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We consider calculation of the mode of compositional data, the data related to each other through a linear constraint. Compositional data arises in various extensions of the fuzzy sets theory (type-2 interval-valued, intuitionistic, hesitant fuzzy sets), biomedicine (relative abundance, genome sequencing, activity recognition), and data analytics (various wealth indices, interval-valued observations, traffic congestion, etc.). Mode is a pre-aggregation function in the case of single variable, used as a classical estimator robust to outliers, but its multivariate extensions face the challenges of high computational complexity and potential oversmoothing. In this work we present several novel techniques for mode estimation on the unit k-simplex representing compositional, interval-valued, and general vector-valued data. We highlight the re-weighted k-nearest neighbours algorithm based on the Choquet integral with respect to a 2-additive fuzzy measure, compare its performance against other approaches based on spatial partitioning, and illustrate its applications to aggregation of real-world interval-valued data sets.

We introduce the Robust Logistic Zero-Sum Regression (RobLZS) estimator, which can be used for a two-class problem with high-dimensional compositional covariates. Since the log-contrast model is employed, the estimator is able to do feature selection among the compositional parts. The proposed method attains robustness by minimizing a trimmed sum of deviances. A comparison of the performance of the RobLZS estimator with a non-robust counterpart and with other sparse logistic regression estimators is conducted via Monte Carlo simulation studies. Two microbiome data applications are considered to investigate the stability of the estimators to the presence of outliers. Robust Logistic Zero-Sum Regression is available as an R package that can be downloaded at https://github.com/giannamonti/RobZS.

Precision mapping of soil texture is critical for hydrological, ecological, environmental, and agricultural modeling and field management. However, the mapping precision is generally restricted by the limited number of soil sampling and insufficient use of available information in spatial interpolation. To map layered soil texture with higher precision, we propose an additive log-ratio (ALR) transformation and exploratory factor analysis (EFA)-based co-kriging (CK) method (ALR-EFA-CK) to study the spatial variability of multi-layered soil particle-size fractions and soil texture. In this method, the ALR transformation is used to reduce the closure effect of soil particle-size fractions as compositional data that are characterized by non-negativity and a constant sum of 100%, and EFA is used to extract common factors from variables related to soil texture that are further used as auxiliary variables of CK. Six interpolation methods, ordinary kriging (OK), traditional CK (CC-CK), and EFA-CK for both the original and ALR transformed data, were evaluated in a case study with data collected at seven soil layers of 108 sampling points in the middle reach of Heihe River basin in Northwest China. CC-CK is superior to OK by including auxiliary data in interpolation, EFA-CK is more effective in improving the interpolation precision by taking full advantages of auxiliary information, and ALR transformation can improve the interpolation precision effectively for soil particle-size fractions as compositional data. Therefore, the proposed ALR-EFA-CK method is beneficial in improving the interpolation precision and recommended to interpolate multi-layered soil texture.

Aim: To describe food scores that can be interpreted as indicators of the potential effects of a food on health outcomes. Methods: The scores consist of weighted averages of the estimates of the association between a health outcome and nutrient balances. Each food contributes a set of weights depending on its nutrient composition. The nutrient balances of a food's constituents are defined in a manner that is similar to those included in the model from which the estimated associations are obtained. As the balances are based on log-ratios, the presence of zero observations is problematic, but observations recorded as trace (rounded zeros) can be suitably replaced by a small value. In order to overcome the essential zero problem, we propose a strategy that exploits the flexibility of sequential binary partitioning, a procedure used to define orthogonal balances, and also describe the estimates of the variance and confidence limits of food scores. Results: Data from an Italian population-based study were used to provide examples regarding serum high-density lipoprotein and triglyceride levels and waist circumference values, and the scores calculated for some foods or groups of foods are shown. Conclusions: Food scores allow foods to be ranked on the basis of their potential effects on health outcomes and, as they are not directly related to the amount of the food consumed, may be less subject to confounding. The accuracy of the estimates depends on the completeness of the food's nutrient composition: that is, the more complete the nutrient composition, the more accurate the estimate.

High‐throughput data representing large mixtures of chemical or biological signals are ordinarily produced in the molecular sciences. Given a number of samples, partial least squares (PLS)...

