On the meaning of mean shape: manifold stability, locus and the two sample test. (English) Zbl 1440.62201
Summary: Various concepts of mean shape previously unrelated in the literature are brought into relation. In particular, for non-manifolds, such as Kendall’s 3D shape space, this paper answers the question, for which means one may apply a two-sample test. The answer is positive if intrinsic or Ziezold means are used. The underlying general result of manifold stability of a mean on a shape space, the quotient due to an proper and isometric action of a Lie group on a Riemannian manifold, blends the slice theorem from differential geometry with the statistics of shape. For 3D Procrustes means, however, a counterexample is given. To further elucidate on subtleties of means, for spheres and Kendall’s shape spaces, a first-order relationship between intrinsic, residual/Procrustean and extrinsic/Ziezold means is derived stating that for high concentration the latter approximately divides the (generalized) geodesic segment between the former two by the ratio \(1:3\). This fact, consequences of coordinate choices for the power of tests and other details, e.g. that extrinsic Schoenberg means may increase dimension are discussed and illustrated by simulations and exemplary datasets.
MSC:
| 62H11 | Directional data; spatial statistics |
| 60D05 | Geometric probability and stochastic geometry |
| 62G10 | Nonparametric hypothesis testing |
| 62R30 | Statistics on manifolds |
Keywords:
intrinsic mean; extrinsic mean; Procrustes mean; Schoenberg mean; Ziezold mean; shape spaces; proper Lie group action; slice theorem; horizontal lift; stratified spacesSoftware:
ishapesReferences:
| [1] | Afsari, B. (2011). Riemannian L p center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139, 655–773. · Zbl 1220.53040 |
| [2] | Anderson, T. (2003). An introduction to multivariate statistical analysis (3rd ed.). New York: Wiley. · Zbl 1039.62044 |
| [3] | Bandulasiri, A., Patrangenaru, V. (2005). Algorithms for nonparametric inference on shape manifolds. Proceedings of JSM 2005 Minneapolis, MN, 1617–1622. |
| [4] | Bhattacharya A. (2008) Statistical analysis on manifolds: A nonparametric approach for inference on shape spaces. Sankhya, Series A, 70(2): 223–266 · Zbl 1193.62079 |
| [5] | Bhattacharya R. N., Patrangenaru V. (2003) Large sample theory of intrinsic and extrinsic sample means on manifolds I. The Annals of Statistics, 31(1): 1–29 · Zbl 1020.62026 · doi:10.1214/aos/1046294456 |
| [6] | Bhattacharya R. N., Patrangenaru V. (2005) Large sample theory of intrinsic and extrinsic sample means on manifolds II. The Annals of Statistics, 33(3): 1225–1259 · Zbl 1072.62033 · doi:10.1214/009053605000000093 |
| [7] | Billera L., Holmes S., Holmes S., Holmes S. (2001) Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4): 733–767 · Zbl 0995.92035 · doi:10.1006/aama.2001.0759 |
| [8] | Bredon, G. E. (1972). Introduction to compact transformation groups. In Pure and applied mathematics (Vol. 46). New York: Academic Press. · Zbl 0246.57017 |
| [9] | Choquet G. (1954) Theory of capacities. Annales de l’Institut de Fourier, 5: 131–295 · Zbl 0064.35101 · doi:10.5802/aif.53 |
| [10] | Dryden I. L., Mardia K. V. (1998) Statistical shape analysis. Wiley, Chichester · Zbl 0901.62072 |
| [11] | Dryden, I. L., Kume, A., Le., H., Wood, A. T. A. (2008). A multidimensional scaling approach to shape analysis (to appear). · Zbl 1437.62447 |
| [12] | Fréchet M. (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. 10(4): 215–310 |
| [13] | Gower J. C. (1975) Generalized Procrustes analysis. Psychometrika, 40(33–51): 40 33–51 · Zbl 0305.62038 |
| [14] | Hendriks H., Landsman Z. (1996) Asymptotic behaviour of sample mean location for manifolds. Statistics and Probability Letters, 26: 169–178 · Zbl 0848.62017 · doi:10.1016/0167-7152(95)00007-0 |
| [15] | Hendriks H., Landsman Z. (1998) Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. Journal of Multivariate Analysis, 67: 227–243 · Zbl 0941.62069 · doi:10.1006/jmva.1998.1776 |
| [16] | Hendriks H., Landsman Z., Ruymgaart F. (1996) Asymptotic behaviour of sample mean direction for spheres. Journal of Multivariate Analysis, 59: 141–152 · Zbl 0864.62036 · doi:10.1006/jmva.1996.0057 |
| [17] | Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V., Skwerer, S. (2012). Sticky central limit theorems on open books. arXiv.org, 1202.4267 [math.PR] [math.MG] [math.ST]. · Zbl 1293.60006 |
| [18] | Huckemann, S. (2010). R-package for intrinsic statistical analysis of shapes. http://www.mathematik.uni-kassel.de/\(\sim\)huckeman/software/ishapes_1.0.tar.gz . |
