Isometric logistic transform (ILR coordinates to the simplex)

Inverse isometric log-ratio map \(\boldsymbol{\psi}:\boldsymbol{z}\mapsto\boldsymbol{p}\) with a Helmert basis \(\mathbf{V}\), $$ \boldsymbol{p} = \operatorname{softmax}(\mathbf{V}\boldsymbol{z}) = \mathcal{C}\bigl(\exp(\mathbf{V}\boldsymbol{z})\bigr). $$ No cell type is pinned as a reference (unlike additive_logistic()). Evaluated with a log-sum-exp shift so large \(\|\boldsymbol{z}\|\) cannot overflow.

$$ \boldsymbol{z} = \mathbf{V}^{\mathsf{T}}\log\boldsymbol{p} = \mathbf{V}^{\mathsf{T}}\operatorname{clr}(\boldsymbol{p}), $$ which is well-defined because \(\mathbf{V}^{\mathsf{T}}\mathbf{1}=0\). Requires a strictly positive composition (open simplex).

Usage

isometric_logistic(z, V = NULL)

isometric_log_ratio(p, V = NULL)

Arguments

z

Numeric vector \(\boldsymbol{z}\in\mathbb{R}^{J-1}\).

V

Optional ILR basis; see isometric_logistic().

p

Numeric vector on the open simplex.

Value

Numeric vector \(\boldsymbol{p}\) on the unit simplex.

Numeric vector \(\boldsymbol{z}\in\mathbb{R}^{J-1}\).

See also

isometric_log_ratio(), helmert_basis(), additive_logistic()

Examples

z <- c(0.2, -0.5)
p <- isometric_logistic(z)
sum(p)
#> [1] 1
isometric_log_ratio(p)
#> [1]  0.2 -0.5
isometric_logistic(c(800, 1200))
#> [1] 1 0 0