Chapter 2

The Euclidian space structure of the simplex

Contents 2.1 Logratio analysis 2.1.1 The additive logratio transformation 2.1.2 The centred logratio transformation 2.1.3 Philosophy of the logratio analysis 2.2 The algebraic-geometric structure of the simplex 2.2.1 Perturbation 2.2.2 Powering 2.2.3 The simplex (SD, ⊕, ⊙) a real vector space 2.2.4 The compositional distance on the simplex 2.2.5 The simplex SD a Euclidean space 2.3 Compositional-linear dependence, basis and coordinates 2.3.1 Compositional linear dependence and independence 2.3.2 C-basis and C-coordinates 2.4 Scale invariant logratios or logcontrasts 2.5 Representation of compositions by orthogonal coordinates. Isometric logratio transformations on the simplex 2.5.1 C-orthonormal basis in the simplex 2.5.2 Coordinates expressed in C-orthonormal basis 2.5.3 Isometric logratio transformations 2.6 C-orthonormal basis associated to a sequential binary partition 2.6.1 Sequential binary partition 2.6.2 Balances

Objectives X To learn how to structure the simplex SD in a Euclidean space of dimension D − 1.

33 34 2. The Euclidian space structure of the simplex

X To introduce the concept of logcontrast on the simplex SD with special em- phasis on the additive and the centred logratio transformations. X To show the procedure for calculating the coordinates of a composition with respect to an orthonormal basis of SD introducing isometric logratio transfor- mations. X To show a procedure for selecting a suitable orthonormal basis that allows the coordinates of a composition to be easily interpreted.

2.1. Logratio analysis †

What has come to be known as logratio analysis for compositional data problems arose in the 1980’s out of the realization of the importance of the principle of scale invariance (see Section 4.1 of Chapter 1) and that its practical implementation required working with ratios of components. This, together with an awareness that logarithms of ratios are mathematically more tractable than ratios, led to the advocacy of a transformation technique involving logratios of the components. In this section we will introduce the two main transformations on the simplex on which this analysis is based.

D 2.1.1. The additive logratio transformation. Let x = [x1, . . . , xD] ∈ S be a typical D-part composition. Then the so-called additive logratio transformation alr : SD → IRD−1 is defined by

(2.1) y = alr x = [log(x1/xD), log(x2/xD),..., log(xD−1/xD)], where the ratios involve the division of each of the first D − 1 components by the final component. The alr transformation is one-to-one. The inverse transformation alr−1 : IRD−1 → SD is −1 x = alr y = C[exp y1,..., exp yD−1, 1], where C denotes the closure operation. Note that the alr transformation takes the composition into the whole of the IRD−1 space and so we have the prospect of using standard unconstrained multi- variate analysis on the transformed data, and because of the one-to-one nature of this transformation, of transferring any inferences back to the simplex and to the components of the composition. One apparent drawback to this technique is the choice of the final component as the divisor, with the frequently asked question: Would we obtain the same inference if we chose another component as divisor, or more generally if we permuted the parts? The answer to this question is yes. We will not go into any details here that prove this assertion, but the interested reader may find these in [Ait86, Chapter 5].

† This section is an adaptation of [Ait03, Sections 2.1-2.2, p. 29-32].