Spatial Modelling of Count Data: A Case Study in Mod- elling Breeding Bird Survey Data on Large Spatial Do- mains
The North American Breeding Bird Survey (BBS) is conducted each breeding season by volunteer observers (e.g. Robbins et al. 1986). The observers count the number of various species of birds along speciﬁed routes. The collected data are used for several purposes, including the study of the range of bird species, and the variation of the range and abundance over time (Link and Sauer 1998). Such studies usually require spatial maps of relative abundance. Traditional methods for producing such maps are somewhat ad hoc (e.g., inverse distance methods) and do not always account for the special discrete, positive nature of the count data (e.g. Sauer et al. 1995). In addition, corresponding prediction uncertainties for maps produced in this fashion are not typically available. Providing such uncertainties is critical as the prediction maps are often used as “data” in other studies and for the design of auxiliary sampling plans. We consider the BBS modeling problem from a hierarchical perspective,
modeling the count data as Poisson, conditional on a spatially varying intensity process. The intensities are then assumed to follow a log-normal distribution with ﬁxed eﬀects and with spatial and non-spatial random eﬀects. Model-based geostatistical methods for generalized linear mixed models (GLMMs) of this type have been available since the seminal work of Diggle et al. (1998). However, implementation is problematic when there are large data sets and prediction is desired over large domains. We show that by utilizing spectral representations of the spatial random eﬀects process, Bayesian spatial prediction can easily be carried out on very large data sets over extensive prediction domains. General discussion of the role of such Bayesian hierachical random eﬀect modelling is given in 1, and approaches to spatio-temporal modelling are found here in 12.