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Modeling and Analysis of Compositional Data presents a practical and comprehensive introduction to the analysis of compositional data along with numerous examples to illustrate both theory and application of each method. Based upon short courses delivered by the authors, it provides a complete and current compendium of fundamental to advanced methodologies along with exercises at the end of each chapter to improve understanding, as well as data and a solutions manual which is available on an accompanying website. Complementing Pawlowsky-Glahn's earlier collective text that provides an overview of the state-of-the-art in this field, Modeling and Analysis of Compositional Data fills a gap in the literature for a much-needed manual for teaching, self learning or consulting.

Issue Title: Advances in Compositional Data We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations.[PUBLICATION ABSTRACT]

We propose a new algorithm for the identification of microbial signatures. These microbial signatures can be used for diagnosis, prognosis, or prediction of therapeutic response based on an individual’s specific microbiota. High-throughput sequencing technologies have revolutionized microbiome research by allowing the relative quantification of microbiome composition and function in different environments. In this work we focus on the identification of microbial signatures, groups of microbial taxa that are predictive of a phenotype of interest. We do this by acknowledging the compositional nature of the microbiome and the fact that it carries relative information. Thus, instead of defining a microbial signature as a linear combination in real space corresponding to the abundances of a group of taxa, we consider microbial signatures given by the geometric means of data from two groups of taxa whose relative abundances, or balance, are associated with the response variable of interest. In this work we present selbal , a greedy stepwise algorithm for selection of balances or microbial signatures that preserves the principles of compositional data analysis. We illustrate the algorithm with 16S rRNA abundance data from a Crohn’s microbiome study and an HIV microbiome study. IMPORTANCE We propose a new algorithm for the identification of microbial signatures. These microbial signatures can be used for diagnosis, prognosis, or prediction of therapeutic response based on an individual’s specific microbiota.

Issue Title: Advances in Compositional Data Amalgamation of parts of a composition has been extensively used as a technique of analysis to achieve reduced dimension, as was discussed during the CoDaWork'03 meeting (Girona, Spain, 2003). It was shown to be a non-linear operation in the simplex that does not preserve distances under perturbation. The discussion motivated the introduction in the present paper of concepts such as group of parts, balance between groups, and sequential binary partition, which are intended to provide tools of compositional data analysis for dimension reduction. Key concepts underlying this development are the established tools of subcomposition, coordinates in an orthogonal basis of the simplex, balancing element and, in general, the Aitchison geometry in the simplex. Main new results are: a method to analyze grouped parts of a compositional vector through the adequate coordinates in an ad hoc orthonormal basis; and the study of balances of groups of parts (inter-group analysis) as an orthogonal projection similar to that used in standard subcompositional analysis (intra-group analysis). A simulated example compares results when testing equal centers of two populations using amalgamated parts and balances; it shows that, in certain circumstances, results from both analysis can disagree.[PUBLICATION ABSTRACT]

Geometry in the simplex has been developed in the last 15 years mainly based on the contributions due to J. Aitchison. The main goal was to develop analytical tools for the statistical analysis of compositional data. Our present aim is to get a further insight into some aspects of this geometry in order to clarify the way for more complex statistical approaches. This is done by way of orthonormal bases, which allow for a straightforward handling of geometric elements in the simplex. The transformation into real coordinates preserves all metric properties and is thus called isometric logratio transformation (ilr). An important result is the decomposition of the simplex, as a vector space, into orthogonal subspaces associated with nonoverlapping subcompositions. This gives the key to join compositions with different parts into a single composition by using a balancing element. The relationship between ilr transformations and the centered-logratio (clr) and additive-logratio (alr) transformations is also studied. Exponential growth or decay of mass is used to illustrate compositional linear processes, parallelism and orthogonality in the simplex.[PUBLICATION ABSTRACT]

Compositional data analysis requires selecting an orthonormal basis with which to work on coordinates. In most cases this selection is based on a data driven criterion. Principal component analysis provides bases that are, in general, functions of all the original parts, each with a different weight hindering their interpretation. For interpretative purposes, it would be better to have each basis component as a ratio or balance of the geometric means of two groups of parts, leaving irrelevant parts with a zero weight. This is the role of principal balances, defined as a sequence of orthonormal balances which successively maximize the explained variance in a data set. The new algorithm to compute principal balances requires an exhaustive search along all the possible sets of orthonormal balances. To reduce computational time, the sets of possible partitions for up to 15 parts are stored. Two other suboptimal, but feasible, algorithms are also introduced: (i) a new search for balances following a constrained principal component approach and (ii) the hierarchical cluster analysis of variables. The latter is a new approach based on the relation between the variation matrix and the Aitchison distance. The properties and performance of these three algorithms are illustrated using a typical data set of geochemical compositions and a simulation exercise.

