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John Aitchison revolutionised in 1982 our way of approaching geochemical data focusing on their relative nature. In this perspective, the investigation of single variables is meaningless due to the entangled structure that links all the parts of a composition. Starting from that time, several developments have characterized the debate within the scientific community, both from the applied and the theoretical point of view. The consequence was that the number of papers where compositional data are consistently and coherently managed increased exponentially. The exploratory phase of compositional data is a very important step in data analysis and modeling. It not only helps to clarify the available sample data structure but also determines the base to develop models to predict time and space changes. Real chemical data along the course of the river Tevere (Tiber) (Italy) and its tributaries are taken to illustrate how compositional techniques help explore compositions and detect patterns and outliers in the data. [Display omitted] • Explorative statistics as a fundamental step of data analysis • Choice of the sample space for compositional data • Investigating the source of compositional variability and outliers • Combining compositional and non-compositional information • Identifying discriminant compositional indices [ABSTRACT FROM AUTHOR]

This paper studies the relative evaluation of young people and the possible benefits associated with three methods of avoiding sexually transmitted infections/AIDS and/or unwanted pregnancies (condoms, contraceptive pills, morning-after pills). A survey evaluating these three methods, with respect to ten different items, was given to 145 undergraduate students (76% females, 24% males). Results show that, compared to pills, condoms are positively valued as protection against sexually transmitted infections/AIDS, couples are interested in using them to protect their health, and they are appreciated for their lack of side effects. Pills are better valued than condoms with respect to feelings of pleasure for both sexes and not as valued in preventing sexually transmitted infections. Females´ assessments are less extreme than male’s for five of the six items, but are more pronounced with respect to the assumption that using condoms shows an interest in the couple taking care of their sexual health. The study shows a general preference towards condoms rather than contraceptive pills and postcoital pills, but men are far more likely to choose this option. The conclusions are based on a compositional analysis of two way tables. [ABSTRACT FROM AUTHOR]

Like the statistical analysis of compositional data in general, spatial analysis of compositional data requires specific tools. A historical overview of their development is presented in three steps: (a) the recognition of the problem, known as spurious spatial covariance, (b) first attempts to use the logratio approach, and (c) the application of the principle of working in coordinates using isometric logratio representations. Also mentioned are the use of matrix-valued variation-variograms as a tool to model crossvariograms, and the simplicial approach to indicator kriging, that solves inconsistencies in the standard approach to indicator kriging. [ABSTRACT FROM AUTHOR]

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]

Compositional data analysis usually deals with relative information between parts where the total (abundances, mass, amount, etc.) is unknown or uninformative. This article addresses the question of what to do when the total is known and is of interest. Tools used in this case are reviewed and analysed, in particular the relationship between the positive orthant of D-dimensional real space, the product space of the real line times the D-part simplex, and their Euclidean space structures. The first alternative corresponds to data analysis taking logarithms on each component, and the second one to treat a log-transformed total jointly with a composition describing the distribution of component amounts. Real data about total abundances of phytoplankton in an Australian river motivated the present study and are used for illustration. [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]

