Multi-species distribution modeling using penalized mixture of regressions

Multi-species distribution modeling, which relates the occurrence of multiple species to environmental variables, is an important tool used by ecologists for both predicting the distribution of species in a community and identifying the important variables driving species co-occurrences. Recently, Dunstan, Foster and Darnell [Ecol. Model. 222 (2011) 955-963] proposed using finite mixture of regression (FMR) models for multi-species distribution modeling, where species are clustered based on their environmental response to form a small number of "archetypal responses." As an illustrative example, they applied their mixture model approach to a presence–absence data set of 200 marine organisms, collected along the Great Barrier Reef in Australia. Little attention, however, was given to the problem of model selection—since the archetypes (mixture components) may depend on different but likely overlapping sets of covariates, a method is needed for performing variable selection on all components simultaneously. In this article, we consider using penalized likelihood functions for variable selection in FMR models. We propose two penalties which exploit the grouped structure of the covariates, that is, each covariate is represented by a group of coefficients, one for each component. This leads to an attractive form of shrinkage that allows a covariate to be removed from all components simultaneously. Both penalties are shown to possess specific forms of variable selection consistency, with simulations indicating they outperform other methods which do not take into account the grouped structure. When applied to the Great Barrier Reef data set, penalized FMR models offer more insight into the important variables driving species co-occurrence in the marine community (compared to previous results where no model selection was conducted), while offering a computationally stable method of modeling complex species-environment relationships (through regularization).