IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v178y2020ics0047259x19305330.html
   My bibliography  Save this article

Parametrising correlation matrices

Author

Listed:
  • Forrester, Peter J.
  • Zhang, Jiyuan

Abstract

Correlation matrices are the sub-class of positive definite real matrices with all entries on the diagonal equal to unity. Earlier work has exhibited a parametrisation of the corresponding Cholesky factorisation in terms of partial correlations, and also in terms of hyperspherical co-ordinates. We show how the two are related, starting from the definition of the partial correlations in terms of the Schur complement. We extend this to the generalisation of correlation matrices to the cases of complex and quaternion entries. As in the real case, we show how the hyperspherical parametrisation leads naturally to a distribution on the space of correlation matrices {R} with probability density function proportional to (detR)a. For certain a, a construction of random correlation matrices realising this distribution is given in terms of rectangular standard Gaussian matrices.

Suggested Citation

  • Forrester, Peter J. & Zhang, Jiyuan, 2020. "Parametrising correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x19305330
    DOI: 10.1016/j.jmva.2020.104619
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X19305330
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2020.104619?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Lewandowski, Daniel & Kurowicka, Dorota & Joe, Harry, 2009. "Generating random correlation matrices based on vines and extended onion method," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1989-2001, October.
    2. Pourahmadi, Mohsen & Wang, Xiao, 2015. "Distribution of random correlation matrices: Hyperspherical parameterization of the Cholesky factor," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 5-12.
    3. Joe, Harry, 2006. "Generating random correlation matrices based on partial correlations," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2177-2189, November.
    4. Kurowicka, Dorota, 2014. "Joint density of correlations in the correlation matrix with chordal sparsity patterns," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 160-170.
    5. Frederick Wong, 2003. "Efficient estimation of covariance selection models," Biometrika, Biometrika Trust, vol. 90(4), pages 809-830, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Flórez, Alvaro J. & Molenberghs, Geert & Van der Elst, Wim & Alonso Abad, Ariel, 2022. "An efficient algorithm to assess multivariate surrogate endpoints in a causal inference framework," Computational Statistics & Data Analysis, Elsevier, vol. 172(C).
    2. Pourahmadi, Mohsen & Wang, Xiao, 2015. "Distribution of random correlation matrices: Hyperspherical parameterization of the Cholesky factor," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 5-12.
    3. Hirofumi Michimae & Takeshi Emura, 2022. "Bayesian ridge estimators based on copula-based joint prior distributions for regression coefficients," Computational Statistics, Springer, vol. 37(5), pages 2741-2769, November.
    4. Madar, Vered, 2015. "Direct formulation to Cholesky decomposition of a general nonsingular correlation matrix," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 142-147.
    5. Soyeon Ahn & John M. Abbamonte, 2020. "A new approach for handling missing correlation values for meta‐analytic structural equation modeling: Corboundary R package," Campbell Systematic Reviews, John Wiley & Sons, vol. 16(1), March.
    6. Davide Delle Monache & Ivan Petrella & Fabrizio Venditti, 2021. "Price Dividend Ratio and Long-Run Stock Returns: A Score-Driven State Space Model," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 1054-1065, October.
    7. Tuitman, Jan & Vanduffel, Steven & Yao, Jing, 2020. "Correlation matrices with average constraints," Statistics & Probability Letters, Elsevier, vol. 165(C).
    8. Kurowicka, Dorota, 2014. "Joint density of correlations in the correlation matrix with chordal sparsity patterns," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 160-170.
    9. Benjamin Poignard & Jean-Davis Fermanian, 2014. "Dynamic Asset Correlations Based on Vines," Working Papers 2014-46, Center for Research in Economics and Statistics.
    10. Wang, Y. & Daniels, M.J., 2013. "Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 130-140.
    11. Saxena, Shobhit & Bhat, Chandra R. & Pinjari, Abdul Rawoof, 2023. "Separation-based parameterization strategies for estimation of restricted covariance matrices in multivariate model systems," Journal of choice modelling, Elsevier, vol. 47(C).
    12. Daniels, M.J. & Pourahmadi, M., 2009. "Modeling covariance matrices via partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2352-2363, November.
    13. Durante Fabrizio & Puccetti Giovanni & Scherer Matthias & Vanduffel Steven, 2017. "The Vine Philosopher: An interview with Roger Cooke," Dependence Modeling, De Gruyter, vol. 5(1), pages 256-267, December.
    14. Falk, Carl F. & Muthukrishna, Michael, 2021. "Parsimony in model selection: tools for assessing fit propensity," LSE Research Online Documents on Economics 110856, London School of Economics and Political Science, LSE Library.
    15. Anindya Bhadra & Arvind Rao & Veerabhadran Baladandayuthapani, 2018. "Inferring network structure in non†normal and mixed discrete†continuous genomic data," Biometrics, The International Biometric Society, vol. 74(1), pages 185-195, March.
    16. Giraud Christophe & Huet Sylvie & Verzelen Nicolas, 2012. "Graph Selection with GGMselect," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(3), pages 1-52, February.
    17. Zhang, Yixiao & Yu, Cindy L. & Li, Haitao, 2022. "Nowcasting GDP Using Dynamic Factor Model with Unknown Number of Factors and Stochastic Volatility: A Bayesian Approach," Econometrics and Statistics, Elsevier, vol. 24(C), pages 75-93.
    18. MohammadAmin Fazli & Parsa Alian & Ali Owfi & Erfan Loghmani, 2021. "RPS: Portfolio Asset Selection using Graph based Representation Learning," Papers 2111.15634, arXiv.org.
    19. Brian Hartley, 2020. "Corridor stability of the Kaleckian growth model: a Markov-switching approach," Working Papers 2013, New School for Social Research, Department of Economics, revised Nov 2020.
    20. Pesaran, M. Hashem & Yamagata, Takashi, 2012. "Testing CAPM with a Large Number of Assets," IZA Discussion Papers 6469, Institute of Labor Economics (IZA).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x19305330. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.