Skip to main content

Copula Functions and Applications in Engineering

  • Chapter
  • First Online:
Logistics, Supply Chain and Financial Predictive Analytics

Part of the book series: Asset Analytics ((ASAN))

Abstract

Copula functions also known as copulas, which connect the marginal distributions to their joint distributions, are useful in simulating the linear or nonlinear relationships among multivariate data in the scientific and engineering studies. Copula is a multivariate distribution function with marginally uniform random variables on [0, 1]. Copula functions have some appealing properties such as they allow scale-free measures of dependence and are useful in constructing families of joint distributions. As seen recently, copulas have been applied in statistics, insurance, finance, economics, survival analysis, image processing, and engineering applications. In this paper, we aim to briefly describe the copula functions, their properties, copula families, simulations, and examples of copula applications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akaike H (1972) Information theory and an extension of the maximum likelihood principle. In: Proceedings of the 2nd international symposium information theory, pp 267–281

    Google Scholar 

  2. Beněs V, Štěpán J (eds) (1997) Distributions with given marginals and moment problems. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  3. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151

    Article  Google Scholar 

  4. Cuadras CM, Fortiana J, Rodr´ıguez Lallena JA (eds) (2002) Distributions with given marginals and statistical modelling. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  5. Dall’Aglio G (1991) Fr´echet classes: the beginnings. In: Advances in probability distributions with given marginals. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  6. Dall’Aglio G, Kotz S, Salinetti G (eds) (1991) Advances in probability distributions with given marginals. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  7. Dobrowolski E, Kumar P (2014) Some properties of the marshall-olkin and generalized cuadras-augé families of copulas. Aust J Math Anal Appl 11(1), Article 2, 1–13

    Google Scholar 

  8. Embrechts P, McNeil A, Straumann D (1997) Correlation and dependence in risk management: properties and pitfalls. Risk 12(5):69–71

    Google Scholar 

  9. Fang K-T, Kotz S, Ng K-W (1987) Symmetric multivariate and related distributions. Chapman & Hall, London

    Google Scholar 

  10. Fisher NI (1997) Copulas. Encyclopedia of statistical sciences, update, vol 1. Wiley, New York, pp 159–163

    Google Scholar 

  11. Frank MJ (1979) On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequ Math 19:194–226

    Article  Google Scholar 

  12. Fréchet M (1951) Sur les tableaux de corr´elation dont les marges son donn´ees. Ann Univ Lyon Sect A 9:53–77

    Google Scholar 

  13. Frees EW, Valdez EA (1998) Understanding relationships using copulas. North Am Actuar J 2:1–25

    Article  Google Scholar 

  14. Harwig D, Ittiwattana W, Castner H (2001) Advances in oxygen equivalence equations for predicting the properties of titanium welds. Weld J 126–136

    Google Scholar 

  15. Genest C (1987) Franks family of bivariate distributions. Biometrika 74:549–555

    Article  Google Scholar 

  16. Genest C, MacKay J (1986) Copules archim´ediennes et familles de lois bidimensionnelles dont les marges sont donn´ees. Canad J Stat 14:145–159

    Article  Google Scholar 

  17. Genest C, MacKay J (1986) The joy of copulas: bivariate distributions with uniform marginals. Am Stat 40:280–285

    Google Scholar 

  18. Genest C, Rivest L-P (1993) Statistical inference procedures for bivariate Archimedean copulas. J Am Stat Assoc 55:698–707

    Google Scholar 

  19. Genest C, Ghoudi K, Rivest L (1995) A semi-parametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82:543–552

    Article  Google Scholar 

  20. Genest C, Quesada Molina JJ, Rodríguez Lallena JA, Sempi C (1999) A characterization of quasi-copulas. J Multivar Anal 69:193–205

    Article  Google Scholar 

  21. Goluksuz, CT, Kumar P (2017) On a new bivariate one parameter archimedean copula function. (Under review)

    Google Scholar 

  22. Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55:698–707

    Article  Google Scholar 

  23. Herath HSB, Kumar Pranesh (2007) New research directions in engineering economics—modeling dependencies with copulas. Eng Econ 52(4):305–331

