Directional Dependence via Copulas: Examples and Applications in Engineering
DOI:
https://doi.org/10.22517/23447214.25954Keywords:
Dependencia direccional, Modelos de cópulas, Dependencia en colas, Eventos extremos, Ingeniería hidráulica, Ingeniería estructural, Modelado multivariadoAbstract
Abstract— In many engineering systems, the variables of interest do not merely exhibit average levels of correlation; rather, their relationship changes markedly under extreme conditions. Directional dependence refers to the ability to capture such asymmetric relationships, particularly in the upper tail (maximal events) or the lower tail (minimal events). Copulas provide a flexible and robust framework for modeling this form of dependence, independently of the marginal distributions of each variable.
This article seeks to elucidate this phenomenon, detailing how it can be quantified, the methodological tools available, and its practical relevance through two real-world applications: one in hydraulic engineering (extreme precipitation events) and another in structural engineering (combined extreme loads). In addition, R code and high-resolution visualizations are provided to facilitate reproducibility and broader adoption of these analyses by engineering practitioners.
Downloads
References
T. Morais, G. A. Barberes, I. V. A. F. Souza, F. G. Leal, J. V. P. Guzzo, and A. L. D. Spigolon, “Pearson correlation coefficient applied to petroleum system characterization: The case study of Potiguar and Recôncavo basins, Brazil,” Geosciences, vol. 13, no. 9, p. 282, 2023.
J. Rymarz, M. Rymarz, T. Uhl, and P. Uhl, “Application of Spearman’s method for the analysis of vehicle downtime and age on availability,” Applied Sciences, vol. 12, no. 6, p. 2921, 2022.
E. A. Sungur and S. Çelebioğlu, “Copulas with directional dependence property,” Gazi University Journal of Science, vol. 24, no. 3, pp. 415–424, 2011.
C. Spearman, “The proof and measurement of association between two things,” American Journal of Psychology, vol. 15, no. 1, pp. 72–101, 1904.
K. Pearson, “Notes on regression and inheritance in the case of two parents,” Proceedings of the Royal Society of London, vol. 58, pp. 240–242, 1895.
P. Schober, C. Boer, and L. A. Schwarte, “Correlation coefficients: Appropriate use and interpretation,” Anesthesia & Analgesia, vol. 126, no. 5, pp. 1763–1768, 2018.
T. Afrin, A. S. Iquebal, M. Karimi, A. Souris, S. Y. Lee, and B. K. Mallick, “Directionally dependent multi-view clustering using copula model,” PLOS ONE, vol. 15, no. 10, e0238996, 2020.
D. Kim and J. M. Kim, “Analysis of directional dependence using asymmetric copula-based regression models,” Journal of Statistical Computation and Simulation, vol. 84, no. 9, pp. 1990–2010, 2014.
E. Liebscher, “Construction of asymmetric multivariate copulas,” Journal of Multivariate Analysis, vol. 99, no. 10, pp. 2234–2250, 2008.
Y. S. Jung, J. M. Kim, and J. Kim, “New approach of directional dependence in exchange markets using generalized FGM copula function,” Communications in Statistics: Simulation and Computation, vol. 37, no. 4, pp. 772–788, 2008.
S. Huang, X. Chen, Y. Zhang, and H. Li, “Copula-based estimation of directional extreme wind speeds,” Science of the Total Environment, In Press, 2024.
X. Zhang, L. Zhi, and X. Li, “Influence of dependence of directional extreme wind speeds when estimating the wind load,” Journal of Wind Engineering & Industrial Aerodynamics, vol. 152, pp. 41–52, 2016.
H. Zhou, J. Liu, C. Gao, W. Li, S. Ou, Y. Zhou, and Q. Luan, “Copula-based joint impact assessment of rainfall and tidal level on flood risk in tidal-influenced plain river network areas, Taihu Lake Basin,” Journal of Hydrology, vol. 653, p. 132785, 2025.
M. Khajehali, A. H. Bagherzadeh, A. R. Karami, and M. Nikoo, “A copula-based multivariate flood frequency analysis,” Scientific Reports, vol. 15, p. 84543, 2025.
C. Yu, D. Wang, V. P. Singh, P. Xu, A. Zhang, Z. Yang, Z. Wang, X. Zeng, J. Jiang, and J. Wu, “An ensemble vine copula quantile regression model with non-stationary margins (EVQR-NS) for soil moisture prediction,” Journal of Hydrology, vol. 659, p. 133248, 2025.
J. Sung, J. Kim, S. Kim, and Y.-S. Jung, “Comparative study of low flow frequency analysis using bivariate copula models,” Water, vol. 11, no. 6, p. 79, 2024.
M.-Z. Lyu, Z.-J. Fei, and D.-C. Feng, “Copula-based cloud analysis for seismic fragility and its application to nuclear power plant structures,” Engineering Structures, vol. 305, p. 117754, 2024.
