A generic all-purpose transformation for multivariate modeling through copulas

  • Manoj Bahuguna
  • Ravindra KhattreeEmail author
Regular Paper


Copulas have been used in various applications in biomedical sciences and finance. We suggest copulas as the generic all-purpose transformations which can enable one to apply various standard multivariate procedures more efficiently and with better statistical properties and results. More specifically, we consider the problem of transformation of any continuous data to multivariate normality using copulas as a device for defining the transformation. Such a transformation effectively enables us to model a variety of problems involving non-normal data using the classical multivariate statistical techniques. We evaluate and illustrate various applications including those in regression, multicollinearity, principal component analysis, factor analysis, partial least square modeling and structural equation modeling where analyses using the appropriate copula transformations result in substantial improvement in implementation, interpretation, prediction as well as in the corresponding models. A great many datasets available in the literature are analyzed which amply demonstrate the power of such an approach.


Copula Gaussian copula Linear models Multivariate modeling Multivariate normality Transformations 



The authors wish to thank the associate editor and two referees for their critical comments leading to the improvement in this work.


  1. 1.
    Abdi, H.: Partial Least Square Regression (PLS Regression), Encyclopedia for Research Methods for the Social Sciences, pp. 792–795. Sage, Thousand Oaks (2003)Google Scholar
  2. 2.
    Atkinson, A., Riani, M., Cerioli, A.: Exploring Multivariate Data with The Forward Search. Springer, New York, NY (2004)CrossRefGoogle Scholar
  3. 3.
    Belsley, D.A.: Conditioning Diagnostics: Collinearity and Weak Data in Regression. Wiley, NewYork, NY (1991)zbMATHGoogle Scholar
  4. 4.
    Bentler, P. M.: EQS, Structural Equations Program Manual, Program Version 5.0, Encino, CA (1995)Google Scholar
  5. 5.
    Bentler, P.M., Bonett, D.G.: Significance tests and goodness of fit in the analysis of covariance structures. Psychol. Bull. 88(3), 588 (1980)CrossRefGoogle Scholar
  6. 6.
    Box, G.E.P., Cox, D.R.: An analysis of transformations. J. R. Stat. Soc. Ser. B (Methodol.) 26(2), 211–252 (1964)zbMATHGoogle Scholar
  7. 7.
    Casella, G., Berger, R.L.: Statistical Inference, 2nd edn. Duxbury, Pacific Grove, CA (2002)zbMATHGoogle Scholar
  8. 8.
    Chatterjee, S., Hadi, A., Price, B.: The Use of Regression Analysis by Example. Wiley, New York, NY (2006)CrossRefGoogle Scholar
  9. 9.
    Cherubini, U., Gobbi, F., Mulinacci, S., Romagnoli, S.: Dynamic Copula Methods in Finance. Wiley, Newyork, NY (2012)Google Scholar
  10. 10.
    Daniel, C., Wood, F.S.: Fitting Equations to Data: Computer Analysis of Multifactor Data. Wiley, New York, NY (1999)zbMATHGoogle Scholar
  11. 11.
    Graybill, F.A., Iyer, H.K.: Regression Analysis Concepts and Applications. Duxbury, Belmont, CA (1994)zbMATHGoogle Scholar
  12. 12.
    Joe, H.: Multivariate Models and Multivariate Dependence Concepts. CRC Press, New York, NY (1997)CrossRefGoogle Scholar
  13. 13.
    Kendall, M.G.: Multivariate Analysis. Macmillan, New York, NY (1980)zbMATHGoogle Scholar
  14. 14.
    Khattree, R.: Antieigenvalues, multicollinearity and influence: a revisit to regression diagnostics (preprint) (2016)Google Scholar
  15. 15.
    Khattree, R., Bahuguna, M.: A revisit to estimation of beta risk and an analysis of stock market through copula transformation and winsorization with S&P 500 index as proxy. J. Index Invest. 8(4), 61–83 (2018)CrossRefGoogle Scholar
  16. 16.
    Khattree, R., Bahuguna, M.: An alternative data analytic approach to measure the univariate and multivariate skewness. Int. J. Data Sci. Anal. (2018) CrossRefGoogle Scholar
  17. 17.
    Khattree, R., Naik, D.N.: Applied Multivariate Statistics with SAS Software, 2nd edn. SAS Institute Inc., Cary, NC (1999)Google Scholar
  18. 18.
    Khattree, R., Naik, D.N.: Multivariate Data Reduction and Discrimination with SAS Software. SAS Institute Inc., Cary, NC (2000)Google Scholar
  19. 19.
    Kiplinger’s Personal Finance,  57(12), 104–123 (2003)Google Scholar
  20. 20.
    Kutner, M.H., Nachtsheim, C.J., Neter, J., Li, W.: Applied Linear Statistical Models. McGraw-Hill, New York, NY (2005)Google Scholar
  21. 21.
    Mai, J., Scherer, M.: Financial Engineering with Copulas Explained. Palgrave-Macmillan, London (2014)CrossRefGoogle Scholar
  22. 22.
    McDonald, G.C., Schwing, R.C.: Instabilities of regression estimates relating air pollution to mortality. Technometrics 15(3), 463–481 (1973)CrossRefGoogle Scholar
  23. 23.
    McDonald, G.C., Ayers, J.: Some applications of the Chernoff faces: a technique for graphically representing multivariate data. In: Wang, P. (ed.) Graphical Representation of Multivariate Data, pp. 183–197. Academic Press, New York, NY (1978)CrossRefGoogle Scholar
  24. 24.
    Nelsen, R.B.: An Introduction to Copulas. Springer, New York, NY (2006)zbMATHGoogle Scholar
  25. 25.
    Rao, C.R.: Test of significance in multivariate analysis. Biometrika 35, 58–79 (1948)MathSciNetCrossRefGoogle Scholar
  26. 26.
    O’Rourke, N., Hatcher, L.: A Step-by-Step Approach to Using SAS for Factor Analysis and Structural Equation Modeling, Second edn. SAS Press, Cary, NC (2013)Google Scholar
  27. 27.
    SAS Institute, SAS/STAT 12.1 User’s Guide: Survey Data Analysis.SAS Institute Inc., Cary, NC (2012)Google Scholar
  28. 28.
    Sklar, A.: Distribution functions of \(n\) dimensions and margins. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959)Google Scholar
  29. 29.
    Sklar, A.: Random variables, distribution functions, and copulas: a personal look backward and forward. IMS Lect. Notes Monogr. Ser. Inst. Math. Stat. 28, 1–14 (1996)MathSciNetGoogle Scholar
  30. 30., TC2000 Software, Version 7 (2010)Google Scholar
  31. 31.
    Wang, J., Wang, X.: Structural Equation Modeling: Applications Using Mplus. Wiley, New York, NY (2012)CrossRefGoogle Scholar
  32. 32.
    Wicklin, R.: Generating a random orthogonal matrix, SAS Blogs. SAS Institute Inc. (2012)
  33. 33.
    Wicklin, R.: Simulating Data with SAS. SAS Institute Inc., Cary, NC (2013)Google Scholar
  34. 34.
    Wright, S.: On the nature of size factors. Genet. Genet. Soc. Am. Bethesda, MD 3(4), 367–374 (1918)Google Scholar
  35. 35.
    Wold, S., Sjöström, M., Eriksson, L.: PLS-regression: a basic tool of chemometrics. Chemom. Intell. Lab. Syst. 58(2), 109–130 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsOakland UniversityRochesterUSA
  2. 2.Department of Mathematics and Statistics, and Center for Data Science and Big Data AnalyticsOakland UniversityRochesterUSA

Personalised recommendations