Summary In microbiome and genomic studies, the regression of compositional data has been a crucial tool for identifying microbial taxa or genes that are associated with clinical phenotypes. To account for the variation in sequencing depth, the classic log-contrast model is often used where read counts are normalized into compositions. However, zero read counts and the randomness in covariates remain critical issues. We introduce a surprisingly simple, interpretable and efficient method for the estimation of compositional data regression through the lens of a novel high-dimensional log-error-in-variable regression model. The proposed method provides corrections on sequencing data with possible overdispersion and simultaneously avoids any subjective imputation of zero read counts. We provide theoretical justifications with matching upper and lower bounds for the estimation error. The merit of the procedure is illustrated through real data analysis and simulation studies.

Longitudinal microbiome studies have been widely used to unveil the dynamics in the complex host-microbial ecosystems. Modeling the longitudinal microbiome compositional data, which is semi-continuous in nature, is challenging in several aspects: the overabundance of zeros, the heavy skewness of non-zero values that are bounded in (0, 1), and the dependence between the binary and non-zero parts. To deal with these challenges, we first extended the work of Chen and Li [1] and proposed a two-part zero-inflated Beta regression model with shared random effects (ZIBR-SRE), which characterize the dependence between the binary and the continuous parts. Besides, the microbiome compositional data have unit-sum constraint, indicating the existence of negative correlations among taxa. As ZIBR-SRE models each taxon separately, it does not satisfy the sum-to-one constraint. We then proposed a two-part linear mixed model (TPLMM) with shared random effects to formulate the log-transformed standardized relative abundances rather than the original ones. Such transformation is called “additive logistic transformation”, initially developed for cross-sectional compositional data. We extended it to analyze the longitudinal microbiome compositions and showed that the unit-sum constraint can be automatically satisfied under the TPLMM framework. Model performances of TPLMM and ZIBR-SRE were compared with existing methods in simulation studies. Under settings adopted from real data, TPLMM had the best performance and is recommended for practical use. An oral microbiome application further showed that TPLMM and ZIBR-SRE estimated a strong correlation structure in the binary and the continuous parts, suggesting models without accounting for this dependence would lead to biased inferences.

Compositional data are nonnegative data carrying relative, rather than absolute, information—these are often data with a constant-sum constraint on the sample values, for example, proportions or percentages summing to 1% or 100%, respectively. Ratios between components of a composition are important since they are unaffected by the particular set of components chosen. Logarithms of ratios (logratios) are the fundamental transformation in the ratio approach to compositional data analysis—all data thus need to be strictly positive, so that zero values present a major problem. Components that group together based on domain knowledge can be amalgamated (i.e., summed) to create new components, and this can alleviate the problem of data zeros. Once compositional data are transformed to logratios, regular univariate and multivariate statistical analysis can be performed, such as dimension reduction and clustering, as well as modeling. Alternative methodologies that come close to the ideals of the logratio approach are also considered, especially those that avoid the problem of data zeros, which is particularly acute in large bioinformatic data sets.

Compositional data are positive multivariate data, constrained to lie within the simplex space. Regression analysis of such data has been studied and many regression models have been proposed, but most of them not allowing for zero values. Secondly, the case of compositional data being in the predictor variables side has gained little research interest. Surprisingly enough, the case of both the response and predictor variables being compositional data has not been widely studied. This paper suggests a solution for this last problem. Principal components regression using the α -transformation and Kulback-Leibler divergence are the key elements of the proposed approach. An advantage of this approach is that zero values are allowed, in both the response and the predictor variables side. Simulation studies and examples with real data illustrate the performance of our algorithm.