D. A. Alexandre, C. Chaudhuri, and J. Gill-Fortin, “Novel extensions to the Fisher copula to model flood spatial dependence over North America,” Hydrology and Earth System Sciences, vol. 28, pp. 5069–5085, 2024.
E. De Amo, B. González, J. Martínez-Moreno, and A. Ortega, “Directional dependence orders of random vectors,” Mathematics, vol. 12, no. 3, p. 419, 2024.
R. B. Nelsen, An Introduction to Copulas, 2nd ed., Springer, 2006.
R Core Team, R: A Language and Environment for Statistical Computing, Vienna, 2025.
A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management, Princeton University Press, 2005.
C. Genest, B. Rémillard, A. C. Favre, and J. Nešlehová, “Copula modeling from Abe Sklar to the present day,” Science of Probability and Statistics, 2024.
P. Jaworski, F. Durante, W. K. Hardle, and T. Rychlik, Copula Theory and Its Applications, Springer, 2010.
H. H. Ahmad, S. A. Bantan, M. Elgarhy, and A. H. Hendi, “Copula-linked modified Fréchet–exponential distributions: Maximum likelihood and IFM methods with real-world applications,” Mathematics, vol. 14, no. 6, p. 431, 2025.
E. M. Almetwally, A. H. Abd El‐Ghaly, M. S. Eliwa, and M. M. Mohie El‐Din, “Advanced copula-based models for type II censored data,” Mathematics, vol. 12, no. 12, p. 1774, 2024.
M. Mohammadi, M. A. Amini, and M. Emadi, “A simulation study of semiparametric estimation in copula models based on minimum alpha-divergence,” Computational Statistics, vol. 36, no. 2, pp. 613–640, 2021.
K. Otieno, L. Chaba, E. Omondi, and B. Omolo, “A hierarchical Archimedean copula model for climatic variables: An application to Kenyan data,” Frontiers in Applied Mathematics and Statistics, vol. 11, 2025.
S. Shrivastava, S. Gairola, A. K. Lohani, and A. Kumar, “Copula-based dependency modelling of hydraulic extremes in precipitation data,” Journal of Hydrology, vol. 603, p. 127006, 2025.
X. W. Zheng, H. K. Lam, Y. Xu, J. Li, and Y. Li, “Hybrid Bayesian-copula-based damage probability of tall buildings under concurrent seismic and strong wind,” Engineering Structures, vol. 300, p. 116753, 2024.
X. S. Tang, Y. F. Liu, W. Zhang, and S. C. Li, “Impact of copula selection on geotechnical reliability under incomplete probability information,” Geotechnical Reliability Engineering, vol. 43, pp. 57–70, 2013.
J. Rózsás, “The effect of copulas on time-variant reliability involving time-continuous stochastic processes,” Structural Safety, vol. 32, no. 4, pp. 335–345, 2010.
D. Meyer, T. Nagler, T. F. Münch, and S. N. Murphy, “Copula-based synthetic data augmentation for machine-learning emulators,” Geoscientific Model Development, vol. 14, pp. 5205–5222, 2021.
T. Nagler, C. Bumann, and C. Czado, “Model selection in sparse high-dimensional vine copula models…,” Journal of Multivariate Analysis, vol. 172, pp. 180–192, 2019.
D. H. Oh and A. J. Patton, “High-dimensional copula-based distributions with mixed frequency data,” Journal of Econometrics, vol. 193, no. 2, pp. 349–366, 2016.
Y. Hu and Y. Hou, “A copula-based approach to modelling and testing for heavy-tailed data with bivariate heteroscedastic extremes, arXiv e-prints, 2024.
H. Joe, Dependence Modeling with Copulas, CRC Press, 2015.
Downloads
-
Vistas(Views): 26
- PDF (Español (España)) Descargas(Downloads): 19
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Scientia et Technica
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
The undersigned authors declare that the article submitted to the journal Scientia et Technica is an original work and that all its content is free of third-party copyright restrictions or has the corresponding authorizations. Consequently, the authors assume responsibility for any litigation or claim related to intellectual property rights, releasing the Technological University of Pereira and the journal Scientia et Technica from any liability.
If the submitted work is accepted for publication, the authors retain copyright to the article and grant the journal Scientia et Technica the right of first publication, as well as a non-exclusive, perpetual license to reproduce, edit, distribute, display, and publicly communicate the article in any medium or format, including print, electronic, databases, repositories, the Internet, or other scientific dissemination systems. The authors agree that the article will be published in open access and distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0).
The journal Scientia will respect in all cases the moral rights of the authors, in accordance with the provisions of article 30 of Law 23 of 1982 of the Republic of Colombia, recognizing the authorship of the work, the right to integrity and the right of disclosure, which are inalienable and non-waivable.