Cramer–Rao / ILR delta-method covariance of $\hat{\boldsymbol{p}}$

Maps the expected Fisher information of unconstrained proportions through the isometric log-ratio (ILR) chart and back to the simplex via the delta method.

Let $\boldsymbol{p}=\operatorname{softmax}(\mathbf{V}\boldsymbol{z})$ with Jacobian $\mathbf{J}_{\boldsymbol{\psi}}=\mathbf{S}(\boldsymbol{p}) \mathbf{V}$ (jacobian_isometric_logistic()). Fisher information transforms as $$ I_{\boldsymbol{z}} = \mathbf{J}_{\boldsymbol{\psi}}^{\mathsf{T}} I(\boldsymbol{p}) \mathbf{J}_{\boldsymbol{\psi}} = \mathbf{V}^{\mathsf{T}} \mathbf{S}(\boldsymbol{p}) I(\boldsymbol{p}) \mathbf{S}(\boldsymbol{p}) \mathbf{V}. $$ The first-order delta method then yields $$ \mathrm{Var}(\hat{\boldsymbol{p}}) \approx \mathbf{J}_{\boldsymbol{\psi}} I_{\boldsymbol{z}}^{-1} \mathbf{J}_{\boldsymbol{\psi}}^{\mathsf{T}}. $$ This simplex covariance is invariant to orthogonal rotations of $\mathbf{V}$. The construction is undefined on the simplex boundary (the log-ratio chart blows up); the function then returns NA with a warning.

Wald intervals in vcov.decovart_fit(), confint.decovart_fit() and deconvolute_ratios() all use this expected-Fisher ILR map. The observed Hessian hessian_loglik_constrained() is for Newton steps only; it is not the Wald covariance.

Usage

vcov_ilr_delta(p, mean_signature_matrix, Sigma, warn = TRUE)

Arguments

p: Numeric proportions on the open simplex.
mean_signature_matrix: Mean signature $\boldsymbol{\mu}$ ($G\times J$).
Sigma: Cell-type covariances $G\times G\times J$.
warn: If FALSE, skip the boundary / singularity warnings (Monte Carlo loops). Default TRUE.

Value

Symmetric $J\times J$ asymptotic covariance of $\hat{\boldsymbol{p}}$, or a matrix of NA if the bound is undefined / singular.

Examples

# Two cell types: a single unconstrained ILR coordinate maps back to
# a rank-1 covariance on the simplex (rows sum to zero).
p <- c(ct1 = 0.6, ct2 = 0.4)
mu <- cbind(ct1 = c(0, 0), ct2 = c(3, 0))
Sigma <- array(c(diag(2), diag(2)), dim = c(2, 2, 2))
v <- vcov_ilr_delta(p, mu, Sigma)
round(v, 4)
#>         ct1     ct2
#> ct1  0.0559 -0.0559
#> ct2 -0.0559  0.0559
abs(rowSums(v)) < 1e-10
#>  ct1  ct2 
#> TRUE TRUE

Cramer–Rao / ILR delta-method covariance of \(\hat{\boldsymbol{p}}\)

Usage

Arguments

Value

See also

Examples