Water is a vital, finite resource whose quantity and quality are deteriorating as the world population increases. The current study aims to investigate the concentration of heavy metals (HM) in surface water for irrigation purposes with associated human health risks and pollution sources near the marble industry in Malakand, Pakistan. Twenty-seven water samples were randomly collected and analyzed for HM concentration by inductively coupled plasma‒optical emission spectrometry (ICP‒OES). pH, electrical conductivity (EC), total dissolved solids (TDS), biological oxygen demand (BOD), and chemical oxygen demand (COD) were measured using standard methods of American Public Health Association (APHA). Irrigation suitability was assessed using specific water quality parameters. The associated health risks from ingestion and dermal exposure to heavy metals were calculated by USEPA health risk indices. Pollution sources and spatial distribution mapping were studied using compositional data analysis (CoDa) and the application of a geographic information system (GIS) to understand the changing behavior of heavy metals in surface waters. The concentrations of BOD (89%), COD (89%), Al (89%), Ca (89%), Cr (56%), Cu (78%), Fe (56%), K (34%) Mg (23%), Mn (56%), Na (89%), Ni (56%), P (89%), and Zn (11%) exceeded the safety limits of National Environmental Quality standards (NEQs) of Pakistan. The results of Kelly's ratio (KR) classified surface water as unsuitable for irrigation. The average daily doses (ADD, mg/kg/day) for Al, Cu, Cr, Fe, Mn, Ni, and Zn were higher in children than in adults. The hazard index (HI) for children and adults was above the threshold (HI > 1), indicating a significant risk of non-carcinogenic toxicity. The carcinogenic risk values for Cr and Ni were above the USEPA limit (1 × 10−6 to 1 × 10−4), suggesting a potential carcinogenic risk for the target population. Principal component analysis (PCA), biplot (CLR), and the CoDa-dendrogram allowed for the identification of elemental associations, and their potential source was anthropogenic rather than natural in origin. Regular monitoring and phytoremediation strategies are proposed to safeguard crops and human health. [ABSTRACT FROM AUTHOR]

Geochemical samples can be restricted to be on a locus in the sample space. When this occurs there is a chemical equilibrium of some kind. Typical examples are chemical equilibrium regulated by the mass action law or stoichiometry given by the structure of crystals. These equilibria can correspond to linear restrictions in the simplex, for which the restriction is on a linear manifold (hyperplane) or, more generally, restriction is on a general manifold (warped hypersurface). Exact equilibrium is seldom observed in geochemistry and so the focus is shifted to measuring distances or deviations from equilibrium. A linear equilibrium locus is defined by a constant logcontrast. When this logcontrast is evaluated on a sample, the difference between the equilibrium constant and the value obtained for each sample is a deviation from equilibrium. It is—in absolute values—the Aitchison distance to equilibrium. The non-linear case is more involved. Here, a linearization technique is proposed. It consists of adding terms to the composition so that the deviation or distance to the equilibrium locus reduces to the linear case. These techniques are illustrated with some examples: chemical reactions producing bicarbonate ions in water (linear), the weathering of microcline and the formation of kaolinite (non-linear), both concerning stream water composition at a European scale. Stoichiometric behaviour of olivines (non-linear) is also analyzed. Finally, a discussion on methods for discovering unknown equilibria in a sample is presented. • Geochemical equilibria described under compositional data analysis • Compositional distances by equilibria as a tool for monitoring natural processes • Geochemical samples can be restricted to be on a locus in the sample space. • Linear and non-linear compositional equilibrium compared [ABSTRACT FROM AUTHOR]

A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive projections of the exact error on two finite-dimensional subspaces. The analysis introduced in this paper is based on a probabilistic approach, that is, the idea is to assess the average value of the effectivity index (the ratio estimated error over exact error) by assuming the randomness of the exact error. The average value characterizes the mean behavior of the estimator and it is found to be related with some geometric properties of the subspaces. These geometric properties are obtained from the standard matrices of the linear systems arising in the formulation of the finite element method. [ABSTRACT FROM AUTHOR]

Statistical and machine learning techniques based on relative abundances have been used to predict health conditions and to identify microbial biomarkers. However, high dimensionality, sparsity and the compositional nature of microbiome data represent statistical challenges. On the other hand, the taxon grouping allows summarizing microbiome abundance with a coarser resolution in a lower dimension, but it presents new challenges when correlating taxa with a disease. In this work, we present a novel approach that groups Operational Taxonomical Units (OTUs) based only on relative abundances as an alternative to taxon grouping. The proposed procedure acknowledges the compositional data making use of principal balances. The identified groups are called Principal Microbial Groups (PMGs). The procedure reduces the need for user-defined aggregation of |$\textrm{OTU}$| s and offers the possibility of working with coarse group of |$\textrm{OTU}$| s, which are not present in a phylogenetic tree. PMGs can be used for two different goals: (1) as a dimensionality reduction method for compositional data, (2) as an aggregation procedure that provides an alternative to taxon grouping for construction of microbial balances afterward used for disease prediction. We illustrate the procedure with a cirrhosis study data. PMGs provide a coherent data analysis for the search of biomarkers in human microbiota. The source code and demo data for PMGs are available at: https://github.com/asliboyraz/PMGs. [ABSTRACT FROM AUTHOR]

