Abstract:
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This paper proposes a data-driven method to identify and exploit parsimony in a covariance matrix. The approach employs a Cholesky decomposition of the inverse of the covariance matrix, coupled with a Bayesian hierarchical model to identify any non-zero, off-diagonal elements of the Cholesky factor. It is convenient to work with the inverse covariance matrix when seeking a parsimonious representation, because its off-diagonal elements are the unnormalized partial correlations. The method is particularly suitable for longitudinal data, because many time-series structures have sparse Cholesky factors of their inverse covariance matrices. We demonstrate the method with substantive examples. In each of these examples, the covariance matrix is large relative to the sample size, but the Cholesky factorization of the inverse is identified as parsimonious. The computation is carried out using a Markov chain Monte Carlo sampling scheme that is more efficient than the Gibbs sampler. The parameters are estimated using their posterior means, which account for the uncertainty associated with the form of the Cholesky factor.
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