Statistical inference of vine copulas
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README.md

VineCopula

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Vine copulas are a flexible class of dependence models consisting of bivariate building blocks (see e.g., Aas et al., 2009). You can find a comprehensive list of publications and other materials on vine-copula.org.

This package is primarily made for the statistical analysis of vine copula models. The package includes tools for parameter estimation, model selection, simulation, goodness-of-fit tests, and visualization. Tools for estimation, selection and exploratory data analysis of bivariate copula models are also provided. Please see the API documentation for a detailed description of all functions.

Table of contents


How to install

You can install:

  • the stable release on CRAN:

    install.packages("VineCopula")
  • the latest development version:

    devtools::install_github("tnagler/VineCopula")

Package overview

Below, we list most functions and features you should know about. As usual in copula models, data are assumed to be serially independent and lie in the unit hypercube.

Bivariate copula modeling: the BiCop-family

  • BiCop: Creates a bivariate copula by specifying the family and parameters (or Kendall's tau). Returns an object of class BiCop. The class has the following methods:

    • print, summary: a brief or comprehensive overview of the bivariate copula, respectively.

    • plot, contour: surface/perspective and contour plots of the copula density. Possibly coupled with standard normal margins (default for contour).

  • BiCopSim: Simulates from a bivariate copula.

  • BiCopEst: Estimates parameters of a bivariate copula with a prespecified family. Estimation can be done by maximum likelihood (method = "mle") or inversion of the empirical Kendall's tau (method = "itau", only available for one-parameter families). Returns an object of class BiCop.

  • BiCopSelect: Estimates the parameters of a bivariate copula for a set of families and selects the best fitting model (using either AIC or BIC). Returns an object of class BiCop.

  • BiCopGofTest: Goodness-of-Fit tests for bivariate copulas.

  • BiCopVuongClarke: Vuong and Clarke tests for model comparison within a prespecified set of copula families.

  • BiCopPar2Tau, BiCopTau2Par, BiCopPar2Beta, BiCopPar2TailDep: Conversion between dependence measures and parameters (for a given family). Functions are vectorized in all arguments.

  • Evaluate functions related to a bivariate copula: BiCopPDF, BiCopCDF, BiCopDeriv, BiCopDeriv2, BiCopHfunc, BiCopHfuncDeriv, BiCopHfuncDeriv2, BiCopHinv. Functions are vectorized in the family, par, and par2 arguments.

  • BiCopKDE: Kernel density plots for copula data.

  • BiCopLambda, BiCopKPlot, BiCopChiPlot: Further plot types for the analysis of bivariate copulas.

For most functions, you can provide an object of class BiCop instead of specifying family, par and par2 manually.

Vine copula modeling: the RVine-family

  • RVineMatrix: Creates a vine copula model by specifying structure, family and parameter matrices. Such matrices have been introduced by Dissman et al. (2013). Returns an object of class RVineMatrix. The class has the following methods:

    • plot: Plots the trees of the the R-vine tree structure. Optionally, you can annotate the edges with pair-copula families and parameters.

    • contour: Creates a matrix of contour plots associated with the pair-copulas.

  • RVineSim: Simulates from a vine copula model.

  • RVineSeqEst: Estimates the parameters of a vine copula model with prespecified structure and families.

  • RVineCopSelect: Estimates the parameters and selects the best family for a vine copula model with prespecified structure matrix.

  • RVineStructureSelect: Fits a vine copula model assuming no prior knowledge. It selects the R-vine structure using Dissmann et al. (2013)'s method, estimates parameters for various families, and selects the best family for each pair.

  • RVineGoFTest: Goodness-of-Fit tests for a vine copula model (c.f., Schepsmeier, 2013, 2015). Related functions are RVineGrad, RVineHessian, RVineStdError, and RVinePIT.

  • RVineVoungTest, RVineClarkeTest: Vuong and Clarke tests for comparing two vine copula models.

  • RVinePar2Tau, RVinePar2Beta: Calculate dependence measures corresponding to a vine copula model.

  • RVinePDF, RVineLogLik, RVineAIC, RVineBIC: Calculate the density, log-likelihood, AIC, and BIC of a vine copula.

Additional features

The functions C2RVine and D2RVine create RVineMatrix objects for C- and D-vine copulas. This is particularly useful for former users of the CDVine package.

Furthermore, bivariate and vine copula models from this packages can be used with the copula package (Hofert et al., 2015). For example, vineCopula transforms an RVineMatrix object into an object of class vineCopula which provides methods for dCopula, pCopula, and rCopula. For more details, we refer to the package manual.

Bivariate copula families

In this package several bivariate copula families are included for bivariate and multivariate analysis using vine copulas. It provides functionality of elliptical (Gaussian and Student-t) as well as Archimedean (Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8) copulas to cover a large range of dependence patterns. For Archimedean copula families, rotated versions are included to cover negative dependence as well.

The Tawn copula is a non-exchangable extension of the Gumbel copula with three parameters. For simplicity, we implemented two versions of the Tawn copula with two parameters each. Each type has one of the asymmetry parameters fixed to 1, so that the corresponding copula density is either left- or right-skewed (relative to the main diagonal). In the manual we will call these two new copulas "Tawn type 1" and "Tawn type 2".

