×

Data analysis on nonstandard spaces. (English) Zbl 07910743


MSC:

62-08 Computational methods for problems pertaining to statistics

References:

[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] Ahidar‐Coutrix, A., Le Gouic, T., & Paris, Q. (2019). Convergence rates for empirical barycenters in metric spaces: Curvature, convexity and extendable geodesics. In Probability theory and related fields (pp. 1-46). Berlin, Heidelberg: Springer.
[3] Ahlfors, L. V. (1966). Complex analysis: An introduction to the theory of analytic functions of one complex variable, New York, London: McGraw Hill.
[4] Allen, J. C., & Healy, D. M. (2003). Hyperbolic geometry, Nehari’s theorem, electric circuits, and analog signal processing. Modern Signal Processing, 46, 1-62. · Zbl 1073.94029
[5] Altis, A., Otten, M., Nguyen, P. H., Rainer, H., & Stock, G. (2008). Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis. The Journal of Chemical Physics, 128(24), 245102.
[6] Anderson, T. (1963). Asymptotic theory for principal component analysis. Annals of Mathematical Statistics, 34(1), 122-148. · Zbl 0202.49504
[7] Arnaudon, M., & Miclo, L. (2014). Means in complete manifolds: Uniqueness and approximation. ESAIM: Probability and Statistics, 18, 185-206. · Zbl 1352.60012
[8] Asta, D. M. (2014). Kernel density estimation on symmetric spaces. arXiv preprint arXiv:1411.4040.
[9] Asta, D. M. (2015). Kernel density estimation on symmetric spaces. In International conference on geometric science of information (pp. 779-787). Cham, Switzerland: Springer. · Zbl 1396.94022
[10] Ball, F. G., Dryden, I. L., & Golalizadeh, M. (2008). Brownian motion and Ornstein-Uhlenbeck processes in planar shape space. Methodology and Computing in Applied Probability, 10(1), 1-22. · Zbl 1145.60327
[11] Barden, D., Le, H., & Owen, M. (2013). Central limit theorems for Fréchet means in the space of phylogenetic trees. Electronic Journal of Probability, 18(25), 1-25. · Zbl 1284.60023
[12] Barden, D., Le, H., & Owen, M. (2018). Limiting behaviour of Fréchet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics, 70(1), 99-129. · Zbl 1394.62153
[13] Basser, P. J., Mattiello, J., & LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66(1), 259-267.
[14] Bertrand, J., & Kloeckner, B. (2012). A geometric study of Wasserstein spaces: Hadamard spaces. Journal of Topology and Analysis, 4(04), 515-542. · Zbl 1263.53026
[15] Bhattacharya, R., & Lin, L. (2017). Omnibus CLTs for Fréchet means and nonparametric inference on non‐Euclidean spaces. Proceedings of the American Mathematical Society, 145(1), 413-428. · Zbl 1353.60019
[16] 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
[17] 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
[18] Bigot, J. (2019). Statistical data analysis in the Wasserstein space. arXiv preprint arXiv:1907.08417.
[19] Billera, L., Holmes, S., & Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4), 733-767. · Zbl 0995.92035
[20] Bookstein, F. L. (1991). Morphometric tools for landmark data: Geometry and biology. Cambridge, England: Cambridge University Press. · Zbl 0770.92001
[21] Bouziane, T. (2005). Brownian motion in Riemannian admissible complexes. Illinois Journal of Mathematics, 49(2), 559-580. · Zbl 1077.60058
[22] Bredon, G. E. (1972). Introduction to compact transformation groups. In Pure and applied mathematics (Vol. 46). New York, NY: Academic Press. · Zbl 0246.57017
[23] Brin, M., & Kifer, Y. (2001). Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes. Mathematische Zeitschrift, 237(3), 421-468. · Zbl 0984.58022
[24] Chang, T. (1986). Spherical regression. The Annals of Statistics, 14, 907-924. · Zbl 0605.62079
[25] Chaudhuri, P., & Marron, J. (1999). Sizer for exploration of structures in curves. Journal of the American Statistical Association, 94(447), 807-823. · Zbl 1072.62556
[26] Chaudhuri, P., & Marron, J. (2000). Scale space view of curve estimation. The Annals of Statistics, 28(2), 408-428. · Zbl 1106.62318
[27] Chavel, I. (1984). Eigenvalues in Riemannian geometry (Vol. 115), New York: Academic press. · Zbl 0551.53001
[28] Cheng, M.‐y., & Wu, H.‐t. (2013). Local linear regression on manifolds and its geometric interpretation. Journal of the American Statistical Association, 108(504), 1421-1434. · Zbl 1426.62402
[29] Chirikjian, G. S., & Kyatkin, A. B. (2000). Engineering applications of noncommutative harmonic analysis: With emphasis on rotation and motion groups, New York: CRC Press.
