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Particle or grain size distributions often play an important role in understanding processes in the geosciences. Functional data analysis allows applying multivariate methods like principal components and discriminant analysis directly to such distributions. These are however often observed in the form of samples, and thus with a sampling error, i.e. each data point is a distribution, but one where the sampling error is present. This additional sampling error changes the properties of the multivariate variance and thus the value, number and direction of the principle components. The result of the principal component analysis becomes an artefact of the sampling error and can negatively affect the following data analysis. Our contribution presents how to compute this sampling error and how to confront it in the context of principal component analysis. We demonstrate the effect of the sampling error and the effectiveness of the correction with a simulated dataset. We show how the interpretability and reproducibility of the principal components improve and become independent of the selection of the basis. We also demonstrate how the correction improves interpretability of the results on a grain size distribution dataset from river sediments.

Compositional data carry their relevant information in the relationships (logratios) between the compositional parts. It is shown how this source of information can be used in regression modeling, where the composition could either form the response, or the explanatory part, or even both. An essential step to set up a regression model is the way how the composition(s) enter the model. Here, balance coordinates will be constructed that support an interpretation of the regression coefficients and allow for testing hypotheses of subcompositional independence. Both classical least-squares regression and robust MM regression are treated, and they are compared within different regression models at a real data set from a geochemical mapping project.

The occurrence of environmental clusters of Chronic Kidney Disease of uncertain aetiology (CKDu), where there is no known cause for the onset of kidney dysfunction, is a concern globally. Waterborne exposure pathways in the environment may result in indirect or direct ingestion of trace elements with potential health risks. This research examines the relationship between Standardised Incidence Rates (SIRs) of CKDu and the log-ratio balances of Potentially Toxic Elements (PTEs) in regional stream water. Compositional elemental balances were created and regression was used to identify the balances most associated with log-transformed SIR CKDu. At the regional scale, a statistically significant relationship was found between log (CKDu SIR) and the elemental balance Al/As which effectively delineates different geological domains across Northern Ireland. Following stratification by basalt bedrock (the dominant bedrock geology for SIR CKDu), the balance Al/Fe was identified as significantly associated with log (CKDu SIR). Superficial deposits, dissolved organic carbon and pH may act as controls on the balance of Al and Fe. With a high proportion of private water supplies registered in these areas, this research highlights the importance of considering bedrock geology and superficial deposits in understanding multi-element interactions of waterborne environmental toxins and potential links with environmental clusters of CKDu. We would like to acknowledge and thank our esteemed colleague and friend Vera Pawlowsky-Glahn who has been instrumental in the development of this research, pioneering the importance of Compositional Data Analysis (CoDA) in the study of health and the environment. For several of us, Vera provided our first introduction to CoDA and over many years has aided our shared understanding and increased awareness of the need to address the compositional nature of data such as soil and water geochemistry in our research. We are pleased to have this work on the

The occurrence of environmental clusters of Chronic Kidney Disease of uncertain aetiology (CKDu), where there is no known cause for the onset of kidney dysfunction, is a concern globally. Waterborne exposure pathways in the environment may result in indirect or direct ingestion of trace elements with potential health risks. This research examines the relationship between Standardised Incidence Rates (SIRs) of CKDu and the log-ratio balances of Potentially Toxic Elements (PTEs) in regional stream water. Compositional elemental balances were created and regression was used to identify the balances most associated with log-transformed SIR CKDu. At the regional scale, a statistically significant relationship was found between log (CKDu SIR) and the elemental balance Al/As which effectively delineates different geological domains across Northern Ireland. Following stratification by basalt bedrock (the dominant bedrock geology for SIR CKDu), the balance Al/Fe was identified as significantly associated with log (CKDu SIR). Superficial deposits, dissolved organic carbon and pH may act as controls on the balance of Al and Fe. With a high proportion of private water supplies registered in these areas, this research highlights the importance of considering bedrock geology and superficial deposits in understanding multi-element interactions of waterborne environmental toxins and potential links with environmental clusters of CKDu. We would like to acknowledge and thank our esteemed colleague and friend Vera Pawlowsky-Glahn who has been instrumental in the development of this research, pioneering the importance of Compositional Data Analysis (CoDA) in the study of health and the environment. For several of us, Vera provided our first introduction to CoDA and over many years has aided our shared understanding and increased awareness of the need to address the compositional nature of data such as soil and water geochemistry in our research. We are pleased to have this work on the

Compositional data carry their relevant information in the relationships (logratios) between the compositional parts. It is shown how this source of information can be used in regression modeling, where the composition could either form the response, or the explanatory part, or even both. An essential step to set up a regression model is the way how the composition(s) enter the model. Here, balance coordinates will be constructed that support an interpretation of the regression coefficients and allow for testing hypotheses of subcompositional independence. Both classical least-squares regression and robust MM regression are treated, and they are compared within different regression models at a real data set from a geochemical mapping project.

