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This paper presents an overview of results for the geostatistical analysis of collocated multivariable data sets, in which the variables form a composition, i.e. where the components inform of the relative important of the parts forming a whole. Such data sets occur most often in mining, hydrogeochemistry and soil science, but the results gathered here are relevant for any regionalised compositional data set. The paper covers the basic definitions, the analysis of the spatial codependence between components, mapping methods of cokriging and cosimulation honoring the compositional constraints, the role of pre- and post-transformations like logratios or multivariate normal score transforms, and block-support upscaling. Proofs of all statements are included in an appendix.

The discrete case of Bayes' formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes' formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.

To analyse compositional data (CoDa) sets it is advisable to apply the principle of working on orthonormal coordinates. Sequential Binary Partition (SBP) is a technique to construct an interpretable basis. The provided coordinates are called balances, and they are complemented with a descriptive tool, the CoDa-dendrogram. The goal of a Principal Balances method is to identify an orthonormal basis of the simplex such that the coordinates are balances approaching the properties of principal component analysis (PCA) of CoDa. PCA is one of the main tools for exploratory analysis and modelling of CoDa. However, the main shortcoming of PC's is the difficulty in interpreting the resulting coordinates because a PC is a function of all the original parts. Balances are log-contrasts resulting from a logratio of two geometric means of two groups of parts. Their interpretation is considerably simpler than the interpretation of PC's. The resulting procedures provide tools improving interpretability and intuitive dimension reduction. The algorithm to compute principal balances requires an exhaustive search along all possible sets of orthonormal balances. The consumption of computational time may be considerable for even a small number of parts. Here, to reduce this time, the sets of possible partitions for up to 15 parts is stored. For comparison, two suboptimal, but feasible, algorithms are introduced: (i) searching for balances following a constraint PC's approach; and (ii) a hierarchical cluster analysis of variables based on the Aitchison distance between parts. The properties and performance of these three algorithms are illustrated using a data set of geochemical composition of glacial sediments. Results obtained corroborate the theoretical properties of the methods: they approximate reasonably well the PC's improving the interpretability. However, the price payed is a smaller amount of variance explained by the first balances and the lack of uncorrelation between the coordinates.

These notes have been prepared as support to a short course on compositional data analysis. The first version dates back to the year 2000. Their aim is to transmit the basic concepts and skills for simple applications, thus setting the premises for more advanced projects. The awareness of problems related to the statistical analysis of compositional data analysis dates back to a paper by Karl Pearson (1897), which title began significantly with the words "On a form of spurious correlation". The aim of these notes is to summarize the last stages of research on this topic, focusing on the practical aspects necessary for a statistical modelling of these kind of data. The notes are based both on the monograph by John Aitchison, "Statistical analysis of compositional data" (1986), and on recent developments that complement the theory developed there. These notes have been updated over the years, and frequent updates will be still required in the near future, as the theory presented here is a field of active research.

A Bayes linear space is a linear space of equivalence classes of proportional -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.

It has been suggested that climate change might modify the occurrence rate and magnitude of large ocean-wave and wind storms. The hypothesised reason is the increase of available energy in the atmosphere-ocean system. Forecasting models are commonly used to assess these effects, given that good quality data series are often too short. However, forecasting systems are often tuned to reproduce the average behavior, and there are concerns on their representativity for extremal regimes. We present a methodology of simultaneous analysis of observed and hindcasted data with the aim of extracting potential time drifts as well as systematic regime discrepancies between the two data sources. The method is based on the Peak-Over- Threshold (POT) approach and the Generalized Pareto Distribution (GPD) within a Bayesian estimation framework. In this context, storm events are considered points in time, and modelled as a Poisson process. Storm magnitude over a reference threshold is modelled with a GPD, a flexible model that captures the tail behaviour of the magnitude distribution. All model parameters, i.e. shape and location of the magnitude GPD and the Poisson occurrence rate, are affected by a trend in time. Moreover, a systematic difference between parameters of hindcasted and observed series is considered. Finally, the posterior joint distribution of all these trend parameters is studied using a conventional Gibbs sampler. This method is applied to compare hindcast and observed series of 10-minute average windspeed at a deep buoy location off the Catalan coast (NE Spain, Western Mediterranean; buoy data from 2001; REMO wind hindcasting from 1958 on). Appropriate scale and domain of attraction are discussed, and the reliability of trends in time are addressed.

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute one, it is possible to use simple algebraic properties to show that it is more convenient to work with a geometry that is not the usual Euclidean geometry in real space, and with a measure which is not the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated both on the positive real line and on the D-part simplex., 21 pages, 8 figures