We propose a new method to estimate the horizontal-to-vertical (H / V) spectral ratio using microtremor measurements. The technique is based on modeling the H / V transfer function by means of an AutoRegressive Moving Average (ARMA) filter. As compared with the conventional, Fourier-based spectra processing routines, this method is efficient in extracting the fundamental resonant frequency with a higher spectral resolution. We demonstrate its usefulness by an application to several sites in northeastern Iberian Peninsula. For the data sets examined, our approach shows improvement over the Fourier transform-based method under adverse experimental conditions. We conclude that the new technique provides robust estimates of the fundamental resonance frequency and can be effective to characterize empirically the unconsolidated sediment thickness and bedrock geometry. • A new method to estimate the H / V spectral ratio using microtremors. • An Autoregressive Moving Average filter models the H / V transfer function. • The ARMA approach is a robust method for estimating soil resonance frequencies. • Engineering bedrock can be more precisely estimated using the ARMA method. [ABSTRACT FROM AUTHOR]

R-mode hierarchical clustering is a method for forming hierarchical groups of mutually exclusive subsets of variables. This R-mode cluster method identifies interrelationships between variables which are useful for variable selection and dimension reduction. Importantly, the method is based on metric elements defined on the sample space of variables. Consequently, hierarchical clustering of compositional parts should respect the particular geometry of the simplex. In this work, the connections between concepts such as distance, cluster representative, compositional biplot, and log-ratio basis are explored within the framework of the most popular R-mode agglomerative hierarchical clustering methods. The approach is illustrated in a paleoecological study to identify groups of species sharing similar behavior. [ABSTRACT FROM AUTHOR]

Abstract The study of the relationships between two compositions is of paramount importance in geochemical data analysis. This paper develops a compositional version of canonical correlation analysis, called CoDA-CCO, for this purpose. We consider two approaches, using the centred log-ratio transformation and the calculation of all possible pairwise log-ratios within sets. The relationships between both approaches are pointed out, and their merits are discussed. The related covariance matrices are structurally singular, and this is efficiently dealt with by using generalized inverses. We develop compositional canonical biplots and detail their properties. The canonical biplots are shown to be powerful tools for discovering the most salient relationships between two compositions. Some guidelines for compositional canonical biplot construction are discussed. A geochemical data set with X-ray fluorescence spectrometry measurements on major oxides and trace elements of European floodplains is used to illustrate the proposed method. The relationships between an analysis based on centred log-ratios and on isometric log-ratios are also shown. Highlights • Canonical correlation analysis is adapted for compositional data sets. • Approach based on the centred log-ratio transformation • Generalized inverses used to deal with structurally singular covariance matrices • Canonical biplots proposed to uncover relationships between compositions • Geochemical applications on major oxides and trace elements of European floodplains [ABSTRACT FROM AUTHOR]

The area east of Varanasi is one of numerous places along the watershed of the Ganges River with groundwater concentrations of arsenic surpassing the maximum value of 10 parts per billion (ppb) recommended by the World Health Organization in drinking water. Here we apply geostatistics and compositional data analysis for the mapping of arsenic and iron to help in understanding the conditions leading to the occurrence of elevated level of arsenic in groundwater. The methodology allows for displaying concentrations of arsenic and iron as maps consistent with the limited information from 95 water wells across an area of approximately 210 km; visualization of the uncertainty associated with the sampling; and summary of the findings in the form of probability maps. For thousands of years, Varanasi has been on the erosional side in a meander of the river that is free of arsenic values above 10 ppb. Maps reveal two anomalies of high arsenic concentrations on the depositional side of the valley, which has started seeing urban development. The methodology using geostatistics combined with compositional data analysis is completely general, so this study could be used as a prototype for hydrochemistry mapping in other areas. [ABSTRACT FROM AUTHOR]

