CDVine-package: Statistical inference of C- and D-vine copulas
In CDVine: Statistical Inference of C- And D-Vine Copulas
Description
Details
Bivariate copula families
C- and D-vine copula models
Acknowledgment
Author(s)
References
Description
Functions for statistical inference of canonical vine (C-vine) and D-vine copulas.
Tools for bivariate exploratory data analysis and for bivariate as well as vine copula selection are provided.
Models can be estimated either sequentially or by joint maximum likelihood estimation.
Sampling algorithms and plotting methods are also included.
Data is assumed to lie in the unit hypercube (so-called copula data).
Details
Package: CDVine
Type: Package
Version: 1.4
Date: 2015-10-29
License: GPL (>=2)
Depends: R (>= 2.11.0)
Imports: MASS, mvtnorm, graphics, igraph, stats
LazyLoad: yes
Bivariate copula families
In this package several bivariate copula families are included for bivariate analysis as well as for 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 bandwidth of possible dependence structures.
For the Archimedean copula families rotated versions are included to cover negative dependence too.
The two parameter BB1, BB6, BB7 and BB8 copulas are however numerically instable for large parameters,
in particular, if BB6, BB7 and BB8 copulas are close to the Joe copula which is a boundary case of these three copula families.
In general, the user should be careful with extreme parameter choices.
The following table shows the parameter ranges of bivariate copula families with parameters par and par2:
Copula family par par2
Gaussian (-1,1) -
Student t (-1,1) (2,∞)
(Survival) Clayton (0,∞) -
(Survival) Gumbel [1,∞) -
Frank R\backslash\{0\} -
(Survival) Joe (1,∞) -
Rotated Clayton (90 and 270 degrees) (-∞,0) -
Rotated Gumbel (90 and 270 degrees) (-∞,-1] -
Rotated Joe (90 and 270 degrees) (-∞,-1) -
(Survival) Clayton-Gumbel (BB1) (0,∞) [1,∞)
(Survival) Joe-Gumbel (BB6) [1,∞) [1,∞)
(Survival) Joe-Clayton (BB7) [1,∞) (0,∞)
(Survival) Joe-Frank (BB8) [1,∞) (0,1]
Rotated Clayton-Gumbel (90 and 270 degrees) (-∞,0) -∞,-1]
Rotated Joe-Gumbel (90 and 270 degrees) (-∞,-1] (-∞,-1]
Rotated Joe-Clayton (90 and 270 degrees) (-∞,-1] (-∞,0)
Rotated Joe-Frank (90 and 270 degrees) (-∞,-1] [-1,0)
C- and D-vine copula models
When specifying C- and D-vine copula models, one has to select an order of the variables.
For a D-vine the order of the variables in the first tree has to be chosen and for a C-vine the root nodes for each tree need to be determined.
Functions for inference of C- and D-vine copula models in this package assume that the order of the variables in the data set under investigation
exactly corresponds to this C- or D-vine order.
E.g., in a C-vine the first column of a data set is the first root node, the second column the second root node, etc.
According to this order arguments have to be provided to functions for C- and D-vine copula inference.
After choosing the type of the vine model,
the copula families (family) and parameters (par and par2) have to be specified as vectors of length d(d-1)/2,
where d is the number of variables.
In a C-vine, the entries of this vector correspond to the following pairs and associated pair-copula terms
(1,2), (1,3), (1,4), ..., (1,d),
(2,3|1), (2,4|1), ..., (2,d|1),
(3,4|1,2), (3,5|1,2), ..., (3,d|1,2),
...,
(d-1,d|1,...,d-2).
Similarly, the pairs of a D-vine are denoted in the following order:
(1,2), (2,3), (3,4), ..., (d-1,d),
(1,3|2), (2,4|3), ..., (d-2,d|d-1),
(1,4|2,3), (2,5|3,4), ..., (d-3,d|d-2,d-1),
...,
(1,d|2,...,d-1).
Acknowledgment
A first version of this package was based on and inspired by code from Daniel Berg
(Norwegian Computing Center; http://www.danielberg.no) provided by personal communication.
We further acknowledge substantial contributions by our working group at Technische Universitaet Muenchen,
in particular by Carlos Almeida and Aleksey Min.
In addition, we like to thank Shing (Eric) Fu, Feng Zhu, Guang (Jack) Yang, and Harry Joe for providing their implementation
of the method by Knight (1966) for efficiently computing the empirical Kendall's tau.
We are especially grateful to Harry Joe for his contributions to the implementation of the bivariate Archimedean copulas.
Author(s)
Ulf Schepsmeier, Eike Christian Brechmann <CDVine@ma.tum.de>
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.
Submitted for publication.
http://mediatum.ub.tum.de/doc/1079276/1079276.pdf.
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 and Data Analysis, 59(1), 52-69.
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.
Chapman and Hall, London.
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.
CDVine documentation built on May 2, 2019, 9:28 a.m.
