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VineCopulaMatlab toolbox

A MATLAB toolbox for vine copulas based on the C++ shared library VineCopulaCPP

Description of the Vine Copulas with C++ toolbox

The toolbox can be used for high-dimensional dependence modeling with vine copula models. A key feature of the toolbox is a framework, which allows to test whether the simplifying assumption is a reasonable assumption for approximating high-dimensional distributions using simplified vine copula models.

Highlights

• Modeling of (high-)dimensional [0,1]-data by C-vine and D-vine copulas.
• 20 different pair-copula families (62 families with rotated pair-copulas).
• The most important object class VineCopula is implemented in MATLAB.
• Functions for simulating from simplified and non-simplified C- and D-vine copulas. In the case of the non-simplified C- or D-vines, the parameters of all conditional bivariate copulas can be specified as functions of the conditioning variables.
• Functions for selecting and (jointly-)estimating C- and D-vine copula models.
• Functions for (jointly-)estimating C- and D-vine copula models.
• A vectorial independence test, which can be used for testing sequentially the simplified assumption for C- and D-vine copulas.
• Most computations are implemented in C++. Parts can be optionally performed parallel.

Hosting and bug reporting

• The VineCopulaMatlab toolbox is hosted at GitHub and can be found under https://github.com/MalteKurz/VineCopulaMatlab.
• The C++ shared library VineCopulaCPP is also hosted at GitHub and can be found under https://github.com/MalteKurz/VineCopulaCPP.
• Please use GitHub issues (VineCopulaMatlab issues and VineCopulaCPP issues for reporting any issues or bugs.

Demo

Please see the demo for further details about the functionality of the VineCPP toolbox.

Remarks

• The central class of the toolbox is the VineCopula class. Working with this class, for most input variables consistency checks are performed.
• In all the other functions (primarily the functions for two-dimensional (pair-)copulas there are not that many consistency checks performed.
• At the moment within this toolbox you can only choose between C-vines and D-vines. The superclass of regular vines (R-vines) is not implemented yet. If you also want to use R-Vines then the R-package VineCopula (cf. Schepsmeier, Stöber, and Brechmann (2013)) is an excellent alternative.

Dependencies

• A C++-compiler being compatible with the used MATLAB release (cf. http://www.mathworks.de/support/compilers for informations about compatible compilers for different releases and opperating systems).
• The C++ libraries boost (cf. http://www.boost.org/). Used are some functions for statistical distributions from the Boost Math Toolkit. Furthermore, the boost libraries are used for random number generation.
• The nonlinear optimization library NLopt (http://ab-initio.mit.edu/wiki/index.php/NLopt).
• OpenMP for parallel computing (http://openmp.org/wp/).
• The Fortran 77 routine MVTDST (file mvtdstpack.f) from (http://www.math.wsu.edu/faculty/genz/software/software.html; Alan Genz). (It is only needed for computing the CDF of the bivariate normal and t copula.)
• The C++ library VineCopulaCPP (https://github.com/MalteKurz/VineCopulaCPP).

References

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

[2] Acar, E. F., C. Genest and J. Neslehová (2012), "Beyond simplified pair-copula constructions", Journal of Multivariate Analysis 110, pp. 74-90.

[3] Balakrishnan, N. and Lai, C.-D. (2009), "Continuous Bivariate Distributions", 2. ed., New York, NY: Springer.

[4] 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 (1), pp. 245-268.

[5] Bedford, T. and R. M. Cooke (2002), "Vines -- A new graphical model for dependent random variables", The Annals of Statistics 30(4), pp. 1031-1068.

[6] 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), R package version 1.1-13, pp. 1-27, url: http://CRAN.R-project.org/package=CDVine.

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

[8] Eschenburg, P. (2013), "Properties of extreme-value copulas", Diploma Thesis, Fakultät für Mathematik, Technische Universität München, url: http://mediatum.ub.tum.de/download/1145695/1145695.pdf.

[9] Genest, C. and A. Favre (2007), "Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask", Journal of Hydrologic Engineering 12(4), pp. 347-368.

[10] Hobaek-Haff, I., K. Aas and A. Frigessi (2010), "On the simplified pair-copula construction -- Simply useful or too simplistic?", Journal of Multivariate Analysis 101(5), pp. 1296-1310.

[11] Joe, H. (1996), "Families of m-Variate Distributions With Given Margins and m(m-1)/2 Bivariate Dependence Parameters", Distributions with Fixed Marginals and Related Topics, ed. by L. Rüschendorf, B. Schweizer, and M. D. Taylor, Hayward, CA: Institute of Mathematical Statistics.

[12] Joe, H. (1997), Multivariate models and dependence concepts, 1. ed., reprint., Monographs on statistics and applied probability; 73, Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC.

[13] Kojadinovic, I. and M. Holmes (2009), "Tests of independence among continuous random vectors based on Cramér-von Mises functionals of the empirical copula process", Journal of Multivariate Analysis 100(6), pp. 1137-1154.

[14] Kosorok, M. R. (2008), Introduction to Empirical Processes and Semiparametric Inference, Springer Series in Statistics, New York, NY: Springer.

[15] Kurowicka, D. and H. Joe (Eds.) (2011), "Dependece Modeling -- Vine Copula Handbook", Singapore: World Scientific Publishing Co. Pte. Ltd.

[16] Nelsen, R. B. (2006), "An introduction to copulas", 2. ed., Springer series instatistics, New York, NY: Springer.

[17] Omelka, M. and M. Pauly (2012), "Testing equality of correlation coefficients in two populations via permutation methods, Journal of Statistical Planning and Inference 142, pp. 1396-1406.

[18] Patton, A. J. (2002), "Applications of Copula Theory in Financial Econometrics", Unpublished Ph.D. dissertation, University of Colifornia, San Diego, url: http://www.amstat.org/sections/bus_econ/papers/patton_dissertation.pdf.

[19] Patton, A. J. (2006), "Modelling asymmetric exchange rate dependence", International Economic Review 47(2), pp. 527-556.

[20] Quessy, J.-F. (2010), "Applications and asymptotic power of marginal-free tests of stochastic vectorial independence", Journal of Statistical Planning and Inference 140(11), pp. 3058-3075.

[21] Rémillard, B. (2013), "Statistical Methods For Financial Engineering", Boca Raton, FL: Chapman & Hall.

[22] Rémillard, B. and O. Scaillet (2009), "Testing for equality between two copulas", Journal of Multivariate Analysis 100(3), pp. 377-386.

[23] Schepsmeier, U., J. Stöber and E. C. Brechmann (2013), VineCopula: Statistical inference of vine copulas, R package version 1.2, url: http://CRAN.R-project.org/package=VineCopula.

[24] Segers, J. (2012), "Asymptotics of empirical copula processes under non-restrictive smoothness assumptions", Bernoulli 18(3), pp. 764-782.

[25] Spanhel, F., Kurz, M.S., 2015. Simplified vine copula models: Approximations based on the simplifying assumption. ArXiv e-prints https://arxiv.org/abs/1510.06971.

[26] Stöber, J., H. Joe and C. Czado (2013), "Simplified pair copula constructions -- Limitations and extensions", Journal of Multivariate Analysis 119, pp. 101-118.

[27] van der Vaart, A. W. and J. A. Wellner (1996), Weak Convergence and Empirical Processes -- With Applications to Statistics, Springer Series in Statistics, New York [u.a.]: Springer.

Author: Malte Kurz