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Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis

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  • Osborne, Daniel
  • Patrangenaru, Vic
  • Ellingson, Leif
  • Groisser, David
  • Schwartzman, Armin

Abstract

This paper addresses much needed asymptotic and nonparametric bootstrap methodology for two-sample tests for means on Riemannian manifolds with a simply transitive group of isometries. In particular, we develop a two-sample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. The new method naturally leads to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average at a specific voxel between corresponding signals in Diffusion Tensor Images (DTIs) in young children with dyslexia when compared to their clinically normal peers, based on data that was previously analyzed using parametric methods.

Suggested Citation

  • Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    • Handle: RePEc:eee:jmvana:v:119:y:2013:i:c:p:163-175
    DOI: 10.1016/j.jmva.2013.04.006
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    References listed on IDEAS

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    1. Bandulasiri, Ananda & Bhattacharya, Rabi N. & Patrangenaru, Vic, 2009. "Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1867-1882, October.
    2. Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
    3. Koo, Ja-Yong & Kim, Peter T., 2008. "Sharp adaptation for spherical inverse problems with applications to medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 165-190, February.
    4. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.
    5. Stephan Huckemann, 2011. "Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(3), pages 424-446, September.
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    Cited by:

    1. Ilenia Lovato & Alessia Pini & Aymeric Stamm & Maxime Taquet & Simone Vantini, 2021. "Multiscale null hypothesis testing for network‐valued data: Analysis of brain networks of patients with autism," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(2), pages 372-397, March.
    2. Armin Schwartzman, 2016. "Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices," International Statistical Review, International Statistical Institute, vol. 84(3), pages 456-486, December.
    3. Vic Patrangenaru & Mingfei Qiu & Marius Buibas, 2014. "Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 485-506, June.
    4. Benoit Ahanda & Daniel E. Osborne & Leif Ellingson, 2022. "Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions," METRON, Springer;Sapienza Università di Roma, vol. 80(3), pages 281-303, December.
    5. Pigoli, Davide & Menafoglio, Alessandra & Secchi, Piercesare, 2016. "Kriging prediction for manifold-valued random fields," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 117-131.

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