zbMATH — the first resource for mathematics

pyvine: the Python package for regular vine copula modeling, sampling and testing. (English) Zbl 1465.62044
Summary: Regular vine copula provides rich models for dependence structure modeling. It combines vine structures and families of bivariate copulas to construct a number of multivariate distributions that can model a wide range dependence patterns with different tail dependence for different pairs. Two special cases of regular vine copulas, C-vine and D-vine copulas, have been extensively investigated in the literature. We propose the Python package, pyvine, for modeling, sampling and testing a more generalized regular vine copula (R-vine for short). R-vine modeling algorithm searches for the R-vine structure which maximizes the vine tree dependence in a sequential way. The maximum likelihood estimation algorithm takes the sequential estimations as initial values and uses L-BFGS-B algorithm for the likelihood value optimization. R-vine sampling algorithm traverses all edges of the vine structure from the last tree in a recursive way and generates the marginal samples on each edge according to some nested conditions. Goodness-of-fit testing algorithm first generates Rosenblatt’s transformed data \(\boldsymbol{E}\) and then tests the hypothesis \(H_0^*:\boldsymbol{E}\sim C_{\perp}\) by using Anderson-Darling statistic, where \(C_{\perp}\) is the independence copula. Bootstrap method is used to compute an adjusted \(p\)-value of the empirical distribution of replications of Anderson-Darling statistic. The computing of related functions of copulas such as cumulative distribution functions, H-functions and inverse H-functions often meets with the problem of overflow. We solve this problem by reinvestigating the following six families of bivariate copulas: Normal, Student \(t\), Clayton, Gumbel, Frank and Joe’s copulas. Approximations of the above related functions of copulas are given when the overflow occurs in the computation. All these are implemented in a subpackage bvcopula, in which subroutines are written in Fortran and wrapped into Python and, hence, good performance is guaranteed.
62E17 Approximations to statistical distributions (nonasymptotic)
62H20 Measures of association (correlation, canonical correlation, etc.)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62-08 Computational methods for problems pertaining to statistics
Full Text: DOI
[1] Aas, K.; Czado, C.; Frigessi, A.; Bakken, H., Pair-copula constructions of multiple dependence, Insur. Math. Econ., 44, 182-198 (2009) · Zbl 1165.60009
[2] Acar, E.; Genest, C.; Nešlehová, J., Beyond simplified pair-copula constructions, J. Multivar. Anal., 110, 74-90 (2012) · Zbl 1243.62067
[3] Bedford, T.; Cooke, RM, Probability density decomposition for conditionally dependent random variables modeled by vines, Ann. Math. Artif. Intell., 32, 245-268 (2001) · Zbl 1314.62040
[4] Bedford, T.; Cooke, RM, Vines—a new graphical model for dependent random variables, Ann Stat, 30, 1031-1068 (2002) · Zbl 1101.62339
[5] Brechmann, EC; Schepsmeier, U., Modeling dependence with C- and D-vine copulas: the R package CDvine, J. Stat. Softw, 52, 1-27 (2013)
[6] Brent, R., Algorithms for Minimization Without Derivatives (1973), Englewood Cliffs: Prentice Hall, Englewood Cliffs · Zbl 0245.65032
[7] Breymann, W.; Dias, A.; Embrechts, P., Dependence structures for multivariate high-frequency data in finance, Quant. Finance, 3, 1-14 (2003) · Zbl 1408.62173
[8] Chapeau-Blondeau, F.; Monir, A., Numerical evaluation of the Lambert \(W\) function and application to generation of generalized Gaussian noise with exponent \(1/2\), IEEE Trans. Signal Process., 50, 2160-2165 (2002) · Zbl 1369.33022
[9] Czado, C.; Schepsmeier, U.; Min, A., Maximum likelihood estimation of mixed C-vines with application to exchange rates, Stat. Model., 12, 229-255 (2012) · Zbl 07257878
[10] Daul, S.; De Giorgi, E.; Lindskog, F.; McNeil, A., Using the grouped t-copula, Risk Mag., 16, 73-76 (2003)
[11] Dißmann, J.; Brechmann, EC; Czado, C.; Kurowicka, D., Selecting and estimating regular vine copulae and application to financial returns, Comput. Stat. Data Anal., 59, 52-69 (2013) · Zbl 1400.62114
