extension for presence-only data analysis.
  • This approach facilitates the adaptation of existing logistic regression extensions to IPP and Maxent models.