[30] Collin, R. E. (1992). Foundations for microwave engineering. In McGraw‐Hill series in electrical engineering: Radar and antennas. New York: McGraw‐Hill.
[31] Cornea, E., Zhu, H., Kim, P., & Ibrahim, J. G. (2017). Regression models on Riemannian symmetric spaces. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(2), 463-482. · Zbl 1414.62177
[32] Denysenkov, V., Prisner, T., Stubbe, J., & Bennati, M. (2006). High‐field pulsed electron-electron double resonance spectroscopy to determine the orientation of the tyrosyl radicals in ribonucleotide reductase. Proceedings of the National Academy of Sciences of the United States of America, 103(36), 13386-13390.
[33] Devilliers, L., Allassonnière, S., Trouvé, A., & Pennec, X. (2017). Inconsistency of template estimation by minimizing of the variance/pre‐variance in the quotient space. Entropy, 19(6), 288.
[34] Di Marzio, M., Panzera, A., & Taylor, C. C. (2014). Nonparametric regression for spherical data. Journal of the American Statistical Association, 109(506), 748-763. · Zbl 1367.62115
[35] Di Marzio, M., Panzera, A., & Taylor, C. C. (2019). Nonparametric rotations for sphere‐sphere regression. Journal of the American Statistical Association (in press), 114(525), 466-476. · Zbl 1418.62155
[36] Downs, T. (2003). Spherical regression. Biometrika, 90(3), 655-668. · Zbl 1436.62194
[37] Dryden, I., Koloydenko, A., & Zhou, D. (2009). Non‐Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3(3), 1102-1123. · Zbl 1196.62063
[38] Dryden, I. L., Kim, K.‐R., Laughton, C. A., & Le, H. (2019). Principal nested shape space analysis of molecular dynamics data. arXiv preprint arXiv:1903.09445.
[39] Dryden, I. L., & Mardia, K. V. (2016). Statistical shape analysis (2nd ed.). Chichester: Wiley. · Zbl 1381.62003
[40] Dubey, P., & Müller, H.‐G. (2019). Fréchet analysis of variance for random objects. Biometrika, 106(4), 803-821. · Zbl 1435.62383
[41] Eltzner, B. (2019). Measure dependent asymptotic rate of the mean: Geometrical and topological smeariness. arXiv preprint arXiv:1908.04233.
[42] Eltzner, B., Galaz‐García, F., Huckemann, S. F., & Tuschmann, W. (2019). Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifolds. arXiv: 1909.00410.
[43] Eltzner, B., Huckemann, S., & Mardia, K. V. (2018). Torus principal component analysis with applications to RNA structure. The Annals of Applied Statistics, 12(2), 1332-1359. · Zbl 1405.62173
[44] Eltzner, B., & Huckemann, S. F. (2019). A smeary central limit theorem for manifolds with application to high‐dimensional spheres. The Annals of Statistics, 47(6), 3360-3381. · Zbl 1436.60032
[45] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics, 19, 1257-1272. · Zbl 0729.62033
[46] Faugeras, O., Luong, Q., & Papadopoulo, T. (2001). The geometry of multiple images (Vol. 2). Cambridge, MA: MIT Press. · Zbl 1002.68183
[47] Feragen, A., Lauze, F., Lo, P., deBruijne, M., & Nielsen, M. (2011). Geometries on spaces of treelike shapes. In Computer Vision - ACCV 2010 (pp. 160-173).
[48] Fisher, R. (1953). Dispersion on a sphere. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 217(1130), 295-305. · Zbl 0051.37105
[49] Fletcher, P., Venkatasubramanian, S., and Joshi, S. (2008). Robust statistics on Riemannian manifolds via the geometric median. In IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR 2008 (pp. 1-8). IEEE.