The human microbiome has emerged as a central research topic in human biology and biomedicine. Current microbiome studies generate high-throughput omics data across different body sites, populations, and life stages. Many of the challenges in microbiome research are similar to other high-throughput studies, the quantitative analyses need to address the heterogeneity of data, specific statistical properties, and the remarkable variation in microbiome composition across individuals and body sites. This has led to a broad spectrum of statistical and machine learning challenges that range from study design, data processing, and standardization to analysis, modeling, cross-study comparison, prediction, data science ecosystems, and reproducible reporting. Nevertheless, although many statistics and machine learning approaches and tools have been developed, new techniques are needed to deal with emerging applications and the vast heterogeneity of microbiome data. We review and discuss emerging applications of statistical and machine learning techniques in human microbiome studies and introduce the COST Action CA18131 “ML4Microbiome” that brings together microbiome researchers and machine learning experts to address current challenges such as standardization of analysis pipelines for reproducibility of data analysis results, benchmarking, improvement, or development of existing and new tools and ontologies.

A large number of families of distributions are available to model multivariate real vectors. On the contrary, for the simplex sample space, we have only a limited number of families arising through the generalization of the Dirichlet family or the logratio normal family. This chapter tries to summarize those models and some generalizations with a special emphasis on the algebraic-geometric structure of the simplex and on the measure which is considered compatible. In particular, the shifted-scaled Dirichlet distribution is studied and the logratio t distribution is rewritten and studied with respect to the Aitchison measure.

The analysis of compositional data using the log-ratio approach is based on ratios between the compositional parts. Zeros in the parts thus cause serious difficulties for the analysis. This is a particular problem in case of structural zeros, which cannot be simply replaced by a non-zero value as it is done, e.g. for values below detection limit or missing values. Instead, zeros to be incorporated into further statistical processing. The focus is on exploratory tools for identifying outliers in compositional data sets with structural zeros. For this purpose, Mahalanobis distances are estimated, computed either directly for subcompositions determined by their zero patterns, or by using imputation to improve the efficiency of the estimates, and then proceed to the subcompositional and subgroup level. For this approach, new theory is formulated that allows to estimate covariances for imputed compositional data and to apply estimations on subgroups using parts of this covariance matrix. Moreover, the zero pattern structure is analyzed using principal component analysis for binary data to achieve a comprehensive view of the overall multivariate data structure. The proposed tools are applied to larger compositional data sets from official statistics, where the need for an appropriate treatment of zeros is obvious.

