How to cite this WIREs title:
WIREs Comp Stat

Data analysis on nonstandard spaces

Advanced Review

Benjamin Eltzner, Stephan F. Huckemann

Published Online: Sep 08 2020

DOI: 10.1002/wics.1526

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Abstract The task to write on data analysis on nonstandard spaces is quite substantial, with a huge body of literature to cover, from parametric to nonparametrics, from shape spaces to Wasserstein spaces. In this survey we convey simple (e.g., Fréchet means) and more complicated ideas (e.g., empirical process theory), common to many approaches with focus on their interaction with one‐another. Indeed, this field is fast growing and it is imperative to develop a mathematical view point, drawing power, and diversity from a higher level of abstraction, for example, by introducing generalized Fréchet means. While many problems have found ingenious solutions (e.g., Procrustes analysis for principal component analysis [PCA] extensions on shape spaces and diffusion on the frame bundle to mimic anisotropic Gaussians), more problems emerge, often more difficult (e.g., topology and geometry influencing limiting rates and defining generic intrinsic PCA extensions). Along this survey, we point out some open problems, that will, as it seems, keep mathematicians, statisticians, computer and data scientists busy for a while. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data

Empirical variances of intrinsic sample means times sample size versus sample size. (a) On the circle. Solid blue: a smeary distribution. Solid green and red: distributions with lower probability density at the antipodal of the mean, their asymptotic variance is dashed. Solid cyan and purple: distributions close to smeary distribution, but zero in around the antipodal of the mean. Solid black: the height of the Euclidean variance where all curves start. (b) On the three‐spider with three points with equal weight. Two lie on different legs at distance 1 from the origin and one lies at distance 2 + 3t on the third leg such that their mean lies at t on the third leg. The number labeling each curve denotes t. The height of the Euclidean variance which all curves eventually slightly overshoot, is solid black

[ Normal View | Magnified View ]

