The Rosenblatt transform (Rosenblatt, 1952) \(U = T(V)\) of a random vector
\(V = (V_1,\ldots,V_d) ~ F\) is defined as
$$
U_1= F(V_1), U_{2} = F(V_{2}|V_1), \ldots, U_d =F(V_d|V_1,\ldots,V_{d-1}),
$$
where \(F(v_k|v_1,\ldots,v_{k-1})\) is the conditional distribution of
\(V_k\) given \(V_1 \ldots, V_{k-1}, k = 2,\ldots,d\). The vector
\(U = (U_1, \dots, U_d)\) then contains independent standard uniform
variables. The inverse operation
$$
V_1 = F^{-1}(U_1), V_{2} = F^{-1}(U_2|U_1), \ldots,
V_d =F^{-1}(U_d|U_1,\ldots,U_{d-1}),
$$
can be used to simulate from a distribution. For any copula \(F\), if
\(U\) is a vector of independent random variables, \(V = T^{-1}(U)\) has
distribution \(F\).
The formulas above assume a vine copula model with order \(d, \dots, 1\).
More generally, rosenblatt()
returns the variables
$$
U_{M[d + 1- j, j]}= F(V_{M[d + 1- j, j]} | V_{M[d - j, j - 1]}, \dots, V_{M[1, 1]}),
$$
where \(M\) is the structure matrix. Similarly, inverse_rosenblatt()
returns
$$
V_{M[d + 1- j, j]}= F^{-1}(U_{M[d + 1- j, j]} | U_{M[d - j, j - 1]}, \dots, U_{M[1, 1]}).
$$