rvinecopulib (version 0.7.2.1.0)

rosenblatt: (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)