Helmert contrast matrix for isometric log-ratio coordinates

Returns the \(J\times(J-1)\) Helmert sub-matrix \(\mathbf{V}\) with $$ \mathbf{V}^{\mathsf{T}}\mathbf{V}=\mathbf{I}_{J-1},\qquad \mathbf{V}^{\mathsf{T}}\mathbf{1}=\mathbf{0}. $$ Column \(k\) contrasts the first \(k\) parts against part \(k+1\). This is the standard ILR basis used with isometric_log_ratio() / isometric_logistic(); any other valid ILR basis is \(\mathbf{V}Q\) for an orthogonal \(Q\), which rotates coordinates but leaves simplex geometry, eigenvalues of quadratic forms, and \(\mathrm{Var}(\hat{\boldsymbol{p}})\) unchanged.

Usage

helmert_basis(n_parts)

Arguments

n_parts

Integer \(J\ge 2\) (number of simplex parts).

Value

Numeric matrix \(J\times(J-1)\).

Examples

V <- helmert_basis(3L)
crossprod(V)
#>              [,1]         [,2]
#> [1,] 1.000000e+00 2.451427e-17
#> [2,] 2.451427e-17 1.000000e+00
drop(crossprod(V, rep(1, 3)))
#> [1] 0 0