Quantile regression is a very important tool to explore the relationship between the response variable and its covariates. Motivated by mean regression with LASSO for compositional covariates proposed by Lin et al. (2014), we consider quantile regression with no-penalty and penalty function. We develop the computational algorithms based on linear programming. Numerical studies indicate that our methods provides the better alternative than mean regression under many settings, particularly for heavy-tailed or skewed distribution of the error term. Finally, we study the fat data using the proposed method., Comment: 10 pages

Human body composition is made up of mutually exclusive and exhaustive parts (e.g. %truncal fat, %non-truncal fat and%fat-free mass) which are constrained to sum to the same total (100%). In statistical analyses, individual parts of body composition (e.g. %truncal fat or %fat-free mass) have traditionally been used as proxies for body composition, and have been linked with a range of health outcomes. But analysis of individual parts omits information about the other parts,which are intrinsically co-dependent because of the constant sum constraint of 100%. Further, body mass may be associated with health outcomes. We describe a statistical approach for body composition based on compositional data analysis. The body composition data are expressed as logratios to allow relative information about all the compositional parts to be explored simultaneously in relation to health outcomes. We describe a recent extension to the logratio approach to compositional data analysis which allows absolute information about the total of the compositional parts (body mass) to be considered alongside relative information about body composition. The statistical approach is illustrated by an example that explores the relationships between adults’ body composition, body mass and bone strength. Refereed/Peer-reviewed

Evaluating the effect of nutritional components on health outcomes is hampered by technical difficulties that are closely related to the compositional nature of dietary data. Compositional data consist of parts of a whole and, on the basis of the assumption that any relevant information they provide is contained in the ratios between them, procedures based on log-ratio transformations have been proposed for their analysis. It has been previously suggested that nutrient compositions can be analyzed using isometric log-ratio transformations which allow the compositions to be represented by a system of orthogonal coordinates corresponding to the log-contrasts between the compositional parts. One particular expression of these contrasts is called balances, and the aim of this note is to provide a step-by-step description of how to construct orthonormal balances and show how they can be applied to dietary data by means of an example based on data coming from an Italian population-based study. Nutrient balances are new variables that can be included as covariates in regression models. After fitting a linear regression model using estimates of relative fat mass as a response variable, scores indicating the obesogenic potential of foods can be defined on the basis of their nutritional contents and the estimated effect of the balances on relative fat mass. This approach to evaluating the relationship between diet and health outcomes provides insights into the roles of specific nutrient ratios and foods within a comprehensive and integrated framework that complies with the essentially compositional nature of dietary data.

A folded type model is developed for analyzing compositional data. The proposed model involves an extension of the $\alpha$-transformation for compositional data and provides a new and flexible class of distributions for modeling data defined on the simplex sample space. Despite its rather seemingly complex structure, employment of the EM algorithm guarantees efficient parameter estimation. The model is validated through simulation studies and examples which illustrate that the proposed model performs better in terms of capturing the data structure, when compared to the popular logistic normal distribution, and can be advantageous over a similar model without folding.

Motivation Cell-type abundance data arising from mass cytometry experiments are compositional in nature. Classical association tests do not apply to the compositional data due to their non-Euclidean nature. Existing methods for analysis of cell type abundance data suffer from several limitations for high-dimensional mass cytometry data, especially when the sample size is small. Results We proposed a new multivariate statistical learning methodology, Compositional Data Analysis using Kernels (CODAK), based on the kernel distance covariance (KDC) framework to test the association of the cell type compositions with important predictors (categorical or continuous) such as disease status. CODAK scales well for high-dimensional data and provides satisfactory performance for small sample sizes (n Availability and implementation CODAK is implemented using R. The codes and the data used in this manuscript are available on the web at http://github.com/GhoshLab/CODAK/. Contact prudra@okstate.edu Supplementary information Supplementary data are available at Bioinformatics Advances online.

The Dirichlet distribution is a well-known candidate in modeling compositional data sets. However, in the presence of outliers, the Dirichlet distribution fails to model such data sets, making other model extensions necessary. In this paper, the Kummer–Dirichlet distribution and the gamma distribution are coupled, using the beta-generating technique. This development results in the proposal of the Kummer–Dirichlet gamma distribution, which presents greater flexibility in modeling compositional data sets. Some general properties, such as the probability density functions and the moments are presented for this new candidate. The method of maximum likelihood is applied in the estimation of the parameters. The usefulness of this model is demonstrated through the application of synthetic and real data sets, where outliers are present.

