Skip to main content

A Fast Metropolis-Hastings Method for Generating Random Correlation Matrices

  • 2080 Accesses

Part of the Lecture Notes in Computer Science book series (LNISA,volume 11314)


We propose a novel Metropolis-Hastings algorithm to sample uniformly from the space of correlation matrices. Existing methods in the literature are based on elaborated representations of a correlation matrix, or on complex parametrizations of it. By contrast, our method is intuitive and simple, based the classical Cholesky factorization of a positive definite matrix and Markov chain Monte Carlo theory. We perform a detailed convergence analysis of the resulting Markov chain, and show how it benefits from fast convergence, both theoretically and empirically. Furthermore, in numerical experiments our algorithm is shown to be significantly faster than the current alternative approaches, thanks to its simple yet principled approach.


  • Correlation matrices
  • Random sampling
  • Metroplis-Hastings

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-03493-1_13
  • Chapter length: 8 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-03493-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.


  1. 1.

  2. 2.


  1. Marsaglia, G., Olkin, I.: Generating correlation matrices. SIAM J. Sci. Stat. Comput. 5(2), 470–475 (1984)

    CrossRef  MathSciNet  Google Scholar 

  2. Holmes, R.: On random correlation matrices. SIAM J. Matrix Anal. Appl. 12(2), 239–272 (1991)

    CrossRef  MathSciNet  Google Scholar 

  3. Fallat, S., Lauritzen, S., Sadeghi, K., Uhler, C., Wermuth, N., Zwiernik, P.: Total positivity in Markov structures. Ann. Stat. 45(3), 1152–1184 (2017)

    CrossRef  MathSciNet  Google Scholar 

  4. Pourahmadi, M., Wang, X.: Distribution of random correlation matrices: hyperspherical parameterization of the Cholesky factor. Stat. Prob. Lett. 106, 5–12 (2015)

    CrossRef  MathSciNet  Google Scholar 

  5. Lewandowski, D., Kurowicka, D., Joe, H.: Generating random correlation matrices based on vines and extended onion method. J. Multivar. Anal. 100(9), 1989–2001 (2009)

    CrossRef  MathSciNet  Google Scholar 

  6. Laurent, M., Poljak, S.: On the facial structure of the set of correlation matrices. SIAM J. Matrix Anal. Appl. 17(3), 530–547 (1996)

    CrossRef  MathSciNet  Google Scholar 

  7. Diaconis, P., Holmes, S., Shahshahani, M.: Sampling from a Manifold, Collections, vol. 10, pp. 102–125. Institute of Mathematical Statistics (2013)

    Google Scholar 

  8. Eaton, M.L.: Multivariate Statistics: A Vector Space Approach. Wiley, Hoboken (1983)

    MATH  Google Scholar 

  9. Mardia, K., Jupp, P.: Directional Statistics. Wiley, Hoboken (1999)

    CrossRef  Google Scholar 

  10. Pukkila, T.M., Rao, C.R.: Pattern recognition based on scale invariant discriminant functions. Inf. Sci. 45(3), 379–389 (1988)

    CrossRef  MathSciNet  Google Scholar 

  11. Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004).

    CrossRef  MATH  Google Scholar 

  12. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2018)

    Google Scholar 

Download references


This work has been partially supported by the Spanish Ministry of Economy, Industry and Competitiveness through the Cajal Blue Brain (C080020-09; the Spanish partner of the EPFL Blue Brain initiative) and TIN2016-79684-P projects; by the Regional Government of Madrid through the S2013/ICE-2845-CASI-CAM-CM project; and by Fundación BBVA grants to Scientific Research Teams in Big Data 2016. I. Córdoba has been supported by the predoctoral grant FPU15/03797 from the Spanish Ministry of Education, Culture and Sports. G. Varando has been partially supported by research grant 13358 from VILLUM FONDEN.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Irene Córdoba .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Córdoba, I., Varando, G., Bielza, C., Larrañaga, P. (2018). A Fast Metropolis-Hastings Method for Generating Random Correlation Matrices. In: Yin, H., Camacho, D., Novais, P., Tallón-Ballesteros, A. (eds) Intelligent Data Engineering and Automated Learning – IDEAL 2018. IDEAL 2018. Lecture Notes in Computer Science(), vol 11314. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-03492-4

  • Online ISBN: 978-3-030-03493-1

  • eBook Packages: Computer ScienceComputer Science (R0)