Additive logistic transform (unconstrained coordinates to the simplex)

Inverse additive log-ratio map $\boldsymbol{\psi}:\boldsymbol{\rho}\mapsto\boldsymbol{p}$ of Aitchison ($\mathrm{alr}^{-1}$): a softmax with the last category $J$ pinned as reference ($\rho_J\equiv 0$). Solvers and vcov.decovart_fit() use the isometric log-ratio chart isometric_logistic() instead. This ALR helper is retained for the vignette appendix.

Recovers the unconstrained additive log-ratio coordinates $\rho_j=\ln(p_j/p_J)$ for $j=1,\ldots,J-1$, with the last part $p_J$ as reference. This is Aitchison's additive log-ratio ($\mathrm{alr}$) transform, equivalently the multinomial-logit link with reference category $J$ (see compositions::alr() and vignette("theory-decovart-generative-model", package = "DeCovarT")).

Usage

additive_logistic(rho)

additive_log_ratio(p)

Arguments

rho: Numeric vector $\boldsymbol{\rho}\in\mathbb{R}^{J-1}$ of unconstrained additive log-ratio coordinates (reference cell type $J$).
p: Numeric vector $\boldsymbol{p}$ on the open simplex.

Value

Numeric vector $\boldsymbol{p}\in\mathbb{R}^{J}$ on the unit simplex ($\mathbf{1}^{\mathsf{T}}\boldsymbol{p}=1$, $\boldsymbol{p}\ge\mathbf{0}$).

Numeric vector $\boldsymbol{\rho}\in\mathbb{R}^{J-1}$.

Details

With $A=\sum_{k=1}^{J-1}\mathrm{e}^{\rho_k}+1$, $$ p_j=\frac{\mathrm{e}^{\rho_j}}{A}\quad(j<J),\qquad p_J=\frac{1}{A}. $$ Equivalently, $\boldsymbol{p}=\boldsymbol{\psi}(\boldsymbol{\rho})$ with $\boldsymbol{\psi}(\boldsymbol{\rho})\propto (\mathrm{e}^{\rho_1},\ldots,\mathrm{e}^{\rho_{J-1}},1)^{\mathsf{T}}$. Jacobians and Hessians of both maps are derived in the package vignette vignette("theory-decovart-generative-model", package = "DeCovarT"). See also compositions::alrInv().

Numerical stability

Evaluated through the log-sum-exp shift $p_i\propto\exp(\tilde{\rho}_i-\max_k\tilde{\rho}_k)$ with $\tilde{\boldsymbol{\rho}}=(\rho_1,\ldots,\rho_{J-1},0)$, which is algebraically identical to the ratio above but never forms $\mathrm{e}^{\rho_i}$ directly. The naive quotient overflows to NaN for $\rho_i\gtrsim 710$, which unconstrained ascent can reach when the MLE approaches a simplex face; the shifted form returns an exact zero instead and keeps $\boldsymbol{\Sigma}(\boldsymbol{p})$ positive definite.

Examples

rho <- c(0.2, -0.5)
p <- additive_logistic(rho)
sum(p)
#> [1] 1
additive_log_ratio(p)
#> [1]  0.2 -0.5
# Stable far out in ALR space, where exp(rho) would overflow.
additive_logistic(c(800, 1200))
#> [1] 1.91517e-174  1.00000e+00  0.00000e+00