Summary The simplex plays an important role as sample space in many practical situations where compositional data, in the form of proportions of some whole, require interpretation. It is argued that the statistical analysis of such data has proved difficult because of a lack both of concepts of independence and of rich enough parametric classes of distributions in the simplex. A variety of independence hypotheses are introduced and interrelated, and new classes of transformed‐normal distributions in the simplex are provided as models within which the independence hypotheses can be tested through standard theory of parametric hypothesis testing. The new concepts and statistical methodology are illustrated by a number of applications.

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1 SummaryThe singular value decomposition and its interpretation as a linear biplot has proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the speci¯c case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is demonstrated on a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.2

Practitioners of many skills face the need to make some realistic statement about the likely outcome of a future 'experiment of interest' on the basis of observed variability of outcomes in previously conducted related experiments. In this book the authors provide the predictor with the data and formulae which will assist in accurate forecasting, and suggest that an effective answer is to be found in the concept of predictive distribution within the framework of statistical prediction analysis. An applied mathematical approach is adopted throughout and the book is aimed at readers with some statistical knowledge, final year undergraduates, numerate scientists, technologists and medical workers interested in predictive techniques.

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