Bivariate copula models

Bivariate copulas are the building blocks of vine models. rvinecopulib can construct a known model, fit or select one from data, evaluate its distribution functions, and summarize its dependence.

Implemented families

Type	Family	Identifier	Parameters
Independence	Independence	`"indep"`	0
Elliptical	Gaussian	`"gaussian"`	1
Elliptical	Student t	`"t"`	2
Archimedean	Clayton	`"clayton"`	1
Archimedean	Gumbel	`"gumbel"`	1
Archimedean	Frank	`"frank"`	1
Archimedean	Joe	`"joe"`	1
Archimedean	BB1, BB6, BB7, BB8	lower-case name	2
Extreme value	Tawn	`"tawn"`	3
Nonparametric	Transformation kernel	`"tll"`	effective count

Unambiguous partial names such as "gauss" and collection aliases such as "par" are accepted. For reusable code, full names make intent clearer.

Construct a known model

bicop_dist() specifies a family, rotation, parameters, and variable types.

cop <- bicop_dist(
  family = "clayton",
  rotation = 90,
  parameters = 2
)
cop
#> Bivariate copula ('bicop_dist'): family = clayton, rotation = 90, parameters = 2, var_types = c,c

Rotations are 0, 90, 180, or 270 degrees. They allow asymmetric Archimedean copulas to represent different corners and negative dependence. Elliptical, Frank, independence, and nonparametric copulas already cover both signs without rotations.

Evaluate and simulate

The density, CDF, and random generator follow R’s d*, p*, and r* conventions.

points <- rbind(c(0.2, 0.7), c(0.8, 0.3))
dbicop(points, cop)
#> [1] 1.562211 1.901324
pbicop(points, cop)
#> [1] 0.08022147 0.13123681
rbicop(4, cop)
#>            [,1]       [,2]
#> [1,] 0.77133621 0.46791844
#> [2,] 0.79602646 0.08946854
#> [3,] 0.07922695 0.35240165
#> [4,] 0.96904452 0.02742706

The same functions also accept family, rotation, and parameters directly:

dbicop(points, "gaussian", 0, 0.6)
#> [1] 0.6267684 0.6267684

Conditional distributions and h-functions

H-functions are the conditional distributions used recursively in vine copulas. If cond_var = 1, the first input variable is conditioned on and the function returns the conditional CDF of the second. With inverse = TRUE, it returns the corresponding conditional quantile.

h <- hbicop(points, cond_var = 1, family = cop)
h
#> [1] 0.4649857 0.6008183
hbicop(cbind(points[, 1], h), cond_var = 1, family = cop, inverse = TRUE)
#> [1] 0.7 0.3

This round trip recovers the second column up to numerical precision. The same conditional-distribution operations drive vine recursion and simulation; see Conditional simulation and Rosenblatt transforms for the multivariate workflow.

Dependence measures

par_to_ktau() converts family parameters to Kendall’s tau, and ktau_to_par() provides the inverse for families where tau determines all parameters. Model objects expose Kendall’s tau through get_ktau().

par_to_ktau("clayton", 90, 2)
#> [1] -0.5
get_ktau(cop)
#> [1] -0.5
blomqvist_beta(cop)
#> [1] -0.5118579
tail_dep(cop)
#>          variable2
#> variable1     lower upper
#>     lower 0.0000000     0
#>     upper 0.7071068     0

tail_dep() reports the four corner tail-dependence coefficients. This is especially useful for rotated and asymmetric families, where a single lower/upper pair is not enough to describe the model.

Fit and select from data

bicop() expects two approximately uniform columns. It fits every requested compatible family and selects the best according to selcrit.

u <- rbicop(300, "gumbel", 180, 2)
fit <- bicop(
  u,
  family_set = c("gaussian", "clayton", "gumbel", "frank"),
  selcrit = "bic"
)
fit
#> Bivariate copula fit ('bicop'): family = gumbel, rotation = 180, parameters = 2.16, var_types = c,c
summary(fit)
#> Bivariate copula fit ('bicop'): family = gumbel, rotation = 180, parameters = 2.16, var_types = c,c
#> Dependence: tau = 0.54; beta = 0.54; tail dependence: LL = 0.62
#> Fit: n = 300; logLik = 126.44; df = 1.00; AIC = -250.87; BIC = -247.17

Useful family collections include:

"all", "parametric", and "nonparametric";
"oneparametric", "twoparametric", and "threeparametric";
"elliptical", "archimedean", "ev", and "bbs";
"itau", the families compatible with Kendall’s tau inversion.

Partial matching makes shorter forms such as "onepar" and "par" available. Explicit character vectors are preferable when the candidate set is part of a scientific specification.

Maximum likelihood is the default for parametric families. Set par_method = "itau" for Kendall’s tau inversion. The transformation-kernel model uses nonpar_method and mult to control the local-likelihood order and smoothing.

Observation weights and missing values

Pass one nonnegative weight per row with weights. The backend standardizes weights internally. Missing or zero-weight observations do not contribute to a pair fit.

weights <- seq_len(nrow(u))
weighted_fit <- bicop(
  u,
  family_set = c("gaussian", "gumbel"),
  weights = weights
)
weighted_fit
#> Bivariate copula fit ('bicop'): family = gumbel, rotation = 180, parameters = 2.1, var_types = c,c

Discrete variables

A discrete copula observation needs both F(x) and F(x-). For two discrete variables, pass four columns: the two CDF values followed by their two left limits. Redundant left-limit columns for continuous variables may be omitted.

counts <- cbind(rpois(80, 2), rpois(80, 3))
u_discrete <- cbind(
  ppois(counts[, 1], 2),
  ppois(counts[, 2], 3),
  ppois(counts[, 1] - 1, 2),
  ppois(counts[, 2] - 1, 3)
)
fit_discrete <- bicop(
  u_discrete,
  var_types = c("d", "d"),
  family_set = "onepar"
)
fit_discrete
#> Bivariate copula fit ('bicop'): family = joe, rotation = 270, parameters = 1.05, var_types = d,d

The discrete-data guide derives the likelihood contribution and covers compact layouts, mixed models, and atoms in detail.

Observation-specific parameters

Evaluation and simulation can use one parameter set per observation without constructing many model objects. A one-parameter family accepts a vector; multi-parameter families use a matrix with one row per observation.

rho <- seq(-0.7, 0.7, length.out = nrow(points))
dbicop(points, "gaussian", 0, parameters = rho)
#> [1] 1.6000932 0.4764093
rbicop(length(rho), "gaussian", 0, parameters = rho)
#>            [,1]      [,2]
#> [1,] 0.56361393 0.3312878
#> [2,] 0.01234443 0.1489253

This interface is intended for conditional or covariate-dependent copula models. Parameters are not recycled, and vectorized evaluation is limited to parametric families.