Structural Risk Minimization: a robust method for density-dependence detection and model selection =Ecography, 2007 (preprint)=

Statistically distinguishing density-dependent from density-independent populations and selecting the best demographic model for a given population are problems of primary importance. Traditional approaches are PBLR (parametric bootstrapping of likelihood ratios) and Information Criteria (IC), such as the Schwarz Information Criterion (SIC), the Akaike Information Criterion (AIC) or the Final Prediction Error (FPE). While PBLR is suitable for choosing from a couple of models, ICs select the best model from among a set of candidates. In this paper, we use the Structural Risk Minimization (SRM) approach. SRM is the model selection criterion developed within the Statistical Learning Theory (SLT), a theory of great generality for modelling and learning with finite samples. SRM is almost unknown in the ecological literature and has never been used to analyze time series. First, we compare SRM with PBLR in terms of their ability to discriminate between the Malthusian and the density-dependent Ricker model. We rigorously repeat the experiments described in Dennis and Taper (1994) and find out that SRM is equally powerful in detecting density-independence and much more powerful in detecting density-dependence. Then, we compare SRM against ICs in terms of their ability to select one of several candidate models; we generate, via stochastic simulation, a huge amount of artificial time series both density-independent and dependent, with and without exogenous covariates, using different dataset sizes, noise levels and parameter values. Our findings show that SRM outperforms traditional ICs, because generally (a) it recognizes the model underlying the data with higher frequency, and (b) it leads to lower errors in out-of-samples predictions. SRM superiority is specially apparent with short time series. We finally apply SRM to the population records of Alpine ibex (Capra ibex ) living in the Gran Paradiso National Park (Italy), already investigated by Jacobson et. al. (2004) via traditional statistical methods; we both analyze the models cited in Jacobson et. al. (2004) and introduce some novel models. We show that models that are best according to SRM show also the lowest leave-one-out crossvalidation error.