We propose the total bootstrap median (TBM) as a robust and efficient estimator of location and scale for small samples. We demonstrate its performance by estimating the mean and variance of a variety of distributions. We also show that, if the underlying distribution is unknown and there is either no contamination or low to moderate contamination, the TBM provides a better estimate of the mean, in mean square terms, than the sample mean or the sample median. In addition, the TBM is a better estimator of the variance of the underlying distribution than the sample variance or the square of the bias-corrected median absolute deviation from the median estimator. We also show that the TBM is an explicit L-estimator, which allows a direct study of its properties. [ABSTRACT FROM PUBLISHER]

In the life sciences, many measurement methods yield only the relative abundances of different components in a sample. With such relative—or compositional—data, differential expression needs careful interpretation, and correlation—a statistical workhorse for analyzing pairwise relationships—is an inappropriate measure of association. Using yeast gene expression data we show how correlation can be misleading and present proportionality as a valid alternative for relative data. We show how the strength of proportionality between two variables can be meaningfully and interpretably described by a new statistic ϕ which can be used instead of correlation as the basis of familiar analyses and visualisation methods, including co-expression networks and clustered heatmaps. While the main aim of this study is to present proportionality as a means to analyse relative data, it also raises intriguing questions about the molecular mechanisms underlying the proportional regulation of a range of yeast genes. [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]

Background: Fecal microbiota transplantation (FMT) has been recently approved by FDA for the treatment of refractory recurrent clostridial colitis (rCDI). Success of FTM in treatment of rCDI led to a number of studies investigating the effectiveness of its application in the other gastrointestinal diseases. However, in the majority of studies the effects of FMT were evaluated on the patients with initially altered microbiota. The aim of our study was to estimate effects of FMT on the gut microbiota composition in healthy volunteers and to monitor its long-term outcomes. Results: We have performed a combined analysis of three healthy volunteers before and after capsule FMT by evaluating their general condition, adverse clinical effects, changes of basic laboratory parameters, and several immune markers. Intestinal microbiota samples were evaluated by 16S rRNA gene and shotgun sequencing. The data analysis demonstrated profound shift towards the donor microbiota taxonomic composition in all volunteers. Following FMT, all the volunteers exhibited gut colonization with donor gut bacteria and persistence of this effect for almost ∼1 year of observation. Transient changes of immune parameters were consistent with suppression of T-cell cytotoxicity. FMT was well tolerated with mild gastrointestinal adverse events, however, one volunteer developed a systemic inflammatory response syndrome. Conclusions: The FMT leads to significant long-term changes of the gut microbiota in healthy volunteers with the shift towards donor microbiota composition and represents a relatively safe procedure to the recipients without long-term adverse events. [ABSTRACT FROM AUTHOR]

Coal proximate analysis is a form of typical compositional data, commonly represented with constant sum. Although the direct geostatistical modeling of compositional data provides apparently reasonable outputs, the results are always exposed to inconsistency and non-optimality. In this paper, we compare the compositional and noncompositional approaches to assess the problems caused by neglecting the compositional nature of data. The ultimate goal is to attain an accurate approach for coal reserve evaluation. The presented compositional approach was executed and validated on field data from the Parvadeh IV coal deposit in Central Iran. The comparison of sum of values maps, Aitchison distance, actual vs. estimated cross-plots and correct classification rate of the mentioned approaches illustrate that the compositional approach is notably more accurate and reliable. To compare tonnage – grade curves, new formulations are proposed to calculate T and T + , two moments of total tonnage, defined based on the random variables of stochastic simulation. Tonnage – grade and mean grade – cut-off grade curves showed that the noncompositional approach has overestimated ash values and underestimated carbon content. This can lead the analyst to misinterpretation and underrating the deposit. Quantitative comparison of tonnage – grade curves of the approaches revealed that at cut-offs of 31% for ash and 52% for fixed carbon, nearly 5 and 3 million tons of coal are being considered as waste by the noncompositional approach. Consequently, neglecting the compositional nature of data will result in deviated outputs, unrealistic models and unreliable evaluations and finally lead to financial losses. Thus, it is strongly recommended to consider the compositional nature of data in reserve evaluations. [ABSTRACT FROM AUTHOR]

