Dataset Information

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.

SUBMITTER: Lan S

PROVIDER: S-EPMC8048134 | biostudies-literature | 2020 Dec

REPOSITORIES: biostudies-literature

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Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices.

Lan Shiwei S Holbrook Andrew A Elias Gabriel A GA Fortin Norbert J NJ Ombao Hernando H Shahbaba Babak B

Bayesian analysis 20191104 4

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 ...[more]

PMID: 33868547

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Project description:This article investigates a form of rank deficiency in phenotypic covariance matrices derived from geometric morphometric data, and its impact on measures of phenotypic integration. We first define a type of rank deficiency based on information theory then demonstrate that this deficiency impairs the performance of phenotypic integration metrics in a model system. Lastly, we propose methods to treat for this information rank deficiency. Our first goal is to establish how the rank of a typical geometric morphometric covariance matrix relates to the information entropy of its eigenvalue spectrum. This requires clear definitions of matrix rank, of which we define three: the full matrix rank (equal to the number of input variables), the mathematical rank (the number of nonzero eigenvalues), and the information rank or "effective rank" (equal to the number of nonredundant eigenvalues). We demonstrate that effective rank deficiency arises from a combination of methodological factors-Generalized Procrustes analysis, use of the correlation matrix, and insufficient sample size-as well as phenotypic covariance. Secondly, we use dire wolf jaws to document how differences in effective rank deficiency bias two metrics used to measure phenotypic integration. The eigenvalue variance characterizes the integration change incorrectly, and the standardized generalized variance lacks the sensitivity needed to detect subtle changes in integration. Both metrics are impacted by the inclusion of many small, but nonzero, eigenvalues arising from a lack of information in the covariance matrix, a problem that usually becomes more pronounced as the number of landmarks increases. We propose a new metric for phenotypic integration that combines the standardized generalized variance with information entropy. This metric is equivalent to the standardized generalized variance but calculated only from those eigenvalues that carry nonredundant information. It is the standardized generalized variance scaled to the effective rank of the eigenvalue spectrum. We demonstrate that this metric successfully detects the shift of integration in our dire wolf sample. Our third goal is to generalize the new metric to compare data sets with different sample sizes and numbers of variables. We develop a standardization for matrix information based on data permutation then demonstrate that Smilodon jaws are more integrated than dire wolf jaws. Finally, we describe how our information entropy-based measure allows phenotypic integration to be compared in dense semilandmark data sets without bias, allowing characterization of the information content of any given shape, a quantity we term "latent dispersion". [Canis dirus; Dire wolf; effective dispersion; effective rank; geometric morphometrics; information entropy; latent dispersion; modularity and integration; phenotypic integration; relative dispersion.].

Dataset Information

Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices.

Publications

Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices.

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