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Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example. [ABSTRACT FROM PUBLISHER]

A Bayes linear space is a linear space of equivalence classes of proportional σ-finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes' rule and substraction by Radon-Nikodym derivatives. The present contribution shows the subspace of square-log-integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred-log-ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square-integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support. [ABSTRACT FROM AUTHOR]

It often occurs in practice that it is sensible to give different weights to the variables involved in a multivariate data analysis—and the same holds for compositional data as multivariate observations carrying relative information. It can be convenient to apply weights to better accommodate differences in the quality of the measurements, the occurrence of zeros and missing values, or generally to highlight some specific features of compositional parts. The characterisation of compositional data as elements of a Bayes space, which is as a natural generalisation of the ordinary Aitchison geometry, enables the definition of a formal framework to implement weighting schemes for the parts of a composition. This is formally achieved by considering a reference measure in the Bayes space alternative to the common uniform measure via the well-known chain rule. Unweighted centred logratio (clr) coefficients and isometric logratio (ilr) coordinates then allow us to express compositions in real space equipped with (unweighted) Euclidean geometry. The resulting elements of real space generated by the clr coefficients or ilr coordinates are invariant to the scale of the original compositions, but the actual scale of the weights matters. In this work, these formal developments are presented and used to introduce a general approach for weighting parts in compositional data analysis. The practical use is demonstrated on simulated and real-world data sets in the context of the earth sciences. [ABSTRACT FROM AUTHOR]