[12] Dobrić, J.; Schmid, F., A goodness of fit test for copulas based on Rosenblatt’s transformation, Comput. Stat. Data Anal., 51, 4633-4642 (2007) · Zbl 1162.62343
[13] Donnelly, T., Algorithm 462: bivariate normal distribution, Commun. ACM, 16, 638 (1973)
[14] Fang, H.; Fang, K.; Kotz, S., The meta-elliptical distributions with given marginals, J. Multivar. Anal., 82, 1-16 (2002) · Zbl 1002.62016
[15] Genz, A., Numerical computation of rectangular bivariate and trivariate normal and t probabilities, Stat. Comput., 14, 151-160 (2004)
[16] Haff, IH; Aas, K.; Frigessi, A., On the simplified pair-copula construction—simply useful or too simplistic?, J. Multivar. Anal., 101, 1296-1310 (2010) · Zbl 1184.62079
[17] Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11-15. Pasadena, CA USA (2008)
[18] Hofert, M., Kojadinovic, I., Maechler, M., Yan, J.: Copula: multivariate dependence with copulas. http://CRAN.R-project.org/package=copula (2012)
[19] Hofert, M.; Maechler, M., Nested Archimedean copulas meet R: the nacopula package, J. Stat. Softw., 39, 9, 1-20 (2011)
[20] Hunter, John D., Matplotlib: a 2D graphics environment, Comput. Sci. Eng., 9, 3, 90-95 (2007)
[21] Joe, H., Parametric families of multivariate distributions with given margins, J. Multivar. Anal., 46, 262-282 (1993) · Zbl 0778.62045
[22] Joe, H.: Families of \(m\)-variate distributions with given margins and \(m(m-1)/2\) bivariate dependence parameters. In: Distributions with Fixed Marginals and Related Topics, L. Rüschendorf, B. Schweizer and M.D. Taylor (eds.), vol. 28, pp. 120-141. Lecture Notes—Monograph Series, Institute of Mathematical Statistics, Hayward, CA (1996)
[23] Joe, H., Multivariate Models and Dependence Concepts (1997), London: Chapmann & Hall, London · Zbl 0990.62517
[24] Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python. http://www.scipy.org/ (2001)
[25] Knight, WR, A computer method for calculating Kendall’s tau with ungrouped data, J. Am. Stat. Assoc., 61, 314, 436-439 (1966) · Zbl 0138.13202
[26] Kojadinovic, I.; Yan, J., Modeling multivariate distributions with continuous margins using the copula R package, J. Stat. Softw., 34, 9, 1-20 (2010)
[27] Kurowicka, D., Cooke, R.M.: Distribution—free continuous bayesian belief nets. In: Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice. Santa Fe, New Mexico (2004) · Zbl 1083.62054
[28] Lambda Foundry: pandas: Python data analysis library. http://pandas.pydata.org/ (2012)
[29] Marsaglia, G.; Marsaglia, J., Evaluating the Anderson-Darling distribution, J. Stat. Softw., 9, 1-5 (2004)
[30] Nápoles, OM; Kurowicka, D.; Joe, H., Chapter 9: Counting vines, Dependence Modeling, 189-218 (2011), Singapore: World Scientific, Singapore
[31] Peterson, P., F2PY: a tool for connecting Fortran and Python programs, Int. J. Comput. Sci. Eng., 4, 4, 296-305 (2009)
[32] Piessens, R.; Doncker-Kapenga, ED; Überhuber, CW, QUADPACK: A Subroutine Package for Automatic Integration (1983), Berlin: Springer, Berlin
[33] Prim, RC, Shortest connection networks and some generalizations, Bell Lab Tech. J., 36, 1389-1401 (1957)
[34] R Core Team: R: a Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/ (2012)
[35] Rosenblatt, M., Remarks on a multivariate transformation, Ann. Math. Stat., 23, 470-472 (1952) · Zbl 0047.13104
[36] Savu, C., Trede, M.: Hierarchical Archimedean copulas. In: International Conference on High Frequency Finance. Konstanz, Germany (2006) · Zbl 1270.91086
[37] Schepsmeier, U., Stoeber, J., Brechmann, E.C., Graeler, B., Nagler, T., Erhardt, T.: VineCopula: statistical inference of vine copulas. https://cran.r-project.org/web/packages/VineCopula/index.html (2018)
[38] Whelan, N., Sampling from Archimedean copulas, Quant. Finance, 4, 339-352 (2004) · Zbl 1409.62108
[39] Yan, J., Enjoy the joy of copulas: with a package copula, J. Stat. Softw., 21, 4, 1-21 (2007)
[40] Zhu, C.; Byrd, RH; Lu, P.; Nocedal, J., Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization, ACM Trans. Math. Softw., 23, 4, 550-560 (1997) · Zbl 0912.65057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.