Vinecop.rosenblatt

Vinecop.rosenblatt(self: pyvinecopulib.Vinecop, u: numpy.ndarray[numpy.float64[m, n]], num_threads: int = 1, randomize_discrete: bool = True, seeds: list[int] = []) → numpy.ndarray[numpy.float64[m, n]]

Evaluates the Rosenblatt transform for a vine copula model.

The Rosenblatt transform converts data from this model into independent uniform variates.

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, Vinecop.rosenblatt() returns the variables

\[U_{M[d - j, j]}= F(V_{M[d - j, j]} | V_{M[d - j - 1, j - 1]}, \dots, V_{M[0, 0]}),\]

where \(M\) is the structure matrix. Similarly, Vinecop.inverse_rosenblatt() computes

\[V_{M[d - j, j]}= F^{-1}(U_{M[d - j, j]} | U_{M[d - j - 1, j - 1]}, \dots, U_{M[0, 0]}).\]

If some variables have atoms, Brockwell (10.1016/j.spl.2007.02.008) proposed a simple randomization scheme to ensure that output is still independent uniform if the model is correct. The transformation reads

\[U_{M[d - j, j]}= W_{d - j} F(V_{M[d - j, j]} | V_{M[d - j - 1, j - 1]}, \dots, V_{M[0,\]

0]}) + (1 - W_{d - j}) F^-(V_{M[d - j, j]} | V_{M[d - j - 1, j - 1]}, dots, V_{M[0, 0]}),

where \(F^-\) is the left limit of the conditional cdf and \(W_1, \dots, W_d\) are are independent standard uniform random variables. This is used by default. If you are interested in the conditional probabilities

\[F(V_{M[d - j, j]} | V_{M[d - j - 1, j - 1]}, \dots, V_{M[0, 0]}),\]

set randomize_discrete = FALSE.

Parameters:

  • u – An \(n \times d\) matrix of evaluation points.

  • num_threads – The number of threads to use for computations; if greater than 1, the function will be applied concurrently to num_threads batches of u.

  • randomize_discrete – Whether to randomize the transform for discrete variables; see Details.

  • seeds – Seeds to scramble the quasi-random numbers; if empty (default), the random number quasi-generator is seeded randomly. Only relevant if there are discrete variables and randomize_discrete = TRUE.

Returns:

An \(n \times d\) matrix of independent uniform variates.