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

Objective Emerging evidence suggests that excessive sitting has negative health effects. However, this evidence largely relies on research using self-reported sitting time, which is known to be biased. To correct this bias, we aimed at developing a calibration model estimating "true" sitting from self-reported sitting. Methods Occupational sitting time was estimated by self-reports (the International Physical Activity Question-naire) and objective measurements (thigh-worn accelerometer) among 99 Swedish office workers at a governmental agency, at baseline and 3 and 12 months afterwards. Following compositional data analysis procedures, both sitting estimates were transformed into isometric log-ratios (ILR). This effectively addresses that times spent in various activities are inherently dependent and can be presented as values of only 0−100%. Linear regression was used to develop a simple calibration model estimating objectively measured "true" sitting ILR (dependent variable) from self-reported sitting ILR (independent variable). Additional self-reported variables were then added to construct a full calibration model. Performance of the models was assessed by root-mean-square (RMS) differences between estimated and objectively measured values. Models developed on baseline data were validated using the follow-up datasets. Results Uncalibrated self-reported sitting ILR showed an RMS error of 0.767. Simple and full calibration models (incorporating body mass index, office type, and gender) reduced this error to 0.422 (55%) and 0.398 (52%), respectively. In the validations, model performance decreased to 57%/62% (simple models) and 57%/62% (full models) for the two follow-up data sets, respectively. Conclusion Calibration adjusting for errors in self-reported sitting led to substantially more correct estimates of "true" sitting than uncalibrated self-reports. Validation indicated that model performance would change somewhat in new datasets and that full models perform no better than simple models, but calibration remained effective.

Compositional data frequently arise when data refer to components which are proportions or fractions of a whole. Within the log-ratio approach, the analysis of compositional data can be conducted i...

This paper introduces area-level compositional mixed models by applying transformations to a multivariate Fay–Herriot model. Small area estimators of the proportions of the categories of a classification variable are derived from the new model, and the corresponding mean squared errors are estimated by parametric bootstrap. Several simulation experiments designed to analyse the behaviour of the introduced estimators are carried out. An application to real data from the Spanish Labour Force Survey of Galicia (north-west of Spain), in the first quarter of 2017, is given. The target is the estimation of domain proportions of people in the four categories of the variable labour status: under 16 years, employed, unemployed and inactive.

34 Páginas.-- 3 Figuras.-- 4 Tablas, The work studies the effect of season, montanera length, and sampling location on the Iberian pig fat, using compositional data (CoDa) analysis and standard statistics. CoDa variation array and logratios involving C18:3: (C18:3/C17:1), (C18:3/C20:0), and (C18:3/18:0) as well as the ilr balances (coordinates), based on C18:3, C17:1, and C20:0, showed the highest variances. Discriminant Analysis (DA) led to similar (conventional/coordinates) correct assignations regarding seasons (69/70%), montanera length (71/70%) and sampling location (68/67%). Re-analyzing the subcomposition of only the major fatty acids (FAs), led to slightly poorer results; therefore, the removed FAs might play a role in segregation. Results are in line with those from other authors and could indicate a partial capacity of pigs to control their FA profiles. Overall, CoDa analysis provided useful information on data variability, the effects of factors and, after conversion into coordinates, allowed applying standard statistics while being respectful with the compositional sampling space., This work was supported by the Spanish Government (project AGL2010-15494/ALI, partially financed by European regional development funds, ERDF), and Junta de Andalucía (through financial support to group AGR-125 and AGR-283).

This paper presents novel methods for robust statistical testing of particle shape data. Shape (the relative lengths of three orthogonal axes) is a key property of sedimentary particles, providing information on provenance, transport history and depositional environment. However, the usefulness of shape data, including the ability to make robust comparisons between samples, has been constrained by the absence of a satisfactory definition of the mean shape for a sample of particles. Such a definition is proposed and used to develop confidence regions for the population mean shape using both parametric (theoretical) and computational (bootstrap) methods. These techniques are based on a transform that permits multivariate statistical methods for the analysis of compositional data to be extended to shape. These techniques are validated with reference to a dataset of 169 clast samples and found to perform well. A statistical test on the mean – using the multivariate extension of Student's t‐test, Hotelling's T2 – is presented. The benefits of the methods presented are demonstrated with reference to a case study.

The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison’s seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain.

