Gaussian Markov random fields and structural additive regression : applications in freshwater fisheries management

Abstract

In this thesis structural additive regression (STAR) models are constructed for two applications in freshwater fisheries management 1) large scale modelling of fish abundance using electrofishing removal data and 2) assessing the effect on stream temperature of tree felling. Both approaches take advantage of the central role Gaussian Markov random fields (GMRFs) play in the construction of structured additive regression components.

The R package mgcv can fit, in principle, any STAR model. In practice, however, several extensions are required to allow a non-specialised user to access this functionality, and a large part of this thesis is the developement of software to allow a general user ready access to a wide range of GMRF models within the familiar mgcv framework. All models are fitted making use of this extension where possible (and practical).

The thesis is divided into three main chapters. Chapter 2 serves to provide background and insight into penalised regression and STAR models and the role that GMRFs play in smoothing. Also presented are the extensions required to fit GMRF models in mgcv.

Chapter 3 presents a two stage model for fish density using electrofishing removal data. The first stage of this model estimates fish capture probability and is not a STAR model, but can utilise aspects of GMRFs through low rank approximations; software to make this available is developed and presented. The second stage is a Poisson STAR model and can therefore be fitted in the extended mgcv framework.

Finally, Chapter 4 presents a model for the impact of a clear felling event on stream temperature. This model utilises cyclic smoothers applied to the functional principal components of daily temperature curves. This allows for a detailed assessment of the effects of felling on stream temperature that is not possible when modelling daily summaries alone.