Minimum message length clustering, environmental heterogeneity and the variable Poisson model

One possible explanation of variation in vegetation is based on the variable Poisson model. In this model, species occurrence is presumed to follow a Poisson distribution, but the value of the Poisson parameter for any species varies from point to point, as a result of environmental variation. As an extreme, this includes dividing the given habitat into areas favourable to a community and areas which are unfavourable, or at least not occupied. The spatial area can then be viewed as a series of patches within which each species follows a Poisson distribution, although different patchesmay have different values for the Poisson parameter for any particular species. In this paper, I use a method of fuzzy clustering (mixture modelling) based on the minimum message length principle to examine the variation in Poisson parameter of individual species. The method uses the difference between the message length for the null, 1-cluster case and the message length for the optimal cluster solution, appropriately normalised, as a measure of the amount of pattern any analysis captures. I also compare the Poisson results with results obtained by assuming the within patch distribution is Gaussian. The Poisson alternative consistently results in a greater capture of pattern than the Gaussian, but at the expense of a much larger number of clusters. Overall, the Gaussian alternative is strongly supported. Other mechanisms that might introduce extra clusters, for example within-cluster correlation or spatial dependency between observations, would presumably apply equally to both models. The variable Poisson model, in the limit, converges on the individualistic model of vegetation, the Gaussian on something like the community unit model. With these data, the individualistic model is strongly rejected. Difficulties with comparing model classes mean this conclusion must remain tentative