Space-Time Zero-Inflated Count Models of Harbor Seal

Environmental data is spatial, temporal, and often comes with many zeros. In this paper, we included space-time random effects in zero-inflated Poisson, or ZIP, and "hurdle" models to investigate haul-out patterns of harbor seals on glacial ice. The data consisted of counts for 18 dates on a lattice grid of samples of harbor seals hauled out on glacial ice in Disenchantment Bay near Yakutat, Alaska. A hurdle model is similar to a ZIP model except it does not mix zeros from the binary and count processes. Both models can be used for zero-inflated data, and we compared space-time ZIP and hurdle models in a Bayesian hierarchical model. Space-time ZIP and hurdle models were constructed by using spatial conditional autoregressive (CAR) models and temporal first-order autoregressive (AR(1)) models as random effects in ZIP and hurdle regression models. We created maps of smoothed predictions for harbor seal counts based on ice density, other covariates, and spatio-temporal random effects. For both models predictions around the edges appeared to be positively biased. The linex loss function is an asymmetric loss function that penalizes overprediction more than underprediction, and we used it to correct for prediction bias to get the best map for space-time ZIP and hurdle models. 

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Jay M. Ver Hoef and John K. Jansen. Published in Environmetrics, 2007; 18:697–712, Wiley InterScience, DOI:10.1002/env.873