Mixed spatial autoregressive (SAR) models with numerical covariates have been well studied. However, as non-numerical data, such as functional data and compositional data, receive substantial amounts of attention and are applied to economics, medicine and meteorology, it becomes necessary to develop flexible SAR models with multiple data types. In this article, we integrate three types of covariates, functional, compositional and numerical, in an SAR model. The new model has the merits of classical functional linear models and compositional linear models with scalar responses. Moreover, we develop an estimation method for the proposed model, which is based on functional principal component analysis (FPCA), the isometric logratio (ilr) transformation and the maximum likelihood estimation method. Monte Carlo experiments demonstrate the effectiveness of the estimators. A real dataset is also used to illustrate the utility of the proposed model.

In this paper we consider the multivariate Dirichlet regression model proposed by Melo et al. (2009), which is tailored to situations where the multivariate response consists of multivariate positi...

The analysis of grain-size distributions has a long tradition in Quaternary Science and disciplines studying Earth surface and subsurface deposits. The decomposition of multi-modal grain-size distributions into inherent subpopulations, commonly termed end-member modelling analysis (EMMA), is increasingly recognised as a tool to infer the underlying sediment sources, transport and (post-)depositional processes. Most of the existing deterministic EMMA approaches are only able to deliver one out of many possible solutions, thereby shortcutting uncertainty in model parameters. Here, we provide user-friendly computational protocols that support deterministic as well as robust (i.e. explicitly accounting for incomplete knowledge about input parameters in a probabilistic approach) EMMA, in the free and open software framework of R. In addition, and going beyond previous validation tests, we compare the performance of available grain-size EMMA algorithms using four real-world sediment types, covering a wide range of grain-size distribution shapes (alluvial fan, dune, loess and floodplain deposits). These were randomly mixed in the lab to produce a synthetic data set. Across all algorithms, the original data set was modelled with mean R2 values of 0.868 to 0.995 and mean absolute deviation (MAD) values of 0.06 % vol to 0.34 % vol. The original grain-size distribution shapes were modelled as end-member loadings with mean R2 values of 0.89 to 0.99 and MAD of 0.04 % vol to 0.17 % vol. End-member scores reproduced the original mixing ratios in the synthetic data set with mean R2 values of 0.68 to 0.93 and MAD of 0.1 % vol to 1.6 % vol. Depending on the validation criteria, all models provided reliable estimates of the input data, and each of the models exhibits individual strengths and weaknesses. Only robust EMMA allowed uncertainties of the end-members to be objectively estimated and expert knowledge to be included in the end-member definition. Yet, end-member interpretation should carefully consider the geological and sedimentological meaningfulness in terms of sediment sources, transport and deposition as well as post-depositional alteration of grain sizes. EMMA might also be powerful in other geoscientific contexts where the goal is to unmix sources and processes from compositional data sets.

Compositional data modeling is of great practical importance, as exemplified by applications in economic and geochemical data analysis. In this study, we investigate the sliced inverse regression (SIR) procedure for multivariate compositional data with a scalar response. We can achieve dimension reduction for the original multivariate compositional data quickly and then conduct a regression on the dimensional-reduced compositions. It is documented that the proposed method is successful in detecting effective dimension reduction directions, which generalizes the theoretical framework of SIR to multivariate compositional data. Comprehensive simulation studies are conducted to evaluate the performance of the proposed SIR procedure and the simulation results show its feasibility and effectiveness. A real data application is finally used to illustrate the success of the proposed SIR-based method.

Different correlation detection methods have been specifically designed for the microbiome data analysis considering the compositional data structure and different sequencing depths. Along with the speedy development of omics studies, there is an increasing interest in discovering the biological associations between microbes and host metabolites. This raises the need of finding proper statistical methods that facilitate the correlation analysis across different omics studies. Here, we comprehensively evaluated six different correlation methods, i.e., Pearson correlation, Spearman correlation, Sparse Correlations for Compositional data (SparCC), Correlation inference for Compositional data through Lasso (CCLasso), Mutual Information Coefficient (MIC), and Cosine similarity methods, for the correlations detection between microbes and metabolites. Three simulated and two real-world data sets (from public databases and our lab) were used to examine the performance of each method regarding its specificity, sensitivity, similarity, accuracy, and stability with different sparsity. Our results indicate that although each method has its own pros and cons in different scenarios, Spearman correlation and MIC outperform the others with their overall performances. A strategic guidance was also proposed for the correlation analysis between microbe and metabolite.