[50] Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’Institut de Henri Poincaré, 10(4), 215-310. · Zbl 0035.20802
[51] Frellsen, J., Moltke, I., Thiim, M., Mardia, K. V., Ferkinghoff‐Borg, J., & Hamelryck, T. (2009). A probabilistic model of RNA conformational space. PLoS Computational Biology, 5(6), e1000406.
[52] Garba, M. K., Nye, T. M., Lueg, J., & Huckemann, S. F. (2020). Information geometry for phylogenetic trees. arXiv preprint arXiv:2003.13004.
[53] Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Vol. 7). Hamburg, Germany: Perthes et Besser.
[54] Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33-51. · Zbl 0305.62038
[55] Haff, L. R., Kim, P. T., Koo, J.‐Y., & Richards, D. S. P. (2011). Minimax estimation for mixtures of Wishart distributions. The Annals of Statistics, 39(6), 3417-3440. · Zbl 1246.62137
[56] Healy, D. M. J., Hendriks, H., & Kim, P. T. (1998). Spherical deconvolution. Journal of Multivariate Analysis, 67, 1-22. · Zbl 1126.62346
[57] Helton, J. W. (1982). Non‐Euclidean functional analysis and electronics. Bulletin of the American Mathematical Society, 7, 1-64. · Zbl 0493.46047
[58] Hendriks, H. (1992). Sur le cut‐locus d’une sous‐variété de l’espace euclidean. négligeabilité. Comptes Rendus de l’Académie des Sciences - Series I, 315, 1275-1277. · Zbl 0766.57019
[59] 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
[60] Hinkle, J., Muralidharan, P., Fletcher, P. T., & Joshi, S. (2012). Polynomial regression on Riemannian manifolds. In Computer Vision - ECCV 2012 (pp. 1-14). Springer.
[61] Hotz, T., & Huckemann, S. (2015). Intrinsic means on the circle: Uniqueness, locus and asymptotics. Annals of the Institute of Statistical Mathematics, 67(1), 177-193. · Zbl 1331.62269
[62] Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J., Miller, E., … Skwerer, S. (2013). Sticky central limit theorems on open books. Annals of Applied Probability, 23(6), 2238-2258. · Zbl 1293.60006
[63] Hotz, T., Kelma, F., & Kent, J. T. (2016). Manifolds of projective shapes. arXiv preprint arXiv:1602.04330.
[64] Hotz, T., Kelma, F., & Wieditz, J. (2016). Non‐asymptotic confidence sets for circular means. Entropy, 18(10), 375.
[65] Hsu, E. P. (2002). Stochastic analysis on manifolds (Vol. 38). Providence, Rhode Island: American Mathematical Society. · Zbl 0994.58019
[66] Huckemann, S. (2011a). 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
[67] Huckemann, S. (2011b). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. The Annals of Statistics, 39(2), 1098-1124. · Zbl 1216.62084
[68] Huckemann, S. (2012). On the meaning of mean shape: Manifold stability, locus and the two sample test. Annals of the Institute of Statistical Mathematics, 64(6), 1227-1259. · Zbl 1440.62201
[69] Huckemann, S. (2014). (Semi‐)intrinsic statistical analysis on non‐Euclidean spaces. In Advances in complex data modeling and computational methods in statistics (pp. 103-118). New York: Springer.
[70] Huckemann, S., & Hotz, T. (2014). On means and their asymptotics: Circles and shape spaces. Journal of Mathematical Imaging and Vision, 50(1-2), 98-106. · Zbl 1310.62066
[71] Huckemann, S., Hotz, T., & Munk, A. (2010a). Intrinsic MANOVA for Riemannian manifolds with an application to Kendall’ space of planar shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(4), 593-603.