    Article  Google Scholar 

  24. Herath HSB, Kumar Pranesh, Amershi AH (2013) Crack spread option pricing with copulas. J Econ Financ 37:100–121

    Article  Google Scholar 

  25. Herath HSB, Kumar P (2014) Using copula functions in Bayesian analysis: a comparison of the lognormal conjugate. Eng Econ J Devot Probl Cap Invest 1–26. https://doi.org/10.1080/0013791x.2014.962719

  26. Hoeffding W (1940) Masstabinvariante Korrelationstheorie. Schriften des Matematischen Instituts und des Instituts fur Angewandte Matematik de Universitat Berlin, vol 5, pp 179–233. [Reprinted as Scale-invariant correlation theory. In Fisher NI, Sen PK (eds) (1994). The collected works of Wassily Hoeffding. Springer, New York, pp 57–107.]

    Google Scholar 

  27. Hoeffding W (1941) Masstabinvariante Korrelationsmasse fur diskontinuierliche Verteilungen. Arkivfr matematischen Wirtschaften und Sozialforschung, vol 7, pp 49–70. [Reprinted as Scale-invariant correlation measures for discontinuous distributions. In Fisher NI, Sen PK (eds) (1994) The collected works of Wassily Hoeffding. Springer, New York, pp 109–133]

    Google Scholar 

  28. Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 73:671–678

    Google Scholar 

  29. Hutchinson TP, Lai CD (1990) Continuous bivariate distributions, emphasising applications. Rumsby Scientific Publishing, Adelaide

    Google Scholar 

  30. Joe H (1997) Multivariate models and dependent concepts. Chapman & Hall, New York

    Book  Google Scholar 

  31. Johnson ME (1987) Multivariate statistical simulation. Wiley, New York

    Book  Google Scholar 

  32. Kapur JN, Kesavan HK (1992) Entropy maximization principles with applications. Academic Press, Cambridge, Massachusetts

    Google Scholar 

  33. Kashanchi F, Kumar P (2013) Copula based aerial image registration. IEEE Trans Image Process

    Google Scholar 

  34. Kashanchi F, Kumar Pranesh (2014) Copulas applications in estimating Value-at-Risk (VaR): Iranian Crude Oil Prices. J Data Sci 12(3):1–24

    Google Scholar 

  35. Kimeldorf G, Sampson AR (1978) Monotone dependence. Ann Stat 6:895–903

    Article  Google Scholar 

  36. Kirk E, Aron J (1995) Correlation in energy markets. In Managing energy price risk. Risk Books, London

    Google Scholar 

  37. Kovács E (2007) On the using of copulas in characterizing of dependence with entropy. Pollack Period Int J Eng Inf Sci

    Google Scholar 

  38. Kruskal WH (1958) Ordinal measures of association. J Am Stat Assoc 53:814–861

    Article  Google Scholar 

  39. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22:79–86

    Article  Google Scholar 

  40. Kumar Pranesh, Shoukri MM (2007) Copula based prediction models: an application to an aortic regurgitation study. BMC Med Res Methodol 7(21):1–9

    Google Scholar 

  41. Kumar Pranesh, Shoukri MM (2008) Evaluating aortic stenosos using the Archimedean copula methodology. J Data Sci 6:173–187

    Google Scholar 

  42. Kumar Pranesh (2009) Applications of the Farlie-Gumbel-Morgenstern copulas in predicting the properties of the Titanium Welds. Int J Math 1(1):13–22

    Google Scholar 

  43. Kumar P (2010) Probabilistic modeling using copula functions based Bayesian approach. In: Gil-Lafuente A, Merigó J (eds) Computational intelligence in business and economics. World Scientific Publishing, 19–26

    Google Scholar 

  44. Kumar P (2010) Probability distributions and estimation of Ali-Mikhail-Haq Copula. Appl Math Sci Theory Appl 4(13–16):657–666