The statistical techniques based on compositional data are applied to investigate the evolution of the traffic share of the container throughput in a multi-port system. Compositional vectors are those which contain relative information of parts of some whole. The application of conventional statistical techniques to compositional data may lead to erroneous conclusions and spurious correlations. Therefore, compositional data (CoDa) should be treated taking into account their own mathematical structure. The so-called log-ratio approach provides a set of transformations that allow to apply conventional statistical techniques to the transformed compositional data samples. Thus, the objective of this paper is double. As a first stage it aims to introduce the CoDa formalism and highlight its potentiality in the port container throughput analysis as example of transport system providing an applied example: the container throughput evolution in the Spanish Mediterranean Ports system during the period 1976–2015. Second, based on the previous analysis, the aim is to characterize the container throughput in SpanishMed ports and its temporal evolution. The CoDa analysis clarifies the interpretation and data association of the container traffic throughput evolution in function of some selected change points: boom of containerization in 1990s and 2008 crisis. This contribution proves that the CoDa methodology is useful to investigate the complexity of the transport disciplines in order to understand and to manage the spatial integration that results from the movement of people and freight.

Chronic kidney disease (CKD), a collective term for many causes of progressive renal failure, is increasing worldwide due to ageing, obesity and diabetes. However, these factors cannot explain the many environmental clusters of renal disease that are known to occur globally. This study uses data from the UK Renal Registry (UKRR) including CKD of uncertain aetiology (CKDu) to investigate environmental factors in Belfast, UK. Urbanisation has been reported to have an increasing impact on soils. Using an urban soil geochemistry database of elemental concentrations of potentially toxic elements (PTEs), we investigated the association of the standardised incidence rates (SIRs) of both CKD and CKD of uncertain aetiology (CKDu) with environmental factors (PTEs), controlling for social deprivation. A compositional data analysis approach was used through balances (a special class of log contrasts) to identify elemental balances associated with CKDu. A statistically significant relationship was observed between CKD with the social deprivation measures of employment, income and education (significance levels of 0.001, 0.01 and 0.001, respectively), which have been used as a proxy for socio-economic factors such as smoking. Using three alternative regression methods (linear, generalised linear and Tweedie models), the elemental balances of Cr/Ni and As/Mo were found to produce the largest correlation with CKDu. Geogenic and atmospheric pollution deposition, traffic and brake wear emissions have been cited as sources for these PTEs which have been linked to kidney damage. This research, thus, sheds light on the increasing global burden of CKD and, in particular, the environmental and anthropogenic factors that may be linked to CKDu, particularly environmental PTEs linked to urbanisation.

Compositional data carry relative information. Hence, their statistical analysis has to be performed on coordinates with respect to a log-ratio basis. Frequently, the modeler is required to back-transform the estimates obtained with the modeling to have them in the original units such as euros, kg or mg/liter. Approaches for recovering original units need to be formally introduced and its properties explored. Here, we formulate and analyze the properties of two procedures: a simple approach consisting of adding a residual part to the composition and an approach based on the use of an auxiliary variable. Both procedures are illustrated using a geochemical data set where the original units are recovered when spatial models are applied.

Environmental risk management consists of making decisions on human activities or construction designs that are affected by the environment and/or have consequences or impacts on it. In these cases, decisions are made such that risk is minimized. In this regard, the forthcoming paper develops a close form that relates risk with cost, hazard, and vulnerability; and then focuses on vulnerability. The vulnerability of a system under an external action can be described by the conditional probability of the degrees of damage after an event. This vulnerability model can be obtained by a simplicial regression of those outputs, as a response variable, on explanatory variables. After a theoretical explanation, the authors present the case study of a nuclear power plant containment building. Once a given overpressure is registered inside the containment building, three possible outputs are to be considered: serviceability, breakdown, and collapse. The study consists of three steps: (a) modelling the containment building using the finite element method; (b) given an overpressure, simulating uncertain parameters related to material constitutive equations in order to obtain the corresponding proportions; (c) performing a simplicial regression to obtain a meaningful vulnerability model. The simulation provides normalized-to-unity outputs under the overpressure conditions. The obtained vulnerability model is in definite correspondence with previous results in nuclear power plant safety analysis reports.