F
F(d)

User interface for FMatrix, as defined by Aitchison.

F is a d by D matrix that can be defined as

F[i, j] = 1, if i==j

F[i, j] = -1, if j==D

F[i, j] = 0, otherwise

Examples

julia> F(3)
julia> F*v
julia> v'*F

G
G(D)

User interface for GMatrix, as defined by Aitchison.

G is an D by D matrix that can be defined as

G[i, j] = I[i, j] - J[i, j] / D

Examples

julia> G(3)
julia> G*v
julia> v'*G

H
H(d)

User interface for HMatrix, as defined by Aitchison.

H is a d by d matrix that can be defined as

H[i, j] = I[i, j] + J[i, j]

Examples

julia> H(3)
julia> H*v
julia> v'*H

J
J(d)

User interface for JMatrix, a d by d matrix of ones.

Examples

julia> J(3)
julia> J*v
julia> v'*J

ALR([refvar])

Additive log-ratio transform following the TableTransforms.jl interface.

Optionally, specify the reference variable refvar for the ratios. Default to the last column of the input table.

Aitchison

Aitchison distance between compositions.

CLR()

Centered log-ratio transform following the TableTransforms.jl interface.

Closure()

The transform that applies the closure operation 𝒞 to all rows of the input table. The rows of the output table sum to one.

CoDaArray(table)

Construct an array of compositional data from table.

Composition(partscomps)
Composition(parts, comps)
Composition(part₁=comp₁, part₂=part₂, ...)
Composition(comps)
Composition(comp₁, comp₂, ...)

A D-part composition as defined by Aitchison 1986.

Examples

A 2-part composition with parts a=0.1 and b=0.8:

julia> Composition(a=0.2, b=0.8)
julia> Composition((a=0.2, b=0.8))
julia> Composition((:a, :b), (0.2, 0.8))

When the names of the parts are not specified, the constructor uses default names part1, part2, ..., partD:

julia> Composition(0.1, 0.8)
julia> Composition((0.1, 0.8))

FMatrix{T}

F matrix, as defined by Aitchison 1986. See also F.

F′Matrix{T}

Lazy adjoint of FMatrix{T}.

GMatrix{T}

G matrix, as defined by Aitchison 1986. See also G.

HMatrix{T}

H matrix, as defined by Aitchison 1986. See also H.

H⁻¹Matrix{T}

Lazy inverse of HMatrix{T}.

ILR([refvar])

Isometric log-ratio transform following the TableTransforms.jl interface.

Optionally, specify the reference variable refvar for the ratios. Default to the last column of the input table.

JMatrix{T}

Square matrix of ones. See also J.

LogRatio

A log-ratio transform following the TableTransforms.jl interface.

See also ALR, CLR, ILR.

Remainder([total])

The transform that takes a table with columns x₁, x₂, …, xₙ and returns a new table with an additional column containing the remainder value xₙ₊₁ = total .- (x₁ + x₂ + ⋯ + xₙ) If the total value is not specified, then default to the maximum sum across rows.

rand(Composition{D}, n=1)

Generates D-part composition at random according to a balanced Dirichlet distribution.

alr(c)

Additive log-ratio transformation of composition c.

alrcov(table)

Return the log-ratio covariance matrix Σ of the table such that:

• Σ[i,j] = cov(log(x[i]/x[D]), log(x[j]/x[D])) for i, j = 1, ..., d

alrinv(x)

Inverse alr transformation of coordinates x.

clr(c)

Centered log-ratio transformation of composition c.

clrcov(table)

Return the centered log-ratio covariance matrix Γ of the table such that:

• Γ[i,j] = cov(log(x[i]/g(x)), log(x[j]/g(x))) for i, j = 1, ..., D,

where g(x) is the geometric mean.

clrinv(x)

Inverse clr transformation of coordinates x.

components(c)

Components in the composition c.

compose(table, cols; keepcols=true, as=:coda)

Convert columns cols of table into parts of a composition and save the result in a CoDaArray. If keepcols is set to true, then save the result as a column in a new table with all other columns preserved.

ilr(c)

Isometric log-ratio transformation of composition c.

ilrinv(x)

Inverse ilr transformation of coordinates x.

lrarray(table)

Return the variation array A of the table such that:

• A[i,j] = E[log(x[i]/x[j])] for i > j
• A[i,j] = Var(log(x[i]/x[j])) for i < j
• A[i,j] = 0 for i = j

parts(array)

Parts in compositional array.

parts(c)

Parts in the composition c.

variation(table)

Return the variation matrix Τ of the table such that:

• Τ[i,j] = Var(log(x[i]/x[j])) for i, j = 1, ..., D

𝓒(x)

Return closure of x.