Greenacre 2023 - Simplex, Closure and Core CoDA Properties
Summary
A composition is a non-negative vector divided by its total; it lives on the simplex with and only relative information is meaningful. Aitchison’s approach is built on scale invariance and subcompositional coherence; two further properties (dominance, distributional equivalence) are discussed and the paper argues dominance is optional.
Overview
Aitchison (1982, read paper; 1986 book) addressed problems of analysing data subject to the unit-sum constraint. Early applications had few parts (10, 4, 3, 7); modern ones (ecology, microbiome, omics) have hundreds or thousands.
Main Content
Simplex and composition
With parts, the simplex is (Eq. 1). Its dimension is because any part is determined by the rest. For (wide matrices) the dimension of the sample is . Closure turns a positive vector into a composition; the original total is its size and the closed vector its shape. Zeros are admitted in the definition (most real data sets contain them, some are structural).
Scale invariance
Results must be identical for and for any diagonal matrix of positive scalars: for all rows. Only the shape matters (except where the size is substantively meaningful, e.g. allometry of fish).
Subcompositional coherence
A subcomposition of parts, re-normalised by its own subtotal, lives in . Coherence requires that relationships among a set of parts be the same whether they are computed within the full composition or any subcomposition. Part ratios satisfy it; raw proportions do not. Incoherence can be measured (e.g. via stress or Procrustes correlation across random subcompositions); “quasi-coherence” is closeness to exact coherence.
The correlation paradox (Tellus, p. 4)
Geochemist A analyses all 52 elements and finds correlation of La and Y equal to 0.479; geochemist C analyses only the 9 rare-earth elements (a subcomposition, renormalised) and finds -0.344. With ratios (log-ratios) all three geochemists have identical data and no paradox.
Subcompositional dominance
Distances between cases computed on a subcomposition should be no larger than on the full composition. This holds if distances are sums of positive logratio terms, but fails for averages. Averaged distances are comparable across studies with different numbers of parts, so the paper concludes dominance need not hold: it depends on the research question.
Distributional equivalence (proportionality)
Two parts are distributionally equivalent if one is a scalar multiple of the other; the analysis should be unchanged if they are amalgamated. Measured by the variance of their logratio . Tellus: Sm/Yb has variance 0.0062 (nearly proportional; Sm Yb; uncentered correlation 0.997) while Br/Nd has variance 5.06. Logratio methods violate this property (splitting a column into 10 equal parts multiplies its variance by 10); weighting parts restores it (link to correspondence analysis).
Definitions of amalgamation and partition. Amalgamations merge parts by summation (e.g. fatty acids into saturated / monounsaturated / polyunsaturated; Tellus mafic = Mg+Fe+Mn, felsic = Na+Si+Al+K); a partition amalgamates all parts into subsets. Aitchison emphasised the hierarchy of parts.
Examples
Tellus: 3,883 zeros (1.1% of values) from 12 of 52 cation elements below detection; no structural zeros assumed; zeros replaced by 2/3 of the element’s minimum positive value (triangular-distribution argument); alternative imputations did not change conclusions.
Connections
- Next: transformations in Greenacre 2023 - Logratios ALR CLR ILR and Log-Contrasts; quasi-coherence in Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence.
- Spurious correlations from closure are the compositional analogue of selecting on a collider; marketing share-of-spend analyses face the same issue.