| [19] | Huckemann S. (2011) Inference on 3D Procrustes means: Tree boles growth, rank-deficient diffusion tensors and perturbation models. Scandinavian Journal of Statistics, 38(3): 424–446 · Zbl 1246.62120 |
| [20] | Huckemann S., Ziezold H. (2006) Principal component analysis for Riemannian manifolds with an application to triangular shape spaces. Advances of Applied Probability (SGSA), 38(2): 299–319 · Zbl 1099.62059 · doi:10.1239/aap/1151337073 |
| [21] | Huckemann S., Hotz T., Munk A. (2010) Intrinsic MANOVA for Riemannian manifolds with an application to Kendall’s space of planar shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(4): 593–603 · doi:10.1109/TPAMI.2009.117 |
| [22] | Huckemann S., Hotz T., Munk A. (2010) Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions (with discussion). Statistica Sinica, 20(1): 1–100 · Zbl 1180.62087 |
| [23] | Jupp P. E. (1988) Residuals for directional data. Journal of Applied Statistics, 15(2): 137–147 · doi:10.1080/02664768800000021 |
| [24] | Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 509–541. · Zbl 0354.57005 |
| [25] | Kendall, D. (1974). Foundations of a theory of random sets. In Stochastic geometry, tribute memory Rollo Davidson (pp. 322–376). New York: Wiley. · Zbl 0275.60068 |
| [26] | Kendall, D. G., Barden, D., Carne, T. K., Le, H. (1999). Shape and shape theory. Chichester: Wiley. · Zbl 0940.60006 |
| [27] | Kendall W. S. (1990) Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proceedings of the London Mathematical Society, 61: 371–406 · Zbl 0675.58042 |
| [28] | Kent, J., Hotz, T., Huckemann, S., Miller, E. (2011). The topology and geometry of projective shape spaces (in preparation). |
| [29] | Klassen, E., Srivastava, A., Mio, W., Joshi, S. (2004, March). Analysis on planar shapes using geodesic paths on shape spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(3), 372–383. |
| [30] | Kobayashi, S., Nomizu, K. (1963). Foundations of differential geometry (Vol. I). Chichester: Wiley. · Zbl 0119.37502 |
| [31] | Kobayashi, S., Nomizu, K. (1969). Foundations of differential geometry (Vol. II). Chichester: Wiley. · Zbl 0175.48504 |
| [32] | Krim, H., Yezzi, A. J. J. E. (2006). Statistics and analysis of shapes. In Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhäuser. · Zbl 1091.00003 |
| [33] | Le H. (2001) Locating Fréchet means with an application to shape spaces. Advances of Applied Probability (SGSA), 33(2): 324–338 · Zbl 0990.60008 · doi:10.1239/aap/999188316 |
| [34] | Le H. (2004) Estimation of Riemannian barycenters. LMS Journal of Computation and Mathematics, 7: 193–200 · Zbl 1054.60011 · doi:10.1112/S1461157000001091 |
| [35] | Lehmann, E. L. (1997). Testing statistical hypotheses. In Springer texts in statistics. New York: Springer. · Zbl 0862.62020 |
| [36] | Mardia, K., Patrangenaru,. V. (2001). On affine and projective shape data analysis. In K. V. Mardia, R. G. Aykroyd (Eds.), Proceedings of the 20th LASR Workshop on functional and spatial data analysis (pp. 39–45). |
| [37] | Mardia K., Patrangenaru V. (2005) Directions and projective shapes. The Annals of Statistics, 33: 1666–1699 · Zbl 1078.62068 · doi:10.1214/009053605000000273 |
| [38] | Matheron, G. (1975). Random sets and integral geometry. In Wiley series in probability and mathematical statistics. New York: Wiley. · Zbl 0321.60009 |
| [39] | Nash J. (1956) The imbedding problem for Riemannian manifolds. The Annals of Mathematics, 63: 20–63 · Zbl 0070.38603 · doi:10.2307/1969989 |
| [40] | Palais, R. S. (1961). On the existence of slices for actions of non-compact Lie groups. The Annals of Mathematics 2nd Series, 73(2), 295–323 · Zbl 0103.01802 · doi:10.2307/1970335 |
| [41] | Schmidt, F. R., Clausen, M., Cremers, D. (2006). Shape matching by variational computation of geodesics on a manifold. In Pattern recognition (Proceedings DAGM), LNCS (Vol. 4174, pp. 142–151). Berlin: Springer. |
| [42] | Small, C. G. (1996). The statistical theory of shape. New York: Springer. · Zbl 0859.62087 |
| [43] | Zahn C., Roskies R. (1972) Fourier descriptors for plane closed curves. IEEE Transactions on Computers, C 21: 269–281 · Zbl 0231.68042 · doi:10.1109/TC.1972.5008949 |
| [44] | Ziezold, H. (1977). Expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Transaction of the 7th Prague conference on information theory, statistical decision function and random processes, A, pp. 591–602. |
| [45] | Ziezold, H. (1994). Mean figures and mean shapes applied to biological figure and shape distributions in the plane. Biometrical Journal, 36, 491–510 · Zbl 0838.62101 · doi:10.1002/bimj.4710360409 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
