Greenacre 2023 - Logratios ALR CLR ILR and Log-Contrasts

Summary

Pairwise logratios (LR), additive logratios (ALR, fixed reference part) and centred logratios (CLR, divide by geometric mean) all have dimension $d=D-1$ and are linear in $\log x$. CLR is exactly isometric to the full logratio geometry but not coherent; ALR is coherent but only approximately isometric; ILR is both but needs ratios of geometric means that are hard to interpret.

Overview

Zeros prevent logarithms, so every logratio transform needs zero replacement or a different tool (see Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence).

Main Content

Pairwise logratio (Eq. 2)

$\mathrm{LR}(j,k)=\log (x_j/x_k)=\log x_j-\log x_k$, $j\ne k$. There are $\frac{1}{2}D(D-1)$ unique LRs (Tellus: $\frac{1}{2}\cdot 52\cdot 51=1326$), linearly interrelated with rank $d=D-1$. LRs map $S^d$ to ${\mathbb{R}}^d$.

Additive logratio, ALR (Eq. 3)

$\mathrm{ALR}(j\mid \text{ref})=\log (x_j/x_{\text{ref}})$, $j\ne \text{ref}$: $d$ logratios with a common denominator (the reference). There are $D$ possible ALRs. If $\log x_{\text{ref}}$ has very low variance, ALRs are practically $\log x_j$ up to a near-constant. Wide data sets often contain such a near-constant reference.

Centred logratio, CLR (Eq. 4)

$\mathrm{CLR}(j)=\log \frac{x_j}{g(\mathbf{x})}=\log x_j-\frac{1}{D}{\sum }_k\log x_k$ with geometric mean $g(\mathbf{x})=(x_1\cdots x_D{)}^{1/D}$. $\mathrm{CLR}(j)=\frac{1}{D}{\sum }_k\mathrm{LR}(j,k)$. The $D$ CLRs sum to zero so have rank $d$; they live in $T^d=\{\mathbf{y}\in {\mathbb{R}}^D:\sum y_j=0\}$ (Eq. 5). Euclidean distances on CLRs equal those on all $\frac{1}{2}D(D-1)$ LRs, so the CLR transformation is isometric.

Log-contrast

${\sum }_j{a}_j\log x_j$ with ${\sum }_j{a}_j=0$ (ensures scale invariance). LR and ALR have coefficients $+1,-1$ on two parts; a CLR has $1-\frac{1}{D}$ on part $j$ and $-\frac{1}{D}$ on the others. If compositions are predictors or responses, all logratio transformations generate the same log-contrasts, so the choice among them is driven by interpretability, with LR/ALR being the most attractive.

Isometry and ILR

Logratio geometry: the set of distances between cases based on all LRs. A set of transformed variables is isometric if it reproduces this geometry exactly up to scale. ILRs use $d$ orthogonal log-contrasts. Example with parts A,B,C,D,E: $h_1=\frac{1}{2}(\log A+\log B)-\frac{1}{3}(\log C+\log D+\log E)=\log \frac{(AB{)}^{1/2}}{(CDE{)}^{1/3}}$ (coefficients $[\frac{1}{2},\frac{1}{2},-\frac{1}{3},-\frac{1}{3},-\frac{1}{3}]$, sum 0); $h_2=\frac{1}{\sqrt{2}}$-type contrast of A vs B, etc. (Eq. 6 matrix: rows $[\frac{1}{2}\frac{1}{2}-\frac{1}{3}-\frac{1}{3}-\frac{1}{3}]$, $[1-1000]$, $[001-\frac{1}{2}-\frac{1}{2}]$, $[0001-1]$ up to normalisation) giving a dendrogram of balances. $h_1$ equals the average of the six LRs between {A,B} and {C,D,E}.

Dispersion in compositions (Aitchison)

The variation matrix $\mathbf{T}=[{\tau }_{jk}]$, ${\tau }_{jk}=\mathrm{Var}\log (x_j/x_k)$, and covariances of LRs follow from it: $\mathrm{Cov}(\mathrm{LR}(j,k),\mathrm{LR}(u,v))=\frac{1}{2}({\tau }_{jk}+{\tau }_{uv}-{\tau }_{ju}-{\tau }_{kv})$. Total logratio variability $=\frac{1}{D}{\sum }_{j<k}{\tau }_{jk}={\sum }_j\mathrm{Var}(\mathrm{CLR}(j))$.

Greenacre’s version, $\text{TotVar}$, divides by $D$ again (weights $1/D^2$); weights on parts can be unequal. Distances are defined as averages rather than sums so they are comparable across numbers of parts (Sec. 3.9).

Transform	# variables	Coherent?	Isometric?	Needs
LR (all pairs)	$\frac{1}{2}D(D-1)$	yes	yes (by definition)	selection of subset
ALR (reference)	$D-1$	yes	approx. (quasi)	choice of reference
CLR	$D$ (rank $D-1$)	no (uses all parts)	yes	singular covariance
ILR / pivot	$D-1$	no	yes	orthonormal basis; ratios of geometric means
SLR (amalgamation)	any	with respect to additions	not a log-contrast	subject-matter amalgams

Examples

Python: ALR, CLR and ILR in numpy

import numpy as np
x = np.array([[0.5, 0.3, 0.2], [0.2, 0.3, 0.5]])          # compositions (rows sum to 1)
alr = np.log(x[:, :-1] / x[:, [-1]])                        # reference = last part
clr = np.log(x) - np.log(x).mean(axis=1, keepdims=True)
# one ILR basis: balance {1,2 | 3} and {1 | 2}
V = np.array([[1/np.sqrt(6), 1/np.sqrt(6), -2/np.sqrt(6)],
              [1/np.sqrt(2), -1/np.sqrt(2), 0]])
ilr = clr @ V.T
# CLR distances equal the "all-LR" distances (isometry)
d_clr = np.linalg.norm(clr[0] - clr[1])

Connections

Logratio response models: Greenacre 2023 - Compositional Outcome Models in PyMC and Dirichlet alternatives Tsagris 2018 - Dirichlet Regression with Zeros - Overview.

See Also

Greenacre 2023 - Aitchison CoDA 40 Years On - Overview