Monotonicity-constrained species distribution models.

Flexible modeling frameworks for species distribution models based on generalized additive models that allow for smooth, nonlinear effects and interactions are of increasing importance in ecology. Commonly, the flexibility of such smooth function estimates is controlled by means of penalized estimation procedures. However, the actual shape remains unspecified. In many applications, this is not desirable as researchers have a priori assumptions on the shape of the estimated effects, with monotonicity being the most important. Here we demonstrate how monotonicity constraints can be incorporated in a recently proposed flexible framework for species distribution models. Our proposal allows monotonicity constraints to be imposed on smooth effects and on ordinal, categorical variables using an additional asymmetric L2 penalty. Model estimation and variable selection for Red Kite (Milvus milvus) breeding was conducted using the flexible boosting framework implemented in R package mboost.