[72] Huckemann, S., Hotz, T., & Munk, A. (2010b). 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
[73] Huckemann, S., Kim, K.‐R., Munk, A., Rehfeldt, F., Sommerfeld, M., Weickert, J., & Wollnik, C. (2016). The circular sizer, inferred persistence of shape parameters and application to early stem cell differentiation. Bernoulli, 22(4), 2113-2142. · Zbl 1349.62195
[74] Huckemann, S., Mattingly, J. C., Miller, E., & Nolen, J. (2015). Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability, 20(78), 1-34. · Zbl 1327.60028
[75] Huckemann, S. F., & Eltzner, B. (2018). Backward nested descriptors asymptotics with inference on stem cell differentiation. The Annals of Statistics, 46(5), 1994-2019. · Zbl 1405.62070
[76] Huckemann, S. F., & Eltzner, B. (2020). Statistical methods generalizing principal component analysis to non‐Euclidean spaces. In Handbook of variational methods for nonlinear geometric data (pp. 317-338). Cham, Switzerland: Springer. · Zbl 1512.62060
[77] Huckemann, S. F., Kim, P. T., Koo, J.‐Y., & Munk, A. (2010). Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. The Annals of Statistics, 38(4), 2465-2498. · Zbl 1203.62055
[78] Hughes, B. (1996). Geometric topology of stratified spaces. Electronic Research Announcements of the American Mathematical Society, 2(2), 73-81. · Zbl 0866.57018
[79] Hundrieser, S., Eltzner, B., & Huckemann, S. F. (2020). Finite sample smeariness of Fréchet means and application to climate. arXiv preprint arXiv:2005.02321.
[80] Jung, S., Dryden, I. L., & Marron, J. S. (2012). Analysis of principal nested spheres. Biometrika, 99(3), 551-568. · Zbl 1437.62507
[81] Jung, S., Schwartzman, A., & Groisser, D. (2015). Scaling‐rotation distance and interpolation of symmetric positive‐definite matrices. SIAM Journal on Matrix Analysis and Applications, 36(3), 1180-1201. · Zbl 1321.15020
[82] Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30(5), 509-541. · Zbl 0354.57005
[83] Kendall, D. G. (1977). The diffusion of shape. Advances in Applied Probability, 9, 428-430.
[84] Kendall, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), 81-121. · Zbl 0579.62100
[85] Kendall, D. G., Barden, D., Carne, T. K., & Le, H. (1999). Shape and shape theory. Chichester: Wiley. · Zbl 0940.60006
[86] Kendall, W. S. (1990a). The diffusion of Euclidean shape. In Disorder in physical systems (pp. 203-217). Oxford: Oxford University Press. · Zbl 0717.60065
[87] Kendall, W. S. (1990b). 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
[88] Kent, J. T., & Mardia, K. V. (1997). Consistency of Procrustes estimators. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59(1), 281-290. · Zbl 0890.62041
[89] Kent, J. T., & Mardia, K. V. (2009). Principal component analysis for the wrapped normal torus model. In Proceedings of the Leeds Annual Statistical Research (LASR) Workshop 2009.
[90] Kent, J. T., & Mardia, K. V. (2015). The winding number for circular data. In Proceedings of the Leeds Annual Statistical Research (LASR) Workshop 2015.
[91] Kim, P. T., & Koo, J.‐Y. (2002). Optimal spherical deconvolution. Journal of Multivariate Analysis, 80(1), 21-42. · Zbl 0998.62030
[92] Kim, P. T., & Richards, D. S. (2001). Deconvolution density estimation on compact Lie groups. Contemporary Mathematics, 287, 155-171. · Zbl 1020.62029
[93] Kim, P. T., & Richards, D. S. P. (2011). Deconvolution density estimation on the space of positive definite symmetric matrices. In Nonparametric statistics and mixture models: A Festschrift in honor of Thomas P Hettmansperger (pp. 147-168). Danvers, MA: World Scientific. · Zbl 1414.62184
[94] Kobayashi, S., & Nomizu, K. (1969). Foundations of differential geometry (Vol. II). Chichester: Wiley. · Zbl 0175.48504
[95] Kume, A., Dryden, I., & Le, H. (2007). Shape space smoothing splines for planar landmark data. Biometrika, 94(3), 513-528. · Zbl 1134.62044
[96] Kume, A., Preston, S. P., & Wood, A. T. (2013). Saddlepoint approximations for the normalizing constant of Fisher-Bingham distributions on products of spheres and Stiefel manifolds. Biometrika, 100(4), 971-984. · Zbl 1452.62998
[97] Kume, A., & Sei, T. (2018). On the exact maximum likelihood inference of Fisher-Bingham distributions using an adjusted holonomic gradient method. Statistics and Computing, 28(4), 835-847. · Zbl 1384.62172
[98] Lang, S. (1999). Fundamentals of differential geometry, New York: Springer. · Zbl 0932.53001
[99] Le Gouic, T., Paris, Q., Rigollet, P., & Stromme, A. (2019). Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space. arXiv preprints, arXiv:1908.00828.
