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Object oriented data analysis: sets of trees. (English) Zbl 1126.62002
Summary: Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non-Euclidean spaces, such as Lie groups and symmetric spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics. This point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree-structured objects.

MSC:
62A01 Foundations and philosophical topics in statistics
62P10 Applications of statistics to biology and medical sciences; meta analysis
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H99 Multivariate analysis
62G99 Nonparametric inference
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References:
[1] Banks, D. and Constantine, G. M. (1998). Metric models for random graphs. J. Classification 15 199–223. · Zbl 0912.62074
[2] Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 733–767. · Zbl 0995.92035
[3] Bullitt, E. and Aylward, S. (2002). Volume rendering of segmented image objects. IEEE Trans. Medical Imaging 21 998–1002. · Zbl 1041.68574
[4] Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis . Wiley, Chichester. · Zbl 0901.62072
[5] Fisher, N. I. (1993). Statistical Analysis of Circular Data . Cambridge Univ. Press. · Zbl 0788.62047
[6] Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data . Cambridge Univ. Press. · Zbl 0651.62045
[7] Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Medical Imaging 23 995–1005. · Zbl 1054.62081
[8] Fletcher, P. T., Lu, C. and Joshi, S. (2003). Statistics of shape via principal geodesic analysis on Lie groups. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition 1 95–101. IEEE, Los Alamitos, CA.
[9] Hastie, T. and Stuetzle, W. (1989). Principal curves. J. Amer. Statist. Assoc. 84 502–516. JSTOR: · Zbl 0679.62048
[10] Mardia, K. V. (1972). Statistics of Directional Data . Academic Press, New York. · Zbl 0244.62005
[11] Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics . Wiley, New York. · Zbl 0935.62065
[12] Larget, B., Simon, D. L. and Kadane, J. B. (2002). Bayesian phylogenetic inference from animal mitochondrial genome arrangements. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 681–693. JSTOR: · Zbl 1067.62115
[13] Locantore, N., Marron, J. S., Simpson, D. G., Tripoli, N., Zhang, J. T. and Cohen, K. L. (1999). Robust principal component analysis for functional data (with discussion). Test 8 1–73. · Zbl 0980.62049
[14] Margush, T. (1982). Distances between trees. Discrete Appl. Math. 4 281–290. · Zbl 0504.06002
[15] Pizer, S. M., Thall, A. and Chen, D. (2000). M-Reps: A new object representation for graphics. Available at midag.cs.unc.edu/pubs/papers/mreps-2000/mrep-pizer.PDF.
[16] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis . Methods and Case Studies . Springer, New York. · Zbl 1011.62002
[17] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[18] Tschirren, J., Palágyi, K., Reinhardt, J. M., Hoffman, E. A. and Sonka, M. (2002). Segmentation, skeletonization and branchpoint matching—A fully automated quantitative evaluation of human intrathoracic airway trees. Proc. Fifth International Conterence on Medical Image Computing and Computer-Assisted Intervention , Part II. Lecture Notes in Comput. Sci. 2489 12–19. Springer, London. · Zbl 1027.68961
[19] Wang, H. (2003). Functional data analysis of populations of tree-structured objects. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina at Chapel Hill.
[20] Wang, H. and Marron, J. S. (2005). Object oriented data analysis: Sets of trees. Available at www.stat.colostate.edu/ wanghn/papers/OODAtree2.pdf. · Zbl 1126.62002
[21] Wikipedia (2006). Lie group. Available at en.wikipedia.org/wiki/Lie_group.
[22] Wikipedia (2006). Riemannian symmetric space. Available at en.wikipedia.org/wiki/Riemannian_symmetric_space.
[23] Yushkevich, P., Pizer, S. M., Joshi, S. and Marron, J. S. (2001). Intuitive, localized analysis of shape variability. Information Processing in Medical Imaging . Lecture Notes in Comput. Sci. 2082 402–408. Springer, Berlin. · Zbl 0982.68672
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