[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 |