Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices

Abstract
Modeling correlation (and covariance) matrices can be challenging due to the positive-definiteness constraint and potential high-dimensionality. Our approach is to decompose the covariance matrix into the correlation and variance matrices and propose a novel Bayesian framework based on modeling the correlations as products of unit vectors. By specifying a wide range of distributions on a sphere (e.g. the squared-Dirichlet distribution), the proposed approach induces flexible prior distributions for covariance matrices (that go beyond the commonly used inverse-Wishart prior). For modeling real-life spatio-temporal processes with complex dependence structures, we extend our method to dynamic cases and introduce unit-vector Gaussian process priors in order to capture the evolution of correlation among components of a multivariate time series. To handle the intractability of the resulting posterior, we introduce the adaptive Δ-Spherical Hamiltonian Monte Carlo. We demonstrate the validity and flexibility of our proposed framework in a simulation study of periodic processes and an analysis of rat’s local field potential activity in a complex sequence memory task.

Citation
Lan, S., Holbrook, A., Elias, G. A., Fortin, N. J., Ombao, H., & Shahbaba, B. (2020). Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices. Bayesian Analysis, 15(4). doi:10.1214/19-ba1173

Acknowledgements
SL is supported by ONR N00014-17-1-2079. AH is supported by NIH grant T32 AG000096. GAE is supported by NIDCD T32 DC010775. NJF is supported by NSF awards IOS-1150292 and BCS-1439267 and Whitehall Foundation award 2010-05-84. HO is supported by NSF DMS 1509023 and NSF SES 1461534. BS is supported by NSF DMS 1622490 and NIH R01 MH115697. We thank Yulong Lu at Duke for discussions on posterior contraction of covariance regression with Gaussian processes.

Publisher
Institute of Mathematical Statistics

Journal
BAYESIAN ANALYSIS

DOI
10.1214/19-BA1173

arXiv
1711.02869

Additional Links
https://projecteuclid.org/journals/bayesian-analysis/volume-15/issue-4/Flexible-Bayesian-Dynamic-Modeling-of-Correlation-and-Covariance-Matrices/10.1214/19-BA1173.full

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