[100] Le, H. (1994). Brownian motions on shape and size‐and‐shape spaces. Journal of Applied Probability, 31(1), 101-113. · Zbl 0797.58093
[101] Le, H. (1998). On the consistency of procrustean mean shapes. Advances of Applied Probability, 30(1), 53-63. · Zbl 0906.60007
[102] Le, H. (2001). Locating Fréchet means with an application to shape spaces. Advances of Applied Probability, 33(2), 324-338. · Zbl 0990.60008
[103] Le, H. (2003). Unrolling shape curves. Journal of the London Mathematical Society, 68(2), 511-526. · Zbl 1040.60009
[104] Le, H., & Kume, A. (2000). The Fréchet mean shape and the shape of the means. Advances in Applied Probability, 32, 101-113. · Zbl 0961.60020
[105] Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603-621.
[106] Lele, S. (1993). Euclidean distance matrix analysis (EDMA): Estimation of mean form and mean form difference. Mathematical Geology, 25(5), 573-602. · Zbl 0970.86539
[107] Lenglet, C., Rousson, M., Deriche, R., & Faugeras, O. (2006). Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. Journal of Mathematical Imaging and Vision, 25(3), 423-444. · Zbl 1478.62387
[108] Lin, B., Monod, A., and Yoshida, R. (2018). Tropical foundations for probability and statistics on phylogenetic tree space. arXiv preprint arXiv:1805.12400.
[109] Lin, B., Sturmfels, B., Tang, X., & Yoshida, R. (2017). Convexity in tree spaces. SIAM Journal on Discrete Mathematics, 31(3), 2015-2038. · Zbl 1370.05040
[110] Lin, L., St. Thomas, B., Zhu, H., & Dunson, D. B. (2017). Extrinsic local regression on manifold‐valued data. Journal of the American Statistical Association, 112(519), 1261-1273.
[111] Lin, Z., & Müller, H.‐G. (2019). Total variation regularized Fréchet regression for metric‐space valued data. arXiv preprint arXiv:1904.09647.
[112] Lindeberg, T. (2011). Generalized Gaussian scale‐space axiomatics comprising linear scale‐space, affine scale‐space and spatio‐temporal scale‐space. Journal of Mathematical Imaging and Vision, 40(1), 36-81. · Zbl 1255.68250
[113] Lott, J. (2006). Some geometric calculations on Wasserstein space. arXiv preprint math/0612562.
[114] Mackenzie, J. (1957). The estimation of an orientation relationship. Acta Crystallographica, 10(1), 61-62.
[115] Mardia, K., & Patrangenaru, V. (2005). Directions and projective shapes. The Annals of Statistics, 33, 1666-1699. · Zbl 1078.62068
[116] Mardia, K. V., & Jupp, P. E. (2000). Directional statistics. New York: Wiley. · Zbl 0935.62065
[117] Mardia, K. V., Kent, J. T., & Bibby, J. M. (1980). Multivariate analysis, London: Academic press.
[118] Masarotto, V., Panaretos, V. M., & Zemel, Y. (2019). Procrustes metrics on covariance operators and optimal transportation of Gaussian processes. Sankhya A, 81(1), 172-213. · Zbl 1420.60048
[119] McKilliam, R. G., Quinn, B. G., & Clarkson, I. V. L. (2012). Direction estimation by minimum squared arc length. IEEE Transactions on Signal Processing, 60(5), 2115-2124. · Zbl 1391.62035
[120] Meister, A. (2009). Deconvolution problems in nonparametric statistics (Vol. 193). Berlin Heidelberg: Springer Science & Business Media. · Zbl 1178.62028
[121] Miolane, N., Holmes, S., & Pennec, X. (2017). Template shape estimation: Correcting an asymptotic bias. SIAM Journal on Imaging Sciences, 10(2), 808-844. · Zbl 1403.62128
[122] Moulton, V., & Steel, M. (2004). Peeling phylogenetic ‘oranges’. Advances in Applied Mathematics, 33(4), 710-727. · Zbl 1067.92045
[123] Nye, T. M., Tang, X., Weyenberg, G., & Yoshida, R. (2017). Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees. Biometrika, 104(4), 901-922. · Zbl 07072335
[124] Nye, T. M., & White, M. (2014). Diffusion on some simple stratified spaces. Journal of Mathematical Imaging and Vision, 50(1-2), 115-125. · Zbl 1306.32023
[125] O’Neill, B. (1966). The fundamental equations of a submersion. The Michigan Mathematical Journal, 13(4), 459-469. · Zbl 0145.18602
[126] Øksendal, B. (2010). Stochastic differential equations: An introduction with applications, Berlin Heidelberg: Springer.
