A model for the diameter-height distribution in an uneven-aged beech forest and a method to assess the fit of such models

Abstract

This paper illustrates the application of a mixture model to describe the bivariate diameter-height distribution of trees growing in a pure, uneven-aged beech forest. A mixture of two bivariate normal distributions is considered but the methodology is applicable to mixtures of other distributions. The model was fitted to diameter-height observations for 1242 beech trees in the protected forest Dreyberg (Solling, Germany). A considerable advantage of the model, apart from the fact that it happens to fit this large data set unusually well, is that the individual parameters all have familiar interpretations. The bivariate Johnson SBB distribution was also fitted to the data for the purpose of comparing the fits. A second issue discussed in this paper is concerned with the general question of assessing the fit of models for bivariate data. We show how a device called "pseudoresidual" enables one to investigate the fit of a bivariate model in new ways and in considerable detail. Attractive features of pseudo-residuals include the fact that they are not difficult to interpret; they can be computed using generally available statistical software and, most important of all, they enable one to examine the fit of a model by means of simple graphs.