Projection pursuit with robust indices for the analysis of ecological data

The aim of projection pursuit (PP) (Friedman and Tukey 1974, Huber 1985, Jones and Sibson 1987) is to find low (1-3)-dimensional projections showing the most interesting views of high-dimensional data. To reach this goal, many projections are calculated and rated by an objective function, a so called projection index I. Dependent on I, this method is able to characterise the high-dimensional data from different points of view (Fig. 6.3.1). To find good projections it is necessary to maximize I. There are several ways to do this: Gradient Algorithms (Friedman 1987), Simulated Annealing (Montanari and Guglielmi 1996) or Genetic Algorithms (Crawford 1991, Guo et al. 2000). The application of PP has some advantages over other methods. Although the projections are linear, the index may be non-linear, facilitating the detection of non-linear connections in the data. By choosing different and suitable indices, PP is able to provide various views of the data. One important feature of the index functions is robustness; this enables PP to be insensitive to outliers. Projection vectors from previous analysis can be used to view and analyse another data set from the same viewpoint. The multivariate analysis method PP can be used to classify data, comparable to supervised and unsupervised learning in artificial neural networks, but with reproducible results. In ecological applications we are faced with the problem of small data sets of high dimensionality, where statistical confidence is not achievable. Even in this situation PP is a suitable tool for data analysis, combining computational power (fast production and preselection of two-dimensional views of the data) with human intuition (detection of regularities and dependencies in two-dimensional diagrams). PP can generate hypotheses on the underlying data and provide insight regarding experimental design for hypotheses testing.