copula-package {copula} R Documentation

## Multivariate Dependence Modeling with Copulas

### Description

The copula package provides (S4) classes of commonly used elliptical, (nested) Archimedean, extreme value and other copula families; methods for density, distribution, random number generation, and plots.

Fitting copula models and goodness-of-fit tests. Independence and serial (univariate and multivariate) independence tests, and other copula related tests.

### Details

The DESCRIPTION file:

 Package: copula Version: 1.1-3 VersionNote: Last CRAN: 1.1-2 on 2023-01-20 Date: 2023-12-07 Title: Multivariate Dependence with Copulas Authors@R: c(person("Marius", "Hofert", role = "aut", email = "marius.hofert@uwaterloo.ca", comment = c(ORCID = "0000-0001-8009-4665")) , person("Ivan", "Kojadinovic", role = "aut", email = "ivan.kojadinovic@univ-pau.fr", comment = c(ORCID = "0000-0002-2903-1543")) , person("Martin","Maechler", role=c("aut","cre"), email="maechler@stat.math.ethz.ch", comment = c(ORCID = "0000-0002-8685-9910")) , person("Jun", "Yan", role = "aut", email = "jun.yan@uconn.edu", comment = c(ORCID = "0000-0003-4401-7296")) , person(c("Johanna", "G."), "Nešlehová", role = "ctb", comment = c("evTestK()", ORCID = "0000-0001-9634-4796")) , person("Rebecca", "Morger", role = "ctb", comment = "fitCopula.ml(): code for free mixCopula weight parameters") ) Depends: R (>= 3.5.0) Imports: stats, graphics, methods, stats4, Matrix (>= 1.5-0), lattice, colorspace, gsl, ADGofTest, stabledist (>= 0.6-4), mvtnorm, pcaPP, pspline, numDeriv Suggests: MASS, KernSmooth, sfsmisc, scatterplot3d, Rmpfr, bbmle, knitr, rmarkdown, abind, crop, gridExtra, lcopula, mev, mvnormtest, parallel, partitions, polynom, qrng, randtoolbox, rugarch, Runuran, tseries, VGAM, VineCopula, zoo SuggestsNote: packages {abind, ..., zoo} (last lines above) are only used in vignettes, demos and a few tests. VignetteBuilder: knitr Enhances: nor1mix, copulaData SystemRequirements: pdfcrop (part of TexLive) is required to rebuild the vignettes. Description: Classes (S4) of commonly used elliptical, Archimedean, extreme-value and other copula families, as well as their rotations, mixtures and asymmetrizations. Nested Archimedean copulas, related tools and special functions. Methods for density, distribution, random number generation, bivariate dependence measures, Rosenblatt transform, Kendall distribution function, perspective and contour plots. Fitting of copula models with potentially partly fixed parameters, including standard errors. Serial independence tests, copula specification tests (independence, exchangeability, radial symmetry, extreme-value dependence, goodness-of-fit) and model selection based on cross-validation. Empirical copula, smoothed versions, and non-parametric estimators of the Pickands dependence function. License: GPL (>= 3) | file LICENCE Collate: AllClass.R Classes.R AllGeneric.R Auxiliaries.R aux-acopula.R asymCopula.R mixCopula.R rotCopula.R Copula.R special-func.R amhCopula.R claytonCopula.R frankCopula.R cop_objects.R nacopula.R dC-dc.R amhExpr.R An.R archmCopula.R cCopula.R claytonExpr.R ellipCopula.R empCopula.R empPsi.R acR.R estimation.R evCopula.R evTests.R exchTests.R fgmCopula.R fitCopula.R fitLambda.R fitMvdc.R fixedPar.R frankExpr.R galambosCopula.R galambosExpr-math.R galambosExpr.R ggraph-tools.R pairsRosenblatt.R prob.R gofTrafos.R gofEVTests.R gofCopula.R graphics.R gumbelCopula.R gumbelExpr.R huslerReissCopula.R huslerReissExpr.R indepCopula.R fhCopula.R lowfhCopula.R upfhCopula.R indepTests.R joeCopula.R K.R logseries.R mvdc.R margCopula.R matrix_tools.R normalCopula.R obs.R opower.R plackettCopula.R plackettExpr.R moCopula.R rstable1.R safeUroot.R schlatherCopula.R stable.R timing.R tCopula.R tawnCopula.R tawnExpr.R tevCopula.R varianceReduction.R wrapper.R xvCopula.R zzz.R Encoding: UTF-8 URL: https://copula.r-forge.r-project.org/, https://r-forge.r-project.org/projects/copula/, https://CRAN.r-project.org/package=copula BugReports: https://r-forge.r-project.org/tracker/?func=add&group_id=2140&atid=5417 Author: Marius Hofert [aut] (), Ivan Kojadinovic [aut] (), Martin Maechler [aut, cre] (), Jun Yan [aut] (), Johanna G. Nešlehová [ctb] (evTestK(), ), Rebecca Morger [ctb] (fitCopula.ml(): code for free mixCopula weight parameters) Maintainer: Martin Maechler

