The Annals of Applied Statistics

Principal arc analysis on direct product manifolds

Sungkyu Jung, Mark Foskey, and J. S. Marron

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We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.

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Ann. Appl. Stat., Volume 5, Number 1 (2011), 578-603.

First available in Project Euclid: 21 March 2011

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Principal Component Analysis nonlinear dimension reduction manifold folded Normal distribution directional data image analysis medial representation


Jung, Sungkyu; Foskey, Mark; Marron, J. S. Principal arc analysis on direct product manifolds. Ann. Appl. Stat. 5 (2011), no. 1, 578--603. doi:10.1214/10-AOAS370.

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  • Bates, D. M. and Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York.
  • 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.
  • Bingham, C. and Mardia, K. V. (1978). A small circle distribution on the sphere. Biometrika 65 379–389.
  • Boothby, W. M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, Orlando, FL.
  • Chernov, N. (2010). Circular and Linear Regression: Fitting Circles and Lines by Least Squares. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Churchill, R. V. and Brown, J. W. (1984). Complex Variables and Applications, 4th ed. McGraw-Hill, New York.
  • Cootes, T. F. and Taylor, C. J. (2001). Statistical models of appearance for medical image analysis and computer vision. In Medical Imaging 2001: Image Processing ( M. Sonka and K. M. Hanson, eds.). Proceedings of the SPIE 4322 236–248. SPIE.
  • Cox, D. R. (1969). Some sampling problems in technology. In New Developments in Survey Sampling ( U. L. Johnson and H. Smith, eds.). Wiley, New York.
  • Elandt, R. C. (1961). The folded normal distribution: Two methods of estimating parameters from moments. Technometrics 3 551–562.
  • Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge Univ. Press, Cambridge.
  • Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1993). Statistical Analysis of Spherical Data. Cambridge Univ. Press, Cambridge.
  • Fletcher, R. (1971). A modified Marquardt subroutine for non-linear least squares. Technical Report AERE-R 6799.
  • Fletcher, P. T. (2004). Statistical variability in nonlinear spaces: Application to shape analysis and DT-MRI. PhD thesis, Univ. North Carolina at Chapel Hill.
  • 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. Med. Imaging 23 995–1005.
  • Gray, N. H., Geiser, P. A. and Geiser, J. R. (1980). On the least-squares fit of small and great circles to spherically projected orientation data. Math. Geol. 12 173–184.
  • Hastie, T. and Stuetzle, W. (1989). Principal curves. J. Amer. Statist. Assoc. 84 502–516.
  • Helgason, S. (2001). Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics 34. Amer. Math. Soc., Providene, RI.
  • Huckemann, S., Hotz, T. and Munk, A. (2010). Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions. Statist. Sinica 20 1–58.
  • Huckemann, S. and Ziezold, H. (2006). Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Adv. in Appl. Probab. 38 299–319.
  • Jeong, J.-Y., Stough, J. V., Marron, J. S. and Pizer, S. M. (2008). Conditional-mean initialization using neighboring objects in deformable model segmentation. In Medical Imaging 2008: Image Processing ( J. M. Reinhardt and J. P. W. Pluim, eds.). Proceedings of the SPIE 6914 69144R.1–69144R.9. SPIE.
  • Johnson, N. L. (1962). The folded normal distribution: Accuracy of estimation by maximum likelihood. Technometrics 4 249–256.
  • Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30 509–541.
  • Krantz, S. G. (1999). Handbook of Complex Variables. Birkhäuser, Boston, MA.
  • Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. in Appl. Probab. 33 324–338.
  • Le, H. and Kume, A. (2000). The Fréchet mean shape and the shape of the means. Adv. in Appl. Probab. 32 101–113.
  • Leone, F. C., Nelson, L. S. and Nottingham, R. B. (1961). The folded normal distribution. Technometrics 3 543–550.
  • Mardia, K. V. and Gadsden, R. J. (1977). A circle of best fit for spherical data and areas of vulcanism. J. Roy. Statist. Soc. Ser. C 26 238–245.
  • Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley Series in Probability and Statistics. Wiley, Chichester.
  • Merck, S. M., Tracton, G., Saboo, R., Levy, J., Chaney, E., Pizer, S. M. and Joshi, S. (2008). Training models of anatomic shape variability. Med. Phys. 35 3584–3596.
  • Moakher, M. (2002). Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24 1–16 (electronic).
  • Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vision 25 127–154.
  • Pizer, S. M., Fletcher, P. T., Fridman, Y., Fritsch, D. S., Gash, A. G., Glotzer, J. M., Joshi, S., Thall, A., Tracton, G., Yushkevich, P. and Chaney, E. L. (2003). Deformable m-reps for 3D medical image segmentation. Int. J. Comput. Vision 55 85–106.
  • Pizer, S. M., Broadhurst, R. E., Levy, J., Liu, X., Jeong, J.-Y., Stough, J., Tracton, G. and Chaney, E. L. (2007). Segmentation by posterior optimization of m-reps: Strategy and results. Unpublished manuscript.
  • Rivest, L.-P. (1999). Some linear model techniques for analyzing small-circle spherical data. Canad. J. Statist. 27 623–638.
  • Scales, L. E. (1985). Introduction to Nonlinear Optimization. Springer, New York.
  • Siddiqi, K. and Pizer, S. (2008). Medial Representations: Mathematics, Algorithms and Applications. Springer, New York.
  • Umbach, D. and Jones, K. N. (2003). A few methods for fitting circles to data. IEEE Transactions on Instrumentation and Measurement 52 1881–1885.