Maximum power entropy method for ecological data analysis

In ecology predictive models of the geographical distribution of certain species are widely used to capture the spatial diversity. Recently a method of Maxent based on Gibbs distribution is frequently employed to have reasonable accuracy of a target distribution of species at a site using environmental features such as temperature, precipitation, elevation and so on. It requires only presence data, which is a big advantage to the case where absence data is not available or unreliable. It also incorporates our limited knowledge into the model about the target distribution such that the expected values of environmental features are equal to the empirical average. Moreover, the visualization of the inhabiting probability of species is easily done with the aid of geographical coordination information from Global Biodiversity Inventory Facility (GBIF) in a statistical software R. However, the maximum entropy distribution in Maxent is derived from the Boltzmann-GibbsShannon entropy, which causes unstable estimation of the parameters in the model when some outliers in the data are observed. To overcome the weak point and to have deep understandings of the relation among the total number of species, the Boltzmann-Gibbs-Shannon entropy and Simpson ’s index, we propose a maximum power entropy method based on beta-divergence, which is a special case of U-divergence. It includes the Boltzmann-Gibbs-Shannon entropy as a special case, so it could have better performance of estimation of the target distribution by appropriately choosing the value of the power index beta. We demonstrate the performance of the proposed method by simulation studies as well as publicly available real data.