Index of help topics:

.pairsCond              Pairs Plot of a cu.u Object (Internal Use)
A..Z                    Sinc, Zolotarev's, and Other Mathematical
Utility Functions
An                      Nonparametric Rank-based Estimators of the
Pickands Dependence Function
Bernoulli               Compute Bernoulli Numbers
Copula                  Density, Evaluation, and Random Number
Generation for Copula Functions
Eulerian                Eulerian and Stirling Numbers of First and
Second Kind
K                       Kendall Distribution Function for Archimedean
Copulas
Mvdc                    Multivariate Distributions Constructed from
Copulas
RSpobs                  Pseudo-Observations of Radial and Uniform Part
of Elliptical and Archimedean Copulas
SMI.12                  SMI Data - 141 Days in Winter 2011/2012
Sibuya                  Sibuya Distribution - Sampling and
Probabilities
absdPsiMC               Absolute Value of Generator Derivatives via
Monte Carlo
acopula-class           Class "acopula" of Archimedean Copula Families
acopula-families        Specific Archimedean Copula Families ("acopula"
Objects)
allComp                 All Components of a (Inner or Outer) Nested
Archimedean Copula
archmCopula             Construction of Archimedean Copula Class Object
archmCopula-class       Class "archmCopula"
beta.                   Sample and Population Version of Blomqvist's
Beta for Archimedean Copulas
cCopula                 Conditional Distributions and Their Inverses
from Copulas
cloud2-methods          Cloud Plot Methods ('cloud2') in Package
'copula'
coeffG                  Coefficients of Polynomial used for Gumbel
Copula
contour-methods         Methods for Contour Plots in Package 'copula'
contourplot2-methods    Contour Plot Methods 'contourplot2' in Package
'copula'
copula-class            Mother Classes "Copula", etc of all Copulas in
the Package
copula-package          Multivariate Dependence Modeling with Copulas
corKendall              (Fast) Computation of Pairwise Kendall's Taus
dDiag                   Density of the Diagonal of (Nested) Archimedean
Copulas
describeCop             Copula (Short) Description as String
dnacopula               Density Evaluation for (Nested) Archimedean
Copulas
ebeta                   Various Estimators for (Nested) Archimedean
Copulas
ellipCopula             Construction of Elliptical Copula Class Objects
ellipCopula-class       Class "ellipCopula" of Elliptical Copulas
emde                    Minimum Distance Estimators for (Nested)
Archimedean Copulas
emle                    Maximum Likelihood Estimators for (Nested)
Archimedean Copulas
empCopula               The Empirical Copula
empCopula-class         Class "empCopula" of Empirical Copulas
enacopula               Estimation Procedures for (Nested) Archimedean
Copulas
evCopula                Construction of Extreme-Value Copula Objects
evCopula-class          Classes Representing Extreme-Value Copulas
evTestA                 Bivariate Test of Extreme-Value Dependence
Based on Pickands' Dependence Function
evTestC                 Large-sample Test of Multivariate Extreme-Value
Dependence
evTestK                 Bivariate Test of Extreme-Value Dependence
Based on Kendall's Distribution
exchEVTest              Test of Exchangeability for Certain Bivariate
Copulas
exchTest                Test of Exchangeability for a Bivariate Copula
fgmCopula               Construction of a fgmCopula Class Object
fgmCopula-class         Class "fgmCopula" - Multivariate Multiparameter
Farlie-Gumbel-Morgenstern Copulas
fhCopula                Construction of Fréchet-Hoeffding Bound Copula
Objects
fhCopula-class          Class "fhCopula" of Fréchet-Hoeffding Bound
Copulas
fitCopula               Fitting Copulas to Data - Copula Parameter
Estimation
fitCopula-class         Classes of Fitted Multivariate Models: Copula,
Mvdc
fitLambda               Non-parametric Estimators of the Matrix of
Tail-Dependence Coefficients
fitMvdc                 Estimation of Multivariate Models Defined via
Copulas
fixParam                Fix a Subset of a Copula Parameter Vector
gasoil                  Daily Crude Oil and Natural Gas Prices from
2003 to 2006
getAcop                 Get "acopula" Family Object by Name
getIniParam             Get Initial Parameter Estimate for Copula
getTheta                Get the Parameter(s) of a Copula
gnacopula               Goodness-of-fit Testing for (Nested)
Archimedean Copulas
gofBTstat               Various Goodness-of-fit Test Statistics
gofCopula               Goodness-of-fit Tests for Copulas
gofEVCopula             Goodness-of-fit Tests for Bivariate
Extreme-Value Copulas
