Abstract
Estimation of correlation matrices is a challenging problem due to the notorious positive-definiteness constraint and high-dimensionality. Reparameterizing Cholesky factors of correlation matrices in terms of angles or hyperspherical coordinates where the angles vary freely in the range [0,π) has become popular in the last two decades. However, it has not been used in Bayesian estimation of correlation matrices perhaps due to lack of clear statistical relevance and suitable priors for the angles. In this paper, we show for the first time that for longitudinal data these angles are the inverse cosine of the semi-partial correlations (SPCs). This simple connection makes it possible to introduce physically meaningful selection and shrinkage priors on the angles or correlation matrices with emphasis on selection (sparsity) and shrinking towards longitudinal structure. Our method deals effectively with the positive-definiteness constraint in posterior computation. We compare the performance of our Bayesian estimation based on angles with some recent methods based on partial autocorrelations through simulation and apply the method to a data related to clinical trial on smoking.
Highlights
Covariance and correlation matrices play a fundamental role in every aspect of multivariate statistics (Anderson, 2003)
We address some Bayesian modeling and inferential challenges in estimating a correlation matrix by introducing suitable priors on the angles which go beyond the traditional use of the inverse-Wishart distribution, the marginal and joint uniform priors in Barnard et al (2000)
The aim of this article is to study and deal with some of the computational challenges in Bayesian estimation of correlation matrices by using its Cholesky decomposition, (Pinheiro and Bates, 1996; Rapisarda et al, 2007) and the ensuing angles as the new parameters varying freely in the range [0, π). This enables us to deal effectively with the positive-definiteness constraint, resulting in faster computation of the posteriors for our proposed selection and shrinkage priors. We show that these angles are directly related to the as yet dormant notion of semi-partial correlations (SPCs)
Summary
Covariance and correlation matrices play a fundamental role in every aspect of multivariate statistics (Anderson, 2003). The aim of this article is to study and deal with some of the computational challenges in Bayesian estimation of correlation matrices by using its Cholesky decomposition, (Pinheiro and Bates, 1996; Rapisarda et al, 2007) and the ensuing angles (hyperspherical coordinates) as the new parameters varying freely in the range [0, π). This enables us to deal effectively with the positive-definiteness constraint, resulting in faster computation of the posteriors for our proposed selection and shrinkage priors.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