References

Afsari,, B. (2011). Riemannian L^p center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139, 655–773.
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.
Ahlfors,, L. V. (1966). Complex analysis: An introduction to the theory of analytic functions of one complex variable, New York, London: McGraw Hill.
Allen,, J. C., & Healy,, D. M. (2003). Hyperbolic geometry, Nehari`s theorem, electric circuits, and analog signal processing. Modern Signal Processing, 46, 1–62.
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.
Anderson,, T. (1963). Asymptotic theory for principal component analysis. Annals of Mathematical Statistics, 34(1), 122–148.
Arnaudon,, M., & Miclo,, L. (2014). Means in complete manifolds: Uniqueness and approximation. ESAIM: Probability and Statistics, 18, 185–206.
Asta,, D. M. (2014). Kernel density estimation on symmetric spaces. arXiv preprint arXiv:1411.4040.
Asta,, D. M. (2015). Kernel density estimation on symmetric spaces. In International conference on geometric science of information (pp. 779–787). Cham, Switzerland: Springer.
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.
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.
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.
Basser,, P. J., Mattiello,, J., & LeBihan,, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66(1), 259–267.
Bertrand,, J., & Kloeckner,, B. (2012). A geometric study of Wasserstein spaces: Hadamard spaces. Journal of Topology and Analysis, 4(04), 515–542.
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.
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.
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.
Bigot,, J. (2019). Statistical data analysis in the Wasserstein space. arXiv preprint arXiv:1907.08417.
Billera,, L., Holmes,, S., & Vogtmann,, K. (2001). Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4), 733–767.
Bookstein,, F. L. (1991). Morphometric tools for landmark data: Geometry and biology. Cambridge, England: Cambridge University Press.
Bouziane,, T. (2005). Brownian motion in Riemannian admissible complexes. Illinois Journal of Mathematics, 49(2), 559–580.
Bredon,, G. E. (1972). Introduction to compact transformation groups. In Pure and applied mathematics (Vol. 46). New York, NY: Academic Press.
Brin,, M., & Kifer,, Y. (2001). Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes. Mathematische Zeitschrift, 237(3), 421–468.
Chang,, T. (1986). Spherical regression. The Annals of Statistics, 14, 907–924.
Chaudhuri,, P., & Marron,, J. (1999). Sizer for exploration of structures in curves. Journal of the American Statistical Association, 94(447), 807–823.
Chaudhuri,, P., & Marron,, J. (2000). Scale space view of curve estimation. The Annals of Statistics, 28(2), 408–428.
Chavel,, I. (1984). Eigenvalues in Riemannian geometry (Vol. 115), New York: Academic press.
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.
Chirikjian,, G. S., & Kyatkin,, A. B. (2000). Engineering applications of noncommutative harmonic analysis: With emphasis on rotation and motion groups, New York: CRC Press.
Collin,, R. E. (1992). Foundations for microwave engineering. In McGraw‐Hill series in electrical engineering: Radar and antennas. New York: McGraw‐Hill.
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.
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.
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.
Di Marzio,, M., Panzera,, A., & Taylor,, C. C. (2014). Nonparametric regression for spherical data. Journal of the American Statistical Association, 109(506), 748–763.
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.
Downs,, T. (2003). Spherical regression. Biometrika, 90(3), 655–668.
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.
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.
Dryden,, I. L., & Mardia,, K. V. (2016). Statistical shape analysis (2nd ed.). Chichester: Wiley.
Dubey,, P., & Müller,, H.‐G. (2019). Fréchet analysis of variance for random objects. Biometrika, 106(4), 803–821.
Eltzner,, B. (2019). Measure dependent asymptotic rate of the mean: Geometrical and topological smeariness. arXiv preprint arXiv:1908.04233.
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.
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.
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.
Fan,, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics, 19, 1257–1272.
Faugeras,, O., Luong,, Q., & Papadopoulo,, T. (2001). The geometry of multiple images (Vol. 2). Cambridge, MA: MIT Press.
Feragen,, A., Lauze,, F., Lo,, P., de Bruijne,, M., & Nielsen,, M. (2011). Geometries on spaces of treelike shapes. In Computer Vision – ACCV 2010 (pp. 160–173).
Fisher,, R. (1953). Dispersion on a sphere. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 217(1130), 295–305.
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.
Fréchet,, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l`Institut de Henri Poincaré, 10(4), 215–310.
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.
Garba,, M. K., Nye,, T. M., Lueg,, J., & Huckemann,, S. F. (2020). Information geometry for phylogenetic trees. arXiv preprint arXiv:2003.13004.