gofTstat                Goodness-of-fit Test Statistics
htrafo                  GOF Testing Transformation of Hering and Hofert
iPsi                    Generator Functions for Archimedean and
Extreme-Value Copulas
indepCopula             Construction of Independence Copula Objects
indepCopula-class       Class "indepCopula"
indepTest               Test Independence of Continuous Random
Variables via Empirical Copula
initOpt                 Initial Interval or Value for Parameter
Estimation of Archimedean Copulas
interval                Construct Simple "interval" Object
interval-class          Class "interval" of Simple Intervals
khoudrajiCopula         Construction of copulas using Khoudraji's
device
khoudrajiCopula-class   Class '"khoudrajiCopula"' and its Subclasses
log1mexp                Compute f(a) = log(1 +/- exp(-a)) Numerically
Optimally
loss                    LOSS and ALAE Insurance Data
margCopula              Marginal copula of a Copula With Specified
Margins
mixCopula               Create Mixture of Copulas
mixCopula-class         Class '"mixCopula"' of Copula Mixtures
moCopula                The Marshall-Olkin Copula
moCopula-class          Class "moCopula" of Marshall-Olkin Copulas
multIndepTest           Independence Test Among Continuous Random
Vectors Based on the Empirical Copula Process
multSerialIndepTest     Serial Independence Test for Multivariate Time
Series via Empirical Copula
mvdc-class              Class "mvdc": Multivariate Distributions from
Copulas
nacFrail.time           Timing for Sampling Frailties of Nested
Archimedean Copulas
nacPairthetas           Pairwise Thetas of Nested Archimedean Copulas
nacopula-class          Class "nacopula" of Nested Archimedean Copulas
nesdepth                Nesting Depth of a Nested Archimedean Copula
("nacopula")
onacopula               Constructing (Outer) Nested Archimedean Copulas
opower                  Outer Power Transformation of Archimedean
Copulas
p2P                     Tools to Work with Matrices
pacR                    Distribution of the Radial Part of an
Archimedean Copula
pairs2                  Scatter-Plot Matrix ('pairs') for Copula
Distributions with Nice Defaults
pairsRosenblatt         Plots for Graphical GOF Test via Pairwise
Rosenblatt Transforms
pairwiseCcop            Computations for Graphical GOF Test via
Pairwise Rosenblatt Transforms
persp-methods           Methods for Function 'persp' in Package
'copula'
plackettCopula          Construction of a Plackett Copula
plackettCopula-class    Class "plackettCopula" of Plackett Copulas
plot-methods            Methods for 'plot' in Package 'copula'
pnacopula               Evaluation of (Nested) Archimedean Copulas
pobs                    Pseudo-Observations
polylog                 Polylogarithm Li_s(z) and Debye Functions
polynEval               Evaluate Polynomials
printNacopula           Print Compact Overview of a Nested Archimedean
Copula ("nacopula")
prob                    Computing Probabilities of Hypercubes
qqplot2                 Q-Q Plot with Rugs and Pointwise Asymptotic
Confidence Intervals
rAntitheticVariates     Variance-Reduction Methods
rF01Frank               Sample Univariate Distributions Involved in
Nested Frank and Joe Copulas
rFFrank                 Sampling Distribution F for Frank and Joe
radSymTest              Test of Exchangeability for a Bivariate Copula
rdj                     Daily Returns of Three Stocks in the Dow Jones
retstable               Sampling Exponentially Tilted Stable
Distributions
rlog                    Sampling Logarithmic Distributions
rnacModel               Random nacopula Model
rnacopula               Sampling Nested Archimedean Copulas
rnchild                 Sampling Child 'nacopula's
rotCopula               Construction and Class of Rotated aka Reflected
Copulas
rstable1                Random numbers from (Skew) Stable Distributions
safeUroot               One-dimensional Root (Zero) Finding - Extra
"Safety" for Convenience
serialIndepTest         Serial Independence Test for Continuous Time
Series Via Empirical Copula
setTheta                Specify the Parameter(s) of a Copula
show-methods            Methods for 'show()' in Package 'copula'
splom2-methods          Methods for Scatter Plot Matrix 'splom2' in
Package 'copula'
tau                     Dependence Measures for Bivariate Copulas
tauAMH                  Ali-Mikhail-Haq ("AMH")'s and Joe's Kendall's
Tau
uranium                 Uranium Exploration Dataset of Cook & Johnson
(1986)
wireframe2-methods      Perspective Plots - 'wireframe2' in Package
'copula'
xvCopula                Model (copula) selection based on 'k'-fold
cross-validation


