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Conditional simulation and Rosenblatt transforms

Source: vignettes/conditional-simulation.Rmd
conditional-simulation.Rmd

A vine is simulated along an order. Variables to condition on must form the tail of an admissible sampling order. rvinecopulib can enforce this during structure selection or transiently reorient a compatible fitted model during simulation.

Select a conditioning-aware structure

Pass variable indices or names through conditioning_set when fitting a copula. The selected structure places those variables at the end of its order.

n <- 180
z <- rnorm(n)
x <- cbind(
  response1 = z + rnorm(n),
  response2 = -0.5 * z + rnorm(n),
  driver1 = z + rnorm(n, sd = 0.5),
  driver2 = 0.4 * z + rnorm(n)
)
u <- pseudo_obs(x)

fit <- vinecop(
  u,
  family_set = "onepar",
  conditioning_set = c("driver1", "driver2")
)
get_structure(fit)
#> 4-dimensional R-vine structure ('rvine_structure')
#> 3 3 3 3
#> 1 4 4  
#> 4 1    
#> 2

Conditioning-aware selection supports fixed and automatically selected truncation levels. It requires an MST tree algorithm ("mst_prim" or "mst_kruskal"); random tree algorithms do not guarantee the required order.

Simulate conditionally on the copula scale

u_cond contains one value per conditioning variable. A vector or one-row matrix is repeated for all simulations; an n-row matrix specifies different conditions for every output row.

condition <- c(0.25, 0.8)
draws <- rvinecop(
  100,
  fit,
  u_cond = condition,
  conditioning_set = c("driver1", "driver2")
)
stopifnot(
  isTRUE(all.equal(draws[, "driver1"], rep(condition[1], 100))),
  isTRUE(all.equal(draws[, "driver2"], rep(condition[2], 100)))
)
head(draws)
#>       response1 response2 driver1 driver2
#> [1,] 0.63870946 0.4514150    0.25     0.8
#> [2,] 0.31615622 0.3556760    0.25     0.8
#> [3,] 0.07066392 0.3225717    0.25     0.8
#> [4,] 0.54276866 0.8714729    0.25     0.8
#> [5,] 0.87804170 0.9514778    0.25     0.8
#> [6,] 0.40480490 0.9892094    0.25     0.8

When conditioning_set is omitted, the columns of u_cond refer to the last variables in the model’s current order.

conditions <- cbind(
  driver1 = seq(0.1, 0.9, length.out = 5),
  driver2 = rep(0.5, 5)
)
row_draws <- rvinecop(
  5,
  fit,
  u_cond = conditions,
  conditioning_set = c("driver1", "driver2")
)
stopifnot(isTRUE(all.equal(row_draws[, 3:4], conditions)))

Condition on the original data scale

vine() exposes the same structure control through copula_controls. rvine() accepts x_cond on the original scale and uses the fitted margins to compute all necessary CDF values and left limits.

full_fit <- vine(
  as.data.frame(x),
  margins_controls = list(family_set = "kde1d"),
  copula_controls = list(
    family_set = "onepar",
    conditioning_set = c("driver1", "driver2")
  )
)

original_condition <- data.frame(driver1 = -0.5, driver2 = 1.25)
original_draws <- rvine(
  6,
  full_fit,
  x_cond = original_condition,
  conditioning_set = c("driver1", "driver2")
)
stopifnot(
  isTRUE(all.equal(original_draws[, "driver1"], rep(-0.5, 6))),
  isTRUE(all.equal(original_draws[, "driver2"], rep(1.25, 6)))
)

Column names are strongly recommended: they make the mapping explicit and protect code from changes in variable order.

Reorient an existing model transiently

Supplying an explicit conditioning_set to rvinecop() or rvine() asks the backend to evaluate an equivalent orientation without modifying the model. Only conditioning sets that can form an admissible tail of the fitted structure are possible. If a specific set is central to the analysis, request it during fitting; selection then guarantees a usable structure.

Discrete conditioning variables

On the copula scale, a discrete conditioning variable needs its value F(x) and left limit F(x-). As with model evaluation, u_cond accepts:

  • a compact layout with one value column per conditioning variable followed by one left-limit column per discrete variable; or
  • an expanded layout with all value columns followed by all left-limit columns, where a continuous variable’s two columns must be equal.

rvine() handles this internally from x_cond, including ordered and zero-inflated margins.

Rosenblatt transforms

The Rosenblatt transform maps observations from a fitted distribution to independent uniforms. For continuous variables in a sampling sequence s1, ..., sd, its components are Zj=Fsj∣s1,…,sj−1(Xsj∣Xs1,…,Xsj−1),j=1,…,d, Z_j = F_{s_j \mid s_1,\ldots,s_{j-1}} \left(X_{s_j}\mid X_{s_1},\ldots,X_{s_{j-1}}\right), \qquad j=1,\ldots,d, where the first component is unconditional. If the fitted model is correct, the Z_j are independent standard uniforms. The inverse transform applies the corresponding conditional quantiles recursively and therefore maps independent uniforms back to the model.

simulated <- rvinecop(100, fit)
independent <- rosenblatt(simulated, fit)
reconstructed <- inverse_rosenblatt(independent, fit)
max(abs(simulated - reconstructed))
#> [1] 7.21645e-16

The transform follows the model’s sampling order. An explicit conditioning_set requests an admissible order ending in those variables, which is useful when conditional quantiles must follow the same orientation as conditional simulation.

transformed <- rosenblatt(
  simulated,
  fit,
  conditioning_set = c("driver1", "driver2")
)
inverse_rosenblatt(
  transformed,
  fit,
  conditioning_set = c("driver1", "driver2")
)[1:3, ]
#>      response1 response2    driver1   driver2
#> [1,] 0.1180315 0.6240082 0.02953261 0.0577602
#> [2,] 0.8016499 0.6850150 0.30646778 0.7316434
#> [3,] 0.1138517 0.4059435 0.29654106 0.8204806

For discrete variables, the transform randomizes within the jump interval by default. See the discrete-data guide for the randomized formula, deterministic upper endpoints, and required copula layouts.

Reproducibility

Conditional simulation and randomized discrete transforms use R’s random-number state. Call set.seed() immediately before the operation when a reproducible draw is required. The original model object is never mutated by a conditional simulation or transform.