README.md

July 30, 2026 · View on GitHub


This package defines a Composition{D} type representing a D-part composition as defined by Aitchison 1986. In Aitchison's geometry, the D-simplex together with addition (a.k.a. pertubation) and scalar multiplication (a.k.a. scaling) form a vector space, and important properties hold:

  • Scaling invariance
  • Pertubation invariance
  • Permutation invariance
  • Subcompositional coherence

In practice, this means that one can operate on compositional data (i.e. vectors whose entries represent parts of a total) without destroying the ratios of the parts.

Installation

Get the latest stable release with Julia's package manager:

] add CoDa

Usage

Basics

Compositions are static vectors with named parts:

julia> using CoDa

julia> c = Composition(CO₂=2.0, CH₄=0.1, N₂O=0.3)
    ┌                                         ┐
CO₂ ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 2.000000
CH₄ ┤■■                               0.100000
N₂O ┤■■■■■                            0.300000
    └                                         ┘

julia> CoDa.parts(c)
(:CO₂, :CH₄, :N₂O)

julia> CoDa.components(c)
3-element StaticArrays.SVector{3, Union{Missing, Float64}} with indices SOneTo(3):
 2.0
 0.1
 0.3

julia> c.CO₂
2.0

Default names are added otherwise:

julia> c = Composition(1.0, 0.1, 0.1)
   ┌                                         ┐
w1 ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 1.000000
w2 ┤■■■                              0.100000
w3 ┤■■■                              0.100000
   └                                         ┘

and serve for internal compile-time checks.

Compositions can be added, subtracted, negated, and multiplied by scalars. Other operations are also defined including dot product, induced norm, and distance:

julia> c = Composition(CO₂=2.0, CH₄=0.1, N₂O=0.3)
    ┌                                         ┐
CO₂ ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 2.000000
CH₄ ┤■■                               0.100000
N₂O ┤■■■■■                            0.300000
    └                                         ┘

julia> cₒ = Composition(CO₂=1.0, CH₄=0.1, N₂O=0.1)
    ┌                                         ┐
CO₂ ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 1.000000
CH₄ ┤■■■                              0.100000
N₂O ┤■■■                              0.100000
    └                                         ┘

julia> -cₒ
    ┌                                         ┐
CO₂ ┤■■■                              0.047619
CH₄ ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.476190
N₂O ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.476190
    └                                         ┘

julia> 0.5c
    ┌                                         ┐
CO₂ ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.620769
CH₄ ┤■■■■■■■                          0.138808
N₂O ┤■■■■■■■■■■■■                     0.240423
    └                                         ┘

julia> c - cₒ
    ┌                                         ┐
CO₂ ┤■■■■■■■■■■■■■■■■■■■■■            0.333333
CH₄ ┤■■■■■■■■■■■                      0.166667
N₂O ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.500000
    └                                         ┘

julia> c ⋅ cₒ
3.7554028908352994

julia> norm(c)
2.1432393747688687

julia> aitchison(c, cₒ) # Aitchison distance
0.7856640352007868

More complex functions can be defined in terms of these operations. For example, the function below defines the composition line passing through cₒ in the direction of c:

julia> f(λ) = cₒ + λ*c
f (generic function with 1 method)

Finally, two compositions are considered to be equal when their closure is approximately equal:

julia> c == c
true

julia> c == cₒ
false

Log-ratio transformations

Currently, the following log-ratio transformations are implemented:

julia> alr(c)
2-element StaticArraysCore.SVector{2, Float64} with indices SOneTo(2):
  1.8971199848858806
 -1.098612288668108

julia> clr(c)
3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  1.6309507528132896
 -1.3647815207406992
 -0.26616923207259097

julia> ilr(c)
2-element StaticArraysCore.SVector{2, Float64} with indices SOneTo(2):
 -2.1183026052494185
 -0.3259894019031434

and their inverses alrinv, clrinv and ilrinv.

The transforms for tables are defined in the TableTransforms.jl package, they are: Compose, Closure, Remainder, ALR, CLR, ILR. These transforms are functors that can be used as follows:

julia> table |> ILR()

Arrays

It is often useful to compose D columns of a table into D-part compositions. The package provides a CoDaArray type that implements the Julia array interface and the Tables.jl interface. We recommend using the function compose(table, cols) to construct such arrays:

julia> table = (a=[1,2,3], b=[4,5,6], c=[7,8,9])
(a = [1, 2, 3], b = [4, 5, 6], c = [7, 8, 9])

julia> ctable = compose(table, (:a,:b))
(c = [7, 8, 9], coda = Composition{2, (:a, :b)}[1.000 : 4.000, 2.000 : 5.000, 3.000 : 6.000])

julia> ctable.coda[1]
  ┌                                         ┐
a ┤■■■■■■■■                         1.000000
b ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 4.000000
  └                                         ┘

Random

D-part compositions can be created at random from a Dirichlet distribution:

julia> rand(Composition{3})
   ┌                                         ┐
w1 ┤■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 0.508093
w2 ┤■■■■■■■■■■■■                     0.183344
w3 ┤■■■■■■■■■■■■■■■■■■■              0.308563
   └                                         ┘

Plots

Separate packages are available for plotting compositional data:

References

This package is heavily influenced by Aitchison's monograph:

  • Aitchison, J. 1986. The Statistical Analysis of Compositional Data

and by other textbooks:

  • den Boogaart, K. & Tolosana-Delgado. 2011. Analyzing Compositional Data with R
  • Pawlowsky-Glahn et al. 2015. Modeling and Analysis of Compositional Data
  • Pawlowsky-Glahn, V. & Buccianti, A. 2011. Compositional Data Analysis - Theory and Applications