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Parametrising correlation matrices

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  • 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
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    References listed on IDEAS

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    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. Frederick Wong, 2003. "Efficient estimation of covariance selection models," Biometrika, Biometrika Trust, vol. 90(4), pages 809-830, December.
    5. 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.
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