Further information is available in the following vignettes:

 AC_Liouville Archimedean Liouville Copulas (source) AR_Clayton MLE and Quantile Evaluation for a Clayton AR(1) Model with Student Marginals (source) GIG Generalized Inverse Gaussian Archimedean Copulas (source) HAXC Hierarchical Archimax Copulas (source) NALC Nested Archimedean Lévy Copulas (source) copula_GARCH The Copula GARCH Model (source) dNAC Densities of Two-Level Nested Archimedean Copulas (source) empiricial_copulas Exploring Empirical Copulas (source) logL_visualization Log-Likelihood Visualization for Archimedean Copulas (source) qrng Quasi-Random Numbers for Copula Models (source) wild_animals Wild Animals: Examples of Nonstandard Copulas (source) Frank-Rmpfr Numerically stable Frank Copulas via Multiprecision (Rmpfr) (source) nacopula-pkg Nested Archimedean Copulas Meet R (source) rhoAMH-dilog Beautiful Spearman's Rho for AMH Copula (source)

The copula package provides

• Classes (S4) of commonly used copulas including elliptical (normal and t; ellipCopula), Archimedean (Clayton, Gumbel, Frank, Joe, and Ali-Mikhail-Haq; ; archmCopula and acopula), extreme value (Gumbel, Husler-Reiss, Galambos, Tawn, and t-EV; evCopula), and other families (Plackett and Farlie-Gumbel-Morgenstern).

• Methods for density, distribution, random number generation (dCopula, pCopula and rCopula); bivariate dependence measures (rho, tau, etc), perspective and contour plots.

• Functions (and methods) for fitting copula models including variance estimates (fitCopula).

• Independence tests among random variables and vectors.

• Serial independence tests for univariate and multivariate continuous time series.

• Goodness-of-fit tests for copulas based on multipliers, and the parametric bootstrap, with several transformation options.

• Bivariate and multivariate tests of extreme-value dependence.

• Bivariate tests of exchangeability.

Now with former package nacopula for working with nested Archimedean copulas. Specifically,

• it provides procedures for computing function values and cube volumes (prob),

• characteristics such as Kendall's tau and tail dependence coefficients (via family objects, e.g., copGumbel),

• efficient sampling algorithms (rnacopula),

• various estimators and goodness-of-fit tests.

• The package also contains related univariate distributions and special functions such as the Sibuya distribution (Sibuya), the polylogarithm (polylog), Stirling and Eulerian numbers (Eulerian).

Further information is available in the following vignettes:

 nacopula-pkg Nested Archimedean Copulas Meet R (../doc/nacopula-pkg.pdf) Frank-Rmpfr Numerically Stable Frank via Multiprecision in R (../doc/Frank-Rmpfr)

For a list of exported functions, use help(package = "copula").

### References

Yan, J. (2007) Enjoy the Joy of Copulas: With a Package copula. Journal of Statistical Software 21(4), 1–21. https://www.jstatsoft.org/v21/i04/.

Kojadinovic, I. and Yan, J. (2010). Modeling Multivariate Distributions with Continuous Margins Using the copula R Package. Journal of Statistical Software 34(9), 1–20. https://www.jstatsoft.org/v34/i09/.

Hofert, M. and Mächler, M. (2011), Nested Archimedean Copulas Meet R: The nacopula Package., Journal of Statistical Software 39(9), 1–20. https://www.jstatsoft.org/v39/i09/.

Nelsen, R. B. (2006) An introduction to Copulas. Springer, New York.

### See Also

The following CRAN packages currently use (‘depend on’) copula: CoClust, copulaedas, Depela, HAC, ipptoolbox, vines.

### Examples

## Some of the more important functions (and their examples) are

example(fitCopula)## fitting Copulas
example(fitMvdc)  ## fitting multivariate distributions via Copulas
example(nacopula) ## nested Archimedean Copulas

## Independence Tests:  These also draw a 'Dependogram':
example(indepTest)       ## Testing for Independence
example(serialIndepTest) ## Testing for Serial Independence



[Package copula version 1.1-3 Index]