Geochemists have long been aware of the problems surrounding estimating correlation coefficients for their analytical data sets. Very often they do not make sense given the mineralogy of the sample material and knowledge of mineral stoichiometry. The problem lies in the nature of geochemical analyses: they are relative measures reported in such units as weight %, parts per million (mg/kg) or micrograms/l, the sum of the parts, individual measures, added to a constant. Because of the relative units it does not matter whether all the parts have been determined in the analysis, the problem remains whatever the number of parts determined, even just two. Today a common approach in multivariate analyses such as principal components or factor analysis is to use a centred log-ratio (clr) of the data set prior to carrying out the analysis. It might seem apparent then to calculate also the correlation coefficients on the clr-transformed data. However, this does not lead to consistent results, because clr variables are driven by their zero sum constraint. As a consequence, a negative bias occurs when correlation analysis in clr variables is performed. A solution to the problem of negative bias of correlation analysis in clr variables has been proposed by Kynčlová et al. and involves the computation of symmetric coordinates, an extension of isometric log-ratios. The symmetric coordinates are computed as weighted log-ratios that take the total number of parts into consideration. The procedure, which has been demonstrated with two large sets of geochemical (environmental) soil data by Reimann et al., is explained and the results discussed for a small set of petrochemical data with familiar mineralogy., Geochemists have long been aware of the problems surrounding estimating correlation coefficients for their analytical data sets. Very often they do not make sense given the mineralogy of the sample material and knowledge of mineral stoichiometry. The problem lies in the nature of geochemical analyses: they are relative measures reported in such units as weight %, parts per million (mg/kg) or micrograms/l, the sum of the parts, individual measures, added to a constant. Because of the relative units it does not matter whether all the parts have been determined in the analysis, the problem remains whatever the number of parts determined, even just two. Today a common approach in multivariate analyses such as principal components or factor analysis is to use a centred log-ratio (clr) of the data set prior to carrying out the analysis. It might seem apparent then to calculate also the correlation coefficients on the clr-transformed data. However, this does not lead to consistent results, because clr variables are driven by their zero sum constraint. As a consequence, a negative bias occurs when correlation analysis in clr variables is performed. A solution to the problem of negative bias of correlation analysis in clr variables has been proposed by Kynčlová et al. and involves the computation of symmetric coordinates, an extension of isometric log-ratios. The symmetric coordinates are computed as weighted log-ratios that take the total number of parts into consideration. The procedure, which has been demonstrated with two large sets of geochemical (environmental) soil data by Reimann et al., is explained and the results discussed for a small set of petrochemical data with familiar mineralogy.

The investigation of species associations is a classical problem in ecology, in order to describe and predict environmental characteristics. In this contribution we apply the log-ratio approach to the analysis of species abundance data through the Aitchison geometry (Aitchison, 1986). The logratio transformations produce acceptable projections of the correlations among species in principal component space that will be used to analyze association among species. An application to a real dataset of the propose procedure is illustrated.

A modified version of the Kolmogorov-Smirnov (KS) test is presented as a tool to assess whether a specified, although arbitrary, probability model is unsuitable to describe the underlying distribution of a set of observations. The KS test computes distances between points of the sample cumulative distribution function and the hypothetical one as absolute differences between them, and then considering the supreme distance as test statistics. The modification here proposed consists of computing the mentioned distances as Aitchison distances of the probabilities as two part compositions. In this contribution, we investigate by simulation the asymptotic distribution of the proposed test statistic, checking the appropriateness of the Gumbel distribution. The properties of the asymptotic distribution are studied for samples coming from generic distributions such as uniform, normal, lognormal, gamma, beta and exponential with different values of the parameters. A brief Monte Carlo investigation is made of the type I error and power of the test.

Single component geochemical maps are the most basic representation of spatial elemental distributions and commonly used in environmental and exploration geochemistry. However, the compositional nature of geochemical data imposes several limitations on how the data should be presented. The problems relate to the constant sum problem (closure), and the inherently multivariate relative information conveyed by compositional data. Well known is, for instance, the tendency of all heavy metals to show lower values in soils with significant contributions of diluting elements (e.g., the quartz dilution effect); or the contrary effect, apparent enrichment in many elements due to removal of potassium during weathering. The validity of classical single component maps is thus investigated, and reasonable alternatives that honour the compositional character of geochemical concentrations are presented. The first recommended such method relies on knowledge-driven log-ratios, chosen to highlight certain geochemical relations or to filter known artefacts (e.g. dilution with SiO2 or volatiles). This is similar to the classical normalisation approach to a single element. The second approach uses the (so called) log-contrasts, that employ suitable statistical methods (such as classification techniques, regression analysis, principal component analysis, clustering of variables, etc.) to extract potentially interesting geochemical summaries. The caution from this work is that if a compositional approach is not used, it becomes difficult to guarantee that any identified pattern, trend or anomaly is not an artefact of the constant sum constraint. In summary the authors recommend a chain of enquiry that involves searching for the appropriate statistical method that can answer the required geological or geochemical question whilst maintaining the integrity of the compositional nature of the data. The required log-ratio transformations should be applied followed by the chosen statistical meth

With large-scale database systems, statistical analysis of data, occurring in the form of probability distributions, becomes an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment of these data still represents a challenging task in functional data analysis. Namely, the usual (Formula presented.) metric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are captured by Bayes spaces of measures. The easiest possibility of expressing density functions in an (Formula presented.) space is to use centred logratio transformation, even though this results in functional data with a constant integral constraint that needs to be taken into account in further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics and with a simulation study.