[127] Oliveira, M., Crujeiras, R. M., & Rodrguez‐Casal, A. (2013). Circsizer: An exploratory tool for circular data. Environmental and Ecological Statistics, 20, 1-17.
[128] Owen, M., & Provan, J. S. (2011). A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 8(1), 2-13.
[129] Panaretos, V. M., & Zemel, Y. (2019). Statistical aspects of Wasserstein distances. Annual Review of Statistics and its Application, 6, 405-431.
[130] Patrangenaru, V., Liu, X., & Sugathadasa, S. (2010). A nonparametric approach to 3d shape analysis from digitial camera images - I, in memory of W.P. Dayawansa. Journal of Multivariate Analysis, 101, 11-31. · Zbl 1177.62059
[131] Pennec, X. (2018). Barycentric subspace analysis on manifolds. The Annals of Statistics, 46(6A), 2711-2746. · Zbl 1410.60018
[132] Pennec, X. (2019). Curvature effects on the empirical mean in Riemannian and affine manifolds: A non‐asymptotic high concentration expansion in the small‐sample regime. arXiv preprints, arXiv:1906.07418.
[133] Pennec, X. (2020). Advances in geometric statistics. In Handbook of variational methods for nonlinear geometric data (pp. 339-360). Cham, Switzerland: Springer.
[134] Petersen, A., & Müller, H.‐G. (2019). Fréchet regression for random objects with Euclidean predictors. The Annals of Statistics, 47(2), 691-719. · Zbl 1417.62091
[135] Rabin, J., Peyré, G., Delon, J., & Bernot, M. (2011). Wasserstein barycenter and its application to texture mixing. In International Conference on Scale Space and Variational Methods in Computer Vision (pp. 435-446). Springer.
[136] Richardson, J. S., Schneider, B., Murray, L. W., Kapral, G. J., Immormino, R. M., Headd, J. J., … Berman, H. M. (2008). RNA backbone: Consensus all‐angle conformers and modular string nomenclature (an RNA ontology consortium contribution). RNA, 14, 465-481.
[137] Rivest, L.‐P. (1989). Spherical regression for concentrated Fisher‐von Mises distributions. The Annals of Statistics, 17, 307-317. · Zbl 0669.62041
[138] Rivest, L.‐P., Baillargeon, S., & Pierrynowski, M. (2008). A directional model for the estimation of the rotation axes of the ankle joint. Journal of the American Statistical Association, 103(483), 1060-1069. · Zbl 1205.62177
[139] Romano, J. P., & Lehmann, E. L. (2005). Testing statistical hypotheses. Berlin: Springer. · Zbl 1076.62018
[140] Rosenthal, M., Wu, W., Klassen, E., & Srivastava, A. (2014). Spherical regression models using projective linear transformations. Journal of the American Statistical Association, 109(508), 1615-1624. · Zbl 1368.62185
[141] Rosenthal, M., Wu, W., Klassen, E., & Srivastava, A. (2017). Nonparametric spherical regression using diffeomorphic mappings. arXiv preprint arXiv:1702.00823.
[142] Sargsyan, K., Wright, J., & Lim, C. (2012). Geopca: A new tool for multivariate analysis of dihedral angles based on principal component geodesics. Nucleic Acids Research, 40(3), e25-e25.
