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Nonparametric Inference on Manifolds
With Applications to Shape Spaces

$42.99 (C)

Part of Institute of Mathematical Statistics Monographs

  • Date Published: April 2015
  • availability: Available
  • format: Paperback
  • isbn: 9781107484313

$ 42.99 (C)

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About the Authors
  • This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important and varied applications in medical diagnostics, image analysis, and machine vision. An early chapter of examples establishes the effectiveness of the new methods and demonstrates how they outperform their parametric counterparts. Inference is developed for both intrinsic and extrinsic Fréchet means of probability distributions on manifolds, then applied to shape spaces defined as orbits of landmarks under a Lie group of transformations – in particular, similarity, reflection similarity, affine and projective transformations. In addition, nonparametric Bayesian theory is adapted and extended to manifolds for the purposes of density estimation, regression and classification. Ideal for statisticians who analyze manifold data and wish to develop their own methodology, this book is also of interest to probabilists, mathematicians, computer scientists and morphometricians with mathematical training.

    • Expository appendices on differentiable manifolds, Riemannian geometry, parametric models and nonparametric Bayes theory
    • Nonparametric Bayes theory is adapted and extended to manifolds for purposes of density estimation, regression, and classification
    • Suitable for special topics courses at the graduate level
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    Reviews & endorsements

    "In the end, I have to say that this is an excellent text that will benefit many students in computer science, mathematics, and physics. However, I must stress that a proper background in differential geometry and differential calculus is needed to fully understand the material, as well as some graduate learning in advanced statistics. A significant plus of the book is the library of MATLAB codes and datasets available for download from the authors’ site."
    Alexander Tzanov, Computing Reviews

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    Product details

    • Date Published: April 2015
    • format: Paperback
    • isbn: 9781107484313
    • length: 252 pages
    • dimensions: 230 x 152 x 13 mm
    • weight: 0.37kg
    • contains: 20 b/w illus.
    • availability: Available
  • Table of Contents

    1. Introduction
    2. Examples
    3. Location and spread on metric spaces
    4. Extrinsic analysis on manifolds
    5. Intrinsic analysis on manifolds
    6. Landmark-based shape spaces
    7. Kendall's similarity shape spaces Σkm
    8. The planar shape space Σk2
    9. Reflection similarity shape spaces RΣkm
    10. Stiefel manifolds
    11. Affine shape spaces AΣkm
    12. Real projective spaces and projective shape spaces
    13. Nonparametric Bayes inference
    14. Regression, classification and testing
    i. Differentiable manifolds
    ii. Riemannian manifolds
    iii. Dirichlet processes
    iv. Parametric models on Sd and Σk2
    Subject index.

  • Authors

    Abhishek Bhattacharya, Indian Statistical Institute, Kolkata
    Abhishek Bhattacharya is currently working as an assistant professor at the Indian Statistical Institute. After gaining BStat and MStat degrees from the Institute in 2002 and 2004 respectively, and a PhD from the University of Arizona in 2008, he was a postdoctoral researcher at Duke University until the end of 2010, before joining ISI in 2011. Before writing this book, he published several articles in areas as diverse as nonparametric frequentist and Bayesian statistics on non-Euclidean manifolds. All those articles can be accessed from his website.

    Rabi Bhattacharya, University of Arizona
    Rabi Bhattacharya is Professor in the Department of Mathematics at the University of Arizona, Tucson.

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