Gauss,, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Vol. 7). Hamburg, Germany: Perthes et Besser.
Gower,, J. C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33–51.
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.
Healy,, D. M. J., Hendriks,, H., & Kim,, P. T. (1998). Spherical deconvolution. Journal of Multivariate Analysis, 67, 1–22.
Helton,, J. W. (1982). Non‐Euclidean functional analysis and electronics. Bulletin of the American Mathematical Society, 7, 1–64.
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.
Hendriks,, H., & Landsman,, Z. (1998). Mean location and sample mean location on manifolds: Asymptotics, tests, confidence regions. Journal of Multivariate Analysis, 67, 227–243.
Hinkle,, J., Muralidharan,, P., Fletcher,, P. T., & Joshi,, S. (2012). Polynomial regression on Riemannian manifolds. In Computer Vision – ECCV 2012 (pp. 1–14). Springer.
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.
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.
Hotz,, T., Kelma,, F., & Kent,, J. T. (2016). Manifolds of projective shapes. arXiv preprint arXiv:1602.04330.
Hotz,, T., Kelma,, F., & Wieditz,, J. (2016). Non‐asymptotic confidence sets for circular means. Entropy, 18(10), 375.
Hsu,, E. P. (2002). Stochastic analysis on manifolds (Vol. 38). Providence, Rhode Island: American Mathematical Society.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Huckemann,, S. F., & Eltzner,, B. (2018). Backward nested descriptors asymptotics with inference on stem cell differentiation. The Annals of Statistics, 46(5), 1994–2019.
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.
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.
Hughes,, B. (1996). Geometric topology of stratified spaces. Electronic Research Announcements of the American Mathematical Society, 2(2), 73–81.
Hundrieser,, S., Eltzner,, B., & Huckemann,, S. F. (2020). Finite sample smeariness of Fréchet means and application to climate. arXiv preprint arXiv:2005.02321.
Jung,, S., Dryden,, I. L., & Marron,, J. S. (2012). Analysis of principal nested spheres. Biometrika, 99(3), 551–568.
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.
Karcher,, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30(5), 509–541.
Kendall,, D. G. (1977). The diffusion of shape. Advances in Applied Probability, 9, 428–430.
Kendall,, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), 81–121.
Kendall,, D. G., Barden,, D., Carne,, T. K., & Le,, H. (1999). Shape and shape theory. Chichester: Wiley.
Kendall,, W. S. (1990a). The diffusion of Euclidean shape. In Disorder in physical systems (pp. 203–217). Oxford: Oxford University Press.
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.
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.
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.
Kent,, J. T., & Mardia,, K. V. (2015). The winding number for circular data. In Proceedings of the Leeds Annual Statistical Research (LASR) Workshop 2015.
Kim,, P. T., & Koo,, J.‐Y. (2002). Optimal spherical deconvolution. Journal of Multivariate Analysis, 80(1), 21–42.
Kim,, P. T., & Richards,, D. S. (2001). Deconvolution density estimation on compact Lie groups. Contemporary Mathematics, 287, 155–171.
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.
Kobayashi,, S., & Nomizu,, K. (1969). Foundations of differential geometry (Vol. II). Chichester: Wiley.
Kume,, A., Dryden,, I., & Le,, H. (2007). Shape space smoothing splines for planar landmark data. Biometrika, 94(3), 513–528.
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.
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.
Lang,, S. (1999). Fundamentals of differential geometry, New York: Springer.
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.
Le,, H. (1994). Brownian motions on shape and size‐and‐shape spaces. Journal of Applied Probability, 31(1), 101–113.
Le,, H. (1998). On the consistency of procrustean mean shapes. Advances of Applied Probability, 30(1), 53–63.
Le,, H. (2001). Locating Fréchet means with an application to shape spaces. Advances of Applied Probability, 33(2), 324–338.
Le,, H. (2003). Unrolling shape curves. Journal of the London Mathematical Society, 68(2), 511–526.
Le,, H., & Kume,, A. (2000). The Fréchet mean shape and the shape of the means. Advances in Applied Probability, 32, 101–113.
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.
Lele,, S. (1993). Euclidean distance matrix analysis (EDMA): Estimation of mean form and mean form difference. Mathematical Geology, 25(5), 573–602.
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.
Lin,, B., Monod,, A., and Yoshida,, R. (2018). Tropical foundations for probability and statistics on phylogenetic tree space. arXiv preprint arXiv:1805.12400.
Lin,, B., Sturmfels,, B., Tang,, X., & Yoshida,, R. (2017). Convexity in tree spaces. SIAM Journal on Discrete Mathematics, 31(3), 2015–2038.
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.
Lin,, Z., & Müller,, H.‐G. (2019). Total variation regularized Fréchet regression for metric‐space valued data. arXiv preprint arXiv:1904.09647.
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.
Lott,, J. (2006). Some geometric calculations on Wasserstein space. arXiv preprint math/0612562.