Purpose: The ability to properly analyze and interpret large microbiome data sets has lagged behind our ability to acquire such data sets from environmental or clinical samples. Sequencing instruments impose a structure on these data: the natural sample space of a 16S rRNA gene sequencing data set is a simplex, which is a part of real space that is restricted to nonnegative values with a constant sum. Such data are compositional and should be analyzed using compositionally appropriate tools and approaches. However, most of the tools for 16S rRNA gene sequencing analysis assume these data are unrestricted.Methods: We show that existing tools for compositional data (CoDa) analysis can be readily adapted to analyze high-throughput sequencing data sets.Results: The Human Microbiome Project tongue versus buccal mucosa data set shows how the CoDa approach can address the major elements of microbiome analysis. Reanalysis of a publicly available autism microbiome data set shows that the CoDa approach in concert with multiple hypothesis test corrections prevent false positive identifications.Conclusions: The CoDa approach is readily scalable to microbiome-sized analyses. We provide example code and make recommendations to improve the analysis and reporting of microbiome data sets. [ABSTRACT FROM AUTHOR]

Mineral reserve evaluation is based on compositional data, reporting in units such as percentages, mg/kg or ppm. The nature of such data implies that the values cannot vary independently and geostatistical modeling of their raw form could be misleading. In this study we propose a compositional approach —based on orthonormal isometric logratio (olr) coordinates— to simulate the in-situ value (in US Dollars) at Sungun Cu − Mo deposit in NW Iran. A dataset of 1422 surface samples, analyzed for Ag, As, Au, Cu, Mo, Sb, Pb and Zn is used. This surface covers the current mine pit and is sampled prior to extraction, as a detailed exploration phase. We assumed the sampling surface as an extraction mine bench and separated 34% of the original samples for validation. Comparing with a noncompositional approach, it is prominent that in all extraction categories and for all Cu and Mo cut-off grades, the compositional values are closer to the real ones. Comparison of in-situ value maps revealed that the compositional approach is better able to delineate intense local variability; especially in high grade parts of the deposit. This superiority is also evident in Euclidean and Aitchison distance-distance diagrams of the real and simulated values. It is concluded that, in contrast with known geostatistical methods, the proposed olr-based method delivers more accurate models and, consequently, an improved and more realistic vision of the deposit value is achieved. • A polymetallic mineral resource evaluation deals with compositional data. • Otrhonormal logratio transform is able to summarize all data in arbitrary balances. • Desired balances can be flexibly defined using a Sequential Binary Partition (SBP). • An auxiliary variable recovers the original units after interpolation. • The compositional approach improves the estimation of in-situ value of ore elements. [ABSTRACT FROM AUTHOR]

Condition is a central concept in evolutionary ecology, but the roles of genetic and environmental quality in condition-dependent trait expression remain poorly understood. Theory suggests that condition integrates genetic, epigenetic and somatic factors, and therefore predicts alignment between the phenotypic effects of genetic and environmental quality. To test this key prediction, we manipulated both genetic (mutational) and environmental (dietary) quality in Drosophila melanogaster and examined responses in morphological and chemical (cuticular hydrocarbon, CHC) traits in both sexes. While the phenotypic effects of diet were consistent among genotypes, effects of mutation load varied in magnitude and direction. Average effects of diet and mutation were aligned for most morphological traits, but non-aligned for the male sexcombs and CHCs in both sexes. Our results suggest the existence of distinct forms of condition dependence, one integrating both genetic and environmental effects and the other purely environmental. We propose a model to account for these observations. [ABSTRACT FROM AUTHOR]