Two more things about compositional biplots: quality of projection and inclusion of supplementary elements

The biplot is a widely and powerful methodology used with multidimensional data sets to describe and display the relationships between observations and variables in an easy way. Compositional data consist of positive vectors each of which is constrained to have a constant sum; due to this property standard biplots can not be performed with compositional data, instead of a previous transformation of the data is performed. Due to this constant sum constraint, a transformation of data is needed before performing a biplot and, consequently, special interpretation rules are required. However, these rules can only be safely applied when the elements of a biplot have a good quality of projection, for which a new measure is introduced in this paper. Also, we extend the compositional biplot defined by Aitchison and Greenacre on 2002, in order to include the display supplementary elements that are not used in the definition of the compositional biplot. Different types of supplementary elements are considered: supplementary parts of the composition, supplementary continuous variables external to the composition, supplementary categorical variables and supplementary observations. The projection of supplementary parts of the composition is done by means of the equivalence of clr and lr biplots. The other supplementary projections are done by classical methodology. Both the qualities of projections and the supplementary projections are explained using real geological data: a sample of 72 observations of soil in an area about 20 km west of Kiev in the area south of Kiev Polessie

water chemistry are new challenges possible from coda compositional data analysis point of view

John Aitchison died in December 2016 leaving behind an important inheritance: to continue to explore the fascinating world of compositional data. However, notwithstanding the progress that we have made in this field of investigation and the diffusion of the CoDA theory in different researches, a lot of work has still to be done, particularly in geochemistry. In fact most of the papers published in international journals that manage compositional data ignore their nature and their consequent peculiar statistical properties. On the other hand, when CoDA principles are applied, several efforts are often made to continue to consider the log-ratio transformed variables, for example the centered log-ratio ones, as the original ones, demonstrating a sort of resistance to thinking in relative terms. This appears to be a very strange behavior since geochemists are used to ratios and their analysis is the base of the experimental calibration when standards are evolved to set the instruments. In this chapter some challenges are presented by exploring water chemistry data with the aim to invite people to capture the essence of thinking in a relative and multivariate way since this is the path to obtain a description of natural processes as complete as possible

New Features of CoDaPack. An Userfriendly Compositional Data Package

Abstract The statistical analysis of compositional data is commonly used in geological studies. As is well-known, compositions should be treated using logratios of parts, which are difficult to use correctly in standard statistical packages. In this paper we describe the new features of our freeware package, named CoDaPack, which implements most of the basic statistical methods suitable for compositional data. An example using real data is presented to illustrate the use of the package