A Stepwise Assessment of Parsimony and Fuzzy Entropy in Species Distribution Modelling.

Alba Estrada,Raimundo Real

doi:10.3390/e23081014

Abstract

Entropy is intrinsic to the geographical distribution of a biological species. A species distribution with higher entropy involves more uncertainty, i.e., is more gradually constrained by the environment. Species distribution modelling tries to yield models with low uncertainty but normally has to reduce uncertainty by increasing their complexity, which is detrimental for another desirable property of the models, parsimony. By modelling the distribution of 18 vertebrate species in mainland Spain, we show that entropy may be computed along the forward-backwards stepwise selection of variables in Logistic Regression Models to check whether uncertainty is reduced at each step. In general, a reduction of entropy was produced asymptotically at each step of the model. This asymptote could be used to distinguish the entropy attributable to the species distribution from that attributable to model misspecification. We discussed the use of fuzzy entropy for this end because it produces results that are commensurable between species and study areas. Using a stepwise approach and fuzzy entropy may be helpful to counterbalance the uncertainty and the complexity of the models. The model yielded at the step with the lowest fuzzy entropy combines the reduction of uncertainty with parsimony, which results in high efficiency.

Highlights

Species distribution models are widely employed by the scientific community to detect species’ suitable areas in a territory [1,2], to understand the environmental variables that define species distributions [3,4], to forecast the effect of global changes on their distributions [5,6], to detect priority areas for conservation [7,8], or to guide conservation programmes [1,9]
We suggest using the favourability function and the fuzzy entropy to assess the uncertainty of the models
Fuzzy entropy does not have anything that makes it worse than Shannon entropy or Akaike Information Criterion (AIC), but it has the advantage that it is comparable for all species and all systems, while neither Shannon nor AIC are

Summary

Introduction

Species distribution models are widely employed by the scientific community to detect species’ suitable areas in a territory [1,2], to understand the environmental variables that define species distributions [3,4], to forecast the effect of global changes on their distributions [5,6], to detect priority areas for conservation [7,8], or to guide conservation programmes [1,9]. There are a wide variety of procedures to perform species distribution models, including Generalized Linear Models [10], Generalized Additive Models [11], Artificial Neural Networks [12], Maximum Entropy Models (MaxEnt) [13], and Classification and Regression Trees [14] Some of these approaches, such as MaxEnt, are based on the concept of entropy [15]. Stepwise Generalised Linear Models, and stepwise logistic regression, a commonly used supervised machine learning algorithm [20,21], can be considered to be related to the concept of entropy In this case, the goal is to find the model that gives the maximum reduction of entropy in each of the steps. This can be computed with the AIC of the steps and with Shannon entropy and fuzzy entropy (see below)

Methods

Results

Conclusion