Uncertainty analysis of species distribution models

Abstract

The maximum entropy model, a commonly used species distribution model (SDM) normally combines observations of the species occurrence with environmental information to predict the geographic distributions of animal or plant species. However, it only produces point estimates for the probability of species existence. To understand the uncertainty of the point estimates, we analytically derived the variance of the outputs of the maximum entropy model from the variance of the input. We applied the analytic method to obtain the standard deviation of dengue importation probability and Aedes aegypti suitability. Dengue occurrence data and Aedes aegypti mosquito abundance data, combined with demographic and environmental data, were applied to obtain point estimates and the corresponding variance. To address the issue of not having the true distributions for comparison, we compared and contrasted the performance of the analytical expression with the bootstrap method and Poisson point process model which proved of equivalence of maximum entropy model with the assumption of independent point locations. Both Dengue importation probability and Aedes aegypti mosquito suitability examples show that the methods generate comparatively the same results and the analytic method we introduced is dramatically faster than the bootstrap method and directly apply to maximum entropy model.

Fig 1. Standard deviation comparison for Dengue importation probability.

(a) Figure shows the point estimates for the import probability ${\widehat{p}}_i$. (b) Figure visually plots the bootstrap standard deviation estimates for p_i across Texas counties. (c) Figure visually plots the analytic standard deviation estimates for p_i across Texas counties. (d) Figure plots the standard deviations of bootstrap vs. analytic and shows a strong equivalence between the two. Each red dot represent the estimations for one county.

Fig 2. Standard deviation comparison for Aedes aegypti.

(a) Figure presents the point estimates p_i. (b) Figure shows standard deviation calculated using bootstrap method. (c) Figure shows standard deviation calculated using analytic method. (d) Figure shows the standard deviation comparison between analytic method and bootstrap method.