Greenacre 2023 - Tellus Application, Variable Selection and Correspondence Analysis
Summary
On the Tellus soil geochemistry data (6,799 samples, 52 elements) the simplest logratio choice — an ALR with a stable reference part (Al, Procrustes correlation 0.990) — reproduces the full logratio geometry; correspondence analysis of Box-Cox power-transformed data gives a zero-tolerant alternative with optimal power 0.5. Variable selection (a few pairwise logratios) is as important as the transformation.
Overview
Data from the Tellus survey of Northern Ireland (second period 2004-2006), XRF analysis: 52 elements (11 major, 18 minor, 14 trace, 9 rare-earth), N=6,799 samples, converted to cation counts and closed; 10 geological age brackets. .
Main Content
4.1 To weight or not to weight
Total logratio variance is the average of the 52 CLR variances. Weights matter when rare parts have excessively high variance; diagnose by plotting CLR variances against mean proportions (Fig. 4). On Tellus there is no tendency for low-mean parts to show excessive variance, so unweighted analysis is acceptable (contrast an archaeometric example in the paper’s ref. 42).
4.2 Selecting an ALR (FINDALR)
Procrustes correlation for ALR choice
For each of the 52 candidate reference parts, compute the ALR configuration of the 6,799 samples and its Procrustes correlation with the full logratio (CLR) geometry (R
FINDALRineasyCODA). Plot against (Fig. 5).
Result: the highest Procrustes correlation is 0.990 with reference Al; the lowest reference-log variance is Si (Al second lowest). Both are geochemically sensible references. ALR distances approximate the logratio distances from below (Fig. 6, a random 10,000 of 23,109,801 pairwise distances).
4.3 Correspondence-analysis alternative
CA double-centres relative abundances; logratio analysis (LRA = PCA of CLRs) double-centres log relative abundances. They are linked by the Box-Cox power transform : as it tends to , and CA of power-transformed data approaches LRA (paper’s refs. 35, 36). CA uses the chi-square distance (Euclidean on parts divided by , the average of part ). On Tellus the optimal power for CA is 0.5 (square-root), requiring no zero replacement; k-means with 3 clusters was run on the ALRs and on the square-root-transformed (chi-square) data and the two solutions were cross-tabulated to compare them.
4.4 Quasi-coherence
Random subcompositions of size 44 down to 4 (out of 52) show that Procrustes correlations of the chi-square geometry of parts with the full composition are plotted for 100 random subcompositions per size (axis from 0.75 to 1.00, Fig. 11); the authors read the high values as “quasi-coherence” (dilution of incoherence as the number of parts grows).
5 Variable selection
Because the logratio space has dimension , a subset of linearly independent LRs can retain all information, but many equivalent subsets exist; data-driven stepwise selection finds small sets of pairwise logratios that explain most logratio variance. Table (selected, increasing coverage of total logratio variance):
| Step | LR | Parts | % variance explained | Procrustes corr. |
|---|---|---|---|---|
| 1 | Cl/Ga | 19, 24 | 44.7 | 0.669 |
| 2 | Ni/Rb | 33, 35 | 66.0 | 0.807 |
| 3 | Si/I | 1, 27 | 70.5 | 0.829 |
| 4 | Nd/Tl | 32, 45 | 74.2 | 0.852 |
| 5 | Mn/S | 5, 11 | 77.0 | 0.869 |
| 10 | Cu/Ta | 23, 42 | 85.9 | 0.917 |
| 15 | Cu/Yb | 23, 50 | 91.3 | 0.941 |
| 22 | Cr/Zn | 21, 51 | 95.2 | 0.957 |
(The table is a transcription of the paper’s selected-LR listing; numbers are as printed.)
Examples
Reading the ALR choice
Suppose you must choose a reference channel for ALR response modelling of spend shares. Compute, for each candidate reference, the Procrustes correlation between ALR geometry and CLR geometry and the variance of the log reference; prefer references with high correlation and low variance. Tellus shows the winner can have rank-2 variance and still give 0.990 agreement.
Connections
Greenacre 2023 - Logratios ALR CLR ILR and Log-Contrasts; PCA/PCoA of compositions is dimension reduction in the same spirit as the manifold PCA reviewed in Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics.