Nonparametric Inference on Manifolds

Erscheinungsjahr 2012
ISBN: 978-1-107-01958-4
Verlag: Cambridge University Press

Buch, Englisch, Band 2, 256 Seiten, Format (B × H): 157 mm x 235 mm, Gewicht: 521 g

Reihe: Institute of Mathematical Statistics Monographs

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.

Autoren/Hrsg.

Bhattacharya, Abhishek

Bhattacharya, Rabi

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Technische Wissenschaften Sonstige Technologien | Angewandte Technik Signalverarbeitung, Bildverarbeitung, Scanning
• Mathematik | Informatik EDV | Informatik Informatik Künstliche Intelligenz Computer Vision

Inhaltsverzeichnis

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 Skm; 8. The planar shape space Sk2; 9. Reflection similarity shape spaces RSkm; 10. Stiefel manifolds; 11. Affine shape spaces ASkm; 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 Sk2; References; Subject index.

Über Autor(innen)

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

Bhattacharya, Abhishek
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.

Abhishek Bhattacharya is currently working as an assistant professor at the Indian Statistical Institute. After gaining BStat and MStat degrees from the Institute in 2004, and a PhD from the University of Arizona in 2008, he was a postdoctoral researcher at Duke University until 2010, before joining ISI in 2011.

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