The following table shows the parameter ranges of bivariate copula families with parameters par and par2 and internal coding family:

Copula family family par par2
Gaussian 1 (-1, 1) -
Student t 2 (-1, 1) (2,Inf)
(Survival) Clayton 3, 13 (0, Inf) -
Rotated Clayton (90 and 270 degrees) 23, 33 (-Inf, 0) -
(Survival) Gumbel 4, 14 [1, Inf) -
Rotated Gumbel (90 and 270 degrees) 24, 34 (-Inf, -1] -
Frank 5 R \ {0} -
(Survival) Joe 6, 16 (1, Inf) -
Rotated Joe (90 and 270 degrees) 26, 36 (-Inf, -1) -
(Survival) Clayton-Gumbel (BB1) 7, 17 (0, Inf) [1, Inf)
Rotated Clayton-Gumbel (90 and 270 degrees) 27, 37 (-Inf, 0) (-Inf, -1]
(Survival) Joe-Gumbel (BB6) 8, 18 [1 ,Inf) [1, Inf)
Rotated Joe-Gumbel (90 and 270 degrees) 28, 38 (-Inf, -1] (-Inf, -1]
(Survival) Joe-Clayton (BB7) 9, 19 [1, Inf) (0, Inf)
Rotated Joe-Clayton (90 and 270 degrees) 29, 39 (-Inf, -1] (-Inf, 0)
(Survival) Joe-Frank (BB8) 10, 20 [1, Inf) (0, 1]
Rotated Joe-Frank (90 and 270 degrees) 30, 40 (-Inf, -1] [-1, 0)
(Survival) Tawn type 1 104, 114 [1, Inf) [0, 1]
Rotated Tawn type 1(90 and 270 degrees) 124, 134 (-Inf, -1] [0, 1]
(Survival) Tawn type 2 204, 214 [1, Inf) [0, 1]
Rotated Tawn type 2 (90 and 270 degrees) 224, 234 (-Inf, -1] [0, 1]

Associated shiny apps

rvinegraph

This small shiny app enables the user to draw nice tree plots of an R-Vine copula model using the package d3Network. Models have to be set up locally in an RVineMatrix object and uploaded as .RData file. The page is still under construction.
Author: Ulf Schepsmeier

https://rvinegraph.shinyapps.io/rvinegraph


References

Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 (2), 182-198.

Bedford, T. and R. M. Cooke (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial intelligence 32, 245-268.

Bedford, T. and R. M. Cooke (2002). Vines - a new graphical model for dependent random variables. Annals of Statistics 30, 1031-1068.

Brechmann, E. C., C. Czado, and K. Aas (2012). Truncated regular vines in high dimensions with applications to financial data. Canadian Journal of Statistics 40 (1), 68-85.

Brechmann, E. C. and C. Czado (2011). Risk management with high-dimensional vine copulas: An analysis of the Euro Stoxx 50. Statistics & Risk Modeling, 30 (4), 307-342.

Brechmann, E. C. and U. Schepsmeier (2013). Modeling Dependence with C- and D-Vine Copulas: The R Package CDVine. Journal of Statistical Software, 52 (3), 1-27. http://www.jstatsoft.org/v52/i03/.

Czado, C., U. Schepsmeier, and A. Min (2012). Maximum likelihood estimation of mixed C-vines with application to exchange rates. Statistical Modelling, 12(3), 229-255.

Dissmann, J. F., E. C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59 (1), 52-69.

Eschenburg, P. (2013). Properties of extreme-value copulas Diploma thesis, Technische Universitaet Muenchen http://mediatum.ub.tum.de/node?id=1145695.

Hofert, M., I. Kojadinovic, M. Maechler, and J. Yan (2015). copula: Multivariate Dependence with Copulas. R package version 0.999-13 https://cran.r-project.org/package=VineCopula

Joe, H. (1996). Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In L. Rueschendorf, B. Schweizer, and M. D. Taylor (Eds.), Distributions with fixed marginals and related topics, pp. 120-141. Hayward: Institute of Mathematical Statistics.

Joe, H. (1997). Multivariate Models and Dependence Concepts. London: Chapman and Hall.

Knight, W. R. (1966). A computer method for calculating Kendall's tau with ungrouped data. Journal of the American Statistical Association 61 (314), 436-439.

Kurowicka, D. and R. M. Cooke (2006). Uncertainty Analysis with High Dimensional Dependence Modelling. Chichester: John Wiley.

Kurowicka, D. and H. Joe (Eds.) (2011). Dependence Modeling: Vine Copula Handbook. Singapore: World Scientific Publishing Co.

Nelsen, R. (2006). An introduction to copulas. Springer

Nagler, T. (2015). kdecopula: Kernel Smoothing for Bivariate Copula Densities. R package version 0.6.0. https://cran.r-project.org/package=kdecopula

Schepsmeier, U. and J. Stoeber (2012). Derivatives and Fisher information of bivariate copulas. Statistical Papers, 55 (2), 525-542. http://link.springer.com/article/10.1007/s00362-013-0498-x.

Schepsmeier, U. (2013) A goodness-of-fit test for regular vine copula models. Preprint. http://arxiv.org/abs/1306.0818.

Schepsmeier, U. (2015) Efficient information based goodness-of-fit tests for vine copula models with fixed margins. Journal of Multivariate Analysis 138, 34-52.

Stoeber, J. and U. Schepsmeier (2013). Estimating standard errors in regular vine copula models. Computational Statistics, 28 (6), 2679-2707 http://link.springer.com/article/10.1007/s00180-013-0423-8.

White, H. (1982) Maximum likelihood estimation of misspecified models, Econometrica, 50, 1-26.