International audience; A compositional linear model (regression of a scalar response on a set of compositions) for areal data is proposed, where observations are not independent and present spatial autocorrelation. Specifically, we borrow thoughts from the spatial Durbin model considering that it produces unbiased coefficient estimates compared to other spatial linear regression models (including the spatial error model, the spatial autoregressive model, the Kelejian-Prucha model, and the spatial Durbin error model). The orthonormal log-ratio (olr) transformation based on a sequential binary partition of compositions and maximum likelihood estimation method are employed to estimate the new model. We also check the proposed estimators on a simulated and a real dataset. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

In this paper we assess two traditional machine learning (ML) methods which can be used for automatic rock type classification: (1) the Self-Organising Map (SOM) with k-means clustering, and (2) Classification and Regression Trees (CART). The dataset used for this paper were chemical compositional data of rocks acquired through X-Ray Fluorescence (XRF) analysis. The ground truth of the dataset was generated by human experts in the field of geology. The complexity of the chosen dataset influenced the evaluation performance of the two ML models. We achieve an overall accuracy of 68.02 % and 62.79 % respectively when using SOM with k-means and CART.

National audience; In an election, the vote shares by party for a given subdivision of a territory form a compositional vector (positive components adding up to 1). Conventional multiple linear regression models are not adapted to explain this composition due to the constraint on the sum of the components and the potential spatial autocorrelation across territorial units. We develop a simultaneous spatial autoregressive model for compositional data that allows for both spatial correlation and correlations across equations. Using simulations and a data set from the 2015 French departmental election, we illustrate its estimation by two-stage and three-stage least squares methods.

The time spent in glucose ranges is a common metric in type 1 diabetes (T1D). As the time in one day is finite and limited, Compositional Data (CoDa) analysis is appropriate to deal with times spent in different glucose ranges in one day. This work proposes a CoDa approach applied to glucose profiles obtained from six T1D patients using continuous glucose monitor (CGM). Glucose profiles of 24-h and 6-h duration were categorized according to the relative interpretation of time spent in different glucose ranges, with the objective of presenting a probabilistic model of prediction of category of the next 6-h period based on the category of the previous 24-h period. A discriminant model for determining the category of the 24-h periods was obtained, achieving an average above 94% of correct classification. A probabilistic model of transition between the category of the past 24-h of glucose to the category of the future 6-h period was obtained. Results show that the approach based on CoDa is suitable for the categorization of glucose profiles giving rise to a new analysis tool. This tool could be very helpful for patients, to anticipate the occurrence of potential adverse events or undesirable variability and for physicians to assess patients’ outcomes and then tailor their therapies This project has been partially supported by the Spanish Government, Ministerio de Economía y Competitividad (MINECO) (Grants DPI2016-78831-C2-1-R, DPI2016-78831-C2-2-R, RTI2018-095518-B-C21), Agencia Estatal de Investigación (PID2019-107722RB-C21/ AEI/10.13039/501100011033), the National Council of Technological and Scientific Development, CNPq Brazil through Grants 202050/2015-7, 207688/2014-1 and EU through FEDER funds

BackgroundMicrobiome and omics datasets are, by their intrinsic biological nature, of high dimensionality, characterized by counts of large numbers of components (microbial genes, operational taxonomic units, RNA transcripts, etc…). These data are generally regarded as compositional since the total number of counts identified within a sample are irrelevant. The central concept in compositional data analysis is the logratio transformation, the simplest being the additive logratios with respect to a fixed reference component. A full set of additive logratios is not isometric in the sense of reproducing the geometry of all pairwise logratios exactly, but their lack of isometry can be measured by the Procrustes correlation. The reference component can be chosen to maximize the Procrustes correlation between the additive logratio geometry and the exact logratio geometry, and for high-dimensional data there are many potential references. As a secondary criterion, minimizing the variance of the reference component’s log-transformed relative abundance values makes the subsequent interpretation of the logratios even easier. Finally, it is preferable that the reference component not be a rare component but well populated, and substantive biological reasons might also guide the choice if several reference candidates are identified. Results: On each of three high-dimensional datasets the additive logratio transformation was performed, using references that were identified according to the abovementioned criteria. For each dataset the compositional data structure was successfully reproduced, that is the additive logratios were very close to being isometric. The Procrustes correlations achieved for these datasets were 0.9991, 0.9977 and 0.9997, respectively. In the third case, where the objective was to distinguish between three groups of samples, the approximation was made to the restricted logratio space of the between-group variance. Conclusions: We show that for high-dimensional compositional data additive logratios can provide a valid choice as transformed variables that are (1) subcompositionally coherent, (2) explaining 100% of the total logratio variance and (3) coming measurably very close to being isometric, that is approximating almost perfectly the exact logratio geometry. The interpretation of additive logratios is simple and, when the variance of the log-transformed reference is very low, it is made even simpler since each additive logratio can be identified with a corresponding compositional component.

