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Discrete, mixed, and zero-inflated data

Source: vignettes/discrete-data.Rmd
discrete-data.Rmd

Copula likelihoods for variables with atoms require two probabilities at every observation: the CDF value F(x) and its left limit F(x-). For an integer-valued variable, F(x-) = F(x - 1). Continuous variables satisfy F(x-) = F(x).

The high-level vine() interface computes these values from fitted margins. The copula-only bicop() and vinecop() interfaces accept them explicitly.

Discrete variables on the original scale

All ordinary discrete variables are assumed to have integer support. Declare a numeric or integer column through var_types = "d".

n <- 100
latent <- rnorm(n)
x <- data.frame(
  value = latent + rnorm(n),
  count = rpois(n, exp(0.5 + 0.3 * latent))
)

fit <- vine(
  x,
  var_types = c("c", "d"),
  copula_controls = list(family_set = "onepar")
)
summary(fit)$margins
#> # A data.frame: 2 x 8 
#>  margin  name family type xmin xmax npars loglik
#>       1 value  kde1d    c -Inf  Inf   5.6   -170
#>       2 count  kde1d    d -Inf  Inf   3.8   -156
dvine(x[1:5, ], fit)
#> [1] 0.086337749 0.070854020 0.005353429 0.024217577 0.096298044

Explicitly discrete columns are checked for finite integer-valued observations. This catches accidental declarations of rounded-looking continuous data.

Ordered factors

Ordered factors are discrete variables whose levels are mapped internally to the integer support 0, 1, .... Their class and levels are restored after simulation.

survey <- data.frame(
  rating = ordered(
    sample(c("low", "middle", "high"), n, replace = TRUE),
    levels = c("low", "middle", "high")
  ),
  score = rnorm(n)
)

ordered_fit <- vine(
  survey,
  copula_controls = list(family_set = "indep")
)
ordered_draws <- rvine(5, ordered_fit)
str(ordered_draws)
#> 'data.frame':    5 obs. of  2 variables:
#>  $ rating: Ord.factor w/ 3 levels "low"<"middle"<..: 3 3 1 3 1
#>  $ score : num  -2.83583 -0.00419 -0.99283 -0.23898 0.62379

No var_types entry is needed because ordered() carries the declaration in the column class.

Zero-inflated variables

zero_inflated() marks a continuous variable with an atom at zero. It behaves like a numeric vector in a data frame and survives ordinary subsetting.

amount <- zero_inflated(c(rep(0, 25), rexp(75)))
zi_data <- data.frame(amount = amount, score = rnorm(100))
inherits(zi_data$amount, "zero_inflated")
#> [1] TRUE

zi_fit <- vine(
  zi_data,
  copula_controls = list(family_set = "indep")
)
summary(zi_fit)$margins[, c("name", "type")]
#> # A data.frame: 2 x 2 
#>    name type
#>  amount   zi
#>   score    c

The atom is always located at zero. A custom zero-inflated margin must return the atom probability from dmargin(0, margin). rvinecopulib then computes the left limit as

pmargin(0, margin) - dmargin(0, margin)

and uses the ordinary CDF away from zero. Alternatively, declare the type with var_types = "zi".

Copula-scale data layouts

Let d be the dimension and k the number of discrete variables. Copula methods accept two layouts:

  • expanded: 2d columns, with all F(x) columns followed by all F(x-) columns;
  • compact: d + k columns, with the F(x) block followed only by left-limit columns for discrete variables, in variable order.

Consider a three-dimensional model with discrete variables 1 and 3.

n <- 80
raw <- data.frame(
  first = rpois(n, 2),
  middle = rnorm(n),
  last = rpois(n, 4)
)

values <- cbind(
  ppois(raw$first, 2),
  pnorm(raw$middle),
  ppois(raw$last, 4)
)
left_discrete <- cbind(
  ppois(raw$first - 1, 2),
  ppois(raw$last - 1, 4)
)
compact <- cbind(values, left_discrete)

left_all <- cbind(
  ppois(raw$first - 1, 2),
  pnorm(raw$middle),
  ppois(raw$last - 1, 4)
)
expanded <- cbind(values, left_all)

copula_fit <- vinecop(
  compact,
  var_types = c("d", "c", "d"),
  family_set = "indep"
)
stopifnot(isTRUE(all.equal(
  dvinecop(compact, copula_fit),
  dvinecop(expanded, copula_fit)
)))

For a mixed bivariate model, this means three columns: the two CDF values and the left limit of the discrete variable. Fully discrete bivariate models need four columns.