Mackenzie,, J. (1957). The estimation of an orientation relationship. Acta Crystallographica, 10(1), 61–62.
Mardia,, K., & Patrangenaru,, V. (2005). Directions and projective shapes. The Annals of Statistics, 33, 1666–1699.
Mardia,, K. V., & Jupp,, P. E. (2000). Directional statistics. New York: Wiley.
Mardia,, K. V., Kent,, J. T., & Bibby,, J. M. (1980). Multivariate analysis, London: Academic press.
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.
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.
Meister,, A. (2009). Deconvolution problems in nonparametric statistics (Vol. 193). Berlin Heidelberg: Springer Science %26 Business Media.
Miolane,, N., Holmes,, S., & Pennec,, X. (2017). Template shape estimation: Correcting an asymptotic bias. SIAM Journal on Imaging Sciences, 10(2), 808–844.
Moulton,, V., & Steel,, M. (2004). Peeling phylogenetic ‘oranges’. Advances in Applied Mathematics, 33(4), 710–727.
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.
Nye,, T. M., & White,, M. (2014). Diffusion on some simple stratified spaces. Journal of Mathematical Imaging and Vision, 50(1–2), 115–125.
O`Neill,, B. (1966). The fundamental equations of a submersion. The Michigan Mathematical Journal, 13(4), 459–469.
Øksendal,, B. (2010). Stochastic differential equations: An introduction with applications, Berlin Heidelberg: Springer.
Oliveira,, M., Crujeiras,, R. M., & Rodrguez‐Casal,, A. (2013). Circsizer: An exploratory tool for circular data. Environmental and Ecological Statistics, 20, 1–17.
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.
Panaretos,, V. M., & Zemel,, Y. (2019). Statistical aspects of Wasserstein distances. Annual Review of Statistics and its Application, 6, 405–431.
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.
Pennec,, X. (2018). Barycentric subspace analysis on manifolds. The Annals of Statistics, 46(6A), 2711–2746.
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.
Pennec,, X. (2020). Advances in geometric statistics. In Handbook of variational methods for nonlinear geometric data (pp. 339–360). Cham, Switzerland: Springer.
Petersen,, A., & Müller,, H.‐G. (2019). Fréchet regression for random objects with Euclidean predictors. The Annals of Statistics, 47(2), 691–719.
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.
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.
Rivest,, L.‐P. (1989). Spherical regression for concentrated Fisher‐von Mises distributions. The Annals of Statistics, 17, 307–317.
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.
Romano,, J. P., & Lehmann,, E. L. (2005). Testing statistical hypotheses. Berlin: Springer.
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.
Rosenthal,, M., Wu,, W., Klassen,, E., & Srivastava,, A. (2017). Nonparametric spherical regression using diffeomorphic mappings. arXiv preprint arXiv:1702.00823.
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.
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.
Schötz,, C. (2019). Convergence rates for the generalized Fréchet mean via the quadruple inequality. Electronic Journal of Statistics, 13(2), 4280–4345.
Semple,, C., & Steel,, M. (2003). Phylogenetics (Vol. 24). Oxford: Oxford University Press.
Sengupta,, D., & Jammalamadaka,, S. (2003). Linear models: An integrated approach, Danvers, MA: World Scientific.
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.
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.
Siddiqi,, K., & Pizer,, S. (2008). Medial representations: Mathematics, algorithms and applications, Cham, Switzerland: Springer Verlag.
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.
Sommer,, S. (2015). Anisotropic distributions on manifolds: Template estimation and most probable paths. In S. Ourselin,, D. C. Alexander,, C.‐F. Westin,, & M. J. Cardoso, (Eds.), Information processing in medical imaging (pp. 193–204). Cham: Springer International Publishing.
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.
Sturm,, K. (2003). Probability measures on metric spaces of nonpositive curvature. Contemporary Mathematics, 338, 357–390.
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.
Taylor,, J. E. (2006). A Gaussian kinematic formula. The Annals of Probability, 34(1), 122–158.
Teets,, D., & Whitehead,, K. (1999). The discovery of Ceres: How Gauss became famous. Mathematics Magazine, 72(2), 83–93.
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.
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.
Terras,, A. (1985). Harmonic analysis on symmetric spaces and applications I. New York: Springer‐Verlag.
Terras,, A. (1988). Harmonic analysis on symmetric spaces and applications II. New York: Springer‐Verlag.
Tran,, D. (2019). Behavior of Fréchet mean and central limit theorems on spheres. arXiv Preprint arXiv:1911.01985.
van der Vaart,, A. (2000). Asymptotic statistics, Cambridge: Cambridge University Press.
von Mises,, R. (1918). Über die “Ganzzahligkeit” der Atomgewichte und verwandte Fragen. Physikalishce Zeitschrift, 19, 490–500.
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.
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).

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Data analysis on nonstandard spaces

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox