Distances to compositional equilibrium
Introduction
Many systems described by compositions, here named compositional systems, are prone to equilibrium in many forms. In geosciences, stoichiometry is a form of crystal equilibrium (Langmuir and Hanson, 1981; Anderson, 2005), as chemical or electrical equilibria govern many processes (Bethke, 2007). In Biology, genetic laws like the Hardy-Weinberg equilibrium or stable ecological relationships can be considered as equilibrium situations (Edwards, 2008; Goodman, 1975; Graffelman and Egozcue, 2011; Lhomme and Winkel, 2002); also in Physics, invariant magnitudes reflect equilibrium (Jeyasingh and Weider, 2007; Leal et al., 2014). Economy and demography are no exception (Malinvaud, 2012; Tschirhart, 2000).
When studying equilibrium in compositional systems, two kinds of problems frequently arise: (a) given a well known state of equilibrium, to decide whether a compositional sample is far from that equilibrium; in other words, to decide whether the equilibrium can be accepted or has to be rejected based on the compositional observations; (b) to detect any instance of equilibrium that can be hidden in a high dimensional set of observations. Good examples of (a) are checking the thermo-dynamical and chemical equilibrium in a known reaction in natural waters; or testing for Hardy-Weinberg genetic equilibrium in a human or animal community (Graffelman and Egozcue, 2011; Graffelman, 2010; Graffelman and Weir, 2016, and references therein). Situation (b) mainly arises in complex systems like, for instance, the water analysis of a river or groundwater. Here, several simultaneous equilibrium relations may be present, and identifying the dominant process characterized by some invariant quantity may be the main goal.
In a dynamic compositional system, equilibrium can be identified as the existence of an invariant feature along the evolution of the system, as the fact that the composition representing the system is restricted to evolve within the limits of some conditions. Here, equilibrium is identified with some restriction in the sample space of the compositional observations. In order to formalize such a situation, the simplex—endowed with the Aitchison geometry (Pawlowsky-Glahn and Egozcue, 2001; Billheimer et al., 2001)—is taken as an appropriate framework. In this setting, compositions are supposed to be arrays of positive numbers carrying relative information, that is, the relevant information is assumed to be contained in the ratios between components (called parts for short). Two of these arrays are considered compositionally equivalent when their components are proportional. The equivalence classes generated in this way are compositions that can be represented in the simplex of D parts or, for short, the D-part simplex, which is the intersection of an hyperplane with the positive orthant of D-dimensional real space characterized by the constant sum of the components or parts of each point (Barceló-Vidal and Martín-Fernández, 2016; Pawlowsky-Glahn and Buccianti, 2011; Pawlowsky-Glahn et al., 2015; Egozcue and Pawlowsky-Glahn, 2018, Egozcue and Pawlowsky-Glahn, 2019).
The Aitchison geometry of the simplex allows representing compositions in coordinates, specifically Cartesian orthogonal coordinates (Egozcue et al., 2003, Egozcue et al., 2011) with respect to an orthonormal basis, which are available in any Euclidean space. Cartesian coordinates in an orthonormal coordinate system are obtained using some isometric logratio (ilr) representation, recently renamed orthonormal logratio coordinates (olr) (Egozcue and Pawlowsky-Glahn, 2019, see discussion and rejoinder). The representation in olr-coordinates allows working with compositions as if they were real vectors (Mateu-Figueras et al., 2011). This context is used in the following sections to characterize equilibrium loci or restrictions of the sample space. Section 2 introduces restrictions for compositions as a model of equilibrium loci and the Aitchison distance from a data point to it. 3 A case of linear chemical equilibrium, 4 Non-linear equilibrium present some examples of analysis of equilibrium using compositional data samples. Section 5 is a discussion on methods for discovering unknown equilibria in a sample data set.
Section snippets
Aitchison distance to a compositional restriction
Compositional restrictions are modelled as a function of a composition being constant across a given sample. In the simplest case, that function is linear in the Aitchison geometry of the simplex and is addressed in Section 2.1. This case is not the most frequent in geochemistry but the methodological discussion inspires the way of treating the non-linear case. Section 2.2 discusses the case in which the equilibrium locus is not a constant logratio but a constant non-linear function.
A case of linear chemical equilibrium
The chemical reaction involving carbon speciesis responsible for the appearance of dissolved ion bicarbonate in surface waters. Studying the equilibrium of this reaction is of interest also in the context of climate change due to the increase of CO2 concentration in the atmosphere (Crowley and Berner, 2001). A data set reporting pH and molar concentration (mol/L) in 798 stream waters in Europe has been extracted from the FOREGS1
A general chemical equilibrium
The alteration of microcline provides an example of chemical equilibrium whose equilibrium locus is non-linear. This chemical reaction is important since it is the main, normally active, way of alteration of silicates. This chemical reaction can be written as
The mass action law for this reaction is formulated as a function of molar concentrations of ions H+ and K+ and the silicic acid. Denoting molar concentrations with
Discovering stoichiometric equilibrium
In previous sections, deviations and distances to an equilibrium were computed. However, the characteristics of the equilibrium were known. In this section, we deal with the problem to discover an equilibrium that is partially unknown. The example of olivine in Section 4.2 is here used to illustrate some techniques for discovering stoichiometric equilibrium. When a logcontrast is constant or nearly so, it indicates a linear equilibrium or, equivalently, a one dimensional restriction of the
Conclusions
The samples from a D-part compositional system span a (D − 1)-dimensional sample space. However, there are cases in which samples are restricted to a lower dimensional locus. This means that samples are (approximately) constrained due to the existence of a (approximately) constant quantity across samples. This is so in geochemical samples which are in chemical equilibrium, or linked by stoichiometric relations in crystals. When the samples satisfy the restriction we say they are in equilibrium.
CRediT authorship contribution statement
Pawlwosky-Glahn, Egozcue and Buccianti: Conceptualization, Methodology, Software, Data curation, Writing-Original draft preparation, Writing-Reviewing and editing.
Declaration of competing interest
The authors declare the absence of conflicts of interest in the proposed research and concerning the utilized funds.
Acknowledgements
The authors appreciate the thoughtful comments and suggestions by three anonymous reviewers. Funding was provided by Ministerio de Economía y Competitividad (MINECO/FEDER (Spain), MTM2015-65016-C2-1-R and MTM2015-65016-C2-2-R) for JJE and VPG and by University of Florence (Italy), funds 2020, for AB.
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