    Google Scholar 

  45. Kumar P (2011) Copulas: distribution functions and simulation. In: Lovric Miodrag (ed) International encyclopedia of statistical science. Springer Science + Business Media LLC, Heidelberg

    Google Scholar 

  46. Kumar P (2011) Copula functions: characterizing uncertainty in probabilistic systems. Appl Math Sci Theory Appl 5(30):1459–1472

    Google Scholar 

  47. Kumar P (2012) Statistical dependence: copula functions and mutual information based measures. J Stat Appl Prob 1(1):1–14

    Article  Google Scholar 

  48. Kumar P (2013) Statistical inference using copulas and application to analyze effect of storage of the Red Blood Cells on Erythrocyte Adenosine Triphosphate levels. In: Akis V (ed) Essays on mathematics and statistics, vol 3, pp 151–160. ATINER

    Google Scholar 

  49. Kumar P, Kashanchi F (2014) Copula based multivariate statistical models using WinBUGS. In Proceedings of the 11th Iranian statistics conference (ISC11), Tehran

    Google Scholar 

  50. Lehmann EL (1966) Some concepts of dependence. An Math Stat 37:1137–1153

    Article  Google Scholar 

  51. Marshall AW, Olkin I (1967) A generalized bivariate exponential distribution. J Appl Prob 4:291–302

    Article  Google Scholar 

  52. Marshall AW, Olkin I (1988) Families of Multivariate Distributions. J Am Stat Assoc 83:834–841

    Article  Google Scholar 

  53. Mercier G (2005) Measures de Dépendance entre Images RSO. GET/ENST Bretagne, Technical report RR-2005003-ITI. https//:perso.enst-bretagne.fr/126mercierg

    Google Scholar 

  54. Nelsen RB (1992) On measures of association as measures of positive dependence. Stat Prob Lett 14:269–274

    Article  Google Scholar 

  55. Nelsen RB (2003) Properties and applications of copulas: a brief survey. https://sites.google.com/a/lclark.edu/nelsen/brazil.pdf

  56. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York

    Google Scholar 

  57. Nelsen RB, Quesada Molina JJ, Rodríguez Lallena JA, Úbeda Flores M (2001) Bounds on bivariate distribution functions with given margins and measures of association. Commun Statist-Theory Meth 30:155–1162

    Google Scholar 

  58. Rüschendorf L, Schweizer B, Taylor MD (eds) (1996) Distributions with fixed marginals and related topics. Institute of Mathematical Statistics, Hayward, CA

    Google Scholar 

  59. Schweizer B (1991) Thirty years of copulas. Advances in probability distributions with given marginals. Kluwer Academic Publishers, Dordrecht, pp 13–50

    Chapter  Google Scholar 

  60. Schweizer B, Sklar A (1983) Probabilistic metric spaces. North-Holland, New York

    Google Scholar 

  61. Schweizer B, Wolff E (1981) On nonparametric measures of dependence for random variables. Ann Stat 9:879–885

    Article  Google Scholar 

  62. Shannon CE (1948) A mathematical theory of communication-an integrated approach. Cambridge University Press, Cambridge, UK

    Google Scholar 

  63. Sklar A (1959) Fonctions de répartition á n dimensional et leurs marges. Publ Inst Stat Univ Paris 8:229–231

    Google Scholar 

  64. Tjøstheim D (1996) Measures of dependence and tests of independence. Statistics 28:249–284

    Article  Google Scholar 

  65. Yao YY (2002) Information-theoretic measures for knowledge discovery. In: Karmeshu (ed) Entropy measures, maximum entropy principles and engineering applications. Springer

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pranesh Kumar .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kumar, P. (2019). Copula Functions and Applications in Engineering. In: Deep, K., Jain, M., Salhi, S. (eds) Logistics, Supply Chain and Financial Predictive Analytics. Asset Analytics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0872-7_15

Download citation

Publish with us

Policies and ethics