Push-forward of a density through a smooth bijective map

$$\nu(y) = \mu(T^{-1}(y)) / |\det DT(T^{-1}(y))|$$ Peyre & Cuturi (2019), Remark 2.6, eq. (2.8), which writes rho_alpha(x) = |det T'(x)| rho_beta(T(x)).

Usage

Otpushfw(mu_grid, T_jac, T_inv_grid)

Arguments

mu_grid

Source density at the preimages.

T_jac

Jacobian determinant of T at those preimages.

T_inv_grid

The preimages, carried through to the result.

Value

Named list with estimate, preimage, jacobian, n, method.

References

Peyre & Cuturi (2019), eq. (2.8).

Examples

Otpushfw(mu_grid = c(1, 2, 3, 4, 5, 6, 7, 8), T_jac = c(1, 2, 3, 4, 5, 6, 7, 8),
  T_inv_grid = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] 1 1 1 1 1 1 1 1
#> 
#> $preimage
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $jacobian
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Push-forward density -- Peyre & Cuturi (2019) eq. (2.8)"
#>