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https://ceur-ws.org/Vol-2540/FAIR2019_paper_20.pdf
Modifying copulas for improved dependence
modelling
Colette le Roux1 and Alta de Waal12
1
Department of Statistics, University of Pretoria
2
Center for Artificial Intelligence Research (CAIR)
1 Introduction
In 2007 and 2008, underestimation of correlations and risks, as well as the misuse
of dependence models, lead to the financial crisis [5]. This highlighted the need
to improve dependence modelling through both the correlation parameter and
choice of model used. Copulas are useful for modelling dependence patterns in
multivariate data, as well as prediction in regression analysis [1].
The problem is that most traditional methods for dealing with complex de-
pendency structures assume a parametric or Gaussian distribution and linear
correlation structure, but these assumptions are often violated in practical ap-
plications [3, 8]. Furthermore, the two main approaches to handling outliers or
missing data are to either remove them, or replace them with some other appro-
priate value, but there are instances, such as in risk-management, where these
anomalous observations are of key importance and cannot be eliminated. In these
cases, appropriate methods are needed to model tail dependencies.
Uncertainty from volatilities, heteroskedasticity, extreme values and missing
observations all contribute to the difficulty of dependency estimation and pre-
diction. While ignoring underlying covariates might yield reasonably accurate
models in some instances, time (as a covariate) has been found to have an influ-
ence on copula parameters when modelling financial data, and could therefore
lead to improved prediction and estimation when taken into account [2]. Vine
copulas can be applied to address these problems in the multidimensional case,
where assumptions to deal with the model complexity are relaxed. A vine copula
is a hierarchical factorisation of a high-dimensional copula into the product of
bivariate copula densities.
The first problem in high-dimensional dependence structure models is that
the computational cost of approximation and parameter estimation increases as
the dimension increases [4], making traditional bivariate copula methods, such as
MLE [7] and MCMC [6] practically infeasible. The second problem is that vine
copulas allow for the analysis of multivariate copulas, but due to the complexity
of calculating conditional copulas, the restrictive truncation and simplification
assumptions are often applied. A vine copula is proposed to relax assumptions
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2 C le Roux, A de Waal
and simplify computation, ultimately leading to a more flexible and reliable
model.
2 Methodology
Given the wide variety of available bivariate copula families specifically designed
to model symmetric and asymmetric distributions with central and tail depen-
dencies, copulas are capable of modelling extreme dependencies between vari-
ables. A copula can be improved by allowing for underlying variables that in-
fluence the strength of dependencies by use of a conditional copula. As a non-
parametric approach, a copula process combines a copula and a Gaussian Process
(GP) to allow for non-Gaussian distributions. The GP in turn uses a Bayesian
framework to deal with missing observations, adding extra flexibility to the cop-
ula density. A Gaussian process conditional copula can now be built to improve
on the conditional copula [2], using Bayesian non-parametrics to learn the latent
functions that specify the shape of the conditional copulas given the condition-
ing variables and thereby simplifying computation.
When working with the multivariate case, the Gaussian copula can easily
capture high-dimensional dependence structures, but is unable to capture asym-
metric tail dependencies, making it less appropriate for complex dependence
structures. A vine-copula can be applied to model complex dependency struc-
tures between multivariate data by decomposing a multivariate copula into a
hierarchy of bivariate copulas. This model provides flexibility in that the bivari-
ate copulas can come from any parametric or non-parametric family and can be
either conditional or unconditional. The decomposition further avoids the com-
putational cost of the dependence optimisation problem in approximations.
The importance of improving the accuracy of dependency modelling in appli-
cations such as finance, econometrics, insurance and meteorology is self-evident,
considering the potential risks involved in erroneous estimation and prediction
results. In this work, we investigate the advantages, limitations and differences
of copulas and vine-copulas in complex dependence structures. Prediction and
estimation of complicated dependence structures is expected to improve when
modifying a copula to a vine copula. It is also expected that relaxing the assump-
tions commonly applied to the vine copula in applications with high-dimensional
dependency structures, such as independence between the conditional copula and
its conditioning variable, will improve model accuracy, since underlying covari-
ates (time in particular) has been found to have an effect on the dependency
structure between the main variables. The investigation of conditional copulas
and copula processes is reserved for future work.
Keywords: Copula · Copula processes · Gaussian processes · Vine copulas ·
Bayesian methods.
Modifying copulas for improved dependence modelling 3
References
1. Acar, E.F., Azimaee, P., Hoque, M.E.: Predictive assessment of copula models.
Canadian Journal of Statistics 47(1), 8–26 (2019)
2. Hernández-Lobato, J.M., Lloyd, J.R., Hernández-Lobato, D.: Gaussian process con-
ditional copulas with applications to financial time series. In: Advances in Neural
Information Processing Systems. pp. 1736–1744. MIT Press (2013)
3. Jaimungal, S., Ng, E.K.: Kernel-based copula processes. In: Joint European Con-
ference on Machine Learning and Knowledge Discovery in Databases. pp. 628–643.
Springer, Springer, Berlin, Heidelberg (Sep 2009)
4. Sabeti, A., Wei, M., Craiu, R.V.: Additive models for conditional copulas. Stat 3(1),
300–312 (Mar 2014)
5. Salmon, F.: The formula that killed wall street. Significance 9(1), 16–20 (Feb 2012)
6. Tekumalla, L.S., Rajan, V., Bhattacharyya, C.: Vine copulas for mixed data: multi-
view clustering for mixed data beyond meta-gaussian dependencies. Machine Learn-
ing 106(9-10), 1331–1357 (2017)
7. Xu, D., Wei, Q., Elsayed, E.A., Chen, Y., Kang, R.: Multivariate degradation model-
ing of smart electricity meter with multiple performance characteristics via vine cop-
ulas. Quality and Reliability Engineering International 33(4), 803–821 (Jun 2017)
8. Zheng, W., Ren, X., Zhou, N., Jiang, D., Li, S.: Mixture of d-vine copulas for chem-
ical process monitoring. Chemometrics and Intelligent Laboratory Systems 169,
19–34 (Oct 2017)