What the density means

dbicop() and dvinecop() return the copula density contribution defined as the joint density or mass divided by the marginal densities or masses. This definition is consistent across continuous, discrete, and mixed models.

For a bivariate model, write uj=Fj(xj),uj−=Fj(xj−),Δj=uj−uj−. u_j = F_j(x_j), \qquad u_j^- = F_j(x_j^-), \qquad \Delta_j = u_j - u_j^-. If both variables are discrete, the copula contribution is the rectangle probability divided by the two marginal masses: c*(u1,u2)=C(u1,u2)−C(u1−,u2)−C(u1,u2−)+C(u1−,u2−)Δ1Δ2. c^*(u_1,u_2) = \frac{C(u_1,u_2)-C(u_1^-,u_2)-C(u_1,u_2^-)+C(u_1^-,u_2^-)} {\Delta_1\Delta_2}. If only the first variable is discrete, the corresponding mixed contribution is c*(u1,u2)=∂2C(u1,u2)−∂2C(u1−,u2)Δ1. c^*(u_1,u_2) = \frac{\partial_2 C(u_1,u_2)-\partial_2 C(u_1^-,u_2)}{\Delta_1}. The continuous density c(u1, u2) is recovered when both jump widths vanish. Vine likelihoods apply these continuous, mixed, or rectangle contributions edge by edge to the appropriate conditional CDF values.

dvine() multiplies that contribution by the fitted marginal densities or masses and therefore returns the full joint density or probability mass on the original scale. In general, fX(x)=c*(u,u−)∏j=1dfj(xj), f_X(x) = c^*(u,u^-)\prod_{j=1}^d f_j(x_j), where each f_j denotes a density or a probability mass as appropriate.

Simulation

rvinecop() always returns one uniform-scale value per variable. It does not invent a marginal distribution for a discrete copula variable. Use rvine() when simulated values should be returned on the original scale with their integer, ordered, or zero-inflated margins.

copula_draws <- rvinecop(4, copula_fit)
dim(copula_draws)
#> [1] 4 3

original_draws <- rvine(4, ordered_fit)
is.ordered(original_draws$rating)
#> [1] TRUE

Rosenblatt transforms with atoms

For an atom, the Rosenblatt transform is an interval rather than a single value. By default, rosenblatt() randomizes uniformly within that interval so the transformed variable is uniform under a correct model. Set randomize_discrete = FALSE to return the deterministic upper endpoint.

More precisely, if G_j(· | x_<j) is the relevant conditional CDF and V_j is an independent standard uniform variable, the randomized component is Zj=Gj(xj−∣x<j)+Vj{Gj(xj∣x<j)−Gj(xj−∣x<j)}. Z_j = G_j(x_j^- \mid x_{<j}) + V_j\{G_j(x_j \mid x_{<j})-G_j(x_j^- \mid x_{<j})\}. The non-randomized version returns the upper endpoint G_j(x_j | x_<j).

set.seed(402)
randomized <- rosenblatt(compact, copula_fit)
upper_endpoint <- rosenblatt(
  compact,
  copula_fit,
  randomize_discrete = FALSE
)
head(randomized)
#>            [,1]      [,2]      [,3]
#> [1,] 0.05289451 0.0958401 0.7055710
#> [2,] 0.33647905 0.7438199 0.7051919
#> [3,] 0.74612047 0.1121317 0.2203118
#> [4,] 0.13426465 0.5942987 0.1240232
#> [5,] 0.73703700 0.6269100 0.8771097
#> [6,] 0.51260896 0.1971273 0.7066150
head(upper_endpoint)
#>           [,1]      [,2]      [,3]
#> [1,] 0.1353353 0.0958401 0.7851304
#> [2,] 0.4060058 0.7438199 0.7851304
#> [3,] 0.8571235 0.1121317 0.2381033
#> [4,] 0.1353353 0.5942987 0.2381033
#> [5,] 0.8571235 0.6269100 0.8893260
#> [6,] 0.6766764 0.1971273 0.7851304

The randomization uses R’s random-number state and is reproducible with set.seed().

Missing observations

Copula fitting discards incomplete observations pair by pair, retaining the maximum information available for each edge. Margin implementations decide how to handle missing values during marginal fitting; built-in KDE margins support the package’s established missing-data workflow. Custom margin_family() fitters should either handle missing values or fail with a useful message.