rosenblatt: (Inverse) Rosenblatt transform

In rvinecopulib: High Performance Algorithms for Vine Copula Modeling

(Inverse) Rosenblatt transform

Description

The Rosenblatt transform takes data generated from a model and turns it into independent uniform variates, The inverse Rosenblatt transform computes conditional quantiles and can be used simulate from a stochastic model, see Details.

Usage

rosenblatt(x, model, cores = 1, randomize_discrete = TRUE)

inverse_rosenblatt(u, model, cores = 1)

Arguments

x	matrix of evaluation points; must be in (0, 1)^d for copula models.
model	a model object; classes currently supported are bicop_dist(), vinecop_dist(), and vine_dist().
cores	if ⁠>1⁠, computation is parallelized over cores batches (rows of u).
randomize_discrete	Whether to randomize the transform for discrete variables; see Details.
u	matrix of evaluation points; must be in (0, 1)^d.

Details

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 - j + 1, j]} | V_{M[d - j, j]}, \dots, V_{M[1, j]}),

where M is the structure matrix. Similarly, inverse_rosenblatt() returns

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

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.

Examples

# simulate data with some dependence
x <- replicate(3, rnorm(200))
x[, 2:3] <- x[, 2:3] + x[, 1]
pairs(x)

# estimate a vine distribution model
fit <- vine(x, copula_controls = list(family_set = "par"))

# transform into independent uniforms
u <- rosenblatt(x, fit)
pairs(u)

# inversion
pairs(inverse_rosenblatt(u, fit))

# works similarly for vinecop models
vc <- fit$copula
rosenblatt(pseudo_obs(x), vc)

rvinecopulib documentation built on April 12, 2025, 2:09 a.m.