Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
Ian L. Dryden, Alexey Koloydenko, and Diwei Zhou
Source: Ann. Appl. Stat. Volume 3, Number 3 (2009), 1102-1123.
Abstract
The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.
Keywords: Anisotropy; Cholesky; geodesic; matrix logarithm; principal components; Procrustes; Riemannian; shape; size; Wishart
Full-text: Access denied (no subscription detected)
In 2007, access to the Annals of Applied Statistics was open. Beginning in 2008, you must hold a subscription or be a member of the IMS to view the full journal. For more information on subscribing, please visit: http://imstat.org/orders.
If you are already an IMS member, you may need to update your Euclid profile following the instructions here: http://imstat.org/publications/eaccess.htm.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoas/1254773280
Digital Object Identifier: doi:10.1214/09-AOAS249
References
Alexa, M. (2002). Linear combination of transformations. ACM Trans. Graph. 21 380–387.
Alexander, D. C. (2005). Multiple-fiber reconstruction algorithms for diffusion MRI. Ann. N. Y. Acad. Sci. 1064 113–133.
Aljabar, P., Bhatia, K. K., Murgasova, M., Hajnal, J. V., Boardman, J. P., Srinivasan, L., Rutherford, M. A., Dyet, L. E., Edwards, A. D. and Rueckert, D. (2008). Assessment of brain growth in early childhood using deformation-based morphometry. Neuroimage 39 348–358.
Amaral, G. J. A., Dryden, I. L. and Wood, A. T. A. (2007). Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. J. Amer. Statist. Assoc. 102 695–707.
Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2007). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29 328–347.
Basser, P. J., Mattiello, J. and Le Bihan, D. (1994). Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. B 103 247–254.
Basu, S., Fletcher, P. T. and Whitaker, R. T. (2006). Rician noise removal in diffusion tensor MRI. In MICCAI (1) (R. Larsen, M. Nielsen and J. Sporring, eds.) Lecture Notes in Computer Science 4190 117–125. Springer, Berlin.
Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1–29.
Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 1225–1259.
Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1 181–242.
Chen, Z. and Dunson, D. B. (2003). Random effects selection in linear mixed models. Biometrics 59 762–769.
Daniels, M. J. and Kass, R. E. (2001). Shrinkage estimators for covariance matrices. Biometrics 57 1173–1184.
Daniels, M. J. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89 553–566.
Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 1643–1665.
Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis. Wiley, Chichester.
Fillard, P., Arsigny, V., Pennec, X. and Ayache, N. (2007). Clinical DT-MRI estimation, smoothing and fiber tracking with log-Euclidean metrics. IEEE Trans. Med. Imaging 26 1472–1482.
Fletcher, P. T. and Joshi, S. (2007). Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87 250–262.
Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10 215–310.
Goodall, C. R. and Mardia, K. V. (1992). The noncentral Bartlett decompositions and shape densities. J. Multivariate Anal. 40 94–108.
Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika 40 33–50.
James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, CA.
Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30 509–541.
Kendall, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16 81–121.
Kendall, D. G. (1989). A survey of the statistical theory of shape. Statist. Sci. 4 87–120.
Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. Wiley, Chichester.
Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. London Math. Soc. (3) 61 371–406.
Kent, J. T. (1994). The complex Bingham distribution and shape analysis. J. Roy. Statist. Soc. Ser. B 56 285–299.
Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. in Appl. Probab. 33 324–338.
Le, H. and Small, C. G. (1999). Multidimensional scaling of simplex shapes. Pattern Recognition 32 1601–1613.
Le, H.-L. (1988). Shape theory in flat and curved spaces, and shape densities with uniform generators. Ph.D. thesis, Univ. Cambridge.
Le, H.-L. (1992). The shapes of non-generic figures, and applications to collinearity testing. Proc. Roy. Soc. London Ser. A 439 197–210.
Le, H.-L. (1995). Mean size-and-shapes and mean shapes: A geometric point of view. Adv. in Appl. Probab. 27 44–55.
Lenglet, C., Rousson, M. and Deriche, R. (2006). DTI segmentation by statistical surface evolution. IEEE Trans. Med. Imaging 25 685–700.
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26 735–747 (electronic).
Pennec, X. (1999). Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements. In Proceedings of IEEE Workshop on Nonlinear Signal and Image Processing (NSIP99) (A. Cetin, L. Akarun, A. Ertuzun, M. Gurcan and Y. Yardimci, eds.) 1 194–198. IEEE, Los Alamitos, CA.
Pennec, X., Fillard, P. and Ayache, N. (2006). A Riemannian framework for tensor computing. Int. J. Comput. Vision 66 41–66.
Pourahmadi, M. (2007). Cholesky decompositions and estimation of a covariance matrix: Orthogonality of variance correlation parameters. Biometrika 94 1006–1013.
R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
Schwartzman, A. (2006). Random ellipsoids and false discovery rates: Statistics for diffusion tensor imaging data. Ph.D. thesis, Stanford Univ.
Schwartzman, A., Dougherty, R. F. and Taylor, J. E. (2008). False discovery rate analysis of brain diffusion direction maps. Ann. Appl. Statist. 2 153–175.
Wang, Z., Vemuri, B., Chen, Y. and Mareci, T. (2004). A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. Med. Imaging 23 930–939.
Zhou, D., Dryden, I. L., Koloydenko, A. and Bai, L. (2008). A Bayesian method with reparameterisation for diffusion tensor imaging. In Proceedings, SPIE conference. Medical Imaging 2008: Image Processing (J. M. Reinhardt and J. P. W. Pluim, eds.) 69142J. SPIE, Bellingham, WA.
The Annals of Applied Statistics
