rvinecopulib (version 0.7.1.1.2)

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 $V=(V_1,\dots ,V_d)F$ is defined as $$U_1=F(V_1),U_2=F(V_2|V_1),\dots ,U_d=F(V_d|V_1,\dots ,V_{d-1}),$$ where $F(v_k|v_1,\dots ,v_{k-1})$ is the conditional distribution of $V_k$ given $V_1\dots ,V_{k-1},k=2,\dots ,d$. The vector $U=(U_1,\dots ,U_d)$ then contains independent standard uniform variables. The inverse operation $$V_1=F^{-1}(U_1),V_2=F^{-1}(U_2|U_1),\dots ,V_d=F^{-1}(U_d|U_1,\dots ,U_{d-1}),$$ 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)