Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

Schwartzman, Armin; Mascarenhas, Walter F.; Taylor, Jonathan

doi:10.1214/08-aos628

Cited by 41 publications

(75 citation statements)

References 51 publications

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“…Schwartzman et al (2008b) derive various LRTs for the mean parameter M when M is restricted to subsets of S p defined in terms of eigenvalues and eigenvectors of M .…”

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

“…Moreover, Schwartzman et al (2008b) show that in many cases this covariance structure is the only one that allows derivation of closed-form expressions that do not depend on estimates of the covariance parameters.…”

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

“…In this article we are interested in applying the LRT statistics derived by Schwartzman et al (2008b) when the true covariance Σ in the data is not necessarily OI.…”

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

“…Computation of the LRT statistic in both the eigenvalue and eigenvector tests involves Schwartzman et al (2008b) showed that estimation of M ∈ M 2,D is analytically tractable and separable from the estimation of Σ if and only if Σ is OI. In such case, the MLE of M is the same that would be obtained if Σ were assumed spherical with variance one, i.e.…”

Section: Lrt Statisticsmentioning

confidence: 99%

“…Because eigenvalues and eigenvectors are highly nonlinear functions of the data, likelihood ratio test (LRT) statistics for these cases have been derived elsewhere assuming an orthogonally invariant (OI) covariance structure between the matrix entries and assuming that the covariances in the two groups are the same (Schwartzman et al, 2008b). The eigenvector test assuming an OI covariance has been useful for identifying white matter tracts affected by necrosis in a brain cancer survivor (Rauschecker et al, 2009).…”

Section: Introductionmentioning

confidence: 99%

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Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

Schwartzman

Dougherty

Taylor

2010

Journal of the American Statistical Association

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Diffusion tensor images (DTI) differ from most medical images in that values at each voxel are not scalars, but 3 × 3 symmetric positive definite matrices called diffusion tensors (DTs). The anatomical characteristics of the tissue at each voxel are reflected by the DT eigenvalues and eigenvectors. In this article we consider the problem of testing whether two groups of DTIs are equal at each voxel in terms of the DT's eigenvalues, eigenvectors, or both. Because eigen-decompositions are highly nonlinear, existing likelihood ratio statistics (LRTs) for testing differences in the set of eigenvalues or the frame of eigenvectors assume an orthogonally invariant covariance structure between the DT entries. While retaining the form of the LRTs, we derive new approximations to their true distributions when the covariance between the DT entries is arbitrary and possibly different between the two groups. The approximate distributions are those of other similar LRT statistics computed at the tangent space to the parameter manifold at the true value of the parameter, but plugging in an estimate for the point of application of the tangent space. The resulting distributions, which are weighted sums of χ 2 s, are further approximated by scaled χ 2 distributions by matching the first two moments. For application to DTI data, a log transformation that converts positive definite matrices into real symmetric matrices is appropriate but not necessary.Voxelwise application of the test statistics leads to a multiple testing problem, which is solved by false discovery rate inference. The above methods are illustrated in a DTI group comparison of boys vs. girls.

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“…Schwartzman et al (2008b) derive various LRTs for the mean parameter M when M is restricted to subsets of S p defined in terms of eigenvalues and eigenvectors of M .…”

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

“…In this article we are interested in applying the LRT statistics derived by Schwartzman et al (2008b) when the true covariance Σ in the data is not necessarily OI.…”

Section: A Gaussian Signal-plus-noise Modelmentioning

confidence: 99%

Section: Lrt Statisticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

Schwartzman

Dougherty

Taylor

2010

Journal of the American Statistical Association

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A Survey of Statistics in the Neurological Sciences with a Focus on Human Neuroimaging

Caffo

Zhao

Eloyan

et al. 2018

Wiley StatsRef: Statistics Reference Online

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Neuroscience is a vast subject; understanding the brain is one of the most complex, deep, and challenging tasks in all of science. In this article, we survey statistical contributions to the field of neuroscience, though focus heavily on human brain imaging. Statistics has made fundamental contributions in the processing and analysis of neuroscience and neuroimaging data. Contributions range from processing the measurements of new technologies to analyzing large groups of subjects and inference on the impact of behavior or disease. Developments in statistical algorithms and signal processing help in the pipeline that takes raw images and converts them to those used for diagnosis or research. Statistics has provided key protections from type I errors for the high‐dimensional spatially correlated and complex data arising in this domain. In addition, novel modeling and testing approaches have allowed researchers to perform inference for this challenging data. We end the article with recommendations for statisticians and other quantitative scientists to get involved in this exciting and rapidly evolving field.

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A spatial Bayesian semiparametric mixture model for positive definite matrices with applications in diffusion tensor imaging

Lan

Reich²,

Bandyopadhyay

2021

Can J Statistics

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Studies on diffusion tensor imaging (DTI) quantify the diffusion of water molecules in a brain voxel using an estimated 3 × 3 symmetric positive definite (p.d.) diffusion tensor matrix. Due to the challenges associated with modelling matrix-variate responses, the voxel-level DTI data are usually summarized by univariate quantities, such as fractional anisotropy. This approach leads to evident loss of information. Furthermore, DTI analyses often ignore the spatial association among neighbouring voxels, leading to imprecise estimates. Although the spatial modelling literature is rich, modelling spatially dependent p.d. matrices is challenging. To mitigate these issues, we propose a matrix-variate Bayesian semiparametric mixture model, where the p.d. matrices are distributed as a mixture of inverse Wishart distributions, with the spatial dependence captured by a Markov model for the mixture component labels. Related Bayesian computing is facilitated by conjugacy results and use of the double Metropolis-Hastings algorithm. Our simulation study shows that the proposed method is more powerful than competing non-spatial methods. We also apply our method to investigate the effect of cocaine use on brain microstructure. By extending spatial statistics to matrix-variate data, we contribute to providing a novel and computationally tractable inferential tool for DTI analysis.

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Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

Cited by 41 publications

References 51 publications

Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

A Survey of Statistics in the Neurological Sciences with a Focus on Human Neuroimaging

A spatial Bayesian semiparametric mixture model for positive definite matrices with applications in diffusion tensor imaging

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