Digital soil mapping of soil particle-size fractions (PSFs) using log-ratio methods is a widely used technique. As a hybrid interpolator, regression kriging (RK) provides a way to improve prediction accuracy. However, there have been few comparisons with other techniques when RK is applied for compositional data, and it is not known if its performance based on different balances of isometric log-ratio (ILR) transformation is robust. Here, we compared the generalized linear model (GLM), random forest (RF), and their hybrid patterns (RK) using different transformed data based on three ILR balances, with 29 environmental covariables (ECs) for the prediction of soil PSFs in the upper reaches of the Heihe River Basin (HRB), China. The results showed that the RF performed best, with more accurate predictions, but the GLM produced a more unbiased prediction. As a hybrid interpolator, RK was recommended because it widened the data ranges of the prediction values, and modified the bias and accuracy of most models, especially the RF. The prediction maps generated from RK revealed more details of the soil sampling points than the other models. Different data distributions were produced for the three ILR balances. Using the most abundant component of the compositional data as the first component of the permutations was not considered to be the right choice because it produced the worst performance. Based on the relative abundance of the components, we recommend that the focus should be on data distribution. This study provides a reference for the mapping of soil PSFs combined with transformed data at the regional scale.

National audience; In the framework of Compositional Data Analysis, vectors carrying relative information, also called compositional vectors, can appear in regression models either as dependent or as explanatory variables. In some situations, they can be on both sides of the regression equation. Measuring the marginal impacts of covariates in these types of models is not straightforward since a change in one component of a closed composition automatically affects the rest of the composition. Previous work by the authors has shown how to measure, compute and interpret thesemarginal impacts in the case of linear regression models with compositions on both sides of the equation. The resulting natural interpretation is in terms of an elasticity, a quantity commonly used in econometrics and marketing applications. They also demonstrate the link between these elasticities and simplicial derivatives. The aim of this contribution is to extend these results to other situations, namely when the compositional vector is on a single side of the regression equation. In these cases, the marginal impact is related to a semi-elasticity and also linked to some simplicial derivative. Moreover we consider the possibility that a total variable is used as an explanatory variable, with several possible interpretations of this total and we derive the elasticity formulas in that case.

Logratio principal component analysis is often one of the first steps in exploring a compositional data set. Compositional biplots based on the first two principal components are frequently used to uncover proportionality between parts or to detect one-dimensional patterns of variability for larger subcompositions. This article argues that this approach is likely to produce false leads and proposes an alternative procedure based on condition indices and low-variance principal components. We advocate the calculation of condition indices, combined with biplots of the last few principal components and lists of subcompositions with large condition numbers, and these are shown to be useful for detecting proportionality and one-dimensional relationships. The detection of such patterns in compositional data sets is shown to be closely related to the analysis of multicollinearity as employed in linear regression. Two example data sets, amino acid compositions in calves and chemical components of coffee aroma, are used as illustrations.

Compositional data represent a specific family of multivariate data, where the information of interest is contained in the ratios between parts rather than in absolute values of single parts. The analysis of such specific data is challenging as the application of standard multivariate analysis tools on the raw observations can lead to spurious results. Hence, it is appropriate to apply certain transformations prior to further analysis. One popular multivariate data analysis tool is independent component analysis. Independent component analysis aims to find statistically independent components in the data and as such might be seen as an extension to principal component analysis. In this paper, we examine an approach of how to apply independent component analysis on compositional data by respecting the nature of the latter and demonstrate the usefulness of this procedure on a metabolomics dataset.