Institute of Mathematical Statistics Collections

Sampling from a Manifold

Persi Diaconis, Susan Holmes, and Mehrdad Shahshahani

Full-text: Open access

Abstract

We develop algorithms for sampling from a probability distribution on a submanifold embedded in $\mathbb{R}^{n}$. Applications are given to the evaluation of algorithms in ‘Topological Statistics’; to goodness of fit tests in exponential families and to Neyman’s smooth test. This article is partially expository, giving an introduction to the tools of geometric measure theory.

Chapter information

Source
Galin Jones and Xiaotong Shen, eds., Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 102-125

Dates
First available in Project Euclid: 23 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1379942050

Digital Object Identifier
doi:10.1214/12-IMSCOLL1006

Zentralblatt MATH identifier
1356.62015

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
manifold conditional distribution geometric measure theory sampling

Rights
Copyright © 2013, Institute of Mathematical Statistics

Citation

Diaconis, Persi; Holmes, Susan; Shahshahani, Mehrdad. Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102--125, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL1006. https://projecteuclid.org/euclid.imsc/1379942050


Export citation

References

  • Andersen, H. and Diaconis, P. (2007). Hit and run as a unifying device. Journal de la Société Française de Statistique 148 5–28.
  • Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.
  • Barton, D. E. (1953). On Neyman’s test of goodness of fit and its power with respect to a particular system of alternatives. Skandinavisk Aktuarietidskrift 36 24-63.
  • Barton, D. E. (1956). Neyman’s $\psi^{2}_{k}$ test of goodness of fit when the null hypothesis is composite. Skandinavisk Aktuarietidskrift 39 216-46.
  • Bélisle, C. J. P., Romeijn, H. E. and Smith, R. L. (1993). Hit-and-run algorithms for generating multivariate distributions. Mathematics of Operations Research 255–266.
  • Beran, R. (1979). Exponential models for directional data. The Annals of Statistics 1162–1178.
  • Besag, J. and Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika 76 633–642.
  • Bhattacharya, A. and Bhattacharya, R. (2012). Nonparametric Inference on Manifolds with Applications to Shape Spaces. IMS, Cambridge University Press, Cambridge, UK.
  • Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. The Annals of Statistics 1–29.
  • Boender, C., Caron, R., McDonald, J., Kan, A. H. G. R., Romeijn, H., Smith, R., Telgen, J. and Vorst, A. (1991). Shake-and-bake algorithms for generating uniform points on the boundary of bounded polyhedra. Operations Research 945–954.
  • Bormeshenko, O. (2009). Walking around by three flipping. unpublished manuscript.
  • Brown, L. D. (1986). Fundamentals of Statistical Exponential Families: With Applications in Statistical Decision Theory. Institute of Mathematical Statistics, Hayworth, CA, USA.
  • Carlsson, E., Carlsson, G. and de Silva, V. (2006). An algebraic topological method for feature identification. International Journal of Computational Geometry & Applications 16 291–314.
  • Ciccotti, G. and Ryckaert, J. P. (1986). Molecular dynamics simulation of rigid molecules. Computer Physics Reports 4 346–392.
  • Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2009). Billiards in a general domain with random reflections. Archive for Rational Mechanics and Analysis 191 497–537.
  • David, F. N. (1939). On Neyman’s “smooth” test for goodness of fit I. Distribution of the criterion $\psi^{2}$ when the hypothesis tested is true. Biometrika 31 191–199.
  • Diaconis, P. (1988). Sufficiency as statistical symmetry. In Proceedings of the AMS Centennial Symposium 15–26. American Mathematical Society, Providence, RI.
  • Diaconis, P. and Holmes, S. (1994). Gray codes for randomization procedures. Statistics and Computing 287–302.
  • Diaconis, P., Khare, K. and Saloff-Coste, L. (2010). Gibbs sampling, conjugate priors and coupling. Sankhya 72 136–169.
  • Diaconis, P., Lebeau, G. and Michel, L. (2010). Geometric analysis for the Metropolis algorithm on Lipschitz domains. Inventiones Mathematicae 1–43.