[143] Schmidt‐Hieber, J., Munk, A., & Dümbgen, L. (2013). Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features. The Annals of Statistics, 41(3), 1299-1328. · Zbl 1293.62104
[144] Schötz, C. (2019). Convergence rates for the generalized Fréchet mean via the quadruple inequality. Electronic Journal of Statistics, 13(2), 4280-4345. · Zbl 1432.62080
[145] Semple, C., & Steel, M. (2003). Phylogenetics (Vol. 24). Oxford: Oxford University Press. · Zbl 1043.92026
[146] Sengupta, D., & Jammalamadaka, S. (2003). Linear models: An integrated approach, Danvers, MA: World Scientific. · Zbl 1049.62080
[147] Severn, K., Dryden, I. L., & Preston, S. P. (2019). Manifold valued data analysis of samples of networks, with applications in corpus linguistics. arXiv preprint arXiv:1902.08290.
[148] Shiers, N., Zwiernik, P., Aston, J. A., & Smith, J. Q. (2016). The correlation space of Gaussian latent tree models and model selection without fitting. Biometrika, 103(3), 531-545. · Zbl 1506.62321
[149] Siddiqi, K., & Pizer, S. (2008). Medial representations: Mathematics, algorithms and applications, Cham, Switzerland: Springer Verlag. · Zbl 1151.00014
[150] Skwerer, S., Bullitt, E., Huckemann, S., Miller, E., Oguz, I., Owen, M., … Marron, J. S. (2014). Tree‐oriented analysis of brain artery structure. Journal of Mathematical Imaging and Vision, 50(1-2), 126-143. · Zbl 1303.92054
[151] Sommer, S. (2015). Anisotropic distributions on manifolds: Template estimation and most probable paths. In S.Ourselin (ed.), D. C.Alexander (ed.), C.‐F.Westin (ed.), & M. J.Cardoso (ed.) (Eds.), Information processing in medical imaging (pp. 193-204). Cham: Springer International Publishing.
[152] Sommer, S., & Svane, A. M. (2017). Modelling anisotropic covariance using stochastic development and sub‐riemannian frame bundle geometry. Journal of Geometric Mechanics, 9(3), 391-410. · Zbl 1367.53031
[153] Sturm, K. (2003). Probability measures on metric spaces of nonpositive curvature. Contemporary Mathematics, 338, 357-390. · Zbl 1040.60002
[154] Su, J., Dryden, I. L., Klassen, E., Le, H., & Srivastava, A. (2012). Fitting smoothing splines to time‐indexed, noisy points on nonlinear manifolds. Image and Vision Computing, 30(6-7), 428-442.
[155] Taylor, J. E. (2006). A Gaussian kinematic formula. The Annals of Probability, 34(1), 122-158. · Zbl 1094.60025
[156] Teets, D., & Whitehead, K. (1999). The discovery of Ceres: How Gauss became famous. Mathematics Magazine, 72(2), 83-93. · Zbl 1005.01007
[157] Telschow, F. J., Huckemann, S. F., & Pierrynowski, M. R. (2016). Functional inference on rotational curves and identification of human gait at the knee joint. arXiv preprint arXiv:1611.03665.
[158] Telschow, F. J., Pierrynowski, M. R., & Huckemann, S. F. (2019). Confidence tubes for curves on SO(3) and identification of subject‐specific gait change after kneeling. arXiv preprint arXiv:1909.06583.
[159] Terras, A. (1985). Harmonic analysis on symmetric spaces and applications I. New York: Springer‐Verlag. · Zbl 0574.10029
[160] Terras, A. (1988). Harmonic analysis on symmetric spaces and applications II. New York: Springer‐Verlag. · Zbl 0668.10033
[161] Tran, D. (2019). Behavior of Fréchet mean and central limit theorems on spheres. arXiv Preprint arXiv:1911.01985.
[162] van derVaart, A. (2000). Asymptotic statistics, Cambridge: Cambridge University Press. · Zbl 0943.62002
[163] vonMises, R. (1918). Über die “Ganzzahligkeit” der Atomgewichte und verwandte Fragen. Physikalishce Zeitschrift, 19, 490-500. · JFM 46.1493.01
[164] Yuan, Y., Zhu, H., Lin, W., & Marron, J. (2012). Local polynomial regression for symmetric positive definite matrices. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 74(4), 697-719. · Zbl 1411.62110
[165] 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). · Zbl 0413.60024
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.