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Description Details Bivariate copula families C- and D-vine copula models Acknowledgment Author(s) References
Description
Functions for statistical inference of canonical vine (C-vine) and D-vine copulas. Tools for bivariate exploratory data analysis and for bivariate as well as vine copula selection are provided. Models can be estimated either sequentially or by joint maximum likelihood estimation. Sampling algorithms and plotting methods are also included. Data is assumed to lie in the unit hypercube (so-called copula data).
Details
| Package: | CDVine |
| Type: | Package |
| Version: | 1.4 |
| Date: | 2015-10-29 |
| License: | GPL (>=2) |
| Depends: | R (>= 2.11.0) |
| Imports: MASS, mvtnorm, graphics, igraph, stats | |
| LazyLoad: | yes |
Bivariate copula families
In this package several bivariate copula families are included for bivariate analysis as well as for 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 bandwidth of possible dependence structures. For the Archimedean copula families rotated versions are included to cover negative dependence too. The two parameter BB1, BB6, BB7 and BB8 copulas are however numerically instable for large parameters, in particular, if BB6, BB7 and BB8 copulas are close to the Joe copula which is a boundary case of these three copula families. In general, the user should be careful with extreme parameter choices.
The following table shows the parameter ranges of bivariate copula families with parameters par and par2:
| Copula family | par | par2 |
| Gaussian | (-1,1) | - |
| Student t | (-1,1) | (2,∞) |
| (Survival) Clayton | (0,∞) | - |
| (Survival) Gumbel | [1,∞) | - |
| Frank | R\backslash\{0\} | - |
| (Survival) Joe | (1,∞) | - |
| Rotated Clayton (90 and 270 degrees) | (-∞,0) | - |
| Rotated Gumbel (90 and 270 degrees) | (-∞,-1] | - |
| Rotated Joe (90 and 270 degrees) | (-∞,-1) | - |
| (Survival) Clayton-Gumbel (BB1) | (0,∞) | [1,∞) |
| (Survival) Joe-Gumbel (BB6) | [1,∞) | [1,∞) |
| (Survival) Joe-Clayton (BB7) | [1,∞) | (0,∞) |
| (Survival) Joe-Frank (BB8) | [1,∞) | (0,1] |
| Rotated Clayton-Gumbel (90 and 270 degrees) | (-∞,0) | -∞,-1] |
| Rotated Joe-Gumbel (90 and 270 degrees) | (-∞,-1] | (-∞,-1] |
| Rotated Joe-Clayton (90 and 270 degrees) | (-∞,-1] | (-∞,0) |
| Rotated Joe-Frank (90 and 270 degrees) | (-∞,-1] | [-1,0) |
C- and D-vine copula models
When specifying C- and D-vine copula models, one has to select an order of the variables.
For a D-vine the order of the variables in the first tree has to be chosen and for a C-vine the root nodes for each tree need to be determined.
Functions for inference of C- and D-vine copula models in this package assume that the order of the variables in the data set under investigation
exactly corresponds to this C- or D-vine order.
E.g., in a C-vine the first column of a data set is the first root node, the second column the second root node, etc.
According to this order arguments have to be provided to functions for C- and D-vine copula inference.
After choosing the type of the vine model,
the copula families (family) and parameters (par and par2) have to be specified as vectors of length d(d-1)/2,
where d is the number of variables.
In a C-vine, the entries of this vector correspond to the following pairs and associated pair-copula terms
(1,2), (1,3), (1,4), ..., (1,d),
(2,3|1), (2,4|1), ..., (2,d|1),
(3,4|1,2), (3,5|1,2), ..., (3,d|1,2),
...,
(d-1,d|1,...,d-2).
Similarly, the pairs of a D-vine are denoted in the following order:
(1,2), (2,3), (3,4), ..., (d-1,d),
(1,3|2), (2,4|3), ..., (d-2,d|d-1),
(1,4|2,3), (2,5|3,4), ..., (d-3,d|d-2,d-1),
...,
(1,d|2,...,d-1).
Acknowledgment
A first version of this package was based on and inspired by code from Daniel Berg (Norwegian Computing Center; http://www.danielberg.no) provided by personal communication. We further acknowledge substantial contributions by our working group at Technische Universitaet Muenchen, in particular by Carlos Almeida and Aleksey Min. In addition, we like to thank Shing (Eric) Fu, Feng Zhu, Guang (Jack) Yang, and Harry Joe for providing their implementation of the method by Knight (1966) for efficiently computing the empirical Kendall's tau. We are especially grateful to Harry Joe for his contributions to the implementation of the bivariate Archimedean copulas.
Author(s)
Ulf Schepsmeier, Eike Christian Brechmann <CDVine@ma.tum.de>
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. Submitted for publication. http://mediatum.ub.tum.de/doc/1079276/1079276.pdf.
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 and Data Analysis, 59(1), 52-69.
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. Chapman and Hall, London.
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.
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