  • Diaconis, P. and Saloff-Coste, L. (1998). What do we know about the Metropolis algorithm? Journal of Computer and System Sciences 57 20–36.
  • Diaconis, P. and Shahshahani, M. (1986). On square roots of the uniform distribution on compact groups. Proceedings of the American Mathematical Society 98 341–348.
  • Diaconis, P. and Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions. The Annals of Statistics 26 363–397.
  • Drton, M., Sturmfels, B. and Sullivant, S. (2009). Lectures on Algebraic Statistics. Birkhauser.
  • Eaton, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York.
  • Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. Journal of the American Statistical Association 674–688.
  • Federer, H. (1996). Geometric Measure Theory. Springer, Berlin.
  • Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1993). Statistical Analysis of Spherical Data. Cambridge University Press.
  • Fixman, M. (1974). Classical statistical mechanics of constraints: A theorem and application to polymers. Proceedings of the National Academy of Science, USA 71 3050–3053.
  • Giné, E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. The Annals of Statistics 3 1243–1266.
  • Goldman, N. and Whelan, S. (2000). Statistical tests of gamma-distributed rate heterogeneity in models of sequence evolution in phylogenetics. Molecular Biology and Evolution 17 975–8.
  • Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods. Methuen, London.
  • Hipp, C. (1974). Sufficient statistics and exponential families. The Annals of Statistics 1283–1292.
  • Hubbard, J. H. and Hubbard, B. B. (2007). Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Matrix Editions, Ithaca, NY.
  • Kallioras, A. G., Koutrouvelis, I. A. and Canavos, G. C. (2006). Testing the fit of gamma distributions using the empirical moment generating function. Communications in Statistics—Theory and Methods 35 527–540.
  • Krantz, S. G. and Parks, H. R. (2008). Geometric Integration Theory. Birkhauser, Boston.
  • Lalley, S. and Robbins, H. (1987). Asymptotically minimax stochastic search strategies in the plane. Proceedings of the National Academy of Sciences, USA 84 2111-2112.
  • Lebeau, G. and Michel, L. (2010). Semi-classical analysis of a random walk on a manifold. The Annals of Probability 38 277–315.
  • Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses. Springer, New York.
  • Letac, G. (1992). Lectures on Natural Exponential Families and Their Variance Functions. Conselho Nacional de Desenvolvimento Científico e Tecnológico, Instituto de Matemática Pura e Aplicada.
  • Lindqvist, B. H. and Taraldsen, G. (2005). Monte Carlo conditioning on a sufficient statistic. Biometrika 92 451–464.
  • Lindqvist, B. H. and Taraldsen, G. (2006). Conditional Monte Carlo based on sufficient statistics with applications. In Advances in Statistical Modeling and Inference: Essays in Honor of Kjell A. Doksum ( V. Nair, ed.) 545–561.
  • Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • Mattila, P. (1999). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge studies in advanced mathematics. Cambridge University Press.
  • Mezzadri, F. (2007). How to generate random matrices from the classical compact groups. Notices of the American Mathematical Society 54 592–604.
  • Milnor, J. W. (1968). Singular Points of Complex Hypersurfaces 61. Princeton University Press.
  • Morgan, F. (2009). Geometric Measure Theory: A Beginner’s Guide, 3rd ed. Academic Press, San Diego, CA.
  • Narayanan, H. and Niyogi, P. (2008). Sampling hypersurfaces through diffusion. Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques 535–548.
  • Neyman, J. (1937). “Smooth” test for goodness of fit. Skandinavisk Aktuartioskr 20 149–199.
  • Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision 25 127–154.
  • Pettitt, A. N. (1978). Generalized Cramer-von Mises statistics for the gamma distribution. Biometrika 65 232–5.
  • Tjur, T. (1974). Conditional Probability Distributions. Lecture notes - Institute of Mathematical Statistics, University of Copenhagen; 2. Institute of Mathematical Statistics, University of Copenhagen, Copenhagen.
  • Watson, G. S. (1983). Statistics on Spheres 6. Wiley-Interscience.
  • Yang, Z. (2006). Computational Molecular Evolution. Oxford University Press, Oxford.