Jacobian of the isometric logistic map

$$ \mathbf{J}_{\boldsymbol{\psi}}(\boldsymbol{z}) = \frac{\partial\boldsymbol{p}}{\partial\boldsymbol{z}^{\mathsf{T}}} = \mathbf{S}(\boldsymbol{p})\mathbf{V}, $$ with \(\mathbf{S}(\boldsymbol{p})=\operatorname{diag}(\boldsymbol{p}) -\boldsymbol{p}\boldsymbol{p}^{\mathsf{T}}\).

Usage

jacobian_isometric_logistic(z, V = NULL)

Arguments

z

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

V

Optional ILR basis; see isometric_logistic().

Value

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

Examples

jacobian_isometric_logistic(c(0.2, -0.5))
#>            [,1]       [,2]
#> [1,]  0.1952773  0.1742416
#> [2,] -0.1704937  0.1313152
#> [3,] -0.0247836 -0.3055568